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Università Cattolica del Sacro Cuore
Sede di Brescia
Facoltà di Scienze Matematiche, Fisiche e Naturali
Corso di Laurea in Fisica
Tesi di Laurea Magistrale
Cooperative effects and many-bodytunnelling
Relatore:
Ch.mo Prof. Fausto Borgonovi
Correlatore:
Ch.mo Prof. Giuseppe Luca Celardo
Candidato:
Guido Farinacci
Matricola: 4813507
Anno Accademico 2019/2020
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Contents
1 Introduction 3
2 Single-body dynamics 6
2.1 Generic double-well potential solution . . . . . . . . . . .
. . . . . . 6
2.2 Symmetric double-well potential . . . . . . . . . . . . . .
. . . . . . 8
2.3 Dynamics of the symmetric potential . . . . . . . . . . . .
. . . . . . 11
2.4 Validity of the tight-binding approximation . . . . . . . .
. . . . . . 14
2.5 Loss dynamics for asymmetric potentials . . . . . . . . . .
. . . . . . 22
3 Many-body dynamics 25
3.1 Interacting N -body double-well dynamics . . . . . . . . . .
. . . . . 25
3.2 Generalization to N interacting bosons . . . . . . . . . . .
. . . . . . 29
3.3 Non-interacting tight-binding dynamics . . . . . . . . . . .
. . . . . 30
3.4 Hubbard model . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 34
3.5 Extended Hubbard model . . . . . . . . . . . . . . . . . . .
. . . . . 42
3.6 Numerical results . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 50
4 Conclusions 57
Appendices
A Computational methods 58
B Numerical interaction matrices 65
Acknowledgements 73
Bibliography 74
2
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Chapter 1
Introduction
It is difficult to deny that tunnelling is one of the most
fascinating phenomena in
quantum physics as it seemingly defies any classical intuition.
First introduced as
an explanation for the overcome of the Coulomb barrier in alpha
decay, it has since
been at the core of many physical theories such as that of
Josephson oscillations, the
spatial flipping of the nitrogen atom in the ammonium molecule
or the dynamics of
a lattice potential, among many other examples.
The phenomenon is well understood in the context of elementary
quantum mechanics
and is a direct consequence of the so-called wave-particle
duality. However, its
implications on the dynamics of many-body systems are still
unclear. In recent
years, it has been argued that the presence of inter-particle
interactions may lead to
cooperative behaviours such as the simultaneous tunnelling of a
few particles as a
single object through a potential barrier ([10], [3]): this may
explain the simultaneous
double ionization that is observed for some atoms (see for
example [9]), which for
the moment remains unexplained.
Previously, the issue has been faced with a more mathematical
approach (see [4] and
[2]); in [3] it has been demonstrated that two electrons in a
double-well potential
should exhibit cooperative effects in the presence of a strong
interaction in the
means of a simultaneous two-body tunnelling contribution to the
dynamics. We
shall instead employ a more physical approach by computing the
exact dynamics of a
double-well potential inhabited by a few particles and check a
posteriori the presence
of co-tunnelling processes in the dynamics. Even if previous
works have hinted at
the presence of a co-tunnelling contribution in the numerically
computed dynamics
3
-
4
([10], [6]), none have actually verified if this corresponds to
an actual two-particle
tunnelling amplitude in the interaction matrix in the lattice
site representation; in
other words, the effect has been only described qualitatively
and not quantitatively,
as we shall instead do. This has the inherent benefit of not
only verifying if such
process is physical, but also to gauge how physically relevant
it is.
Another open question is whether many-body tunnelling processes
may exist for
more than two particles: as we will show, the presence of a
two-body interaction
alone suggests the absence of such N -body effect. Other
mechanisms should be
probably adopted to see such an effect.
