Superconducting Charge Qubits Denzil Anthony Rodrigues H. H. Wills Physics Laboratory University of Bristol A thesis submitted to the University of Bristol in accordance with the requirements of the degree of Ph.D. in the Faculty of Science November 2003 Word Count: 36, 000
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Superconducting
Charge Qubits
Denzil Anthony Rodrigues
H. H. Wills Physics Laboratory
University of Bristol
A thesis submitted to the University of Bristol in
accordance with the requirements of the degree of
Ph.D. in the Faculty of Science
November 2003
Word Count: 36, 000
Abstract
In this thesis we discuss a particular type of superconducting qubit, the charge qubit.
We review how the Hamiltonian for a single qubit can be constructed by quantising the
Josephson relations as if they were classical equations of motion.
We examine the charging energy of various qubit circuits, in particular focussing on
the effect of connecting a superconducting reservoir to small islands. We establish the
correct form for the charging energies and show that the naive method of constructing
a quantum Hamiltonian by adding the charging energy of the circuit and the tunnelling
energy for each junction leads to the correct Josephson relations for each junction.
We describe a coupled two-qubit system and describe the operations necessary to test
a Bell inequality in this system. We show how passing a magnetic flux through the circuit
leads to oscillations as a function of both time and flux and show that these oscillations can
be considered as being caused by interference between virtual tunnelling paths between
states.
We describe the strong coupling limit of the BCS Hamiltonian, where the sums over
electron creation and annihilation operators can be replaced by operators representing
large quantum spins. We then solve this Hamiltonian in both the exact and mean-field
limits. We show that the spin solutions are equal to the full solutions in the limit where
the interaction energy is much larger than the cutoff energy.
We show how this simple model can be used to easily investigate various phenomena.
We describe an analogue of the quantum optical effects of superradiance where the cur-
rent through an array of Cooper pair boxes is proportional to the square of the number
of boxes. We also describe an analogue of quantum revival, where the coherent oscilla-
tions of a Cooper pair box coupled to a reservoir decay, only to revive again later. We
describe a similar two-qubit effect, where the oscillations of two entangled Cooper pair
boxes decay and revive. If the entanglement between the boxes is calculated, we find that
the entanglement also decays and then revives.
We describe how quantum spins can be represented using a wavefunction that is a
function of phase. This representation is used to find a microscopic derivation of the
charge qubit Hamiltonian in terms of phase. We find that in the limit of large size, the
Hamiltonian found by quantising the Josephson equations is recovered. Corrections to
this phase Hamiltonian are described.
To my family.
Acknowledgments
I have been lucky enough to have two supervisors to whom I could turn in times of
distress; I would like to thank Balazs for instructing me in basic physics as if I was stupid,
and Tim for reminding me that I’m not. As well as technical advice and encouragement,
both have provided instruction by example in the appropriate attitude to research, and in
how to fit research in to a (relatively) normal life.
My fellow PhD students have been of great importance to me over the last three years.
I would like to thank Ben and Mark for proving that it was possible to survive a PhD
and to thank Andy for giving me something to aim for. I thank my roommate Danny for
asking me stupid questions, both ones that I knew the answer to and those I did not. I
thank David and Maria for stopping me when I was talking rubbish and Jamie and Emma
for not stopping me when I was talking rubbish.
Finally I thank Liz giving me something to look forward to at the end of the day.
Authors Declaration
I declare that the work in this thesis was carried out in accordance with the regulations of
the University of Bristol. The work is original except where indicated by special reference
in the text and no part of the thesis has been submitted for any other degree. Any
views expressed in the thesis are those of the author and in no way represent those of
the University of Bristol. The thesis has not been presented to any other University for
examination either in the United Kingdom or overseas.
4.6 The Probability of the Two Qubit system being in the state |00〉 (upperplot) and |01〉 (lower plot). The horizontal axis is time and the vertical axis
is the flux through the loop in units Φ02π . White represents high probability
of occupation and black represents low probability. . . . . . . . . . . . . . . 82
4.7 The Probability of the Two Qubit system being in the state |10〉 (upperplot) and |11〉 (lower plot). The horizontal axis is time and the vertical axis
is the flux through the loop in units Φ02π . White represents high probability
of occupation and Black represents low probability. . . . . . . . . . . . . . . 83
4.8 The Probability of the Two Qubit system being in each of the four charging
4.13 The Probability of the Two Qubit system being in the state |01〉 or |11〉 i.e.the probability that the first qubit is occupied regardless of the occupation
of the second. The z axis represents increasing probability. . . . . . . . . . 91
First order terms are eliminated by setting Ht + [(H1 + H2), K] = 0. If we take the
expectation value of this condition with a complete set of states of H0, |n〉, we find:
〈n|K|m〉 = 〈n|Ht|m〉/(Em − En) (1.8.2)
From this we find the matrix elements of the second order term 12 [Ht, K], which are:
1.8 The Pair Tunnelling Hamiltonian 21
1
2
∑
p
〈n|Ht|p〉〈p|Ht|m〉(
1
Em − Ep− 1
Ep − En
)(1.8.3)
If a pair is transferred from one side of the junction to the other, the final state has
no quasiparticle excitations and therefore has the same energy as the initial state. In
the intermediate state |p〉, an electron has been removed from state −q on one side and
placed in state k on the other, creating a broken pair on either side. The energy of the
intermediate state is then given by Ep − En = Ek + Eq, where Ek/q are the quasiparticle
energies Ek/q =√
(εk/q − µ)2 +∆k/q:
∑
p
〈n|Ht|p〉〈p|Ht|m〉Ek + Eq
(1.8.4)
As states |n〉 and |m〉 are both states with no broken pairs and state |p〉 is a state
with an extra electron in state k and extra hole in state −q, with 〈p|Ht|m〉 the matrix
element for the creation of this state, then 〈n|Ht|p〉 is the matrix element for the transfer
of an electron from the state q to the state −k, and eq. 1.8.4 is the matrix element for the
transfer of a pair from q,−q on one side to k,−k:
Tk,q = − tk,qt−k,−qEk + Eq
= − |tk,q|2
Ek + Eq(1.8.5)
This derivation justifies the use of the pair tunnelling Hamiltonian used in the rest
of the thesis. Note that this matrix element is proportional to the square of the single-
electron tunnelling element, as is expected - the single-electron tunnelling element has to
act twice to transfer a pair across the junction.
We note that this tunnelling element is not independent of k, q. However we shall
approximate by an independent tunnelling element in the same way as we have approx-
imated the BCS interaction Vk,k′ by one that is k - independent, V . We now take the
Hamiltonian for the junction to be:
H = H1 + H2 + HT (1.8.6)
22 Superconductivity and The Josephson Effect
Where H1,2 are the Hamiltonians describing each superconductor (eq. 1.6.1) and the
pair tunnelling is described by HT :
HT = −T∑
k,q
c†kc†−kc−qcq + c†qc
†−qc−kck (1.8.7)
1.9 Small Superconductors
When superconductivity is generally discussed (eg. the BCS wavefunction), the standard
condensed matter assumption of infinite size is made. However, at very small sizes of
island, various effects come into play due to the departure from bulk behaviour.
Recent fabrication techniques have allowed the fabrication of extremely small particles
of aluminium, with radii of order 5nm. This allowed Ralph, Black and Tinkham [43] to
study the nature of superconductivity in ultrasmall grains. The grains were studied by
constructing a Single Electron Transistor (SET) by placing the grain between two leads,
and applying a gate voltage to the grain. The current through the grain can then be
studied as a function of the bias voltage between the two leads for different applied gate
voltages.
These particles were so small that the single-electron energy level spacing d = 1/N (εF )
needed to be regarded as discrete; in particular, d becomes comparable to the bulk super-
conducting gap, ∆. Furthermore, as the number of electrons on the island was well-defined,
a canonical rather than grand-canonical description was required. An extensive review of
the theory of finite size effects can be found in [44]
The spectroscopic gap of the grains in these experiments is particularly significant.
For grains with radii & 5nm, a spectroscopic gap could be observed when an even number
of electrons was present on an island, but could not be observed for islands with an odd
number of electrons. This rather striking parity effect is a clear indication that super-
conducting pairing correlations are taking place in these grains. For smaller grains, there
was no observed spectroscopic gap, even for particles with an even number of electrons.
Thus, these experiments show both the existence of superconducting pairing correlations
in small grains, and the disappearance of a spectroscopic gap as the size of the grains is
reduced even further and so a description of these grains requires an understanding of how
superconductivity changes from the bulk to grains that are so small that superconductivity
is suppressed.
1.9 Small Superconductors 23
1.9.1 The Parity Effect
One of the more startling effects in small metallic grains is the parity effect; despite the
fact that the number of electrons is of order 109, there are measurable differences in the
properties of grains with odd and even numbers of electrons. The main difference is
that the condensation energy of the two is different. This can be simply understood in
the bulk limit; in an odd grain there will always be one unpaired electron, and so the
difference in the condensation energies will be ∆, the energy of a single quasiparticle. In
fact, this energy difference is observed in Coulomb blockade phenomena [45, 46], where
the periodicity of the Coulomb oscillations in the gate voltage changes from e to 2e.
As well as a change in condensation energy, the pairing parameter (∆P , where the
P = 0, 1 indicates the parity of the grain) is also parity dependent. The bulk limit (first
order in d/∆) of the parity dependence has been studied by Janko et al. [47] and von
Delft et al. [48, 49].
Starting with the BCS Hamiltonian for a finite system:
H =∑
k,σ
εkc†k,σck,σ − V
∑
k,k′
c†k′↑c†−k′↓c−k↓ck↑ (1.9.1)
where the labels k,−k refer to generic time-reversed single electron states. In ultrasmall
grains, the number of electrons is well defined, and so the canonical distribution should
really be used. However, the grand-canonical distribution is used to approximate the
canonical distribution, and can in fact be regarded as a saddle-point approximation to
it. Von Delft et al. treated the problem using a parity-projected grand canonical distri-
bution. In this, the exact number of electrons present can vary, but the parity cannot.
This technique allows the use of mean-field techniques, but ensures that parity effects are
included.
ZGP (µ) ≡ TrG1
2[1± (−)NL ]e−β(HL−µNL) (1.9.2)
The procedure followed by von Delft et al. was to use the parity-projected partition
function (eq. 1.9.2) to calculate the finite-temperature properties of the superconducting
grain. Essentially, the grains are described by two gap equations, one for even grains and
one for odd grains.
24 Superconductivity and The Josephson Effect
1
λ= d
∑
q<ωc/d
1
2EP(1− 2fqP ) (1.9.3)
where the Fermi distribution function fqP = 〈γ†qγq〉P also depends on the parity of the
grains. The chemical potential must be chosen to give either an odd or even number
of electrons on the grain, which determines the Fermi function. At zero temperature,
fqp = δj0δP0 i.e. there are no quasiparticles in the even case and only one, at the chemical
potential, in the odd case. Equation 1.9.3 can be solved numerically at finite temperatures,
but at T = 0 it can be evaluated exactly:
1
λd=
ωc∫
0
dω
Eeven,ωtanh(
πEe,ωd
) (1.9.4)
for even, and:
1
λd=
ωc∫
0
dω
Eodd,ω
1
tanh(πEodd,ω
d )− d
Eodd,ω(1.9.5)
for odd. In the limit where the level spacing d is much smaller than the bulk gap, ∆, the
form for the parity-projected pairing parameters is:
∆even(d, T = 0) = ∆(1−√
∆/de−2π∆/d) (1.9.6)
∆odd(d, T = 0) = ∆− d/2 (1.9.7)
We have two different values for the pairing parameter ∆P depending on whether there
are an odd or even number of electrons on the island, which also leads to two different
values for the condensation energy. We find that the forms for ∆P described in eqs.
1.9.6, 1.9.7 lead to the intuitive result that, in the limit d/∆ → 0, the difference in the
condensation energies is equal to the energy caused by having one unpaired quasiparticle.
We also see from these results that the pairing parameter for an odd grain is always
smaller than for an even grain, implying that there is a size region in which an even grain is
superconducting, but an odd grain is not. In fact, as the level spacing becomes larger, the
mean-field approach is no longer adequate. Nevertheless, these results serve to show that
there can be a significant difference in the pairing parameters and condensation energies
of grains with odd or even number of electrons.
1.9 Small Superconductors 25
Whilst these results were originally derived to treat ultrasmall superconducting grains,
they are present at larger grain sizes, just smaller in magnitude. In terms of supercon-
ducting qubits, the parity effect means that there may be fluctuations in the values of EC
and EJ between the values for odd and even grains. Some possible effects of this kind of
fluctuation are discussed in section 2.6.2.
Although a range of techniques for treating ultrasmall superconducting grains have
been developed, there is in fact an exact solution of the BCS Hamiltonian for finite islands,
known as the Richardson solution.
1.9.2 The Richardson Exact Solution
Recently it has come to the attention of the condensed matter community that the reduced
BCS Hamiltonian for a finite superconductor:
H =∑
k,σ
εkc†k,σck,σ − V
∑
k,k′
c†k′↑c†−k′↓c−k↓ck↑ (1.9.8)
has an exact solution, which reproduces the BCS results in the bulk limit [50, 51]. This
solution was discovered by Richardson as far back as 1963 in the context of nuclear physics
[52]. The existence of an exact solution to this Hamiltonian came as a great surprise to
the community and we will take a while to discuss it.
We can write the above Hamiltonian in terms of the Nambu spins [53]. The creation,
annihilation and number operators for each electron pair k are represented as spin half
operators. These are defined as:
σZk =1
2(c†k↑ck↑ + c†−k↓c−k↓ − 1)
σ+k = c†k↑c†−k↓
σ−k = c−k↓ck↑ (1.9.9)
(1.9.10)
The labels k are generic electron spin labels, rather than wavevectors. We also note
that σZk = σ+k σ−k − 1/2, and write the reduced BCS Hamiltonian:
∑
k,k′
(2εkδk,k′ − V )σ+k σ′k (1.9.11)
26 Superconductivity and The Josephson Effect
Our task is to diagonalise this Hamiltonian. We write the diagonalised Hamiltonian
and eigenstates in terms of the operators B+Jν , and our job is now to find these operators.
H =
N∑
ν=1
EJνB+JνB
−Jν
|ψN 〉 =N∏
ν=1
B+Jν |0〉 (1.9.12)
Note that these eigenstates of the Hamiltonian are number eigenstates, and there are
many different eigenstates that all have the same number of Cooper pair on the island, N .
We shall see that the form for the operators B+Jν that diagonalises the Hamiltonian is:
B+Jν =
∑
k
σ+k2εk − EJν
(1.9.13)
Our task is now to find the parameters EJν . To do this we note that, if the form for the
eigenstates is correct, then H
(N∏ν=1
B+Jν |0〉
)=
N∑ν=1
EJν
(N∏ν=1
B+Jν |0〉
)= ε
(N∏ν=1
B+Jν |0〉
)
HN∏
ν=1
B+Jν |0〉 =
N∑
ν=1
EJν
N∏
ν=1
B+Jν |0〉
[H,N∏
ν=1
B+Jν ] |0〉+
N∏
ν=1
B+JνH|0〉 =
N∑
ν=1
EJν
N∏
ν=1
B+Jν |0〉
[H,N∏
ν=1
B+Jν ] |0〉 = ε
N∏
ν=1
B+Jν |0〉 (1.9.14)
We need to calculate the commutator of the Hamiltonian with the product over all the
B+J operators. Calculating the commutator with just a single operator:
[H, B+Jν ] = EJνB
+Jν + S+
(1− V
∑
k
1
2εk − EJν
)
+S+V∑
k
2σ+k σ−k
2εk − EJν(1.9.15)
Following [54] we notice that this is a generalisation of the boson case. Our Nambu spin
operators (sometimes referred to as hard-core boson operators) differ from true bosons by
one of the three commutation relations. In true bosons, we find [bk, b†k] = 1, whereas for
1.9 Small Superconductors 27
our hard-core bosons, we have [σ−k , σ+k ] = 1− 2σ+k σ
−k = −2σZk . If we were discussing true
bosons, then the commutator above (1.9.15 would only consist of the first two terms. We
could eliminate the second term by setting the condition for the EJν parameters:
0 =
(1− V
∑
k
1
2εk − EJν
)
(1.9.16)
This would mean that the commutator would become [H, B+Jν ] = EJνB
+Jν and therefore
that
HN∏
ν=1
B+Jν |0〉 = [H,
N∏
ν=1
B+Jν ] |0〉
=∑
µ
∏
ν 6=µ[H, B+
Jν ] |0〉
=
(∑
µ
EJµ
)N∏
ν=1
B+Jν |0〉 (1.9.17)
That is, in the true boson case, choosing the parameters EJν so that the condition
1.9.16 is satisfied means that the operators B+Jν diagonalise the Hamiltonian.
We, however, wish to consider the full Nambu spin case. This means that the final
term in 1.9.15 is not zero, and must be taken into account. Returning to the commutator
of the Hamiltonian with the product over B+Jν operators:
[H,N∏
ν=1
B+Jν ] =
N∑
ν=1
ν−1∏
η=1
B+Jη
[H, B+
Jν ]
N∏
µ=ν+1
B+Jµ
(1.9.18)
Examining our expression (eq. 1.9.15) for [H, B+Jν ], we see that the first two terms
(EJνB+Jν and S+
(1− V ∑
k
12εk−EJ
)) commute with the products over the remaining B+
J
operators, and so we can simply move these terms past the products in eq. 1.9.20. We
need to commute the third term of eq. 1.9.15 past the products as well. To do this, we
first need to know its commutator with the individual operators:
28 Superconductivity and The Josephson Effect
[S+V∑
k
2σ+k σ−k
2εk − EJν, B+
Jµ] = S+2V∑
k
1
2εk − EJνσ+k
2εk − EJµ
=1
EJν − EJµ
(σ+k
2εk − EJν− σ+k
2εk − EJµ
)
= S+2VB+Jν −B+
Jµ
EJν − EJµ(1.9.19)
We can now move the third term of eq. 1.9.15 past the products:
N∑
ν=1
ν−1∏
η=1
B+Jη
(S+V
∑
k
2σ+k σ−k
2εk − EJν
)N∏
µ=ν+1
B+Jµ =
N∑
ν=1
ν−1∏
η=1
B+Jη
N∑
µ=ν+1
µ−1∏
ρ=ν+1
B+Jρ
(S+2V
B+Jν −B+
Jµ
EJν − EJµ
)ν−1∏
τ=µ+1
B+Jτ
(1.9.20)
All the B+Jν and S+ operators commute, so we can collect up all the products, re-order
the sums in the second term and then interchange the dummy indices µ, ν in the sums in
the first term:
=N∑
ν=1
N∑
µ=ν+1
2S+VN∏η 6=µ
B+Jη
EJν − EJµ−
N∑
ν=1
N∑
µ=ν+1
2S+VN∏η 6=ν
B+Jη
EJν − EJµ
=N∑
ν=1
N∑
µ=ν+1
2S+VN∏η 6=µ
B+Jη
EJν − EJµ−
N∑
µ=1
µ−1∑
ν=1
2S+VN∏η 6=ν
B+Jη
EJν − EJµ
=N∑
µ=1
N∑
ν=ν+1
2S+VN∏η 6=ν
B+Jη
EJµ − EJν−
N∑
µ=1
µ−1∑
ν=1
2S+VN∏η 6=ν
B+Jη
EJν − EJµ
=N∑
µ,ν 6=µ
2S+VN∏η 6=ν
B+Jη
EJµ − EJν(1.9.21)
We now have the third term of eq. 1.9.20. Writing out the commutator of the Hamil-
tonian with the product over B+J operators in full, we get:
1.9 Small Superconductors 29
[H,N∏
ν=1
B+Jν ] =
N∑
ν=1
EJν
N∏
ν=1
B+Jν
+N∑
ν=1
S+
(1− V
∑
k
1
2εk − EJν
)N∏
η 6=νB+Jη
+N∑
ν=1
S+
N∑
µ6=ν
2V
EJµ − EJν
N∏
η 6=νB+Jη (1.9.22)
From eq. 1.9.14, we know that the B+J operators diagonalise the Hamiltonian if the
second two terms above vanish. That is, we have the correct diagonal form for the Hamil-
tonian if, for all ν:
1− V∑
k
1
2εk − EJν+∑
µ6=ν
2V
EJµ − EJν= 0 (1.9.23)
Chapter 2
The Superconducting Charge
Qubit
The progress in constructing superconducting qubits to date is reviewed. A phenomenologi-
cal theoretical description of superconducting charge qubits is formulated, and the dynamics
of such systems examined. We look at the effect of noise in the qubit parameters on the
evolution of the system, and describe new work on how discrete noise in the qubit param-
eters affects the system.
In section 1.1 we mentioned briefly why small superconducting circuits might be well
suited for use as quantum bits. As with all solid-state realisations, each qubit is well-
defined as each is physically separated, and techniques of circuit fabrication suggest that
large numbers of interacting qubits could be manufactured. In addition, superconducting
qubits have the advantage over other solid-state implementations in that the supercon-
ducting gap protects the condensate from the environment to some extent.
Whilst these arguments are suggestive, we require a physical theory of superconducting
circuits in order to describe how they can be used as qubits, and where any possible
problems arise.
2.1 Experimental Progress
Although far behind such implementations as ion trap [11, 16, 17] and NMR-based [55,
56, 57, 18] quantum computing realisations, superconducting charge qubits have developed
31
32 The Superconducting Charge Qubit
quickly in recent years, as have other types of superconducting qubit. As an indication of
the rapid growth of this field, superconducting charge qubits were first suggested in 1997
by Shnirman et al. [58], and by 2001 coherent oscillations were being observed [59].
A superconducting charge qubit consists of a small island of superconducting material
(usually aluminium) connected via a Josephson tunnel junction to a superconducting
reservoir. The island is small enough (the capacitance low enough) that the charging
energy of a single Cooper pair becomes significant - it costs an appreciable amount of
energy to add extra Cooper pairs to the island. The two qubit states |0〉 and |1〉 are the
states where there are 0 or 1 excess Cooper pairs on the island. The separation of the
two charging energy states is controlled by a gate voltage, which gives a diagonal term in
the Hamiltonian (σZ in the spin-half notation where |0〉 = ( 01 ), |1〉 = ( 10 ) and σX , σY , σZ
are the Pauli matrices). The Josephson coupling to the reservoir allows Cooper pairs to
tunnel on and off the island, and provides an off-diagonal term (σX).
Oscillations between the two charge states were observed in 1999 by Nakamura et
al. [59]. These oscillations are an indication that the system is behaving quantum me-
chanically and can be used to infer the existence of a superposition of charge states at a
midpoint of the oscillation. Similar oscillations have since been seen by several groups in
various superconducting systems [60, 61, 62], and spectroscopic measurements made on a
range of others including those based on flux [63, 64] and phase [65, 66]
Coupling between two qubits can be by use of capacitive or Josephson coupling between
the superconducting islands, or through a common (or ‘bus’) inductance [67] or LC-
oscillator circuit [68].
Two-qubit oscillations have now been observed in charge qubits [69] and there is spec-
troscopic evidence for coupling between superconducting flux qubits separated by a dis-
tance of 0.7mm [70]. Recently, the first implementation of a controlled not gate (CNOT)
was performed between two coupled charge qubits [71]
As well as demonstrations of basic quantum behaviour, experiments illustrating the
possibility of more sophisticated manipulations such as spin-echo have been performed
[72]. This experiment in particular gives a clear indication of phase coherent evolution,
and suggests that NMR-style techniques to reduce the effects of decoherence and noise
may be viable.
If superconducting qubits are to be used in quantum computers, they must also satisfy
the final DiVincenzo criteria, which requires that the qubits be individually measurable.
2.2 The Josephson Equations for Small Circuits 33
In particular, in most of the current experiments, an ensemble measurement is made of the
state of the qubit. The Nakamura experiments measure the charge on the qubit by placing
a readout lead close to the qubit island. If the qubit is in the state with an extra Cooper
pair on the island, the Cooper pair will decay through quasiparticle tunnelling to the
lead. Individual Cooper pair decays cannot be detected in this manner, so each oscillation
is repeated many times during one measurement so that there are enough quasiparticle
decays to be detected as a current. The current on the readout lead is then proportional to
the probability of the state |1〉 being occupied. A further problem with this measurement
scheme is that it is ‘always on’; the tunnelling to the leads is present during the evolution
of the qubit, and relies on the fact that the quasiparticle decay time is much longer than
the oscillation time to ensure that the evolution is unaffected. Whilst this is adequate for
demonstrations of coherent oscillations, over a long period of evolution a coupling to a
measurement device will cause decoherence. Ideally, we would like to be able to decouple
(or ‘turn off’) the measurement whilst the quantum evolution of the qubit takes place.
A quantum computer will require single-shot measurements, i.e. measurements re-
turning 1 or 0, rather than a probability. In theory these sorts of measurements could be
done using a single electron transistor (SET) [58, 73]. In such a scheme, the presence of
a Cooper pair on the qubit modifies the voltage on the central island of the transistor,
and allows current to flow. A further advantage of this readout scheme is that if no bias
voltage is applied to the SET, no measurement takes place, and thus the qubit dynamics
are not affected.
Single shot measurements are being performed in ‘quantronium’ qubit systems [60],
and experiments using SETs performed [61], but as yet SETs have not been used to
perform single-shot experiments on charge qubits.
2.2 The Josephson Equations for Small Circuits
Figure 2.1 shows a simple schematic of a circuit that could be used as a charge qubit. A
small island of superconductor is connected via a capacitive Josephson tunnel junction to
a superconducting reservoir. If the island was a bulk region, then the evolution of the
phase and voltage across the Junction would be adequately described by the Josephson
equations ( 1.7.6, 1.7.9 ). Although the Josephson effect is an intrinsically quantum effect,
derived from the quantum behaviour of electrons, the equations of motion 1.7.6, 1.7.9
34 The Superconducting Charge Qubit
q
-q
C IC
Figure 2.1: A simple circuit diagram for a charge qubit consisting of two regions of super-
conductor connected by a Josephson junction with critical current IC whose capacitance
is represented by a capacitor of capacitance C
are classical - the superconductors on either side of the junction are considered infinitely
large, and the capacitance of the junction can be ignored. The superconductors can be
in a superposition of different number eigenstates and therefore the classical equations
for phase are adequate. As the size of the island becomes small, the capacitance of the
junction becomes small and the charging energy large. A large charging energy means
that states with too many or too few Cooper pairs are unfavourable. We can no longer
sum over all number states, and so the phase is not a well-defined variable, and we need
to take into account its quantum mechanical nature. In particular, we need to account for
the conjugate nature of the phase and number operators.
As has been stated, the Josephson equations describe a large junction that can be
adequately described classically. To describe non-classical effects, there are two possible
approaches we could take: a microscopic, or ‘bottom up’ approach, and a semiclassical, ‘top
down’ approach. The microscopic approach is to return to the full quantum description
of the superconductors, and try to re-derive a Josephson equation without making the
assumption of infinite size made in 1.7.
In this chapter, and in chapters 3 and 4, we consider the semiclassical approach. The
motivation for this is the fact that the classical Josephson equations describe very well the
behaviour of large Josephson junctions, and that effects due to the finite size of the circuit
will be a modification to these. In the semiclassical approach, we ignore the quantum
2.2 The Josephson Equations for Small Circuits 35
origin of the Josephson equations, and treat them as purely classical equations of motion.
We then quantise these ‘classical’ equations in the same way as we would for any other
classical equations.
As this semiclassical treatment, and its differences from the microscopic theory will
feature heavily in later chapters of this thesis, we shall spend some time on quantising the
simplest quantum Josephson circuit, shown in figure 2.1. Later we shall show (chapter 8)
how these semiclassical results can be derived as a limiting form of a microscopic theory.
2.2.1 The Classical Lagrangian
We would like a quantum description of the circuit shown in figure 2.1, which consists
of two regions of superconductor connected by a Josephson junction of critical current
IC and capacitance C. The charge on the capacitor is equal to the excess charge on the
island, q, and so the voltage across the capacitor is given by V = q/C.
The first stage in quantising any classical equations of motion is to write the equations
in terms of Lagrangian mechanics (see eg. [74]). That is, we need to find a classical
Lagrangian that leads to the equations of motion we wish to quantise. The Josephson
equations derived in 1.7 are:
I = IC sinφ (2.2.1)
dφ
dt=
2eV
~(2.2.2)
We note that the voltage across the junction is given by V = q/C, and that the current
across the junction is the rate of change of charge on the island. This leads to an equation
of motion for the phase difference:
d2φ
dt2=
2e
~CIC sinφ (2.2.3)
We need to find a Lagrangian that corresponds to this equation. The Euler - Lagrange
equations of the desired Lagrangian give eq. 2.2.3. Using educated guesswork, we try the
Lagrangian:
L =1
2
~2C4e2
(dφ
dt
)2
+~IC2e
cosφ (2.2.4)
36 The Superconducting Charge Qubit
Putting this form for the Lagrangian into the Euler-Lagrange equations for φ, ∂∂t
(∂L∂φ
)−
∂L∂φ = 0 indeed returns eq. 2.2.3.
2.2.2 The Classical Hamiltonian and the Conjugate Variables
If we choose φ to be the canonical position variable, then we find the canonical momentum,
π to be:
π =∂L∂φ
=~2C4e2
φ (2.2.5)
From eqs. 2.2.4 and 2.2.5 we get the classical Hamiltonian:
H = πφ− L
= EC1
~2π2 − EJ cosφ (2.2.6)
Let us examine this Hamiltonian and the conjugate variables further. We know the
physical meaning of the canonical position, φ, is the phase difference between the two
superconducting condensates on either side of the junction. From eq. 2.2.2, we find that
the canonical momentum π can be rewritten:
π =~2C4e2
φ =~ CV2e
=~q2e
= ~N (2.2.7)
The phase and the number of Cooper pairs are conjugate variables. EC = 4e2
2C is the
charging energy of a single Cooper pair, i.e. the energy required to increase the number
of Cooper pairs on the island by one. EJ = ~IC2e is the Josephson tunnelling energy.
We can calculate the equations of motion for φ and π (or N) from the Hamiltonian
directly, just as we did from the Lagrangian. However, as we intend to quantise this
system, let us calculate the equations of motion in another way - in terms of Poisson
brackets.
2.2 The Josephson Equations for Small Circuits 37
Poisson brackets in classical mechanics are analogous to commutators in quantum
mechanics. The Poisson bracket,A,B, of two functions of the canonical positions and
momentums of the system, A,B, is defined as:
A,B =∑
i
(∂A
∂qi
∂B
∂πi− ∂A
∂πi
∂B
∂qi
)(2.2.8)
If we assume that A,B have no explicit time dependence, then the equations of motion
can be derived simply using:
dA
dt= A,H (2.2.9)
Substituting A = φ,A = π and the Hamiltonian in eq. 2.2.6 into eq. 2.2.9 gives back
the Josephson relations (2.2.1,2.2.2) and confirms that we have the correct Hamiltonian
for the system.
2.2.3 Quantising the Classical Josephson Equations
Now we have the correct classical Hamiltonian that described the circuit, we obtain a
quantum description by promoting the conjugate variables (φ, π) to be operators (φ, π).
H = ECN2 − EJ cos φ (2.2.10)
As these operators are conjugate, they have the commutation relation [φ, π] = i~.
Using the number operator N = π/~, we have the relation [φ, N ] = i (see also [75, 76]).
We describe the state of the system by a wavefunction ψ(φ) which is a complex function
of the canonical position φ. The operators are represented by differential operators that
give the correct commutation relations:
φ = φ
N = id
dφ(2.2.11)
This leads to a differential form for the Hamiltonian:
38 The Superconducting Charge Qubit
H = EC
(id
dφ
)2
− EJ cosφ (2.2.12)
Having found our quantum Hamiltonian, we note that we can recover the classical
equations of motion for the circuit through the use of Ehrenfest’s theorem, which gives
the expectation value of the rate of change of an operator O in terms of its commutation
relation with the Hamiltonian:
i~
⟨dO
dt
⟩=⟨[H, O]
⟩(2.2.13)
This leads to a set of equations of motion for the expectation values of current and
phase difference across the junction:
〈I〉 = 2e
⟨dN
dt
⟩=
2e
~EJ〈sin φ〉
⟨dφ
dt
⟩=
2e
~2e
C〈N〉 (2.2.14)
where 2eC 〈N〉 is identified as the voltage across the capacitor, V .
Having found the correct classical Hamiltonian, quantised it, and found a convenient
differential form for the operators, we go on to investigate how a two-level system can be
extracted.
2.3 Tuneable Josephson Energy
In the previous section we formulated a quantum mechanical description of our super-
conducting circuit. For this system to be used as a qubit, it must behave as a two level
system, and we must be able to adjust the relative charging and coupling energies in order
to perform controlled operations.
In order to be able adjust the charging and coupling energies EC and EJ , we replace
the circuit shown in figure 2.1 with a slightly more complicated one (figure 2.2). This
circuit can be described by a Hamiltonian with a gate voltage dependent charging energy,
and a Josephson energy that depends on the flux. We shall leave the dependence on the
gate voltage until chapter 3, where it will be discussed in some detail. For the present, we
2.3 Tuneable Josephson Energy 39
Vg
Island
Reservoir
Cg
IC ICC
Figure 2.2: A simple circuit diagram for a charge qubit with a gate voltage applied and
the single Josephson junction replaced by a magnetic flux dependent double-junction.
shall examine how a double-junction through which a magnetic flux is passed can act like
a single junction with a controllable Josephson energy.
The line integral of the vector potential A around a closed contour is equal to the flux
passing through that contour:
Φ =
∫
c
A.ds (2.3.1)
We write our integral around the double junction 2.3 as a sum of the contribution of
two paths ca and cb. Note that the contour cb is in the opposite sense to contour ca and
thus its contribution is negative.
Φ =
∫
ca
A.ds−∫
cb
A.ds (2.3.2)
The contour passes through the centre of the superconducting electrodes and crosses
the junction. We can make the junctions infinitely thin, so that their contribution to the
integral is essentially zero. The path goes through the centre of the electrodes, where the
supercurrent density given in eq. 1.3.6 is zero, i.e. , where (∇φ − 2πA/Φ0) = 0, with φ
the phase of the order parameter ψ = |ψ|eiφ. We can write the integral as:
40 The Superconducting Charge Qubit
1 2
ca
cb
Figure 2.3: A double junction consisting of a ring of superconductor broken by two junction
and threaded by a flux Φ. The two paths from point 1 to point 2 are labelled ca and cb.
Φ =
∫
ca
Φ0
2π∇φ.ds−
∫
cb
Φ0
2π∇φ.ds (2.3.3)
Doing the integrals, we find that the difference in phase acquired by going from 1 to
2 by following path ca instead of cb is determined by the magnetic flux threading the two
paths:
2πΦ
Φ0= ∆φca −∆φcb (2.3.4)
We can now calculate the current from point 1 to point 2, passing through both paths.
I = Ica sin
(φca +
πΦ
Φ0
)+ Icb sin
(φca −
πΦ
Φ0
)(2.3.5)
If we assume the junctions are equal, Ica = Icb, then:
I = 2Ica sin
(∆φca +
πΦ
Φ0
)cos
(πΦ
Φ0
)(2.3.6)
If we consider the double junction as a whole, we see that we have a critical Josephson
current that is dependent on the magnetic flux through the loop Ic = 2Ica cos(πΦΦ0
).
2.4 The Two-Level System 41
Substituting this into the expression for Josephson energy and assuming the inductance of
the circuit is negligible, we have a Josephson energy that can be adjusted by an external
control parameter, the magnetic field.
EJ(Φ) =~Icae
cos
(πΦ
Φ0
)(2.3.7)
Note that while we have derived the flux dependence of the current using Ginzburg
- Landau theory for a particular junction configuration, this dependence is essentially
nothing more than the Aharonov-Bohm effect [77], and as such is a much more general
phenomena than may be suggested by the above derivation. The phase of an electron
wavefunction is changed by:
exp
(ie
~c
∫A.ds
)(2.3.8)
The charge carrier in a superconductor is a Cooper pair, and so the phase changes by
twice this amount.
This dependence of the behaviour of the qubit on the applied magnetic field has been
impressively demonstrated in the experiments of Nakamura et al. [59, 69].
2.4 The Two-Level System
The Hamiltonian of the system as a function of gate voltage and magnetic flux can be
written:
H = EC
(id
dφ− ng
)2
− EJ(Φ) cosφ (2.4.1)
Where Φ is the magnetic flux through the double-junction, and we have redefined
the origin of phase compared to 2.3.7 so that the cosine term has no flux dependence.
For now, we note that the interaction energy between a charge −2eN and an external
voltage Vg will be of the form −2eNVg = −EcNng and assume that a gate voltage can be
incorporated into the Hamiltonian as above with −2eng = CgVg defining a ‘gate charge’;
this shall discussed in detail in chapter 3.
42 The Superconducting Charge Qubit
Adjusting the gate voltage allows us to control the ‘zero position’ of charge on the
circuit; the energy depends on how far N is from ng. In effect, ng determines at which
value of N the island will be electrically neutral.
In a superconducting charge qubit, EC À EJ and (for most values of ng) the charge
states are a good approximation to the energy eigenstates.
If we consider an island with a fixed number of Cooper pairs N , then we see that the
charging energy EC(N−ng)2 varies quadratically with ng; a plot of EC against ng produces
a parabola centred on N . This is only for a fixed number of Cooper pairs, however, and
the Josephson junction allows Cooper pairs to tunnel on and off the island. If ng is set
close to N , then the charging energy is small and the state |N〉 will be the ground state.
As ng increases, however, the charging energy goes up, until the state |N +1〉 has a lower
energy and becomes the ground state. At this point, if the system is allowed to relax, a
Cooper pair will tunnel from the reservoir to the island.
This tunnelling means that the charging energy of the ground state is a periodic func-
tion of ng. If we plot the charging energy for all states |N〉 against ng (figure 2.4), the
charging energy for each is a parabola centred on ng = N , and the parabolas cross at
half-integer values of ng, where the states N and N + 1 are degenerate.
The effect of the Josephson energy on this picture is to break the degeneracy of the
charge states N,N + 1 at the points ng = N + 1/2 (figure 2.5).
Around these points, the two lowest energy eigenstates will be the symmetric and anti-
symmetric combinations of the N and N + 1 states (figure 2.6), and the other states will
be separated in energy by EC . If EC is large compared to EJ , then only the N , N + 1
states will be occupied and we can regard the Cooper pair box as a two level system. That
is, if we write the Hamiltonian 2.2.12 in terms of the number eigenstates:
H = EC∑
N
|N〉(N − ng)2〈N | −EJ2
∑
N
(|N〉〈N + 1|+ |N + 1〉〈N |) (2.4.2)
We keep only the states N , N + 1 as the other are all suppressed by the charging
energy and obtain a two-level system with the Hamiltonian:
H = EC(1/2− ng) (|1〉〈1| − |0〉〈0|)
+EJ2
(|1〉〈0|+ |0〉〈1|) (2.4.3)
2.4 The Two-Level System 43
0
0.2
0.4
0.6
0.8
1
Ec
1 2ng
Figure 2.4: The charging energy as a periodic function of ng. The plot is a series of
parabolas centred on ng = N + 0, ng = N + 1, etc. At half-integer values of ng, the
parabolas cross and the ground state changes from N to N + 1.
44 The Superconducting Charge Qubit
0
5
10
15
20
3 4 5 6
Figure 2.5: The qubit energy levels as a periodic function of ng. The degeneracy of the
charging energies at half-integer values of ng is broken by the tunnelling energy EJ . The
solid line represents the ground state energy, whilst the first and second excited states are
represented by the dashed and dotted lines respectively.
2.4 The Two-Level System 45
–1
–0.5
0
0.5
1
Ec
0.2 0.4 0.6 0.8 1ng
Figure 2.6: The avoided crossing at half-integer values of ng. The degeneracy of the
charge eigenstates (dotted lines) is broken by the Josephson tunnelling, and the energy
eigenstates become the symmetric and anti-symmetric combinations of the charge states
(solid lines).
46 The Superconducting Charge Qubit
Where we have discarded a term proportional to the identity. We can conveniently
write this in matrix form, or in terms of the Pauli matrices:
H =
Ech/2 −EJ/2−EJ/2 −Ech/2
=Ech2
σZ +EJ2σX
Where Ech = EC(1 − 2ng) is the effective charging energy, as determined by the gate
voltage.
The eigenvalues E± and eigenvectors |v±〉 of this Hamiltonian are:
E± = ±1
2
√E2ch + E2
J
|v±〉 =
(1
Ech−2E±EJ
)(2.4.4)
Our gate voltage allows us to manipulate the Hamiltonian and change the eigenstates
and eigenvalues. Two cases of particular interest are the case where we set Ech À EJ ,
and the case where Ech ¿ EJ (but EC À EJ , so the two-level approximation is still
valid). If the charging energy EC À EJ , then as long as we set ng away from 1/2, then
the Hamiltonian will be diagonal and the eigenstates will be the charge states(10
)and
(01
)with eigenvalues ±Ech. If, however, we set ng = 1/2, then the two charge states are
degenerate. This degeneracy is broken by the Josephson coupling, and the eigenstates are
1√2
(11
)and 1√
2
(1−1)with eigenvalues ±EJ/2.
This ability to manipulate the Hamiltonian by adjusting the gate voltage (and also
the flux through the double junction, discussed later in section 2.3) allows us to perform
controlled quantum operations, or quantum logic gates, by applying a series of pules to
the gate capacitor. We shall discuss an example of this in section 2.5.
2.5 Qubit Dynamics and the NOT Gate
From a quantum algorithmic point of view, a not gate is trivial. This is a gate that takes
a qubit that is initially in the state |0〉 =(01
)and returns the state |1〉 =
(10
)(figure 2.7).
This is perhaps the simplest quantum gate, acting on single qubit alone, and from a
algorithmic perspective merits no more than a mention. However, the notion of a ‘gate’ is
2.5 Qubit Dynamics and the NOT Gate 47
1
0
0
1
Figure 2.7: A not gate takes a state |0〉 and returns a state |1〉, and vice versa.
useful precisely because it is an abstract concept independent of the physics of the process
that implements it. In order to even consider discussing ‘gates,’ we must understand the
dynamics of our system thoroughly. Only when we have a complete understanding of the
physics can we ignore it!
In terms of the dynamics of our two-level system, implementing a not gate is not
as trivial as figure 2.7 would suggest, and is certainly challenging experimentally, as the
coherent quantum oscillation of a qubit must be observed. Perhaps the most famous
experimental paper on superconducting charge qubits is the 1999 paper by Nakamura et
al. [59] in which the coherent evolution of a charge qubit was first demonstrated. The
procedure described in this paper is essentially that required to perform a not gate.
In order to illustrate the difficulties and subtleties involved in even the simplest quan-
tum operation, we shall describe this experiment in some detail.
The operation can be broken down into three separate parts: Initialisation, evolution
and readout. It will turn out that the complete operation can be achieved by applying a
single voltage pulse to the capacitor.
In the first part of the operation, initialisation, we wish to set the state of the system to
be a charge eigenstate. This is done by adjusting the parameters of the Hamiltonian 2.4.3
so that one of the charge eigenstates, |0〉 is the ground state. We do this by setting the
gate charge far from the degeneracy point, ng = 1/2. If EC À EJ then the Hamiltonian
will be essentially diagonal in the charge basis and the state |0〉 will be the ground state.
At zero temperature, if we wait long enough the system will decay into this state. At low
but finite temperature, the system will decay to an equilibrium state that is close to the
ground state. A true NOT gate must invert any general input state, but for demonstration
purposes it is easier to use a charge eigenstate as an input state.
We now wish to control the evolution of the system so that the state of the system
oscillates between the states |0〉 and |1〉. We do this by applying a voltage pulse that takes
48 The Superconducting Charge Qubit
the gate charge to ng = 1/2 for the period t0 < t < t0 +∆t. If the gate voltage is ramped
rapidly enough, we can use the sudden approximation and assume that the system remains
in the state |0〉 at the start of the pulse.
The time evolution of a quantum system is governed by a unitary operator that follows
directly from the Schrodinger equation:
|ψ(t)〉 = exp
− i
~T
t∫
t0
H(t′)dt′
|ψ(t0)〉 (2.5.1)
If the Hamiltonian is constant, we can ignore the time-ordering operator T and do the
integral:
|ψ(t)〉 = e−i~ H(t−t0)|ψ(t0)〉 (2.5.2)
With ng set to 1/2, the diagonal parts of the Hamiltonian vanish, and the eigenstates
are the states 1√2
(11
)and 1√
2
(1−1). We write the state of the system at t0 in terms of these
states. Each of these eigenstates then picks up a phase factor when acted on by the time
evolution operator. This phase factor depends on the energy eigenvalues which in this
case are ±EJ/2:
|ψ(t)〉 = e−i~ H(t−t0)
((1
1
)−(
1
−1
))
= e+i~EJ2(t−t0)
(1
1
)− e− i
~EJ2(t−t0)
(1
−1
)
=
(e+
i~EJ2(t−t0) − e− i
~EJ2(t−t0)
e+i~EJ2(t−t0) + e−
i~EJ2(t−t0)
)(2.5.3)
At the end of a pulse, at time t = t0 +∆t, the gate voltage is again moved away from
ng = 1/2, and the Hamiltonian is again diagonal in the charge basis.
The final stage of the operation is the readout stage. This is implemented simply
by keeping the Hamiltonian in the charge basis, so that the probability of being in each
charge state does not change. If the box is in the state with an extra Cooper pair, then
this Cooper pair will decay after a time tqp via quasiparticle tunnelling to the lead, which
can be detected. This decay also has the advantage of re-initialising the system to the
state |0〉.
2.5 Qubit Dynamics and the NOT Gate 49
0
0.2
0.4
0.6
0.8
1
Prob
1 2 3 4(Ej t) /(2 h)
Figure 2.8: Oscillations in the probability of the qubit being in state 1〉 (solid line) or state
|0〉 (dotted line). The times required to perform a not (π/2) and to prepare a superposition
(π/4) are indicated by vertical dashed lines.
In theory the individual quasiparticle decays could be detected, for instance by means
of a single-electron transistor and of course, a single-shot measurement like this is essential
for a quantum computing implementation. If instead we are interested in demonstrating
coherent quantum oscillations, we only wish to know the probability that a Cooper pair
was present, and furthermore would like the measurement to be as simple (experimentally)
as possible.
With this in mind, the oscillation is repeated many times leading to many quasiparticle
decays, to get a (macroscopic) current proportional to the probability of there being an
extra Cooper pair on the box. The beauty of this experiment is that a whole sequence
of measurements can be performed by applying a sequence of voltage pulses of length ∆t
and separated by a time trest > tqp, and that the probability of a Cooper pair being on
the box can be found by making a current measurement on the lead.
At the end of each pulse, the probabilities of the system being in the states |0〉 and|1〉 are given by |〈0|ψ(t0 +∆t)〉|2 and |〈1|ψ(t0 +∆t)〉|2 which evaluate to cos2(EJ2~ ∆t) and
sin2(EJ2~ ∆t).
These probabilities oscillate with an angular frequency EJ~ (figure 2.8.) In Nakamura’s
experiment [59], the value of EJ was altered by manipulating the magnetic flux through
the double junction (as in section 2.3), and the change in the oscillation frequency was
observed.
50 The Superconducting Charge Qubit
To perform a not gate, we just need to set the length of the pulse to ∆t = π~/EJ . From
eq. 2.5.3, the state of a system starting in state |0〉 =(01
)is then |1〉 =
(10
)and vice versa.
It is also worth noting that the system can be put into an equal superposition of the charge
states by setting ∆t = π~/2EJ , in which case the state of the system becomes 1√2
(i1
)if it
was in the state(01
)at the start of the pulse. This can be thought of as the square root
of a NOT gate (√NOT ) in the sense that performing this gate twice implements a NOT
gate. There is no analogue to this gate in classical computing [78].
The oscillations between the two charge states can be taken as evidence that the system
at some point passes through a superposition of the two states [79].
2.6 Decoherence and Noise
We discussed the DiVincenzo checklist in section 1.1, and mentioned that one of the ad-
vantages of solid-state qubits is that they are easily identifiable by their location and
should be scalable to large numbers by means of microfabrication. Their main disadvan-
tage is that they are particularly prone to decoherence. Although the superconducting
energy gap goes some way to reducing this, it does not eliminate decoherence entirely.
One source of error that is intrinsic to many types of qubit is the truncation of the state
space. Many realisations of qubits approximate a two level system by ensuring that there
is a large separation between two low-lying energy levels (the qubit states or computa-
tional subspace) and other states of the system. In atomic or ionic implementations, two
low-lying electronic energy levels are chosen from the entire spectrum. In the case of our
superconducting charge qubits, the computational subspace is the two charge states clos-
est to the gate charge, and the other states are those with greater (or smaller) numbers of
Cooper pairs on the island. The effect of this approximation has been considered in detail
in [80], in which the number of one-qubit gates feasible before decoherence is around 4000
when the ratio of the Josephson and charging energies is 0.02. For two-qubit gates, the
number of operations possible is less, but measurement on the non-computational space
may reduce this error. Some techniques to reduce this problem are considered in [81].
Other sources of decoherence in charge qubits include the fact the voltage pulses applied
to perform the gates will not be perfectly square but will have a finite rise time [82],
coupling to charge noise in the environment, and non-zero inductances [58, 83, 84, 85, 86].
A further effect that needs consideration is the effect of noise in the control parame-
2.6 Decoherence and Noise 51
ters. In subsection 2.6.1 we review the effect of Gaussian noise, which could arise due to
imperfect control of the external control parameters, and in subsection 2.6.2 we present
a calculation demonstrating the effect of discrete jumps in the tunnelling energy, which
could occur due to physical processes such as the parity effect (1.9.1) leading to jumps in
the tunnelling energy with quasiparticle tunnelling.
2.6.1 Gaussian Noise
A simple example of the effect that noise in the parameters has on the dynamics of a qubit
is the effect of Gaussian noise in the Josephson energy EJ . When the gate voltage is set
to the degeneracy point (which is the relevant voltage to demonstrate oscillations, or to
perform a not gate), the evolution in the presence of noise can be calculated analytically.
From eq. 2.5.3, the probability of a system initialised in the state |0〉 evolving to the
state |1〉 after a time ∆t is given by P (1) sin2(EJ2~ ∆t). For this calculation, we consider
a Josephson energy that fluctuates on an longer timescale that of the voltage pulse, i.e.
tfluc > ∆t, so that the Josephson energy can be considered constant for each evolution.
We can then find the current measured by the lead by calculating the evolution for a
particular EJ and then averaging over the ensemble of all possible values of EJ , which we
take to have a Gaussian distribution.
The time evolution of a charge qubit set at the degeneracy point ng = 1/2 is given in
eq. 2.5.3, and the probability |〈1|ψ(t)〉|2 that the system will be in state |1〉 is given by
P (1) = sin2(EJ2~ ∆t). To calculate the evolution in the presence of noise, we average the
probability over all EJ :
Pav(1) =1√
2π∆EJ
∞∫
−∞
e− (EJ−EJ )
2
2(∆EJ )2 sin2
(EJ2~
∆t
)dEJ (2.6.1)
where we have assumed that EJ has a Gaussian distribution of width ∆EJ centred on EJ .
We do the integral by completing the square and find:
Pav =1
2
(1− e−
(∆t∆EJ
~
)2cos
EJ∆t
~
)(2.6.2)
After averaging, we have oscillations with a frequency EJ/~, determined by the average
Josephson energy, that decay exponentially with a time constant ∆EJ/~, determined by
the amount of noise.
52 The Superconducting Charge Qubit
0
0.2
0.4
0.6
0.8
1
Prob
2 4 6 8 10 12 14(Ej t) /(2 h)
Figure 2.9: The probability of a Cooper pair box being in the state |1〉 as a function of
time, with Gaussian noise in the Josephson energy. The exponential decay envelope is
plotted as dotted lines.
Although these analytic results are only for the case ng = 1/2, an exponential decay
over time is a generic feature of the time evolution when the parameters have a Gaussian
distribution over their average value. Numeric analysis of the effect of Gaussian noise in
EJ and EC can be found in [87].
2.6.2 Discrete Noise in the Josephson Energy
Whilst Gaussian noise is a good generic model of noise in the control parameters, there
are other types of noise which may have different effects on the dynamics. in this section,
we present new work on a simple model of discrete noise.
In this model we consider the case where the Josephson energy can fluctuate, but only
between two discrete values. Again, we consider the case when the gate charge is set to
ng = 1/2.
We model the fluctuation in EJ by assuming that the distribution in the number of
jumps between the two Josephson energy values follows the Poissonian distribution, i.e.
that over a period of time t, the probability of there being j jumps in the Josephson energy
is given by e−βt (βt)j
j! .
The calculation of the evolution of the system is found by calculating the evolution for
a particular number of jumps j, and then averaging over all j.
For this calculation it will be helpful to use a density matrix formulation ρ(t). The
2.6 Decoherence and Noise 53
time evolution of the density matrix is given by ρ(t) = U(t)ρ(0)U †(t) where U(t) is the
time evolution operator that operates on the wavefunction in eq. 2.5.1. If ρch is the density
matrix in the charge basis, then we can make a transformation to the energy eigenbasis,
ρeg:
ρeg =1
2
1 1
1 −1
ρch
1 1
1 −1
ρeg =1
2
1 1
1 −1
ρ00 ρ01
ρ10 ρ11
1 1
1 −1
(2.6.3)
As before, we make the sudden approximation, so that for a given series of jumps at
times τ1, τ2 etc. the time evolution of the density matrix in the energy eigenbasis is given
by:
ρeg(t) = U(t)ρeg(0)U†(t)
= Uj(t− τj) · · ·U1(τ2 − τ1)U0(τ1)
ρeg(0) U†0(τ1)U
†1(τ2 − τ1) · · ·U †j (t− τj) (2.6.4)
The off-diagonal elements become:
ρ+/−(t) = ρ+/−(0)e−iEJ0τ1~ e−i
EJ1(τ2−τ1)
~ · · · e−iEJj(t−τj)
~ (2.6.5)
and its complex conjugate.
We have specified the number of jumps, and need to average over all possible times τ
of those jumps by integrating over all possible times for the jumps, and then dividing by
the integration time:
ρav1+/−(t) = ρ+/−(0)
t∫
0
dτ1t
t∫
τ1
dτ2t− τ1
· · ·t∫
τj
dτjt− τj
e−iEJ0τ1
~ e−iEJ1(τ2−τ1)
~ · · · e−iEJj(t−τj)
~
ρav+/−(t) = χ(t)ρ+/−(0) (2.6.6)
where χ denotes the series of integrals over t. This means that we can write the density
matrix in the charge basis as:
54 The Superconducting Charge Qubit
ρch =1
2
1 1
1 −1
ρ+/+(0) ρ+/−(0)χ(t)
ρ−/+(0)χ∗(t) ρ−/−(0)
1 1
1 −1
(2.6.7)
The problem of the time averaged evolution is now reduced to the problem of calcu-
lating the integral χ(t).
For instance, in the case of a single jump.
χ(t) =1
t
t∫
0
dτ1e−iEJ0τ1~ e−i
EJ1(t−tau1)
~
=
(e−i
EJ0t
~ − e−iEJ1t
~
−i(EJ0 − EJ1)t/~
)(2.6.8)
If we started the system in the state |0〉 =(01
), we find that the average probability of
being in the state |1〉 after a time t given that a single jump has occurred is:
ρav111 (t) =1
2− ~
4(EJ0 − EJ1)t
(sin
2EJ0t
~− sin
2EJ1t
~
)(2.6.9)
So the overall time averaged evolution, including the case where there is no jump, and
the case with only one jump (we assume 2 or more jumps are unlikely, i.e. β ¿ 1) becomes
(figure 2.10):
ρav0,111 (t) = e−βt(1− sin2
EJ0~
)
+(e−βtβt)
(1
2− ~
4(EJ0 − EJ1)t
(sin
2EJ0t
~− sin
2EJ1t
~
))(2.6.10)
There are two timescales in this averaging; 1/β, which determines the time between
jumps, and 1/(EJ0 − EJ1)/~. At times longer than the latter timescale, the average
evolution reduces to the exponential decay seen in section 2.6.1.
Further terms can be found by calculating the integrals χ(t) (eq. 2.6.6) over more
jumps, and the evolution of the density matrix calculated. The evolution of the density
matrix element ρ11, which represents the probability that the system will be found in the
state |1〉, is given by:
2.6 Decoherence and Noise 55
10 20 30 40
0.2
0.4
0.6
0.8
1
Figure 2.10: The probability of a Cooper pair box being in the state |1〉 as a function of
time with discrete noise in the Josephson energy. One jump is included, with parameters
EJ0 = 1, EJ1 = 0.5, β = 0.1. The Gaussian decay is plotted as dashed lines with the
ideal, non-decaying oscillations denoted by dotted lines.
ρav11(t) =∑
j
e−βt(βt)j
j!
(1
2− 1
2(ρ+/−(0)χ(t) + ρ−/+(0)χ
∗(t))
)(2.6.11)
Chapter 3
The Josephson Circuit
Hamiltonian
In this chapter we examine in greater detail the construction of the qubit Hamiltonian for
superconducting charge qubits. We first examine the circuit diagrams of individual and
multiple qubits, and consider the capacitive charging energy of these circuits. We confirm
that the capacitive term found is correct by deriving the Josephson equations of motion for
each junction, and finally produce a quantum Hamiltonian for each circuit.
3.1 The Capacitance of Qubit Circuits
In section 2.2 we derived a Hamiltonian for a single Josephson junction (2.2.10). This
Hamiltonian was written in terms of phase. In order to derive it, we took the Josephson
junction relations, i.e. the equations of motion for the phase difference between the two
superconductors, found a classical Hamiltonian and a set of conjugate variables from which
these equations of motion could be derived using Poisson brackets, and then elevated the
conjugate variables to operators.
A realistic superconducting charge qubit will be much more complicated than a single
Josephson junction. For instance, a single qubit consisting of an island coupled to a
reservoir will not only contain a Josephson junction, but will also need a gate voltage
applied to the island in order to adjust the relative energies of the two charge states. The
island we are discussing is small, and so we will have to consider its self-capacitance as
well as the capacitance between it and the reservoir.
57
58 The Josephson Circuit Hamiltonian
When quantising the Josephson junction relations, we write down equations for motion
of phase and voltage difference. In order to derive the Josephson relations from single-
electron operators, it will be useful to have a charging term which is written in terms
of total charge (number of Cooper pairs) on an island, rather than the charge stored on
a capacitor. In this, we are following the spirit of [68, 47] in calculating the effective
capacitance of a quantum circuit. However, we go further in that we show that this leads
to a quantum Hamiltonian that reproduces the correct Josephson equations for the circuit
as its equations of motion. We also show a general method of performing such calculations.
Finally, a single qubit consists of an island coupled to an infinite reservoir. We would
like to be sure we are taking the correct limit as we allow the size of the reservoir to
become large.
3.1.1 Single Junction with Gate Voltage
We wish to consider more complicated circuits including gate voltages and self-capacitance.
As the circuits become more complex, it will become convenient to solve the circuit equa-
tions using a computer. As with any calculation done on computer, we need to be especially
clear of the procedure used, and so we shall go through a simpler problem, in order to
illustrate the details of the calculation.
After a single Josephson junction, the most simple case to consider is that of two
regions of superconductor coupled by a capacitive Josephson junction, with a gate voltage
between the two regions. For the present, we ignore any self capacitance.
The regions of superconductor are labelled 1 · · · i for the small islands, and r for the
reservoir (in this case 1, r). The charges on these regions are labelled with a single index
qi. The capacitors between the regions i, j have capacitances Cij . The charge on the
capacitor Cij is qij (with a double index) and the voltage across it is Vij . The gate voltage
and their capacitances are labelled with a g. The sign of the charges on the capacitors
and voltage sources is arbitrary, but needs to be consistent within each calculation.
We wish to write the capacitive energy of the circuit in terms of the charges on the
superconductors. We start with Kirchoff’s laws and have one equation (3.1.1) representing
the fact that the sum of the voltages around the circuit must sum to zero, and another
(3.1.2) that indicates that the charge on each island is given by the sum of the charges on
the capacitors connected to it.
3.1 The Capacitance of Qubit Circuits 59
q1
qr
q1r-q1r
q1g
-q1g
q1g
-q1gV1g
Figure 3.1: Single Voltage with Gate Voltage Circuit Diagram
q1rC1r
+q1gC1g
+ V1g = 0 (3.1.1)
q1r − q1g = q1 (3.1.2)
Note that we have no equation for the charge on the reservoir. In this calculation we
are assuming that the circuit is electrically neutral overall, and so q1 = −qr. We shall
relax this restriction in the next calculation.
We solve equations 3.1.1 and 3.1.2 for q1r, q1g:
q1r =
(C1rC1g
C1r + C1g
)(q1C1g− V1g
)
q1g = −(
C1rC1g
C1r + C1g
)(q1C1r
+ V1g
)(3.1.3)
We write the energy due to the charges in the circuit in terms of the potential energy
of the charges on the voltage source and the work done in placing the charges on the
capacitors:
EC =q21r2C1r
+q21g2C1g
+ V1gq1g (3.1.4)
We now substitute the expressions for the charges on the capacitor (eqns. 3.1.3) into
the above expression for the charging energy:
60 The Josephson Circuit Hamiltonian
q1
qr
q1r-q1r
q1g
-q1gq1g
-q1g V1g
q1w-q1w
qrw -qrw
qw
Figure 3.2: Single junction with gate voltage and self-capacitance, which is represented
by a capacitive coupling to a third region, labelled w (for ‘world’).
EC =
(C1rC1g
C1r + C1g
)2( 1
2C1r(q1/C1g − V1g)2 +
1
2C1g(q1/C1r − V1g)2
)+ V1gq1g
=1
2(C1r + C1g)(q1 − V1gC1g)
2 −V 21gC1g
2(3.1.5)
This result shows that the gate voltage has had the effect we expected; the junction
can be described with an effective capacitance, given by 1/Ceff = 1/(C1r + C1g), and a
term that effectively resets the zero of capacitance. This term we call the gate charge, and
denote qg = V1gC1g.
3.1.2 Single Junction with Gate Voltage and Self-Capacitance
As mentioned earlier, in the previous section we have made the restriction that our circuit
is electrically neutral, i.e. that q1 + qr = 0. As we want to consider the region r as a
reservoir, this is an undesirable assumption. In order to deal with this, we introduce
a third region that represents ‘the rest of the world.’ This means that we can use the
restriction that the entire circuit (including the ‘world,’ w), is neutral, and our qubit
circuit itself (1+ r) does not have to be. The self-capacitance of the regions is represented
by their coupling to w.
This time we have two voltage loops around the circuit, and two charges on the regions
1, r. Again, the charge on w gives us no new information as we take the entire circuit to
3.1 The Capacitance of Qubit Circuits 61
be charge neutral.
q1rC1r
+q1gC1g
+ V1g = 0
q1wC1w
+qrwCrw
− q1rC1r
= 0
q1g − q1r − q1w = q1
q1r + qrw − q1g = qr (3.1.6)
We have four equations, which is enough to give the four charges on the capacitors in
terms of q1, qr. Solving four simultaneous equations by hand would be time consuming,
but can easily be solved on a computer, using a package such as Maple or Mathmatica
(as long as we are careful - this is why we have gone through the previous calculations in
such detail). As an example, the expression found for q1g is:
are similar expressions for the other capacitor charges. We can substitute these into the
equation for charging energy, and then rearrange to get a charging energy in terms of q1
and qr.
EC =q21r2C1r
+q21w2C1w
+q2rw2Crw
+q21g2C1g
+ V1gq1g
=(C1r + C1g + Crw)
2C2Σ
(q1 − V1gC1g)2 +
(C1r + C1g + C1w)
2C2Σ
(qr + V1gC1g)2
+(C1r + C1g)
C2Σ
(q1 − V1gC1g)(qr + V1gC1g) +K (3.1.8)
where K represents constant terms. We see that we now have three terms, a term pro-
portional to q21, a term proportional to q2r , and a cross term. Again, the ‘zero position’ is
determined by the gate charge.
We have specifically included a region, w, representing the external world, allowing us
to consider the case where the charge on the circuit is non-zero. Now this has been done
with general capacitances, we can take the limit where C1w → 0 and Crw → ∞, i.e. the
62 The Josephson Circuit Hamiltonian
limit where the island is totally unconnected to the outside world, and the reservoir, r is
strongly connected.
In this limit, the coefficients of q2r and q1qr go to zero. The limit of the q21 coefficient
is:
(C1r + C1g + Crw)
2C2Σ
→ Crw2(CrwC1g + CrwC1r + CrwC1w)
→ 1
2(C1g + C1r)(3.1.9)
By comparing this result with eq. 3.1.5 we see that we have regained the previous
result. This confirms that eq. 3.1.5 is indeed a valid expression for the charging energy of
an isolated island connected via a capacitive tunnel junction to a reservoir.
We have one further task remaining; the energy is given in terms of the charge on the
island, q1, and the Josephson equations are given in terms of phase and voltage differences.
This was unimportant in the previous case, as we were only considering two isolated
islands, and therefore q1 = −qr = 1/2(q1 − qr). To express our charging energy for this
system in terms of charge differences, we substitute qs = (q1 + qr)/2 and q1r = (q1− qr)/2into the expression for charging energy (eq. 3.1.8) before we take the limit.
3.1.3 The 2 Qubit Circuit
Now we have confirmed the form for the charging energy of a single junction, we can move
onto a larger system. We wish to find the charging energy of a two-qubit system. This
consists of two islands coupled to a large superconductor. As before, we wish to find a
result when the reservoir has a finite size and is coupled to the outside world, and only
then take the limit of an infinitely large reservoir. We also include a gate voltage on each
of the two islands.
Examining the circuit diagram (fig. 3.3), we see that we can define five loop equations
and three equations for the charges on the islands. This gives us enough information to
calculate the eight unknown capacitor charges, qij .
As before, inserting these values for the capacitances into the expression for charging
energy (eq. 3.1.10) gives us the charging energy in terms of the charges on the islands q1,
q2, qr.
3.1 The Capacitance of Qubit Circuits 63
q1 q2
qr
q2w
qw
q12
q1g
q1w
qrw
q1r
q2gq2r
-q1w -q2w
-qrw
q1g
-q1g
-q1g
-q1r
-q2r
-q2g
q2g
-q2g
V2gV1g
-q12
Figure 3.3: Two Qubit circuit diagram. The system consists of two islands (1, 2) connected
to a reservoir (r), with gate voltages applied to each. We wish to solve the problem for a
reservoir of general size, and so a further region, w is included to represent the rest of the
world.
64 The Josephson Circuit Hamiltonian
EC =∑
i,j 6=i
q2ij2Cij
+∑
i6=gVigqig (3.1.10)
As before, we are considering region r to be a large reservoir, and so we now wish
to take the limit where q1w, q2w → 0 and qrw → ∞. We find, as before, that the terms
containing the charge on the reservoir qr vanish. We find that the charging energy only
depends on the charges on the two islands.
EC =(q1 − q1g)22Ceff 1
+(q2 − q2g)22Ceff 2
+(q1 − q1g)(q2 − q2g)
2Ceff 12(3.1.11)
We have a term containing only q1, one containing only q2, and a cross term which
is the capacitive coupling between the qubits. Again, the gate charges qg modify the
zero-position of charge. The effective capacitances are given by:
1
Ceff 1=
(C2g + C2r + C12)
C2Σ
1
Ceff 2=
(C1g + C1r + C12)
C2Σ
1
Ceff 12=
2C12
C2Σ
C2Σ = C2gC12 + C2gC1g + C2gC12 + C12C2r
+C2rC1g + C1rC2r + C1gC12 + C1rC12 (3.1.12)
3.1.4 Multiple Islands
The same process as above can be applied to three or more islands. We find very similar
results; there is a quadratic term dependent on the charge on each island alone, and a
cross term between any two charges:
EC =∑
i
(qi − qig)2Ceff i
+∑
i,j 6=i
(qi − qig)(qj − qjg)Ceff ij
(3.1.13)
We examine some of the effective capacitances of a three-island system, in particular
the cross terms between islands:
3.2 Quantising the Josephson Relations 65
1
Ceff 12=
2C12(C3g + C23 + C3r)
C3Σ
1
Ceff 13=
2C12C23
C3Σ
(3.1.14)
where C3Σ is a term with units of capacitance cubed. Every island in the system has an
interaction with every other. However, if we consider the interaction capacitances to be
much smaller than the capacitances between the islands and the reservoir, we find that
effective capacitances between islands that are not adjacent are correspondingly reduced.
3.2 Quantising the Josephson Relations
In section 2.2 we quantised the equations of motion of a single capacitive Josephson junc-
tion between two islands. In this section we wish to ensure that this procedure is still
valid when we consider more complicated circuits. We expect the Hamiltonian of a circuit
to consist of a capacitive term and a Josephson tunnelling term. If we have the correct
Hamiltonian and conjugate variables, we should find that the equations of motion given
by the Poisson brackets (classically) or commutators (quantum mechanically) are just the
Josephson equations for that circuit. Again, we shall consider the simplest case in detail
first before we examine more complicated circuits.
3.2.1 Single Junction with Gate Voltage
In order to find a classical Hamiltonian for the circuit in 3.1, we shall guess a Hamiltonian
based on the charging energy found in 3.1.1 and a Josephson tunnelling energy, and a
pair of conjugate variables. We shall then check if this Hamiltonian is indeed correct by
comparing the equations of motion derived from it to the Josephson equations for the
circuit.
We consider the Hamiltonian:
H =(q1 − V1gC1g)
2
2(C1r + C1g)− 2T cos (φ1 − φr) (3.2.1)
The conjugate variables are the phase φ1, and the conjugate momentum to this, N1 =
~q1/2e. The Poisson brackets for these two variables give:
66 The Josephson Circuit Hamiltonian
φ1, H =dH
dπ
=2e
~(q1 − C1gV1g)
2(C1r + C1g)(3.2.2)
Next we note that the voltage across the junction is given by V1r = q1r/C1r:
V1r =C1rC1g
C1r + C1g
q1/C1g − V1gC1r
=q1 − V1gC1g
C1r + C1g(3.2.3)
Inserting this value for the voltage into equation 3.2.2, we find:
d (φ1 − φr)dt
= φ1, H
=1
~(2e)V1r (3.2.4)
This is the expected Josephson equation for rate of change of phase difference, i.e. that
it is proportional to the voltage across the junction.
Similarly, the rate of change of charge is given by its Poisson bracket:
d q1dt
=2e
~π1, H
=−2T (2e)
~− dH
dφ1
=−2T (2e)
~sin(φ1 − φr) (3.2.5)
That is, the rate of change of charge on the island is given by the current onto the
island.
We have found that the Hamiltonian (eq. 3.2.1) for a single junction with a gate
charge leads to the equations of motion for the charge and phase difference given by the
Josephson relations for that junction. This confirms that this indeed the correct classical
Hamiltonian. We obtain the quantum Hamiltonian by elevating the variables N1 and φ1 to
be operators with the commutation relation [φ1, N1] = i. We can now use the commutation
relations to find the equations of motion for the expectation values of the operators.
3.2 Quantising the Josephson Relations 67
d 〈(φ1 − φr)〉dt
=2e
~2e〈N1〉 − C1gV1g
C1r + C1g
d 2e〈N1〉dt
=−2T (2e)
~〈sin(φ1 − φr)〉 (3.2.6)
We now move on to more complicated circuits.
3.2.2 Single Junction with Gate Voltage and Self-Capacitance
In section 3.1.2 we considered two regions of superconductor coupled to each other and to
a third region representing the rest of the world. We found the charging energy in terms
of the charges on the islands, and then took the limit C1w → 0, Crw → ∞. We can also
check that the Hamiltonian gives the correct equations of motion before the limit is taken.
Using the Hamiltonian H = EC−2T cos(φ1−φr) (we drop the hats on operators Ni, φi
for convenience) with the EC found in section 3.1.2, we find the commutator for the phase
where Ech1 = EC1(1/2− n1,g) is a charging energy term that depends on the gate charge
n1,g = C1gV1g. The four basis states are the charge states |00〉, |01〉, |10〉, |11〉 respectively.Note that if both charging energies are set to 1/2 the four charge states are degenerate.
The interaction between the qubits is:
HI =
0 0 0 0
0 −EC12n2g −T12 0
0 −T12 −EC12n1g 0
0 0 0 EC12(1− n1g − n2g)
(4.1.3)
4.1.1 Energy Levels
The energy levels and of the Hamiltonian HS + HI are plotted in for the case when
the inter-qubit tunnelling is zero (figure 4.1) and when it is included (figure 4.2). The
parameters are chosen to be experimentally reasonable, considering those quoted in [69].
When the gate voltages are set far from the degeneracy point, the eigenstates are close to
4.1 The Two Qubit Circuit 73
–100
–50
0
50
100
0 0.2 0.4 0.6 0.8 1
Figure 4.1: The energy levels of a two qubit system as a function of n1g = n2g, when the
tunnelling between the islands is zero. The parameters of the system are EC1 = EC2 = 100,
EC12 = 10, T1 = 10, T2 = 10
the charge states. The energies of the states |00〉 and |11〉, where either both islands have
an excess Cooper pair or neither do, are well separated in energy. The states with only
one island having an extra Cooper pair are degenerate if the boxes are identical.
When the gate charges are set to the degeneracy position, n1g = n2g = 1/2, all four
charge states of the non-interacting system would be degenerate. The degeneracy between
|00〉, |11〉 and the two states |01〉 and |10〉 is broken by the tunnelling energies T1, T2. The
additional degeneracy between |01〉 and |10〉 is broken by the charging interaction EC12.
The degeneracy in figure 4.2 is due to the fact that we have treated the islands as
identical. Any difference in the tunnelling energies T1, T2 will break the degeneracy.
4.1.2 Two-Qubit Gates
Universal quantum computation can be achieved if we have a complete set of one-qubit
gates and a two-qubit gate that can produce a maximally entangled state. In this section
we show how an entangling two-qubit gate can be performed. As in section 2.5, the
parameters of the qubits are set in such a way that the time evolution of the system
performs the gate. We are considering a system with no tunnelling between the islands,
74 The Dynamics of a Two Qubit System
–100
–50
0
50
100
0 0.2 0.4 0.6 0.8 1
Figure 4.2: The energy levels of a two qubit system as a function of n1g = n2g, when
the tunnelling between the islands is included. The parameters of the system are EC1 =
EC2 = 100, EC12 = 10, T1 = 10, T2 = 10, T12 = 30
i.e. T12 = 0, and equal tunnellings from the two islands to the reservoir T1 = T2. The
evolution is plotted in figure 4.3.
If the system is initialised in the state |01〉 then it will oscillate to the state |10〉 aftera time determined by the differences in the eigenenergies (denoted e2 and e4 in appendix
B).
Halfway through the oscillation, the state will be:
|ψ〉 = 1
2((1 + i)|01〉 − (1− i)|10〉) (4.1.4)
Which is a maximally entangled state. This is an alternative to the square root of
imaginary swap, (√iSWAP ) gate described in [88].
4.2 A Bell Inequality
For a small superconducting island to function as a true qubit, we must be able to perform
many complicated operations on it. One set of operations of particular interest is a set
4.2 A Bell Inequality 75
0
0.2
0.4
0.6
0.8
1
0.2 0.4 0.6 0.8 1 1.2 1.4t
Figure 4.3: Oscillations between state |01〉 (solid line), |10, 〉 (dashed line) and the states
|00〉 and |11〉 (dotted lines). Halfway between oscillations, the system is in a maximally
entangled state (vertical lines).
of operations to test a Bell inequality [89]. As an example of the level of control that we
must have over the charge qubit, we shall go through the operations in detail.
An example of an experiment to test a Bell inequality that is particularly easy to
visualise is that of a pair of spin-half particles being emitted from a source to widely
separated locations. The particles are in a spin singlet state |s〉 that has zero total angular
momentum:
|s〉 = 1√2(| ↑〉A| ↓〉B − | ↑〉B| ↓〉A) (4.2.1)
where | ↑〉 represents the state when a particle is in the spin-up eigenstate of the σZ
operator, and | ↓〉 represents spin down.
At the two locations (A and B), there are devices to measure the spin along a given
direction (SA(θA), SB(θB)). We measure the spin along two directions, θ, θ′ at each
location and take the correllators, C(θA, θB), between them.
76 The Dynamics of a Two Qubit System
z
y
z+yz-y
Figure 4.4: The Bell inequality is violated if a pair of particles in the singlet state are
measured along the bases shown. Note that in three of the combinations of measurements,
the bases are at 45 deg to each other, and in one of the measurements (y, (z−y)/√2) they
The measurements at A are made in the perpendicular directions z and y (with the
corresponding operators SZA and SYA ) , and the measurements atB are made along 1√2(z+y)
and 1√2(z − y), (figure 4.4).
4.2 A Bell Inequality 77
C(z,1√2(z + y)) = 〈s|SZA ,
1√2(SZB + SYB )|s〉
= −1/√2
C(z,1√2(z − y)) = −1/
√2
C(y,1√2(z + y)) = −1/
√2
C(y,1√2(z − y)) = +1/
√2 (4.2.4)
We find that these correllators violate the Bell inequality:
C(z,1√2(z + y)) + C(z,
1√2(z − y)) + C(y,
1√2(z + y))− C(y, 1√
2(z − y))
= | − 1/√2− 1/
√2− 1/
√2− (+1/
√2)|
£ 2 (4.2.5)
The violation of the Bell inequality shows in a very clear and simple way that quantum
mechanics produces results that are fundamentally different from any possible classical
results. As such, a demonstration of a violation of a Bell inequality in a system is an
indication of quantum behaviour. Furthermore, recent experiments have offered hints of
entanglement in superconducting qubit systems, [69, 62], and the ultimate proof of the
existence of entanglement is the violation of a Bell inequality.
Although a coupled pair of charge qubits is a very different physical system from the
two spin-half particles described above (in particular, they are not spatially separated in
the relativistic sense and so cannot violate relativistic locality) we can use Bell inequality
demonstration as an example of the kind of control necessary to implement quantum
computing protocols.
In the above discussion, we stated that the particles began in a spin singlet state. We
need to describe how we can produce such a state. We initialise the system in the state |01〉by setting the gate charges to n1g = 1, n2g = 0 so that Ech1 = EC1(1/2− n1g) = −EC1/2and Ech2 = EC2(1/2−n2g) = EC2/2. With the tunnelling energies much smaller than the
charging energies, the Hamiltonian of the system becomes:
78 The Dynamics of a Two Qubit System
HS =
EC1−EC22 0 0 0
0 −EC1−EC22 0 0
0 0 EC1+EC22 0
0 0 0 −EC1+EC22
(4.2.6)
The ground state of the system is now |01〉 if we wait long enough, the system will decay
to this state. If we now suddenly set n1g = n2g = 1/2 and set the individual tunnelling
energies T1, T2 to zero, then only the tunnelling between the two islands remains. If we
allow the system to evolve (under the evolution operator U = e−iHt~ ) for time t = π~
4T12the
system evolves to the state (|01〉−i|10〉)√2
. We need to perform a further pulse to change this
state to the singlet state. If we suddenly set n1g = 1/2 and n2g = 1, and evolve for a time
t = π~Ech2
then the system evolves to be in the singlet state.
We now wish to make measurements of the four operators SZA , SYA ,
SZB+SYB
2 andSZB−SYB
2 .
In a system of superconducting charge qubits however, we can only measure in the SZ
(charge) basis. To get round this we must ‘rotate’ the state of each qubit so that measuring
SZ is equivalent to measuring in some other basis. We achieve this by setting the tunnelling
energies to the values given in eq. 4.2.7.
SZA : T1t~ = 0
SYA : T1t~ = −π
4SZB + SYB√
2: T2t
~ = −π8
SZB − SYB√2
: T2t~ = +
π
4(4.2.7)
For example, if we wish to measure qubit A in the y basis, and qubit 2 in the zB+yB√2
basis, we would set the tunnelling energies to T1t~ = −π
4 and T2t~ = −π
8 and evolve for a
time t.
After this rotation, the system will be in one of four states:
4.2 A Bell Inequality 79
U(z,1√2(z + y))|s〉 =
−i(1−√2)
1
−1i(1−
√2)
1
2√
2−√2
U(z,1√2(z − y))|s〉 =
i(1−√2)
1
−1−i(1−
√2)
1
2√
2−√2
U(z,1√2(z + y))|s〉 =
i(1−√2)
1
−1−i(1−
√2)
1
2√
2−√2
U(z,1√2(z − y))|s〉 =
−i−(1−
√2)
(1−√2)
i
1
2√
2−√2
(4.2.8)
We now measure both qubits in the z basis. The probabilities of each result are given
by the squares of the elements in the vectors representing the states after rotation given
in eq. 4.2.8, eg. P (00) = |〈00|U(z, 1√2(z + y))|s〉|2. These, when inserted into 4.2.2 lead
to the values of the correllators given in eq. 4.2.4. Again, we find that the Bell inequality
is violated: 2√2 £ 2.
We have given a brief description of how a Bell inequality violation could be observed
in superconducting charge qubits. It is apparent from even this simple analysis that the
series of gates that need to be applied is complicated. Current experiments are now at the
point where a single 2-qubit gate is possible, and so a simple signature of 2-qubit quantum
behaviour would be useful experimentally. In particular, it would be useful if we could
observe a phenomenon that is intrinsically quantum mechanical but does not require the
complex control of a demonstration of the violation of a Bell inequality.
80 The Dynamics of a Two Qubit System
T1
EC12
EC1
T12
EC2 T2
Φ
1 2
Figure 4.5: A two qubit circuit with tunnelling between the qubits and a magnetic flux
passed through the circuit
4.3 Virtual Tunnelling
In section 4.1.2 we considered a two-qubit system with no qubit-qubit tunnelling, and
looked at oscillations between states that occurred when the gate voltages were set to the
degeneracy positions (figure 4.3)
An simple extension to the investigation of these oscillations is to consider the oscilla-
tions when tunnelling between the qubits is allowed and a magnetic field is passed through
the circuit (figure 4.5).
When a Cooper pair completes a circuit it picks up a phase equal to the line integral of
the vector potential A around that circuit (which is equal to the flux through the circuit
- see section 2.3). In terms of our Hamiltonian describing the two two-level systems, we
ensure that the correct phase is picked up by including a phase factor ei 2πΦ0
∫c
A.ds= e
i 2πΦΦ0
in one of the tunnelling matrix elements, T12. Note that it does not matter which of the
tunnelling elements the phase is included in as long as the correct phase is picked up on
4.3 Virtual Tunnelling 81
completing the loop - the overall phase is unimportant.
For simplicity, we consider the gate charges to be set to 1/2, where the charge states
We wish to investigate the evolution of the system over time, and in particular, how
this evolution is affected by the magnetic field. As usual, we calculate the eigenvectors of
the Hamiltonian, write the initial state in terms of these eigenvectors, and multiply each
eigenvector by a phase factor dependent on the eigenenergy. Unfortunately, the above
Hamiltonian cannot be solved exactly (unless we force T12 to be real) and so the time
evolution must be calculated numerically.
Starting in the initial state |00〉, the evolution of the system is as shown in Figs. 4.6 -
4.7:
We note that not only do the probabilities oscillate in time as seen in recent papers
(eg. [69]), but also with the magnetic flux through the loop.
Whilst it is easy to calculate the evolution of the system numerically, we would like to
find a framework for understanding these oscillations.
In order to gain an understanding of this system, we set the charging interaction
between the qubits EC12 to be much larger than the tunnelling either between the islands
and reservoir, or between the two islands themselves. In this case, the charge states form
two pairs, each of which is degenerate. The states where only one of the island has a
charge on (|01〉 and |01〉) have lower energy, −EC12/2. The other pair of degenerate states(|00〉 and |11〉) is higher in energy.
The tunnelling elements break the degeneracies, and so the eigenstates of the Hamil-
tonian are then (to first order) given by the symmetric and anti-symmetric combinations
of each set of two states. That is:
|v1,4〉 =|00〉 ± |11〉
2
|v2,3〉 =|01〉 ± |10〉
2(4.3.2)
82 The Dynamics of a Two Qubit System
0
1
2
3
4
5
6
Flux
Time
0
1
2
3
4
5
6
Flux
Time
Figure 4.6: The Probability of the Two Qubit system being in the state |00〉 (upper plot)and |01〉 (lower plot). The horizontal axis is time and the vertical axis is the flux through
the loop in units Φ02π . White represents high probability of occupation and black represents
low probability.
4.3 Virtual Tunnelling 83
0
1
2
3
4
5
6
Flux
Time
0
1
2
3
4
5
6
Flux
Time
Figure 4.7: The Probability of the Two Qubit system being in the state |10〉 (upper plot)and |11〉 (lower plot). The horizontal axis is time and the vertical axis is the flux through
the loop in units Φ02π . White represents high probability of occupation and Black represents
Figure 4.8: The Probability of the Two Qubit system being in each of the four charging
states. The solid line is the probability of being in |00〉, the dashed line state |11〉, andthe probability of occupation of the other two states are just visible above the time axis.
It is easy to see that if we start the system in the charge eigenstate |01〉 it will oscillatebetween this state and the state |10〉. Similarly, if the system starts in the state |00〉, theoscillations will be between |00〉 and |11〉. This oscillation is plotted in figure 4.8.
It is interesting to note two facts about this oscillation: the system never has a signif-
icant probability of being in state |01〉 or |10〉, and there is no tunnelling element in the
Hamiltonian between the states |00〉 and |11〉. We have an apparent contradiction - if we
were to think classically, we would say that the system must move through either |01〉 or|10〉, but at no time is there any probability of being in either state.
This is an example of the quantum mechanical phenomenon of virtual tunnelling where
a system ‘passes through’ a state without occupying that state. If we think of this process
physically, the system starts with no charge on either island i.e. the two excess Cooper
pairs are both in the reservoir. The system tunnels to the state where there is a Cooper
pair on each island without passing through the intermediate step of having only one
island occupied by a Cooper pair.
The frequency of these oscillations can be calculated, and is given by the difference of
4.3 Virtual Tunnelling 85
the two energy eigenvalues e1, e4. Because the Hamiltonian (eq. 4.3.1) cannot be solved
analytically, we have to calculate this numerically in general. We do find an analytic
solution, however, when all the tunnelling elements are real. This means the eigenvectors
and eigenvalues can be found when the phase on T12 is 0 or π, i.e. when there is an integer
or half-integer number of flux quanta through the circuit. The eigenvalues are:
e1,4 = −1
4EC12 ∓
1
2T12 +
1
4
√(EC12 ± 2T12)2 + 16(T1 ± T2)2
e2,3 = −1
4EC12 ∓
1
2T12 −
1
4
√(EC12 ± 2T12)2 + 16(T1 ± T2)2 (4.3.3)
To find the frequency of the oscillations between |00〉 and |11〉, we need to find e1− e4.To first order in T/EC12, this is zero. To third order, we find:
e1 − e4 = −8T1T2EC12
− 8T12(T21 + T 2
2 )
E2C12
+O(
T
EC12
)4
(4.3.4)
The oscillation between |01〉 and |10〉 has a frequency that is non-zero to first order in
T/EC12:
e2 − e3 = −2T12 +8T1T2EC12
+O(
T
EC12
)3
(4.3.5)
These differences give a good estimate of the frequencies of the oscillations, as is shown
in figure 4.9.
Examining the terms in the oscillation frequencies, we can see that they give an in-
dication of the physical processes at work. There is a direct tunnelling matrix element
between |01〉 and |10〉, and T12 appears to first order in the oscillation frequency. There
is no direct tunnelling between |00〉 and |11〉, and so the first non-zero term is 8T1T2EC12
. This
can be thought of representing the virtual tunnelling processes of a Cooper pair tunnelling
through one of the intermediate states |01〉 or |10〉.
a) |11〉 = T2|01〉 = T2(T1|00〉)
b) |11〉 = T1|10〉 = T1(T2|00〉)
where T1 represents that part of the Hamiltonian describing tunnelling from the reser-
voir to island 1. The labels a and b correspond to the diagrams fig. 4.10. We can interpret
86 The Dynamics of a Two Qubit System
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5t
Figure 4.9: The change in frequency of the oscillations in the probability of being in
the state |00〉. The solid line is the oscillation with T12 = +10 and the dashed line is
the oscillation with T12 = −10. The vertical lines are the periods of these oscillations
calculated from the energy difference in eq. 4.3.4.
4.4 Path Interference 87
1 2 2 1
1
2
3 1
2
3
a)
d)c)
b)
Figure 4.10: The two second order (a, b) and third order (c, d) tunnelling processes in-
volved in the oscillation |00〉 → |11〉.
the other terms in the frequencies in a similar manner; the third order term in e1 − e4 is8T12(T 2
1+T22 )
E2C12
. This represents the two third-order tunnelling processes whereby the state
|00〉 can tunnel to |11〉:
c) |11〉 = T1|10〉 = T1(T12|01〉) = T1(T12(T1|00〉))
d) |11〉 = T2|01〉 = T2(T12|10〉) = T2(T12(T2|00〉))
These processes can be illustrated graphically (fig. 4.10):
We see that the oscillation frequencies consist of terms representing all possible tun-
nelling paths. This observation can give us some insight into the behaviour of the oscilla-
tions as a function of the magnetic flux through the circuit.
4.4 Path Interference
We can view the interference fringes in fig. 4.6, 4.7 as being due to interference between
all possible paths the Cooper pairs could take to go from the initial to the final state.
Unfortunately, the Hamiltonian can not be solved analytically for a general phase, and
so we must go to perturbation theory. Here again we have a problem, as the charging
Hamiltonian on its own has doubly-degenerate energy eigenvalues, and so considering the
tunnelling as a perturbation on the charging-only Hamiltonian will not be simple! Instead,
88 The Dynamics of a Two Qubit System
we can take our unperturbed Hamiltonian as including both the charging interaction
and the tunnelling to the reservoir from the islands, and then considering the tunnelling
between islands as a perturbation on this.
H = H0 +H ′
=
0 −T1 −T2 0
−T1 −EC12/2 0 −T2−T2 0 −EC12/2 −T10 −T2 −T1 0
+
0 0 0 0
0 0 −T ∗12 0
0 −T12 0 0
0 0 0 0
(4.4.1)
The unperturbed energy eigenvalues are:
e01,4 = −1
4EC12 +
1
4
√(EC12)2 + 16(T1 ± T2)2
e02,3 = −1
4EC12 −
1
4
√(EC12)2 + 16(T1 ± T2)2 (4.4.2)
The eigenvectors can be simply written in terms of the eigenenergies, for example:
|v01〉 =
1
−e01/(T1 + T2)
−e01/(T1 + T2)
1
1(2 + 2
(e01)2
(T1+T2)2
)1/2 (4.4.3)
We can now do perturbation theory in T12. This allows us to investigate the oscillations
as a function of phase on T12, i.e. as a function of magnetic flux through the circuit. We
can, for instance, find the oscillations by expanding the energy differences in all three tun-
nelling energies. Using perturbation theory to first order in T12 and expanding to second
order in T1, T2, we find that the eigenvalues, including the secound order perturabtions
e′, are:
(e01 − e04) + (e′1 − e′4) = −8T1T2EC12
− 8T12(T21 + T 2
2 )
E2C12
(e02 − e03) + (e′2 − e′3) =8T1T2EC12
− (T12 + T ∗12)− 8(T12 + T ∗12)T1T2E2C12
(4.4.4)
4.4 Path Interference 89
2
1
1
3
2 1
3
2
a)
c)b)
3
Figure 4.11: The three third order tunnelling processes involved in the oscillation |01〉 →|10〉.
These expressions agree with those given above (eqns. 4.3.4, 4.3.5) for the case when
T12 is real. It should be noted we have not done a full expansion to third order in T/EC12,
but have instead expanded to different orders in the different matrix elements. To get a
full third order expansion, we would have to expand to third order in T1, T2 and do third
order perturbation theory in T12.
Nevertheless, these calculations indicate that we can indeed think of the oscillations
as interference between different tunnelling paths, and also suggests that a diagrammatic
approach may be used to ‘keep track’ of the terms in the expansion. For instance, we
know that the energy difference e1 − e4 = (e01 − e04) + (e′1 − e′4) above (eq. 4.4.4) is correct
to third order in all tunnelling matrices as there are no other possible diagrams going from
the state |00〉 to |11〉. We also know that the correction to e2 − e3 = (e02 − e03) + (e′2 − e′3)is incomplete, as there is a (T12)
3 diagram missing (figure 4.11).
It is also worth noting that this idea of using diagrams to keep track of terms does
not only apply to the degeneracy point (n1g, n2g = 1/2), but would work for any pertur-
bation theory about the charging-only Hamiltonian. Away from the degeneracy point, of
course, we also have the advantage that the charge states are not degenerate, making the
perturbations easier to calculate.
When the condition T ¿ EC12 is relaxed, more and more paths contribute to the
frequencies, but the flux dependence can still be observed. We also observe oscillations
90 The Dynamics of a Two Qubit System
0
0.2
0.4
0.6
0.8
1
0.2 0.4 0.6 0.8 1t
Figure 4.12: The oscillations of the two qubit system when the system is not in the limit
T ¿ EC12. The solid line is probability of state |00〉, the dashed line is the probability
of |11〉 and the grey lines are the probability of being in state |01〉 or |10〉. The vertical
lines are the periods of these oscillations calculated from the energies in eq. 4.3.3. The
parameters are: EC12 = 50, T1 = T2 = T12 = 10
4.4 Path Interference 91
0
1
2
3
4
5
6
00.2
0.40.6
0.81
1.21.4
00.20.40.60.8
1
Figure 4.13: The Probability of the Two Qubit system being in the state |01〉 or |11〉 i.e.the probability that the first qubit is occupied regardless of the occupation of the second.
The z axis represents increasing probability.
from |00〉 into the states |01〉 and |10〉, with a frequency given by e1 − e2 (figure 4.12).
These oscillations are in the probabilities of the system being in the two-qubit states.
To calculate them, all four states must be distinguished from each other. This would
require instantaneous measurements of the charge on each island at the same time, for a
range of times over the evolution.
These single shot measurements may be difficult to perform at present, so the tun-
nelling current off each island may be measured instead. This measures the probability
of a Cooper pair being on each island independently. In effect these currents will be mea-
suring P (01) + P (11) and P (10) + P (11), where P (00) is the probability of finding the
system in state |00〉, etc.The interference fringes can still be observed, as in fig. 4.13. The importance of this
is that the interference fringes are a signature of coherent quantum tunnelling that can
be detected with an ensemble (current) measurement on an individual qubit, and as such
may be much a more experimentally accessible method of inferring quantum behaviour
than, for example, measuring a Bell inequality.
Chapter 5
Finite Superconductors as Spins:
The Strong Coupling Limit
In this chapter we discuss a microscopic model of a finite region of superconductor. This
model is known as the strong coupling approximation where the interaction potential is
much larger than the cutoff energy. This allows us to write the well-known BCS Hamilto-
nian in terms of large but finite quantum spins. After recapping the quantum mechanics
of spins, we look at the solution of this Hamiltonian, both with and without the mean field
approximation, which we compare to the standard BCS solution and to the Richardson
solution.
5.1 The Strong Coupling Approximation
We wish to consider the behaviour of small islands of superconductor, which are small
enough for finite size effects to come into play. Eventually we will wish to consider the
interactions between such islands, and interactions of these islands with a superconducting
reservoir, as has already been done in chapter 3, by quantising the Josephson equations
for each circuit. We can describe a single isolated island using the BCS Hamiltonian [27]:
H =∑
k, σ
εkc†k, σck, σ −
∑
k, k′
Vk, k′ c†k↑c
†−k↓c−k′↓ck′↑ (5.1.1)
As we are discussing a finite superconducting island, the label k does not refer to a
free electron wavevector but to a generic single-electron eigenstate.
93
94 Finite Superconductors as Spins
A common approximation to this equation is made by assuming the pairing potential
Vk, k′ is equal for all k, k′ in a region around the Fermi energy determined by the cutoff
energy ~ωc and zero outside this region. That is, Vk, k′ = V for |εk−εF | < ~ωc and Vk, k′ = 0
otherwise. This greatly simplifies the above equation whilst retaining the essential physics.
H =∑
k, σ
εkc†k, σck, σ − V
εF+~ωc∑
k, k′
εk=εF−~ωc
c†k↑c†−k↓c−k′↓ck′↑ (5.1.2)
To render the problem tractable, we make a further approximation; we take all the
single electron energy levels εk within the cutoff region around the Fermi energy to be
equal to the Fermi energy εF . This crude approximation turns out to be equivalent to
taking the strong coupling limit [90] as will be discussed further in 5.4.4. We can write
the kinetic energy term as:
Ho =∑
k, σεk<εF−~ωc
εkc†k, σck, σ +
εF+~ωc∑
k, σεk=εF−~ωc
εkc†k, σck, σ +
∑
k, σεk>εF+~ωc
εkc†k, σck, σ (5.1.3)
where k is no longer a wavevector, but is a generic label for the single electron energy
levels, with −k representing the time-reversed state. Outside the cutoff region, there is no
scattering and the Hamiltonian is diagonal in k. Considering only the terms within the
cutoff region (the dashes on the sums indicate they are taken over the region εF − ~ωc <εk < εF − ~ωc), we have:
Ho =′∑
k, σ
(εF + (εk − εF )) c†k, σck, σ (5.1.4)
The term (εk − εF ) never has a magnitude greater than ~ωc. In the strong coupling
limit, ~ωc ¿ V , so (εk − εF ) can be discarded. We are left with a Hamiltonian acting on
terms only within the cutoff region:
H = εF
′∑
k
c†k, σck, σ − V′∑
k, k′
c†k↑c†−k↓c−k′↓ck′↑ (5.1.5)
Having made this approximation, we now define operators that correspond to the sums
in the Hamiltonian:
5.2 Quantum Spins 95
SZ =1
2
′∑
k
c†k↑ck↑ + c†−k↓c−k↓ − 1
S+ =′∑
k
c†k↑c†−k↓
S− =′∑
k
c−k↓ck↑ (5.1.6)
These sums obey the usual commutation relations for quantum spins (see section 5.2)
and are essentially Cooper pair number, creation and annihilation operators. Therefore,
we can express our approximated Hamiltonian solely in terms of quantum mechanical spin
operators. Discarding constants, the Hamiltonian becomes:
Hsp = 2εFSZ − V S+S− (5.1.7)
We have now rewritten our Hamiltonian in a vastly simplified form. The strong cou-
pling limit has been studied since early days of BCS theory [91, 92, 93], for materials such
as lead. More recently it has been studied, in particular, as a treatment of high tempera-
ture superconductivity (see eg. [94]) and as a description of nanoscopic superconducting
grains [95] where the symmetry of the grain can cause the degeneracy of the energy levels
to be very large.
Obviously, we need to check the validity of this approximation and its relation to well
known existing results. Before doing this, however, we shall recap the theory of quantum
spins.
5.2 Quantum Spins
We begin our discussion of quantum spins with the commutation relations of our three
operators (5.1.6). All the properties of quantum spins can be derived from these relations
without reference to their physical ‘spin’ nature - i.e. once we have the commutation
relations, we do not need any further physical intuition to find their properties. It is
important to emphasise this as we are discussing a system which is not a real quantum
spin, but merely shares the same commutation relations.
From our superconducting perspective, these commutation relations follow directly the
definition of the operators in terms of electron creation and annihilation operators.
96 Finite Superconductors as Spins
[SZ , S−] =′∑
k, k′
[1
2(c†k↑ck↑ + c†−k↓c−k↓ − 1), c−k′↓ck′↑]
=
′∑
k, k′
1
2[c†k↑ck↑, c−k′↓ck′↑] +
1
2[c†−k↓c−k↓, c−k′↓ck′↑]
=′∑
k, k′
1
2δk, k′(−c−k′↓ck′↑ − c−k′↓ck′↑)
= −′∑
k
c−k↓ck↑
= −S− (5.2.1)
Similar calculations give the other two commutation relations.
[SZ , S+] = S+
[SZ , S−] = −S−
[S+, S−] = 2SZ (5.2.2)
Next we define another operator, S2:
S2 = (SZ)2 +
(S+ + S−
2
)2
+
(S+ − S−
2i
)2
(5.2.3)
In the interpretation of these operators as quantum spin operators, this operator mea-
sures the total spin l2(
l2 + 1) and can be written S2 = (SZ)2 + (SX)2 + (SY )2, with the
Hermitian operators SX = (S+ + S−)/2 and SY = (S+ − S−)/2i. We do not need to
make this physical interpretation yet however, and instead note that [S2, SZ ] = 0. This
means that the operators share a common set of eigenvectors:
S2|l, N〉 = α|l, N〉
SZ |l, N〉 = β|l, N〉 (5.2.4)
We have given the eigenstates these labels because we are anticipating their physical
meaning. The label l will turn out to be the number of levels within the cutoff region
around the Fermi energy εF ± ~ωc. In terms of physical spins, l2 gives the size of the spin,
5.2 Quantum Spins 97
sometimes denoted J . N will turn out to be the number of Cooper pairs on the island and
relates to the eigenvalue of SZ through N − l2 = m. For now though, l and N are merely
labels of the eigenstates with unknown eigenvalues α and β. We take the expectation
value of S2 with one of these common eigenstates:
〈l, N |S2|l, N〉 = 〈l, N |(SZ)2|l, N〉+ 〈l, N |(SX)2|l, N〉+ 〈l, N |(SY )2|l, N〉 (5.2.5)
We have defined SX and SY to be Hermitian, and therefore the expectation values of
these operators are the magnitudes of the vectors SX |l, N〉 and SY |l, N〉. The magnitude
of a vector is always ≥ 0, and so we get:
〈l, N |S2|l, N〉 ≥ 〈l, N |(SZ)2|l, N〉
α ≥ β
This means that there is a state with a minimum SZ eigenvalue. We call this state |l, 0〉and define − l
2 as its eigenvalue. Again, the eigenvalue has been chosen in anticipation of
its physical meaning.
As well as the minimum SZ eigenstate, we wish to find the other eigenstates of SZ .
We use a proof by induction to show that the states (S+)N |l, 0〉 are eigenstates of the
operator SZ . If |l, N − 1〉 is an eigenstate with eigenvalue (N − 1)− l2 , then:
SZ |l, N〉 = SZ(S+)N |l, 0〉
= ([SZ , S+] + S+SZ)(S+)N−1|l, 0〉
= (S+ + S+SZ)(S+)N−1|l, 0〉
= (1 + (N − 1)− l
2)(S+)N−1|l, 0〉 (5.2.6)
As the state |l, 0〉 is an eigenstate of SZ with eigenvalue − l2 , we have proved:
SZ |l, N〉 = (N − l
2)|l, N〉 (5.2.7)
Similarly, we can use another proof by induction to find the correct normalisation
constants. Consider the action of the S− operator on the state |l, N + 1〉. Assuming that
98 Finite Superconductors as Spins
S−(S+)N−1|0〉 = ((N−1)l−(N−2)(N−1))(S+)N−2|0〉, which is trivially true for N = 1,
we find:
S−(S+)N |0〉 = ([S−, S+] + S+S−)(S+)N−1|l, 0〉
= (−2SZ + S+S−)(S+)N−1|l, 0〉
= (l − 2N + 2)(S+)N−1|l, 0〉
+S+((N − 1)l − (N − 2)(N − 1))(S+)N−2|l, 0〉
= [Nl −N(N − 1)](S+)N−1|l, 0〉 (5.2.8)
So the final line is true for all N . We use this to find the magnitude of (S+)N |0〉.
Thus, the assumption 〈l, N−1|l, N−1〉 = l!(N−1)!(l−(N−1))! coupled with the observation that
〈l, 0|l, 0〉 = 1 proves that the magnitudes of the un-normalised states is given by:
〈l, 0|(S−)N (S+)N |l, 0〉 = (N)!l!
(l −N)!(5.2.10)
With this value for the normalisation constant, we can define the orthonormal eigen-
states of the SZ operator:
|l, N〉 = (S+)N |l, 0〉√
(l −N)!
l!N !(5.2.11)
and the previous calculations also allow us to calculate the action of the spin operators:
SZ |l, N〉 = (N − l
2)|l, N〉
S+|l, N〉 =√
(N + 1)(l −N)|l, N + 1〉
S−|l, N〉 =√N(l − (N − 1))|l, N − 1〉
S2|l, N〉 =l
2(l
2+ 1)|l, N〉 (5.2.12)
5.3 The Majorana Representation 99
In summary, we have a set of operators that act on the state space |l, N〉. For each
eigenvalue of S2, l2 , we have l states |l, N〉 where N runs from zero to l, with eigenvalues
N − l2 .
It is worth taking a moment to re-connect these results, which we have been thinking
of in terms of quantum spins, with the original problem of superconductivity in terms of
single-electron creation and annihilation operators. An intuitive way of doing this can be
done be relating our problem to a representation of quantum spins due to Majorana. This
is the topic of the next section.
5.3 The Majorana Representation
5.3.1 Spin l2in terms of Spin 1
2
In his 1932 paper [96], Majorana noted that the Hilbert space of a spin l2 system could be
represented by the product of l spin 12 systems. In the description of the system above,
this has a very clear meaning.
The spin operators defined in 5.1.6 can be viewed as a sum of operators labelled k,
where k runs from 1 to l:
σZk =1
2(c†k↑ck↑ + c†−k↓c−k↓ − 1)
σ+k = c†k↑c†−k↓
σ−k = c−k↓ck↑ (5.3.1)
(5.3.2)
Each of the k sets of operators acts like a set of spin-half operators, e.g.:
σ+k |0〉k↑|0〉−k↓ = c†k↑c†−k↓|0〉k↑|0〉−k↓
σ+k | ↓〉k = | ↑〉kσZk|0〉k↑|0〉−k↓ =
1
2(c†k↑ck↑ + c†−k↓c−k↓ − 1)
= −1
2|0〉k↑|0〉−k↓
σZk| ↓〉k = −1
2| ↓〉k (5.3.3)
100 Finite Superconductors as Spins
The vacuum state |l, 0〉 refers to the product of the l vacuum states |0〉k↑|0〉−k↓, i.e.the Hilbert space of our spin J operators can be represented as the product of l spin-half
down spaces, where each spin-half space refers to the absence or presence of a pair in a
particular energy level.
The action of the S+ operator is to add a Cooper pair to the system, in such a way
that there is an equal amplitude of the pair being in any of the l levels. In spin language,
The complex numbers z∗k are the coefficients of spin-up for each of the spin-half states.
They will be of particular interest in a special set of states: coherent states.
5.3.2 The Coherent State
When considering the harmonic oscillator we define a state as the eigenstate of the de-
struction operator a†. This is the coherent state (see eg. [97]):
5.3 The Majorana Representation 101
|α〉 =∞∑
N=0
αN (a†)N |0〉N !
(5.3.6)
We can define a similar state for our system. This is usually referred to as a (spin)
coherent state [97], but it should be noted that whilst the harmonic oscillator coherent
state (eq. 5.3.6) is defined as an eigenstate of the destruction operator, the analogous spin
state is not an eigenstate because the sum does not run to infinity.
|l, z〉 = 1√2l
l∑
N=0
(z∗S+)N |l, 0〉N !
(5.3.7)
The coherent state is a state where all the component spin-half particles are pointing in
the same direction, i.e. all the points on the Majorana sphere (see section 5.3.3) converge
to one location. The coherent state with z = cot(θ2
)eiφ is the state with maximal spin l
2
along the axis θ, φ. That is, if we define an operator that gives the component of spin in
the direction (θ, φ), then:
Sθ,φ|l, z〉 =l
2|l, z〉 (5.3.8)
We can easily see that this is true for the axis z, i.e. θ = 0 or θ = π. These cases
correspond to the limits |z| → 0 and |z| → ∞ respectively, and examining the definition
above (eq. 5.3.7) show that in these cases the coherent state tends to |l, 0〉 or |l, l2〉.In a real spin, these states are important states; they are the maximal eigenstates of
spin in each direction. The states |l, 0〉 and |l, l2〉 are only two such states, determined by
the definition of the z-axis, and they can be transformed into any other coherent state by
a rotation of co-ordinate basis.
5.3.3 The Majorana Sphere
The values zk in eq. 5.3.5 have a geometric interpretation. Consider the Bloch Sphere, as
a representation of a spin-half particle. Here every pure state |l = 1, ψ〉 (i.e. |j = 12 , ψ〉)
can be uniquely represented by a point on the sphere, with the north pole corresponding
to spin up, the south pole to spin down, and all other states lying on the surface of the
sphere.
Each state can also be specified by a complex number z. If the complex plane passes
through the equator of the sphere, and a line is passed from a point on the sphere to the
102 Finite Superconductors as Spins
a) b)
Figure 5.1: The Majorana Sphere. Each state is represented by a series of points on the
surface of a sphere. In the coherent state, all the points coincide (a), whereas for ZZ
eigenstates, they are distributed around the sphere at a constant ‘latitude.’
north pole, then it crosses the plane at a single point z. Thus, a complex number can
represent a point on the sphere, and thus a spin-half state.
|l = 1, ψ〉 = a1| ↑〉+ a0| ↓〉
= cosθ
2e−i
φ2 | ↑〉+ sin
θ
2ei
φ2 | ↓〉 (5.3.9)
z =1
tan θ2
eiφ
=a0a1
(5.3.10)
In the Majorana representation [96, 98], a spin l2 state is represented by l spin-half
states, i.e. l points on the sphere (figure 5.1), or l projections, zk, onto the complex plane
(figure 5.2).
Any state, then, can be represented by a series of points on a sphere. In particular,
for the coherent state, all the zk are equal, and all the points on the Majorana Sphere
coincide. This point corresponds to the direction in which the state is a maximal spin
state.
As mentioned above, in the case of the maximal and minimal eigenstates of the SZ
operator, the value of z is zero or infinity. The point z = 0 on the complex sphere
corresponds to the south pole, and z =∞ corresponds to the north pole. Thinking about
the states in this way makes it very clear that |l, 0〉 and |l, l2〉 are merely coherent states,
and occupy a special position only in reference to a chosen co-ordinate basis.
5.4 Mean Field Spin Solution 103
z
Figure 5.2: Each point on the Majorana sphere can be represented by its projection onto
the complex plane, z.
5.4 Mean Field Spin Solution
We wish to ensure that our approximation to eq. 5.1.2 is valid. One way to increase
our confidence in the model is to compare its solutions to solutions to the full model.
In particular, we will find the mean-field solution of 5.1.7 and compare it to the mean-
field solution of 5.1.2. When making the mean field approximation, we assume that the
operators S± are close to their expectation values. Writing S± = 〈S±〉+(S± − 〈S±〉), wecan rewrite the Hamiltonian 5.1.7. The mean field Hamiltonian is derived by discarding
terms to second order or higher in (S± − 〈S±〉).
Hsp = 2(εF − µ)SZ − V(〈S+〉+
(S+ − 〈S+〉
)) (〈S−〉+
(S− − 〈S−〉
))(5.4.1)
≈ 2(εF − µ)SZ − V(〈S+〉S− + 〈S−〉S+ − 〈S+〉〈S−〉
)
= 2(εF − µ)SZ −∆S+ −∆∗S− +|∆|2V
(5.4.2)
The term 〈S+〉〈S−〉 is a constant, and will be unimportant for most purposes, and we
shall discard it below when we find the mean field solution. We will need to take note of
it later, however, when we compare the mean field solution to the exact solution (section
5.5).
HMF = 2(εF − µ)SZ −∆S+ −∆∗S−
= 2ξf SZ − SX(∆ +∆∗)− SY i(∆−∆∗) (5.4.3)
Now the use of this spin-model becomes evident. Observing the above Hamiltonian,
104 Finite Superconductors as Spins
ZoZn
Yn
Xn
Yo Xo
o n
Figure 5.3: Rotation of basis of spin operators. The operators SZo , S+o , S
−o refer to a
particular basis o. We can consider another basis, n with three operators SZn , S+n , S
−n . The
operators corresponding to any basis can be written as a linear combination of any other
basis.
we see it is a linear combination of SZ , SY and SX operators. This means that the whole
Hamiltonian is proportional to a spin operator in a different direction. We have reduced
the problem of finding the eigenvalues of eq. 5.4.3 to the problem of rotating a quantum
mechanical spin.
5.4.1 Rotation of Spin Operators
The operators SZ , S+, S− respectively represent a measurement, raising and lowering of
angular momentum with respect to a particular frame of reference, say o. For clarity, we
call these three original operators SZo , S+o , S
−o .
Other unit vectors, n, define other frames of reference, and therefore each direction n
has three operators, SZn , S+n , S
−n , associated with it. An operator in one frame of reference
can be expressed as a combination of the three operators in any other frame of reference.
We want to find a diagonal representation for our Hamiltonian; we want to find a
direction n, such that HMF = γSZn . If we can do this, the eigenstates of the Hamiltonian
are the spin states |l, Nn〉.We need to find the raising and lowering operators for this new reference frame. Again
5.4 Mean Field Spin Solution 105
we express the new operator S+n in terms of the old operators SZo , S
+o , S
−o , with d, e, f the
unknown parameters.
S+n = dSZo + eS+
o + fS−o (5.4.4)
Inserting the expressions for SZn and S+n (5.4.4, 5.4.3) into the commutation relation
for spins will allow us to find d, e and f :
[SZn , S+n ] = S+
n
[1
γ(2ξFS
Zo −∆S+
o −∆∗S−o ), (dSZo + eS+
o + fS−o )] = dSZo + eS+o + fS−o
(2∆∗e− 2∆f)SZo + (2ξF e+∆d)S+o + (−∆∗d− 2ξF f)S
−o = γ(dSZo + eS+
o + fS−o )
(5.4.5)
Equating the coefficients in the commutation relation gives three simultaneous equa-
tions:
(2∆∗e− 2∆f) = γd
(2ξF e+∆d) = γe
(−∆∗d− 2ξF f) = γf (5.4.6)
Solving these gives:
S+n = d(SZ +
−∆2ξF − γ
S+ +−∆∗
2ξF + γS−)
γ =√
(2ξF )2 + 4|∆|2 (5.4.7)
Where we note that γ is equal to twice the quasiparticle energy at the Fermi level, EF .
A similar process gives an expression for S−n (the Hermitian conjugate of S+n ). Finally, we
can use the commutation relation [S+n , S
−n ] = 2SZn to find the parameter d.
Summarising, we found that we could express a mean-field Hamiltonian in terms of
spin operators SZo , S+o , S
−o . This Hamiltonian is then recognized as a spin operator in
another direction, n. The spin commutation relations allow us to find the raising and
lowering operator as well.
106 Finite Superconductors as Spins
γSZn = 2ξFSZo − S+
o ∆− S−o ∆∗
S+n = dSZo + eS+
o + fS−o
|d| = |∆|/EFe =
−d∆2ξF − 2EF
f =−d∆∗
2ξF + 2EF(5.4.8)
We can write the above transformation as a matrix. Using the definitions for d, e, and
f given in eq. 5.4.8, we have:
2ξF /γ ∆/γ ∆∗/γ
d e f
d∗ f∗ e∗
SZo
S+o
S−o
=
SZn
S+n
S−n
(5.4.9)
Note that this matrix is not unitary. This is due to the fact the matrix acts on the
operators SZ , S+, S− rather than on SZ , SX , SY . The inverse of the above matrix is
given by:
2ξF /γ d∗/2 d/2
2∆∗/γ e∗ f
2∆/γ f∗ e
SZn
S+n
S−n
=
SZo
S+o
S−o
(5.4.10)
The fact that the above matrix is not unitary is just a normalisation problem that
arises because the raising and lowering operators are defined as S± = (SX ± iSY ) withouta normalisation constant of 1/
√2.
For completeness, we can write the rotation operators in terms of the operators SZ ,
SX , SY .
2ξF /γ (∆ +∆∗)/γ i(∆−∆∗)/γ(d+d∗)
2(e+f+e∗+f∗)
2 i (e−f−e∗+f∗)
2(d−d∗)
2i(e+f−e∗−f∗)
2i(e−f+e∗−f∗)
2
SZo
SXo
SYo
=
SZn
SXn
SYn
(5.4.11)
This matrix is indeed unitary, as expected.
5.4 Mean Field Spin Solution 107
5.4.2 The Ground State
Our identification of the S operators as spin operators allows us to recognise that the
mean-field Hamiltonian is proportional to an operator that measures the spin in some
direction, n. We can simply rewrite the Hamiltonian (eq. 5.4.3) as:
HMF = 2(εF − µ)SZo −∆S+o −∆∗S−o
= γSZn (5.4.12)
From this, it is obvious that the ground state will be the minimum eigenstate of spin
in the direction n and that the eigenvalue will be −γ l2 . This does not tell us how to write
the ground state in terms of our original operators SZ,+,−o , however. To do this we utilise
the results of section 5.3.2, and note that the minimum spin state in the direction n will be
a coherent state in terms of the operators SZ,+,−o . We still do not know what the direction
n is, and thus do not know the parameter z in equation 5.3.7.
We find the parameter z (and therefore find n) by noting that if a state is the minimal
spin state along the direction n, it will be destroyed by the lowering operator S−n along
that direction i.e. S−n |l, z〉 = 0.
S−n |l, z〉 = 0
=
(SZo +
∆
2ξF + γS+o +
∆∗
2ξF − γS−o
)1√2l
l∑
N=0
(z∗S+o )
N |l, 0〉N !
=l∑
N=0
((N − l
2)(z∗)N
N !+
∆(z∗)N−1
(2ξF + γ)(N − 1)!
+∆∗(N + 1)(l −N)(z∗)N+1
(2ξF − γ)(N + 1)!(S+o )
N
)|l, 0〉 (5.4.13)
If this state is to be equal to zero, the coefficients must be zero for all N . For example,
considering the coefficient of the l = 0 term:
− l2+
∆∗z∗l2ξF − γ
= 0
l
2=
∆∗z∗l2ξF − γ
z∗ =2ξF − γ2∆∗
(5.4.14)
108 Finite Superconductors as Spins
Therefore we have found that the ground state of eq. 5.4.3 is a coherent state (eq.
5.3.7) with z = 2ξF−γ2∆ . Directly calculating the action of SZn on this state confirms it is
indeed the ground state.
HMF |l, z〉 = γSZn |l, z〉
= − l2γ|l, z〉
= − l2
√4(εF − µ)2 + 4∆2|l, z〉
= −l√
(εF − µ)2 +∆2|l, z〉 (5.4.15)
Remembering that we wish to compare the solution to the mean-field spin Hamiltonian,
it may not be immediately obvious how the spin coherent state above relates to the BCS
state (eq. 1.6.6). We can, however, rewrite the coherent state so that the connection is
clear:
|l, z〉 =1√2l
l∑
N=0
(z∗S+)N |l, 0〉N !
=1√
(1 + |z|2)l∏
k
(1 + z∗ c†k↑c†−k↓)|l, 0〉 (5.4.16)
Above we see that the coherent state is equal to the BCS state when all the parameters
uk and vk are equal.
5.4.3 Self Consistency
With the ground state of the mean-field Hamiltonian found, we need to choose the param-
eters ∆ and µ to give self-consistency and also the correct value for the average number
of particles.
We want to choose µ such that the ground state α gives 〈l, z|N |l, z〉 = N . The section
above showed that the state |l, z〉 is the ground state of the Hamiltonian. We calculate
the expectation value of the (pair) number operator, N = SZo + l2 with this state:
N = 〈l, z|N |l, z〉
N =l
2
(1− εF − µ√
(εF − µ)2 − |∆|2
)(5.4.17)
5.4 Mean Field Spin Solution 109
We also have to chose ∆/V to be equal to the expectation value of the S+ operator:
∆∗ = V 〈l, z|S+|l, z〉
= Vl
2
∆∗√(εF − µ)2 − |∆|2
(5.4.18)
In full BCS theory, these equations involve a sum over levels k, and we have two
integral equations. For our simplified spin model, the fact that the εk are all equal means
that we can trivially sum over k, giving two coupled algebraic relations instead of integral
equations. Thus we can evaluate ∆ and µ exactly. We find:
|∆|2 = V 2N(l − N) (5.4.19)
εF − µ = V (l/2− N) (5.4.20)
With these values we have the mean field solution for a given average number N .
Including the constant that we discarded from 5.4.1, we have:
HSMF = 2(εF − µ)SZo −∆S+o −∆∗S−o +
|∆|2V
= γSZn +|∆|2V
(5.4.21)
When we insert 5.4.19 and 5.4.20 into γ and ∆, we obtain:
HSMF |l, z〉 =
(γSZn +
|∆|2V
)|l, z〉
=
(− l2γ +|∆|2V
)|l, z〉
=
(−V l
2
2+ V N(l − N)
)|l, z〉
(5.4.22)
5.4.4 Comparison to the BCS Theory
In order to check the viability of neglecting the differences between electron energy levels
εk, we solve the gap equations for a general BCS case, and then take the limit where
110 Finite Superconductors as Spins
all the levels have the same energy. To take this limit, we consider a top-hat density of
states centred around εF with width w and height l/w. This allows the limit where the
bandwidth w → 0 to be taken whilst keeping a constant number of levels, l. As shown in
appendix B, at zero temperature the integrals can be calculated exactly. We find:
∆2 = (ξFr)2
((x+ 1)2
(x− 1)2− 1
)+(w2
)2((x− 1)2
(x+ 1)2− 1
)(5.4.23)
Nr =lrw
(w
2−√
(ξFr +w
2)2 +∆2 +
√(ξFr −
w
2)2 +∆2
)(5.4.24)
With x = e2wV lr and n = Nr
lr. These two equations can be solved to give ∆ and ξFr.
When the limit V/w →∞ (the strong coupling limit) is taken, we find that we obtain the
same results as given by the spin model, (5.4.19, 5.4.20).
Evidently, the fact that this spin representation reproduces the BCS results, albeit only
in the strong coupling limit, lends credibility to our simplified model, and allows us to con-
sider the situations we wish to describe, namely those of small regions of superconductor,
or ‘Cooper Pair Boxes’, coupled to a superconducting reservoir.
Before we discuss these systems, let us examine the results we obtain for our single
isolated island if we do not make the mean field approximation.
5.5 Solving the Spin Hamiltonian Exactly
Examining the mean field solution of our Hamiltonian confirms that, at least in the mean
field, our spin Hamiltonian behaves regularly, and the results are those that would be
obtained by solving the full (k-dependent) problem and then taking the limit ~ω ¿ V .
However, we do not need to take the mean field approximation to solve the Hamiltonian.
In fact the solution is rather simple.
Hsp = 2εFSZ − V S+S− (5.5.1)
Equation 5.5.1 is the strong coupling limit of the BCS Hamiltonian. It is easy to see
that this is diagonal in |l, N〉, the eigenstates of SZ , as the S− operator reduces by one
the number of Cooper Pairs (N) on the island, and the raising operator increases it by
one. Equations 5.2.12 allow us to find the effect of the operator S+S− on an eigenstate of
5.5 Solving the Spin Hamiltonian Exactly 111
SZ . This is: S+S−|l, N〉 = N(l − N + 1)|l, N〉. With this we can find the action of the
Hamiltonian on its eigenstates:
Hsp|l, N〉 = EN |l, N〉 (5.5.2)
EN = 2(εF − µ)(N −l
2)− V N(l −N + 1) (5.5.3)
We need to find which of these eigenstates is the ground state. We find the lowest
energy solution by differentiating the eigenvalue with respect to N and setting equal to 0:
dENdN
= 2(εF − µ)− V (l + 1) + 2V N = 0
2(εF − µ) = V (l + 1)− 2V N (5.5.4)
This determines the chemical potential for a given N . If we insert this value back into
the Hamiltonian, we have a Hamiltonian which is restricted so that the expectation of the
number operator in its ground state is always equal to a specified value, N .
Hsp = (V (l + 1)− 2N)SZ − V S+S− (5.5.5)
Hsp|l, N〉 = (V (l + 1)− 2N)(N − l
2)− V N(l − N + 1) |l, N〉
=
(−V l
2(l + 1) + V N(l − N)
)|l, N〉
(5.5.6)
If we compare this to the mean-field result:
HSMF |l, z〉 =
(−V l
2
2+ V N(l − N)
)|l, z〉
(5.5.7)
We see that the mean field result has a term l2
2 which in the exact result is replaced
by l2(l + 1). This difference is due to quantum fluctuations, and becomes especially clear
at half-filling (N = l2). The classical energy of a spin J is J2 (=
(l2
)2), but quantum
fluctuations mean the energy is given by S2, i.e. J(J + 1) (= l2
(l2 + 1
)). The quantum
fluctuations are not captured by the mean field theory but are described by the exact
solution.
112 Finite Superconductors as Spins
The mean-field solution gives a non-zero pairing parameter 〈l, z|S−|l, z〉 = ∆V = l
2 . The
exact solution gives 〈l, l2 |S−|l, l2〉 = 0. Thus, the exact solution has zero pairing parameter,
and therefore cannot be said to be superconducting in the usual sense. This is only to be
expected, as we are dealing with a finite system. None the less, the system does possess
pairing fluctuations:
〈l, l2|S+S−|l, l
2〉 − 〈l, l
2|S+|l, l
2〉〈l, l
2|S−|l, l
2〉 =
l
2
(l
2+ 1
)(5.5.8)
5.5.1 Broken Symmetry: Finite size
We have established that the solution to the exact Hamiltonian has zero pairing parameter.
It may be however, that we could find a solution with broken symmetry that has a lower
energy. To find if this is true, we consider a Hamiltonian which includes a symmetry-
breaking field, η (see eg. [99]). We find the ground state of this Hamiltonian, and the
pairing parameter in this ground state. We then let the magnitude of the field go to zero,
and see if we still have a non-zero pairing parameter. This would indicate a spontaneously
broken symmetry.
Take the broken symmetry Hamiltonian:
HBS = 2(εF − µ)SZ − V S+S− − ηS+ − η∗S− (5.5.9)
To investigate this we consider the symmetry-breaking terms η, η∗ as a perturbation
on the exact spin Hamiltonian 5.1.7 (since η → 0 is the limit of interest).
From eq. 5.5.2 we have:
E(0)N = 2(εF − µ)(N −
l
2)− V N(l −N + 1) (5.5.10)
Using time-independent perturbation theory, we find the first order perturbation in
the energy:
E(1)N = 〈l, N |H ′|l, N〉
= 〈l, N | − ηS+ − η∗S−|l, N〉
= 0 (5.5.11)
5.5 Solving the Spin Hamiltonian Exactly 113
So there is no first-order correction to the energy. The first-order correction to the
eigenstates is:
|ψ(0)N 〉 = |l, N〉
|ψ(1)N 〉 =
∑
N ′ 6=N
H ′N ′,N
E(0)N − E
(0)N ′
|ψ(0)N 〉 (5.5.12)
where the matrix elements H ′N ′,N of the perturbing Hamiltonian H ′ = −ηS+−η∗S− are:
H ′N ′,N = 〈l, N ′| − ηS+ − η∗S−|l, N〉
= −η√
(N + 1)(l −N)δN ′,N+1
−η∗√N(l −N + 1) δN ′,N−1 (5.5.13)
and the difference in the energy eigenstates EN of the unperturbed Hamiltonian is given
by:
E(0)N − E
(0)N+1 = −2(εF − µ) + V (l + 1)− 2NV − V
= −2(N −N)− V
E(0)N − E
(0)N−1 = 2(εF − µ)− V (l + 1) + 2NV − V
= −2(N − N)− V
(5.5.14)
Substituting 5.5.14 and 5.5.13 into 5.5.12 gives the first order corrections to the eigen-
states:
|ψ(1)N 〉 = −η
√(N + 1)(l −N)
−2(N −N)− V |ψ(0)N+1〉
−η∗√N(l −N + 1)
−2(N − N)− V |ψ(0)N−1〉 (5.5.15)
The second order correction to the energy is:
E(2)N =
∑
N ′ 6=N
|H ′N ′,N |2
E(0)N − E
(0)N ′
(5.5.16)
114 Finite Superconductors as Spins
Inserting N = N into 5.5.13 and 5.5.14 allows us to calculate the correction to the
ground state:
E(2)N = −η
2(N + 1)(l −N)
V− η2(N)(l −N + 1)
V(5.5.17)
By looking at 5.5.15 and 5.5.17 we see that the broken symmetry solution does have
a lower ground state energy than the non-broken symmetry solution, but that when we
take the η → 0 limit, we find that the correction to the states disappears; we do not
have spontaneous symmetry breaking. The symmetry will break, however, if we have an
external term of the form −ηS+ − η∗S−. This happens when we connect the system to a
larger superconductor, as we shall see later.
5.5.2 Broken Symmetry: Infinite Size
Above we have considered a finite spin, and checked to see if there is a broken symmetry
solution. We find that there is not, for any size of the spin. If, however, we allow the size
of the spin to become infinite first, and only then check for a broken symmetry solution,
we may find a different result.
Starting from eq. 5.5.9 we allow the size to go to infinity. By examining the differential
forms of S+, S−, we see that in the l→∞ limit, S+S− becomes l2
4 . This can also be seen
without recourse to the differential forms if we recall that S+S− = S2 − SZ(SZ + 1) and
make the assumption that SZ ¿ l2 in the infinite limit. Thus, 5.5.9 becomes:
HBS = 2(εF − µ)SZ − Vl2
4− ηS+ − η∗S− (5.5.18)
We see then that this Hamiltonian has the same form as the mean-field Hamiltonian.
We can just read off the result, inserting the relevant parameters.
The ground state is then the coherent state |l, z〉, with:
HBS |l, z〉 =
(γSZn − V
l2
4
)|l, z〉
=
(− l2
(√4(εF − µ)2 + 4|η|2
)− V l
2
4
)|l, z〉
(5.5.19)
As η → 0, the energy at half filling becomes simply −V l2
4 , so we have confirmed
that the broken symmetry solution is indeed degenerate with the non-broken symmetry
solution.
5.6 Expanding the Richardson Solution to First Order 115
We need to examine ground states, to see if the broken symmetry ground state reverts
to the non-broken symmetry ground state as η → 0. If it does not, we have a spontaneous
broken symmetry.
z∗ =2(εF − µ)−
√4(εF − µ)2 + 4|η|22η∗
(5.5.20)
If we are not at half filling, then (εF −µ) is either negative or positive. If it is negative,then as η → 0 z∗ goes to infinity. If (εF − µ) is positive, then we use l’Hopital’s rule and
differentiate both the denominator and the numerator. As η → 0 we get:
z∗ =−4η∗√
4(εF − µ)2 + 4|η|2(5.5.21)
which goes to zero. So, in the limit, z∗ goes to either 0 or infinity. A coherent state with
z = 0 is the state |J,m = −J〉, and a coherent state with z =∞ is the state |J,m = +J〉,so the system is only interesting precisely at half-filling.
At half filling,
z∗ =−√
+4|η|22η∗
=|η|η∗
= eiφ (5.5.22)
|l, z〉 =1√2l
l∑
N=0
(eiφS+o )
N
N !|l, 0〉 (5.5.23)
=1√2l
∏
k
(1 + eiφc†k↑c
†−k↓
)|0〉 (5.5.24)
i.e. the ground state is independent of the magnitude of η in the infinite limit, and the
symmetry remains broken for η → 0. Note that the phase of η still enters, and it is this
that gives the phase of the coherent state.
5.6 Expanding the Richardson Solution to First Order
In section 5.4 we compared the mean field solution of our spin model with the strong-
coupling limit of the mean field solution of the BCS Hamiltonian. In this section we shall
116 Finite Superconductors as Spins
derive the exact solution found in section 5.5 as a large-coupling limit of the Richardson
Solution described in section 1.9.2.
− 1
V+
N∑
ν=1
2
EJµ − EJν=
l∑
k=1
1
EJµ − 2εk(5.6.1)
We shall be taking the strong-coupling limit, that is where λ→∞, where λ = V l/ωc
is the dimensionless coupling constant and we write V = λd, with d = ωc/l the mean level
spacing.
In what follows, we reproduce the derivation of Yuzbashyan et al. [90], with a small
but significant difference. In [90], Yusbashyan et al. were concerned with only isolated
dots, and so after taking the limit, where εk → εF , could disregard the kinetic energy part
of the Hamiltonian (essentially setting εF to zero). We will be discussing coupling between
more than one island, and so will need to keep the kinetic energy term.
As λ → ∞, a number of the Richardson parameters EJ diverge (proportional to V ).
The number which diverge is related to the symmetry of the solution. For any given
number of Cooper pairs on the island, the ground state is the symmetrised solution |l, N〉,and the number of diverging parameters for this state is N , i.e. all EJ diverge.
As stated above, we wish to retain the kinetic energy term. We can regard (εk−εF )/Vas zero, because |εk − εF | < ωc, but we cannot discard εF /V as negligible. We expand the
Richardson equations (eq. 5.6.1) in 2(εk − εF )/(EJµ − 2εF ), where we have not made any
assumptions about the size of εF /V .
− 1
V+
N∑
ν=1
2
EJµ − EJν=
l
(EJµ − 2εF )+
l∑
k=1
2(εk − εF )(EJµ − 2εF )2
(5.6.2)
We discard the second term on the right as negligible, multiply by EJµ−2εF , and sum
over the N parameters EJµ
−EJµ − 2εFV
+N∑
ν=1
2EJµEJµ − EJν
−N∑
ν=1
4εFEJµ − EJν
= l
−N∑
µ=1
EJµ − 2εFV
+N(N − 1) + 0 = Nl (5.6.3)
5.6 Expanding the Richardson Solution to First Order 117
where the double sums over µ, ν have either vanished, or gone toN(N−1) due to symmetry.
Finally, we recall that the energy of the Richardson solution is given by a sum over EJ ,
and rewrite 5.6.3 to get the energy of an island containing N Cooper pairs:
EN = N2εF − V N(l −N + 1) (5.6.4)
This should be compared with the energy as calculated directly from the spin Hamil-
tonian:
((SZ +
l
2)2εF − V S+S−
)|l, N〉 = EN |l, N〉
EN =
(N2εF − V N(l −N + 1)
)(5.6.5)
This derivation shows that the spin Hamiltonian can be thought of as the first term in
an expansion in ωc/V . Including all the terms in the expansion allows an exact treatment
of the BCS Hamiltonian in terms of the spin operators SZ , S+, S−.
Chapter 6
The Super Josephson Effect
The spin model for a superconducting system described in chapter 5 allows the easy inves-
tigation of various phenomena. Here we apply the model to an analogue of a well-known
quantum optical effect known as superradiance, in which the intensity of the radiation
emitted by the atoms in a cavity is proportional to the square of the number of atoms.
We shall investigate an analogue of a quantum optical effect known as superradiance.
In superradiance, a group of atoms in a quantised electromagnetic field (i.e. a cavity)
are placed in a particular state. These atoms then emit light with an intensity that is
proportional to the square of the number of atoms [100]. This is an interesting effect, as
its observation relies on the fact that the atoms are placed in a highly entangled state,
but is observable simply by measurement of the intensity of light emitted.
In this chapter, we will describe a similar effect that occurs in an array of supercon-
ducting islands coupled to a superconducting reservoir, in which the current out of the
array is proportional to the square of the number of islands. This work is presented in
[101].
6.1 The Cooper Pair Box Array
The system we wish to describe is an array of Cooper pair boxes, each of which is coupled
individually to a superconducting reservoir. In this chapter, we assume that we can
represent the boxes as two level systems, and we consider the reservoir to be a quantum
spin, the size of which we allow to go to infinity at the end of the calculation.
119
120 The Super Josephson Effect
Figure 6.1: An array of lb superconducting islands, or Cooper Pair Boxes, that are indi-
vidually coupled to a superconducting reservoir, r, through capacitive Josephson tunnel
junctions (with no inter-box coupling).
The Hamiltonian of this system is:
H = Hb +Hr +HT (6.1.1)
where the individual Hamiltonians refer to the array of Cooper pair boxes, the reservoir,
and the tunnelling between the two respectively. The first term is the Hamiltonian of the
boxes:
Hb =
lb∑
i=0
Echi σzi (6.1.2)
There are lb Cooper pair boxes, each of which is described as a two-level system where
the two states are zero or one excess Cooper pair on the box, with Echi the charging energy
of each, as described in chapter 3.
The second term Hr is the well known BCS Hamiltonian [27] (see section 1.6), and the
final term:
HT = −∑
k,i
Ti,k (σ+i c−k↓ck↑ + σ−i c
†k↑c
†−k↓) (6.1.3)
6.1 The Cooper Pair Box Array 121
describes the tunnelling of Cooper pairs from each of the boxes to the reservoir. The
charging energy of each box can effectively be tuned by applying a gate voltage to that
box, and a tuneable tunnelling energy can be achieved by adjusting the flux through a
system of two tunnel junctions in parallel ([68], section 2.3).
In chapter 5 we used a set of finite spin operators to represent a finite superconductor.
We can also use this formalism to describe the array of boxes. If the charging energies Echi
and tunnelling elements Ti,k of each box are all adjusted to be the same, then the charging
energy for all the boxes acts like the z-component of a quantum spin with J = lb/2.
Similarly the sum over all the raising (lowering) operators acts like a large-spin raising
(lowering) operator.
lb∑
i=0
σZi → SZb
lb∑
i=0
σ±i → S±b (6.1.4)
If we consider transitions caused by only these operators, then the symmetrised number
states form a complete set. The eigenstates of SZb represent states where a given number
Nb of the boxes are occupied, with SZb |Nb〉 = (Nb − lb/2)|Nb〉. We now have a description
of the Cooper pair boxes in terms of a single quantum spin:
Hb = Echb S
Zb
HT =∑
k
−Tk(S+b c−k↓ck↑ + S−b c
†k↑c
†−k↓) (6.1.5)
As mentioned above, superradiance is observed when the atoms are placed in a par-
ticular initial state. The initial state is one of the form:
The time dependence is determined by the energy difference between adjacent number
eigenstates, i.e. by Echb and 2ξFr. If we evaluate the expectation values, we find terms like
Nb(lb −Nb)(lr − 2Nr) which is quadratic in lb and linear in lr (and vice versa, i.e. terms
quadratic in lr and linear in lb). In particular, to have the largest effect, we set Nb = lb/2
i.e. half the boxes are occupied.
〈ψ(t)|I|ψ(t)〉 =−2T 2
(i~)2sin((Ech
b − 2ξFrt/~)
((Ech
b − 2ξFr/~)
(lb2
(lb2+ 1
)(lr − 2Nr)
)
(6.2.7)
6.3 Superradiant Tunnelling: Number State to Coherent State 125
We see that the current is proportional to l2b , the square of the number of Cooper Pair
boxes. This is in contrast to the situation where the initial state of the boxes is a product
state of lb independent wavefunctions, in which any current can be at most linear in the
number of boxes. The current is also proportional to (nr − 1/2)lr, where nr = Nr/lr, i.e.
how far away the large superconductor is from half filling.
6.3 Superradiant Tunnelling Between a Number Eigenstate
and a Coherent State
The second case we consider is when the Cooper Pair boxes are in an eigenstate of the
number operator, and the large superconductor is in a coherent, that is to say BCS-like,
state. We ensure the coherent state of the large superconductor by introducing a pairing
parameter into the unperturbed Hamiltonian:
H = Echb S
Zb + 2ξFrS
Zr −∆rS
+r −∆∗rS
−r (6.3.1)
The ground state of the large superconductor is a coherent state and ∆r = 〈S−r 〉 cannow be found self-consistently (section 5.4).
We make a basis transform (section 5.4.1) and write the Hamiltonian in terms of this
new basis, n:
H = Echb S
Zb + 2EFrS
Zr,n (6.3.2)
This basis transform makes the unperturbed Hamiltonian simpler, but at the expense
of making the perturbation more complicated in the new basis:
HT = −T (S+b S
−r + S−b S
+r )
= −TS+b (
2∆r
EFrSZr,n + f∗r S
+r,n + erS
−r,n)
−TS−b (2∆∗rEFr
SZr,n + e∗rS+r,n + frS
−r,n) (6.3.3)
Again, the tunnelling current is zero to first order in T, as the boxes are in a number
eigenstate. With this basis transform, we can easily calculate the second-order current.
Following the same procedure as before, we get:
126 The Super Josephson Effect
〈ψ(t)|I|ψ(t)〉 =−2T 2
(i~)2|∆r|2E2Fr
〈0r,n|SZr,nSZr,n|0r,n〉t∫
0
dt′ cosEchb t′/~
×(〈Nb|S+b S
−b |Nb〉 − 〈Nb|S−b S+
b |Nb〉)
− 2T 2
(i~)2〈0r,n|S−r,nS+
r,n|0r,n〉 ×
|e|2t∫
0
dt′ cos (2EFr − Echb ) t′/~〈Nb|S+
b S−b |Nb〉
−|f |2t∫
0
dt′ cos (2EFr + Echb ) t′/~〈Nb|S−b S+
b |Nb〉 (6.3.4)
The terms e and f are as defined in 5.4.8.
The time dependence is again given by the level spacing, but for the large supercon-
ductor this is now given by EFr =√ξ2Fr − |∆|2 rather than (εFr − µ).
〈ψ(t)|I|ψ(t)〉 =−2T 2
(i~)2sinEch
b t/~Echb /~
|∆|2E2Fr
(lr2
)2
(lb − 2Nb)
+2T 2
(i~)2sin (Ech
b + γ2ξF )t/~(Ech
b + γ2ξF )/~lr|f |2〈Nb|S−b S+
b |NB〉
− 2T 2
(i~)2sin (Ech
b − γ2ξF )t/~(Ech
b − γ2ξF )/~lr|e|2〈Nb|S+
b S−b |NB〉 (6.3.5)
If we assume the levels of the large superconductor are much more finely spaced than
those of the boxes, i.e. EFr ¿ Echb , then we obtain the expression below. If we do not
make this assumption, we find a similar result (current proportional to l2b ), but with a
more complicated time dependence.
〈ψ(t)|I|ψ(t)〉 (6.3.6)
=−2T 2
(i~)2sinEch
b t/~Echb /~
|∆|2E2Fr
(lr2
)2
(lb − 2Nb)
− 2T 2
(i~)2sinEch
b t/~Echb /~
lr
ξFrEFr
Nb(lb −Nb) + |e|2Nb − |f |2(lb −Nb)
Remembering that lrξFr/EFr = lr/2 − Nr, (where Nr is the expectation value of
the reservoir number operator, see section 5.4.17) and setting Nb = lb/2, we find this is
proportional to:
6.4 Superradiant Tunnelling Between Coherent States 127
lr(1− 2nr)lb2
(lb2+ 1
)(6.3.7)
This is very similar to the case described in (6.2.7) where both the boxes and the
reservoir are in number eigenstates. In both cases, the effect is largest when the boxes are
at half-filling, and the large superconductor is far from half-filling. The difference is that
here it is the average number Nr that is relevant, not the number eigenvalue.
6.4 Superradiant Tunnelling Between Coherent States
The third case is where both the boxes and the large superconductors are in the coherent
state. Pairing parameters are introduced for each:
H = 2ξFbSZb −∆bS
+b −∆∗bS
−b
+2ξFrSZr −∆rS
+r −∆∗rS
−r (6.4.1)
The ground state of this Hamiltonian is for both superconductors to be in a coherent
state, which again is determined self consistently. As before we make a change of basis,
this time for both the boxes and the large superconductor, which makes the unperturbed
Hamiltonian:
H = 2EFbSZb,n + 2EFrS
Zr,n (6.4.2)
The tunnelling Hamiltonian becomes:
HT = −T (S+b S
−r + S−b S
+r ) (6.4.3)
= −T ( ∆∗b
EFbSZb,n + e∗bS
+b,n + fbS
−b,n)(
∆r
EFrSZr,n + f∗r S
+r,n + erS
−r,n)
−T ( ∆b
EFbSZb,n + f∗b S
+b,n + ebS
−b,n)(
∆∗rEFr
SZr,n + e∗rS+r,n + frS
−r,n)
If we insert this into (6.2.5) we find that we do have a non-zero term linear in T , i.e.
the expectation value of I with |ψ(0)〉. To second order in T we have terms proportional
to lr and quadratic in lb and vice versa, and terms linear in both, i.e.
128 The Super Josephson Effect
〈ψ(t)|I|ψ(t)〉 = 〈ψ(0)|I|ψ(0)〉+ 〈ψ(t)|I|ψ(t)〉l2blr
+〈ψ(t)|I|ψ(t)〉lbl2r + 〈ψ(t)|I|ψ(t)〉lblr (6.4.4)
The first term is just the expectation value of I with the ground state:
〈ψ(0)|I|ψ(0)〉 =T
i~|∆b|2EFb
|∆r|2EFr
lb lr 2 sin (φb − φr)
=T
i~|∆b|Vb
|∆r|Vr
2 sin (φb − φr) (6.4.5)
where φb, φr are the phases of ∆b,∆r. This is the usual Josephson effect.
The term quadratic in lb is:
〈ψ(t)|I|ψ(t)〉l2blr
=−T 2
(i~)2l2b lr
sin 2EFrt/~2EFr/~
|∆b|22E2
Fb
ξFrEFr
− T 2
(i~)2l2b lr
cos 2EFrt/~2EFr/~
|∆b|24E2
Fb
|∆r|24E2
Fr
2 sin 2(φb − φr) (6.4.6)
The first term is proportional to |∆b|2/E2Fb and ξFr/EFr, and so is largest when the
boxes are half occupied and the large superconductor is far from half occupied. This term
is also phase-independent and is like the effects seen in sections 6.2 and 6.3
The second term is largest when both are half occupied, and also depends on the phase
difference of the two pairing parameters with a frequency twice that of the usual Josephson
effect. As such, it can be considered a higher harmonic in the Josephson current. The
term quadratic in lr is identical apart from an anti-symmetric exchange of the labels, b, r.
Finally, the term that is linear in both contains a phase - independent term that is
largest when both the boxes and the reservoir are far from half-filling, and a term that is
phase dependent with twice the Josephson frequency that is largest at half filling.
〈ψ(t)|I|ψ(t)〉lblr=−T 2
(i~)2lblr
sin 2(EFb + EFr)t/~2(EFb + EFr)/~
(ξ2Fb4E2
Fb
ξFrEFr
− ξ2Fr4E2
Fr
ξFbEFb
)
+T 2
(i~)2lb lr
cos 2(EFb + EFr)t/~2(EFb + EFr)/~
( |∆b|24E2
Fb
|∆r|24E2
Fr
2 sin 2(φb − φr))
(6.4.7)
6.4 Superradiant Tunnelling Between Coherent States 129
In summary we conclude that tunnelling into or out of a Cooper pair reservoir from a
coherent ensemble of Cooper Pair Boxes can lead to a tunnelling current proportional to
the square of the number of ‘boxes.’ Interestingly, an observation of such scaling, in turn,
could be taken as a demonstration of coherence and entanglement of the ‘boxes’ as such
states are necessary to produce the phenomena.
Chapter 7
Revival
A further quantum optical effect that can be observed in an analogous superconducting
system is the effect of revival, where the state of a superconducting island connected to a
reservoir, initially in a pure state, decays to a mixed state but returns to a pure state at
a later time. The fact that the island is in a pure state is indicated by oscillations in the
probability of occupation of the charge states. The effect can be extended to the case of
multiple islands connected to a reservoir
7.1 The Superconducting Analogue of the Jaynes-Cummings
Model
In the previous chapter we investigated an analogue of superradiance. Another well-known
quantum optical phenomena is that of quantum revival (e.g. [102]), which occurs when
an atom is coupled to a single mode of the electromagnetic field. The initial pure state of
the atom decays into a mixed state, but reappears at a later time. This apparent decay
and subsequent revival occur because the environment of the atom is a discrete quantum
field. Here we examine a Josephson system analogue.
As in the last chapter, we consider an array of Cooper pair boxes individually connected
to a superconducting reservoir (see fig. 6.1). Again, we treat each of the boxes as a
simple two-level system and treat the reservoir as a superconductor represented by a large
quantum spin (see chapters 5, 6). We consider a Hamiltonian (eq. 6.3.1) discussed in
section 6.3.
131
132 Revival
H = Echb S
Zb + 2ξFrS
Zr −∆S+
r −∆∗S−r
HT = −T (S+b S
−r + S−b S
+r ) (7.1.1)
As in section 6.3, we make a basis transform so that the reservoir is written as a
quantum spin in another direction.
H = Echb S
Zb + 2EFrS
Zr,n
HT = −TS+b (
2∆r
EFrSZr,n + f∗r S
+r,n + erS
−r,n)
−TS−b (2∆∗rEFr
SZr,n + e∗rS+r,n + frS
−r,n) (7.1.2)
We can write a general state of the system in terms of the number eigenstates:
|ψ(t)〉 =lr,lb∑
Nr,Nb=0
aNr,Nb(t)|Nr〉|Nb〉 (7.1.3)
For a single two level system, lb = 1, but we can also do the calculations for a general
number of symmetrised boxes. We can calculate the time evolution of the system using
the Schrodinger equations:
i~∑
Nr,Nb
d aNr,Nb(t)
dt|Nr〉|Nb〉 =
∑
Nr,Nb
aNr,Nb(t)
(Echb (Nb −
lb2) + 2ξFr(Nr −
lr2)
)|Nr〉|Nb〉
−∑
Nr,Nb
aNr,Nb(t)TS+b (
2∆r
EFrSZr,n + f∗r S
+r,n + erS
−r,n)|Nr〉|Nb〉
−∑
Nr,Nb
aNr,Nb(t)TS−b (
2∆∗rEFr
SZr,n + e∗rS+r,n + frS
−r,n)|Nr〉|Nb〉
(7.1.4)
More explicitly, we have a set of coupled differential equations for the coefficients
aNr,Nb(t):
7.1 The Superconducting Analogue of the Jaynes-Cummings Model 133
i~d aNr,Nb(t)
dt= aNr,Nb(t)
(Echb (Nb −
lb2) + 2ξFr(Nr −
lr2)
)
−aNr,Nb+1(t)T√
(Nb + 1)(lb −Nb)2∆∗rEFr
(Nr − lr/2)
−aNr−1,Nb+1(t)T√
(Nb + 1)(lb −Nb)e∗r
√Nr(lr −Nr + 1)
−aNr+1,Nb+1(t)T√
(Nb + 1)(lb −Nb)fr√
(Nr + 1)(lr −Nr)
−aNr,Nb−1(t)T√Nb(lb −Nb + 1)
2∆r
EFr(Nr − lr/2)
−aNr−1,Nb−1(t)T√Nb(lb −Nb + 1)f∗r
√Nr(lr −Nr + 1)
−aNr+1,Nb−1(t)T√Nb(lb −Nb + 1)er
√(Nr + 1)(lr −Nr)
(7.1.5)
We consider an approximation that allows a simple analytical model.
The ∆ terms in eq. 7.1.1 mean that the total particle number Nr+Nb is not conserved.
We note that whilst the commutator of the total number operator [Nr + Nb, H] = ∆S+r −
∆∗S−r is not zero, its expectation value in the coherent state is. With this in mind, we
approximate and assume no fluctuations in the total number (although we of course retain
fluctuations in Nr −Nb) and discard any terms in eq. 7.1.5 that change the total number.
With these terms discarded, we have a set of coupled differential equations in which
each coefficient aNr,Nb(t) only couples to aNr−1,Nb+1(t) and aNr+1,Nb−1(t).
i~d aNr,Nb(t)
dt= aNr,Nb(t)
(Echb (Nb −
lb2) + EFr(Nr −
lr2)
)
−aNr+1,Nb−1(t)T√Nb(lb −Nb + 1)
√(Nr + 1)(lr −Nr)
−aNr−1,Nb+1(t)T√
(Nb + 1)(lb −Nb)√Nr(lr −Nr + 1)
(7.1.6)
These equations can be restated in terms of an eigenvalue problem. Each level Nr has
a set of eigenvectors for the boxes. If the boxes and the reservoir are initially in a product
state, the probability to be in a given value of Nb (regardless of Nr) is:
PNb =
lb∑
Nb=0
|aNr(0)|2∣∣∣∣∣∑
i
〈Nb|νi,Nr〉e−iEi,Nr t/~〈νi,Nr |ψNb(0)〉∣∣∣∣∣
2
(7.1.7)
134 Revival
where |νi,Nr〉 (Ei,Nr) are the eigenvectors (values) of the Nb-level system associated with
the reservoir level |Nr〉, aNr(0) are the initial amplitudes of |Nr〉, and |ψNb(0)〉 is the initialstate of the boxes.
This may not be exactly solvable in general, but can be solved for few-level systems,
lb. For example, if we have a single two level system and specify the initial state to be a
product state of the box state |0b〉 with some state∑aNr(0)|Nr〉 of the reservoir, we have
the probabilities that the box and reservoir are in a given state:
P1,Nr−1(t) = |aNr(0)|2T ′Nr
2
ΩNr2 sin
2(ΩNr t/~)
P0,Nr(t) = |aNr(0)|2 (1− P1,Nr−1(t)) (7.1.8)
with T ′Nr = T√Nr(lr −Nr + 1) and ΩNr =
√(Ech
b − EFr)2/4 + T ′2. This is the same
result as in quantum optics, except that the photon creation/ annihilation operators give
T ′Nr = T√Nr .
7.1.1 Quantum Revival of the Initial State
In quantum optics, the phenomenon of revival is seen when the electromagnetic field is
placed in a coherent state (i.e. |aN |2 = exp(−N)NN/N !, with the sum over N running to
infinity). We place the reservoir in the spin coherent state, i.e.
|aNr(0)|2 = |α|2Nrlr!
(lr −Nr)!Nr!(7.1.9)
where α is determined by the average number |α| = Nr/√
(lr − Nr)Nr.
The probability of the box being in the state |0b〉 at a given time is:
P0(t) =
lr∑
Nr
|aNr(0)|2P0,Nr(t) (7.1.10)
For simplicity we consider the case when Echb = EFr, i.e. ΩNr = T ′Nr . This is shown
in Figure 7.1, for the values Nr = 10, lr = 50. The initial state dies away, to be revived at
a later time. We have chosen the above values for Nr and lr in order to show the revival
clearly, and also to show the difference between the spin and quantum optics cases. If the
value of lr is too small (compared to Nr) the oscillations never fully decay. On the other
hand, if lr À Nr, the system behaves identically to the quantum optics case. The effects
7.1 The Superconducting Analogue of the Jaynes-Cummings Model 135
0
0.2
0.4
0.6
0.8
1
P0(t)
1 2 3 4 5 6time
Figure 7.1: Spin Coherent State Revival for Nr = 10, lr = 50 (solid line). The faint line
indicates the usual quantum optical coherent state revival with Nr = 10, scaled along the
t axis by√lr and the vertical dash-dot line indicates the calculated revival time, where
the time is in units of 2π/EJ .
of revival are much more generic than this, and also can be observed in systems with, for
example, non-identical parameters for the Cooper pair boxes.
We can calculate the linear entropy of the system over time (fig. 7.2). This is defined as
2(1−Tr ρ2) and is a measure of how mixed the state is, with 0 indicating a pure state, and
1 indicating a maximally mixed qubit state. As expected, the entropy [9] increases (purity
decreases) as the oscillations decay, and then the entropy is reduced (purity increased) as
the oscillations revive.
The revival occurs when the terms in the sum are in phase. We can make an estimate
of this by requiring the terms close to Nr to be in phase:
2π = 2T√
(Nr + 1)(lr − Nr) trev − 2T√Nr(lr − Nr + 1) trev
trev =2π
T
n1/2r (1− nr)1/2(1− 2nr)
(7.1.11)
where, as before nr = Nr/lr. Note that the revival time remains finite in the lr → ∞limit, as long as Nr/lr is finite.
136 Revival
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6
Figure 7.2: Decay and revival of a single Cooper pair island state, showing the degree
of linear entropy, which increases at first, and is then reduced. The oscillations are the
probability that the island is in state |0〉 (solid line) or state |1〉 (dashed line) against time
in units of 2π/EJ .
7.1 The Superconducting Analogue of the Jaynes-Cummings Model 137
0
0.2
0.4
0.6
0.8
1
P0(t)
0.2 0.4 0.6 0.8 1 1.2 1.4time
Figure 7.3: Analytic asymptotic form for decay oscillations in the probability against time
in units of 2π/EJ(solid line) compared with the numeric evolution (dashed line) Nr = 10,
lr = 100
We can find an analytic form for the initial decay, and also demonstrate the equivalence
of this ‘spin revival’ to the well known quantum optics revival, in the limit lr → ∞,
lr/Nr → ∞. Note that this is an unrealistic limit for our case, where we would keep
the filling factor Nr/lr constant, but it allows us to make a connection with the quantum
optics case.
The coefficients, aNr(0) for the coherent and spin-coherent states are the Poisson and
binomial distributions respectively, and both can be approximated by a Gaussian distri-
bution.
exp(−Nr)NNrr /Nr! ' (2πNr)
−1/2 exp[−(Nr − Nr)
2/2Nr
]
|α|2Nr lr!
(lr −Nr)!Nr!' (2πNr)
−1/2 exp[−(Nr − Nr)
2/2Nr
](7.1.12)
For the coherent state we have taken the limit Nr À 1. For the spin-coherent case we
have also assumed lr À Nr. The Gaussian factor suppresses any terms not around the
138 Revival
average Nr, so we can expand Nr ' Nr + (Nr − Nr).
In the limit, the spin coherent state gives:
P0(t) =1
2+
1
2(2πN)1/2
∫dN exp
(−(Nr − Nr)
2
2Nr
)
× cos
[2T (lrNr)
1/2t
(1 +
(Nr − Nr)2
2Nr
)](7.1.13)
This is the same as for the coherent state apart from the factor of l1/2r in the frequency.
Doing the integral gives:
P0(t) =1
2+
1
2cos(2T (lrNr)
1/2t) exp(−(T lrt)2
2) (7.1.14)
This is plotted in Figure 7.3.
Thus the phenomenon of quantum revival has a direct analogy in our ‘spin supercon-
ductor’ model. In both cases, the revival is due to constructive interference between the
terms in the sum over the reservoir (field) number states.
7.2 Revival of Entanglement
Revival is usually considered for single two-level systems, but the formula in (7.1.7) gives
the state for a general system. In particular, the dynamics of a pair of two-level systems
can be easily calculated, either numerically, or analytically, remembering that the singlet
state is uncoupled due to radiation trapping. The Hamiltonian, 7.1.1, does not cause
transitions between the triplet states and the singlet state, so if we start the system in
any of the triplet states, we essentially have only a three level system to solve.
Starting in the state |00〉, we find a very similar effect to the single-island case. Figure
7.4 shows the oscillations of one Cooper pair box other traced over, compared to the
oscillations of a single box. Note that whilst the decays in the single and double box
systems are in phase, the first revival of the double box is out of phase with that of the
single box, and the second revival is in phase.
If we do not trace over one island, but consider the revival of the state |00〉, we find
oscillations in the probability of returning to this state earlier than in the case of a single
island.
7.2 Revival of Entanglement 139
1 2 3 4 5
0.2
0.4
0.6
0.8
Figure 7.4: Two Cooper Pair Box Spin Coherent State Revival for Nr = 10, lr = 50. The
solid line represents the probability that, when the system is initialised in the state |00〉,one of the Cooper pair boxes is in the state |0〉 at time t (in units of 2π/EJ), after tracing
over the other box. The dashed line indicates the revival of a single isolated Cooper Pair
box connected to a reservoir. Note that the revivals occur at the same time, but out of
phase.
To investigate the revival of entanglement, we start the system in the state (|01〉 +|10〉)/
√2. As before we note that the oscillations in the probability of being in the state
(|01〉+ |10〉)/√2 decay and then revive. It is interesting to note, however, that in this case
the initial state is an entangled one. One measure of entanglement is the negativity [103],
i.e. the sum of the negative eigenvalues of the partially transposed density matrix. If we
plot the negativity over time, we see that it dies away at first, as do the oscillations. As
the oscillations revive, we also see an revival in the negativity of the two islands (Figure
7.5).
This revival of entanglement, an initially counterintuitive phenomenon, is of course due
to the fact that we have considered the entanglement between the two islands only, just
as when we considered a single island, we calculated the purity of the island as an isolated
system and not the entropy of the whole system. When the entanglement ‘disappears,’
it is only the entanglement between the islands that has disappeared. If we were to
consider the whole system, including the reservoir, the evolution would be unitary and the
entanglement would remain constant.
It is worth noting that this ‘entanglement revival’ is not a unique feature of our spin
model, and would be expected whenever two-level systems are coupled to a field with (a
140 Revival
0.2
0.4
0.6
0.8
1
1 2 3 4
Figure 7.5: Two Cooper Pair Box Spin Coherent State Revival for Nr = 10, lr = 50. The
dashed line represents the probability that the Cooper Pair boxes have returned to their
initial state (|01〉 + |10〉)/√2 at time t,in units of 2π/EJ . The dashed line indicates the
negativity of the boxes as a function of time.
7.2 Revival of Entanglement 141
possibly infinite number of) discrete energy levels.
Chapter 8
An Electronic Derivation of The
Qubit Hamiltonian
In this chapter we examine in greater detail the derivation of the tunnelling term of the
qubit Hamiltonian for superconducting charge qubits. We consider a rigorous derivation
of the qubit Hamiltonian from single-electron creation and annihilation operators using the
spin model described in chapter 5.
8.1 Introduction
In chapter 3 we derived a Hamiltonian for the qubit circuit from the Josephson relations.
These equations give the rate of change of phase and the current across each junction. To
attempt to capture the behaviour of junctions with small capacitances, we quantise these
relations. A Lagrangian is found from which the Euler-Lagrange equations reproduce the
Josephson equations. This leads to a Hamiltonian for the system, and tells us that the
phase and charge are conjugate variables.
This procedure is somewhat empirical, in the sense that the equations of motion for
the phase and charge are treated as if they were the equations of motion for any classical
system. These equations are, of course, derived from a microscopic theory of the electrons
in the superconductor. In the derivation, it is assume that the superconductors are bulk so
that BCS theory is the correct description. In particular, taking the infinite limit means
that the phase becomes a well defined classical variable.
In the course of this derivation, then, we have taken a microscopic quantum description
143
144 An Electronic Derivation of The Qubit Hamiltonian
(superconductivity theory), taken its classical limit (the Josephson relations) and then
‘re-quantised’ it (the qubit Hamiltonian) in order to attempt to capture the finite size
behaviour. If nothing else, this procedure is inelegant, and it may be that important
features have been neglected. Furthermore, quantising the Josephson relations in this way
implies the existence of a ‘phase operator’ analogous to the position operator X, which is
problematic. We would like to go directly from an electronic description of the system to
the qubit Hamiltonian. The model described in chapter 5 will allow us to do this.
8.2 The Phase Representation
Quantum mechanics is usually first taught to undergraduates in terms of wavefunctions
(the co-ordinate representation). The position wavefunction ψ(x) gives the amplitude that
a particle will be found between x and x+ dx. The whole of the function ψ(x) defines the
state of the particle, and we may have a set of such states ψn(x) that are of interest to
us, for example that are energy eigenstates. Operators OC are represented by differential
operators that act on these states.
An alternative representation is ket notation. In this, each state is denoted by an
abstract ket, |n〉, and the operators OK are defined by their actions on these states.
We can easily translate between these two equivalent representations. The position
wavefunction is given in terms of ket notation by ψn(x) = 〈x|n〉, where |x〉 is an eigenket
of the position operator X with eigenvalue x (see eg. [39]). The operators are represented
by the differential operators in a way that is consistent with this.
OK |ψa〉 = |ψb〉
OC〈x|ψa〉 = 〈x|ψb〉 (8.2.1)
If we have a complete set of eigenstates |n〉 of the operator OK with eigenvalue λn, we
can fully define the operators in terms of their actions on these states:
OK |ψn〉 = λn|ψn〉
OC〈x|ψn〉 = λn〈x|ψn〉 (8.2.2)
In going from an electronic description of the qubit to one in terms of phase, we have
to go from ket to coordinate representation.
8.2 The Phase Representation 145
8.2.1 Phase Representation of Spins
The model set up in chapter 5 describes a small region of superconductor in terms of a
large but finite quantum spin. Before returning to our superconducting system, we shall
examine how to describe quantum spins in terms of phase.
A quantum spin of size l2 has a complete set of l states, |l, N〉. Operations on this
system can be written in terms of the diagonal z-operator, SZ , and the raising and lowering
operators, S±. The action of these operators on the states l, N〉 is:
SZ |l, N〉 = (N − l
2)|l, N〉
S+|l, N〉 =√
(N + 1)(l −N)|l, N + 1〉
S−|l, N〉 =√N(l − (N − 1))|l, N − 1〉 (8.2.3)
We wish to write both the states and the operator in some coordinate representation,
specifically a phase coordinate representation.
We define a state |φ〉:
|φ〉 =∑
eiφN |l, N〉 (8.2.4)
We now follow the procedure described above. An SZ eigenstate |l, N〉 is now described
by a wavefunction ψN (φ) given by ψN (φ) = 〈φ|l, N〉 = e−iφN . The wavefunction of any
general state |a〉 can be expressed by a linear combination of these eigenfunctions:
|a〉 =∑
N
aN |l, N〉
ψa(φ) =∑
N
aNψN (φ)
=∑
N
aNe−iφN (8.2.5)
We now need the coordinate representation of the operators. Our task is to find
differential operators which are equivalent to our spin operators (eq. 8.2.3) in the sense
defined by eq. 8.2.2.
The action of SZ is reproduced by id/dφ− l/2:
146 An Electronic Derivation of The Qubit Hamiltonian
SZ〈φ|l, N〉 =
(id
dφ− l
2
)e−iφN
=
(N − l
2
)e−iφN
SZ〈φ|l, N〉 =
(N − l
2
)〈φ|l, N〉 (8.2.6)
So our coordinate representation of SZ is id/dφ − l/2. We now want to do the same
thing for the raising and lowering operators. Examining the action of the raising and low-
ering operators on the state |l, N〉 (eq. 8.2.3) tell that the that the appropriate differentialoperator form is:
S+ = e−iφ
√(id
dφ+ 1
)(l − i d
dφ
)
S− = eiφ
√id
dφ
(l − i d
dφ+ 1
)(8.2.7)
We write these in a form that will be more convenient. For example, it is not imme-
diately obvious from the above form that S+ and S− are Hermitian conjugates of each
other.
S+ = e−iφ
√(l
2+
(id
dφ− l
2
)+ 1
)√(l
2−(id
dφ− l
2
))
(8.2.8)
We now want to commute the e−iφ term past the first square root. To do this we
expand the square root in(i ddφ − l
2
).
√(l
2+
(id
dφ− l
2
)+ 1
)=
√l
2
∞∑
p=0
1
p !
(id/dφ− l/2 + 1
l/2
)p(8.2.9)
Note that this is an infinite expansion, and so it is valid regardless of the size of the
term we are expanding in. We have to commute e−iφ past terms like (SZ + 1)p, where
we have just used SZ as a shorthand for id/dφ − l/2. Using the commutation relation
[e−iφ, i ddφ ] = −e−iφ, it is easy to show that e−iφ(SZ + 1)p = (SZ)pe−iφ. We find that we
can write the coordinate representation of the raising and lowering operators as:
8.2 The Phase Representation 147
S+ =
√(l
2+
(id
dφ− l
2
))e−iφ
√(l
2−(id
dφ− l
2
))
S− =
√(l
2−(id
dφ− l
2
))eiφ
√(l
2+
(id
dφ− l
2
))
(8.2.10)
This form is much more convenient; the two operators are obviously Hermitian conju-
gates of each other, and the action of S+S− is much clearer. As an example of how these
operators act on a phase wavefunction, we can see how the raising operator acts on the
wavefunction ψl(φ) for the highest spin state, where N = l:
S+ψl(φ) =
√(l
2+
(id
dφ− l
2
))e−iφ
√(l
2−(id
dφ− l
2
))e−il
=
√(l
2+
(l + 1− l
2
))√(l
2−(l − l
2
))e−i(l+1)
=√
(l + 1)√
(l − l) e−i(l+1)
= 0 (8.2.11)
The raising operator destroys the N = l wavefunction, as we would expect. A similar
calculation confirms that the lowering operator destroys the N = 0 wavefunction, and that
these forms act on all the N wavefunctions in the manner required by eq. 8.2.3.
We also require that the coordinate forms for the operators have the commutation
relations given in eq. 5.2.2:
[SZ , S+] =
[SZ ,
√l
2− SZe−iφ
√l
2+ SZ
]
= SZ
(√l
2− SZe−iφ
√l
2+ SZ
)−(√
l
2− SZe−iφ
√l
2+ SZ
)SZ
= SZ
(√l
2− SZe−iφ
√l
2+ SZ
)−√l
2− SZ
(SZe−iφ − e−iφ
)√ l
2+ SZ
=
√l
2− SZe−iφ
√l
2+ SZ
[SZ , S+] = S+ (8.2.12)
148 An Electronic Derivation of The Qubit Hamiltonian
So the above commutation relation matches the form we desired. A similar calculation
confirms that the other two commutation relations match also.
8.2.2 The Limit of Large Size
We would like to find the form of these operators in the limit where the size of the spin (or
the size of the superconductor) becomes infinite, l2 →∞. We have written our operators
in a form that will make this easy.
The limit we will be taking in order to make the expansion is the limit SZ ¿ l2 . In
a spin, this refers to the z-value of the spin always being close to zero, i.e. the spin lies
close to the x − y plane. In a superconducting island, this corresponds to the number of
Cooper pairs being close to l2 . Specifically, we assume:
|〈l, N |ψ〉| ∼ 0 for |N − l
2| &
(l
2
) 12
(8.2.13)
For the states |ψ〉 that the system occupies. As a particular example of a state that
obeys this condition, we consider a coherent state:
|〈l, N |l, z〉|2 = 1
(1 + |z|2)ll!
(l −N)! N !|z|2 (8.2.14)
We recognise that this expression for |〈l, N |ψ〉|2 is just the binomial coefficient,
C lNpN (1 − p)l−N , with p = |Z|2. Recalling the well known large-l limit for the bino-
mial coefficient (see eg. [104]):
|〈l, N |l, z〉|2 ' 1√2πlp(1− p)
e−(N−lp)2/2lp(1−p) (8.2.15)
If the average number N is set to l2 , this is a Gaussian centred around l
2 with width(l2
)1/2, and satisfies our condition for expansion. We shall take the full expansion later,
but for now we only take the leading term. In this case, Our three operators have the
simple form:
8.2 The Phase Representation 149
SZ → (id
dφ− l
2)
S+ →(l
2
)e−iφ
S− →(l
2
)eiφ (8.2.16)
The commutation relation between the raising and lowering operators is now zero:
[S+, S−] = 0
We can use this representation to show how the qubit Hamiltonian described in chapter
3 can be derived from electronic theory, in the limit where the size of both the island
and reservoir become large. If they are not both infinite, we shall need to calculate the
corrections to this.
We take an expansion of the differential forms of the raising and lowering operators.
For the case of an infinitely large spin, we took the limit described in eq. 8.2.13. Here we
find the next order terms by expanding in SZ
l/2 .
S+ =
(l
2+ SZ
) 12
e−iφ(l
2− SZ
) 12
= l/2
(1 +
SZ
l− (SZ)2
2l2+ . . .
)e−iφ
(1− SZ
l− (SZ)2
4l2+ . . .
)
= l/2
(1 +
SZ
le−iφ − e−iφS
Z
l− (SZ)2
4l2e−iφ − (SZ)2
4l2− e−iφ
−(SZe−iφSZ)l2
. . .
)
(8.2.17)
To combine these terms we must commute the exponential past the SZ operators.
S+ = l/2
(e−iφ(1 +
1
l− (SZe−iφSZ)
l2/2+e−iφ
4l2+ . . .)
)(8.2.18)
Our original expansion was an expansion in SZ
l/2 . Commuting the exponential has given
terms like 1l/2 , so we need to consider the limit. The assumption in eq. 8.2.13 means that
the only states that contribute are those such that |N − l2 | ≤
(l2
) l2 . We therefore consider
N − l2 to be of order O((l/2)
12 ) and discard terms of order O((l/2)−2).
150 An Electronic Derivation of The Qubit Hamiltonian
S+ ' (l
2+
1
2)e−iφ − (SZe−iφSZ)
l+O(
2
l)
S− ' (l
2+
1
2)eiφ − (SZeiφSZ)
l+O(
2
l) (8.2.19)
8.2.3 Phase Representation of BCS States
Instead of treating the superconductors as spins, we consider the BCS state. As has been
discussed in chapter 5, the ω ¿ V limit of the BCS state is simply the spin coherent state
|l, z〉 (eq. 5.3.7). Instead of considering the superconductor to be in a state in the Hilbert
space spanned by |l, N〉 (or equivalently, by |l, z〉), we consider the space spanned by the
BCS states.
The BCS states, like the spin-coherent states, are parameterised by a phase φ, and can
be written:
|BCS, φ〉 =∏
k
(uk + vkeiφ)|l, 0〉 (8.2.20)
By integrating over φ we can extract a number eigenstate from the BCS states:
|BCS,N〉 =
2π∫
0
|BCS, φ〉e−iφNdφ
=
(∏
k
uk
)(S+BCS)
N |l, 0〉 (8.2.21)
where S+BCS is an operator that raises the number of Cooper pairs by one but does not
distribute the extra Cooper pair over all levels k equally.
S+BCS =
∑
k
vkukc†k↑c
†−k↓ (8.2.22)
The number states described above are eigenstates of the operator SZ , and the operator
S+BCS raises the number of the state. As in the case of spin states, we have a phase and
number state, and it might be desirable to describe the superconductor in terms of these
states rather than the spin states, which do not take into account the distribution functions
uk, vk.
8.2 The Phase Representation 151
In order to investigate whether this would be a sensible description, we first need to
calculate the commutation relations of these operators.
[SZ , S+BCS ] =
′∑
k, k′
[1
2(c†k′↑ck′↑ + c†−k′↓c−k′↓ − 1),
vkukc†k↑c
†−k↓]
=∑
k
vkukc†k↑c
†−k↓
= S+BCS (8.2.23)
We find that two of the commutation relations are as desired: [SZ , S±BCS ] = ±S±BCS .However, when we calculate the third, we find:
[S+BCS , S
−BCS ] =
′∑
k, k′
[vkukc†k↑c
†−k↓,
v∗ku∗kc−k′↓ck′↑]
=∑
k
|vk|2|uk|2
(c†k′↑ck′↑ + c†−k′↓c−k′↓ − 1)
6= 2SZ (8.2.24)
The operators do not form a closed basis, and so we cannot form an exact representation
as we have done for the spin operators. We can, however, attempt to find an approximate
representation.
We wish to produce a phase coordinate representation for these states, as described
in sections 8.2 and 8.2.1. Following the procedure we used before, we examine the matrix
elements of the operators SZ , S+, S− with BCS states with different φ.
〈BCS, φ|SZ |BCS, φ′〉 =
(id
dφ− l
2
)〈BCS, φ|BCS, φ′〉
〈BCS, φ|S+|BCS, φ′〉 =∑
k
ukvke−iφ〈BCS, φ|BCS, φ′〉+D
〈BCS, φ|S−|BCS, φ′〉 =∑
k
ukvkeiφ〈BCS, φ|BCS, φ′〉+D
D =
∑k
v4k
(∑k
ukvk)2(8.2.25)
This shows that, as before, we can make the connection:
152 An Electronic Derivation of The Qubit Hamiltonian
SZ ⇒ (id
dφ− l
2) (8.2.26)
In addition, we can make the approximate connection:
S+ ∼(∑
k
ukvk
)e−iφ
S− ∼(∑
k
ukvk
)eiφ (8.2.27)
where D is a measure of the error incurred by making this approximation.
We can estimate the size of this error term. We use the definition for |vk|2, and make
the usual approximation of the sum by an integral:
∑
k
v4k =∑
k
1
4
(1− 2ξk
Ek+ξ2kEk
)
= N (0)V ol
~ωc∫
−~ωC
dξ1
4
(1− 2ξ
(ξ2 +∆2)1/2)+
ξ2
(ξ2 +∆2)1/2)
)
= N (0)V ol
(~ωc −
∆
2tan−1
~ωc∆
)(8.2.28)
with V ol being the volume of the system. From the gap equation, we know∑ukvk = ∆/V .
In the weak coupling limit, we find:
D = V 2N (0)V ol~ωc∆2
=d
∆λ2~ωc∆
(8.2.29)
where in the last line we have written D in terms of the dimensionless coupling constant
λ = V olN (0)V and the level spacing d = 1/N (0)V ol.
This term goes to zero as the superconductor becomes larger (as 1/V olume), so in the
bulk limit the above forms for the operators are exact.
If we consider D in the strong coupling limit, we expand tan−1 in ~ωc/V , and find:
D =V 2N (0)V ol
2
~ωc∆2
=1
l(8.2.30)
8.3 Island - Reservoir Tunnelling 153
where we have use ~ωc/d = l/2 to regain the spin result.
So we see that the term indicating the size of the error in expressing the operators
S+, S− as differential operators acting on a phase, D, goes to zero when we take the size
of the system to infinity, even in the weak coupling limit (as it also does in the strong
coupling limit). We are therefore justified in saying that we can approximate the operators
S+, S− as proportional to e−iφ, eiφ, and that this approximation gets better as the size of
the superconductor becomes larger. In particular, we note that this representation is not
a vagary of the spin representation or strong coupling limit, but a general feature. The
large size limit of the operators in the spin case (eq. 8.2.16) and the BCS case (eq. 8.2.26
8.2.27 are identical up to a constant.
8.3 Island - Reservoir Tunnelling
With the spin operators written in terms of differential operators, we are now ready to
write the Hamiltonian for our superconducting charge qubit in a differential operator form.
Initially we consider tunnelling between one finite and one bulk superconductor. We write
the Hamiltonian in four parts:
H = H1 +Hr +HC +HT (8.3.1)
where H1 is the standard BCS Hamiltonian for the island,
H1 =∑
k,σ
(εk − µ)c†k,σck,σ − V1kF+ωc∑
k,k′=kF−ωcc†k′↑c
†−k′↓c−k↓ck↑ (8.3.2)
The term Hr is the standard BCS Hamiltonian for the reservoir. The Charging Hamil-
tonian measures the electrostatic energy due to charges on the island, as discussed in
chapter 3.
HC =4e2
2C(N1 − ng)2 (8.3.3)
where N1 represents the number of pairs on the island, and ng allows us to incorporate a
gate voltage.
The final term is the pair - tunnelling term derived in section 1.8:
154 An Electronic Derivation of The Qubit Hamiltonian
HT = −T∑
k,q
c†kc†−kc−qcq + c†qc
†−qc−kck
(8.3.4)
where the subscripts k and q refer to the island and reservoir superconductors respectively.
If we represent both superconductors by spins, the Hamiltonian can be written in terms
of two sets of spin operators.
H1 = 2(εF1 − µ1)SZ1 − V1S+1 S
−1
Hr = 2(εFr − µr)SZr − VrS+r S
−r
HC =4e2
2C(SZ1 − ng)2
HT = −T(S+1 S
−r + S−1 S
+r
)(8.3.5)
We would like to consider the case where we have one small superconductor (the island,
1) and one bulk superconductor (the reservoir, r). We can treat the large superconductor in
two ways; as the large - spin limit of our spin representation, or as a BCS superconductor.
As described in section 8.2, we get the same result up to a multiplicative constant. When
the reservoir is considered infinitely large, we can replace the raising and lowering operators
S+r , S
−r by the differential forms (eq. 8.2.16). The tunnelling operator then becomes:
HT = −T lr2(S+
1 eiφr + S−1 e
−iφr) (8.3.6)
The effect of considering the reservoir as a BCS superconductor is just to replace the
term lr2 with
∑k
ukvk. Combining the tunnelling Hamiltonian with the charging energy
found in section 3.1.2, we find the Hamiltonian for a single qubit is:
H = E′C(N1 − n′g)2
−T lr2(S+
1 eiφr + S−1 e
−iφr) (8.3.7)
where E′C and n′g are the renormalised charging energy and gate charge which take into
account the terms in H1 linear and quadratic in SZ1 , and N1 = SZ1 − l12 .
8.3 Island - Reservoir Tunnelling 155
We see that this has the same form as the broken-symmetry Hamiltonian discussed
in 5.5.1; the presence of the bulk superconductor has broken the symmetry of the small
one. The eigenstates are no longer the |l1, N1〉 states, and there will be a non-zero pairing
parameter in the ground state. In the limit where the size of the island as well as the
reservoir becomes large, the raising and lowering operators for the island also take the
form 8.2.16, and eq. 8.3.7 becomes:
H = E′C(N1 − n′g)2
−T lrl14
(ei(φ1−φr) + e−i(φ1−φr))
= E′C(N1 − n′g)2 − Tlrl12
cos(φ1 − φr) (8.3.8)
We have regained the Hamiltonian (eq. 2.2.12) obtained by quantising the Josephson
relations. Thus we see that eq. 2.2.12 is the semiclassical limit of 8.3.7, which becomes
valid as the island becomes large as well as the reservoir.
The corrections to this can be found by using the form for S+1 , S
−1 found in section
8.2.2:
H = E′C(N1 − n′g)2 − Tlr2
(l1 + 1) cos(φ1 − φr)
−2(N1 − l12 )
2
l1cos(φ1 − φr)−
2(N1 − l12 )
l1isin(φ1 − φr)
(8.3.9)
We have written the correction terms above as cosine and sine terms with velocity (SZ)
dependent pre-factors. Note that in this form it is not so obvious that the Hamiltonian
is Hermitian (it is) but this form shows the sine and cosine dependence on phase of the
correction terms explicitly.
These corrections destroy the periodicity with ng in the energy levels (figure 8.1). It
then becomes relevant what the absolute value of ng is (rather than just its value modulo
1). We can consider the extra terms in the expansion to be corrections to the existing
cosine term. In this case, using the analogy of a particle in one dimension, the extra terms
give rise to a change in the cosine potential which depends on SZ = N1 − l12 , i.e. , these
terms are a velocity dependent potential. The tunnelling rate is then dependent on the
value of SZ , i.e. the rate of tunnelling depends on the number of Cooper pairs on the
156 An Electronic Derivation of The Qubit Hamiltonian
–2
0
2
4
6
8
10
2 4 6 8 10 12 14
Figure 8.1: Energy levels of a charge qubit as a function of ng including the finite size
correction to the tunnelling energy. The dashed lines indicate the uncorrected energy
levels. The parameters are EC = 10 EJ = 1 and l = 100
8.4 Inter - Island Tunnelling 157
island. This can be seen in figure 8.1, where the energy gap between the ground and first
excited states depends on ng.
Alternatively, we can consider the terms to be corrections to the charging energy -
they represent a position dependent mass (a phase - dependent capacitance). Although
this interpretation is perhaps less natural, as the correction terms are proportional to the
tunnelling matrix element T , it might be useful when EJ À EC .
The variation of the level splitting with ng, as well as being of interest from a funda-
mental point of view, offers the possibility of control of the effective Josephson tunnelling
with ng, and as such may be of interest in quatum computing applications.
8.4 Inter - Island Tunnelling
The next case we consider is that of two small superconductors with tunnelling between
them. We treat both as finite quantum spins. The tunnelling Hamiltonian contains terms
like:
S+1 S
−2 =
√l12+ SZ1 e
−iφ1√l12− SZ1
√l22− SZ2 e+iφ2
√l22+ SZ2 (8.4.1)
We re-arrange this:
S+1 S
−2 =
l1l24
√1 +
2
l1SZ1 −
2
l2SZ2 −
4
l1l2SZ1 S
Z2
e−iφ1e+iφ2
l1l24
√1− 2
l1SZ1 +
2
l2SZ2 −
4
l1l2SZ1 S
Z2 (8.4.2)
As before, we expand and combine the terms, discarding those of order O( 1l2) and
higher (assuming l1 and l2 are of the same order).
S+1 S
−2 =
l1l24
e−iφ + (
1
l1+
1
l2)e−iφ
− 2
l21(SZ1 e
−iφSZ1 )−2
l22(SZ2 e
−iφSZ2 )
(8.4.3)
where we have written φ = φ1 − φ2. The tunnelling Hamiltonian is then given by:
158 An Electronic Derivation of The Qubit Hamiltonian
HT = −T1
2(l1l2 + l1 + l2) cosφ
−( l2l1(SZ1 )
2 +l1l2(SZ2 )
2)cosφ−
( l2l1iSZ1 +
l1l2iSZ2)sinφ
(8.4.4)
As before, the tunnelling energy is no longer constant with respect to SZ1 and SZ2 , but
varies as the number of Cooper pairs on each island changes.
Chapter 9
Conclusion
In this thesis we have discussed some of the advantages of quantum computing and dis-
cussed some of the problems that the construction of a quantum computer might face.
We have described how constructing qubits from superconductors might help with these
problems.
We have detailed theories of superconductivity, both phenomenological and micro-
scopic and shown how these lead to the Josephson effect, where the current across a
superconducting tunnel junction is proportional to the sine of the difference between the
phases of the superconducting condensates on either side of the junction. We have also
described how for small superconducting grains, finite size effects can lead to the parity
effect, where the condensation energy and pairing parameter are dependent on whether
there is an odd or even number of electrons on the grain.
Experimental progress on superconducting qubits has been reviewed, and the basic
circuit layout of a charge qubit described. A Hamiltonian for such a circuit was derived,
by treating the Josephson equations as classical equations of motion, and then quantising
them in the usual manner by constructing a classical Hamiltonian that leads to those
equations, and then elevating the conjugate variables to operators. We explained how if the
charging energy is much larger than the tunnelling energy, the circuit can function as a two-
level system, and investigated the dynamics of this two level system. We briefly discussed
the effect of noise on a charge qubit, and presented a general method for evaluating the
effect of discrete noise in the Josephson energy.
We examined the charging energy of various relevant circuits, in particular focussing
on the effect of connecting a reservoir to small islands. We found a general computational
159
160 Conclusion
method to establish the correct form for the charging energy by taking the limit where the
capacitance between the islands and the outside world goes to zero and the capacitance
between the reservoir and the outside world goes to zero. These detailed calculations al-
lowed us to construct a quantum Hamiltonian from the charging energy and the tunnelling
energy across each junction. We have shown that this naive method indeed produces the
correct equations of motion for each junction, and confirmed that the gate voltages applied
to each island act as independent tunings to the ‘zero position’ of the individual charging
energies, as is generally assumed.
We described the energy levels of a coupled two-qubit system and described the opera-
tions necessary to test a Bell inequality in this system. We showed how passing a magnetic
flux through the circuit lead to oscillations in both time and flux. We showed that these
oscillations can be considered as being caused by interference between virtual tunnelling
paths between states and showed how diagrams could be a useful visual aid in ‘keeping
track’ of perturbative expansions of the occupation probabilities of each state.
We described the strong coupling limit of the BCS Hamiltonian, where the sums over
electron creation and annihilation operators can be replaced by operators representing
large quantum spins. We then solved this Hamiltonian in both the exact and mean-
field limits, where we found the ground state was the spin coherent state. As part of
this solution, we investigated the rotation of finite spin operators. We investigated the
relationship between the solutions of the spin Hamiltonian and the solutions of the full
Hamiltonian, exact and mean-field. We increased our confidence in the model by showing
that the spin solutions are equal to the full solutions in the limit where the interaction
energy is much larger than the cutoff energy.
We showed how this simple model could be used to easily investigate various phenom-
ena. We described an analogue of the quantum optical effects of superradiance where the
current out of an array of Cooper pair boxes is proportional to the square of the number
of boxes. We showed that this effect remains present when the reservoir is in the spin co-
herent state. This effect illustrates clearly the quantum mechanical nature of the Cooper
pair boxes, and does so using only current measurements, rather than the experimentally
more difficult single-shot measurements.
We also described an analogue of quantum revival, where the coherent oscillations of a
Cooper pair box coupled to a reservoir decay, only to revive again later. The revival is due
to the fact that the Cooper pairs are discrete, and so the phase information is transferred
Conclusion 161
to the levels of the reservoir. We described a similar two-qubit effect, where the oscillations
of two entangled Cooper pair boxes decay and revive. If the entanglement between the
boxes is calculated, we find that the entanglement also decays and then revives. This
effect not only demonstrates the quantum nature of the Cooper pair boxes, but also of the
reservoir, and it is possible that this behaviour could be used as a probe of the reservoir.
We described how quantum spins can be represented using a wavefunction that is
a function of phase. Rather than the standard derivation which treats the Josephson
relations as (classical) equations of motion to be quantised, we derived these equations
directly from a microscopic electronic Hamiltonian, using the spin-Hamiltonian described
in chapter 5. The form for the spin operators in the phase representation were found.
It was found that in the limit of large size, the Hamiltonian found by quantising the
Josephson equations was regained. Corrections to this phase Hamiltonian were described.
In this thesis we have developed a simple model of superconductors that can be easily
applied to the study of various interesting aspects of superconducting systems. We have
used this model to derive the standard macroscopic qubit Hamiltonian directly from a
quantum description of the electrons. This effectively justifies the use of the ‘quantised
Josephson junction Hamiltonian’ that is commonly used to study these systems. The
microscopic derivation allowed us to calculate the corrections to this Hamiltonian, and
show how these corrections affect the energy levels of the system. We have also investigated
some interesting new phenomena (superradiance and revival) in superconducting systems,
which in principle could be observed by measuring the occupation probabilities of the
Cooper pair boxes rather than requiring single - shot measurements. Given that coherent
oscillations between two qubits have been observed, it is hoped that the effects described
above could be observed and illuminate the quantum behaviour of these systems.
Superconductivity remains a fascinating and broad subject nearly a century after its
first discovery. In particular the macroscopic quantum coherent nature of the electron
condensate offers an intriguing glimpse of the connections between the microscopic world
of quantum mechanics and the large-scale everyday macroscopic world. This is clearly
seen in superconducting qubits, where superpositions of macroscopically differing states
can be prepared.
For these reasons it is clear that mesoscopic superconducting systems such as charge
qubits will continue to be of great interest for quantum computing as well as condensed
matter physics, and also for demonstrating and investigating basic aspects of quantum
162 Conclusion
mechanics.
Appendix A
Eigenstates of the Two Qubit
Hamiltonian
In this Appendix we examine the eigenstates and eigenvalues of a coupled two-qubit system.
In particular we find these exactly for the case when all the tunnelling matrix elements are
real and use perturbation theory to generate approximate expressions when the tunnelling
matrix elements are complex, up to third order in T/EC12.
A.1 Exact Eigenstates with Real T12
We are interested in the Hamiltonian of the two - qubit system when the gate charges
are set to the degeneracy point. We can solve this analytically when all the tunnelling
matrix elements are real, i.e. when there is an integer or half integer number of flux quanta