We shall begin by reviewing the theory of a double-well
potential in the single-
body case: we will find the exact eigenfunctions and eigenvalues
(section 2.2) and
then compute the dynamics in the symmetric potential case
(section 2.3). We shall
then find some analytical predictions for the dynamics by means
of a tight-binding
approximation: in section 2.4 we will discuss in detail the
construction of a site
localized basis, which is a debated topic for double-well
potentials. Finally, we
shall briefly review the dynamics of strongly asymmetrical
double-wells (section
2.5), where one can see the transition from usual Rabi
oscillations to a particle loss
regime (as observed in [10]).
Subsequently, we shall move to analysing the dynamics of the
many-body system:
first, we will provide a generalization of the single-body
tight-binding model to the
many-body non-interacting case (section 3.3). Then, in section
3.4 we will review
the Hubbard model ([5]), which is the most commonly used
approximation in the
study of the dynamics of lattice potentials (of which the
double-well represents a
special case): in the strongly-interacting regime analytical
solutions to the dynamics
can be found for two particles. The full exact dynamics of the
system will then be
computed (section 3.6) as a comparison in the case of a δ-style
“contact” potential to
find the shortcomings of the Hubbard approximation; as a
middle-ground, in section
3.5 we will propose an extension to the Hubbard model which
considers the exact
contributions of the interaction to the system, but is limited
to the first few states
in the many-body spectrum. Its validity can be defined in terms
of the interaction
strength, both attractive and repulsive: in particular it should
be weak enough to
give rise to a negligible probability of occupation of the high
energy states.
Our results confirm the presence of two-particle simultaneous
tunnelling processes
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5
in the dynamics, but also show that their amplitude is several
orders of magnitude
smaller compared to the single-body tunnelling terms, implying
that the overall
effect on the dynamics is small. This is in agreement with the
findings of [6] where
it has been observed that for symmetric double-wells the
dynamics are dominated
by sequential single-particle tunnelling processes, which are
faster due to the larger
coupling. Still, we see that the overall effect is appreciable
and leads to a slightly
different behaviour when compared to the traditional Hubbard
model.
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Chapter 2
Single-body dynamics
2.1 Generic double-well potential solution
In this introductory section we will focus on the single-body
dynamics of a one-
dimensional double-well potential: this will constitute the
groundwork of our later
investigation of the many-body dynamics of such system.
By one-dimensional double-well potential we assume any potential
defined as follows:
V (x) :=
∞ for x < x0,
a for x0 ≤ x ≤ x0 + l,
a+ V0 for x0 + l < x < x0 + l + b,
a for x0 + l + b ≤ x ≤ x0 + l + b+ r,
∞ for x > x0 + l + b+ r.
(2.1)
It comes natural to refer to l and r respectively as the left
and right well sizes, while
b and V0 respectively are the barrier size and height. As any
potential can be defined
up to an arbitrary constant, it is common to choose a = 0; one
is also free to translate
the coordinate system so that x0 = 0 to further simplify
calculations. It must be
admitted that this is not the only possible formulation for a
double-potential well,
as one may as well choose non-square wells, or even have the two
wells at different
energies; the proposed model is the simplest and the most
general.
Naturally, the next step is to solve the Schrödinger equation
for such a potential; to
write our results in a more streamlined way we get rid of the
constants by setting
6
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2.1 ∼ Generic double-well potential solution 7
Figure 2.1: example of a double-well potential with parameters l
= 5, b = 2, r = 7, V0 = 5.
~ = 2m = 1, with m being the mass of the particle that lives
inside our one-
dimensional world. For stationary states of the system, which
form a basis for the
whole Hilbert space of the possible solutions, the equation
reads:(− ∂
2
∂x2+ V (x)
)ψ(x) = Eψ(x) (2.2)
In general, such equation admits infinitely many solutions ψ(x):
however, abundance
is not a particularly appreciated quality, especially in the
context of numerical sim-
ulations. Therefore, we chose to discard a part of the Hilbert
space (alas, infinitely
large) by setting an upper cut-off on the energy spectrum so
that we only consider
the stationary states whose energy is lower than that of the
potential barrier, V0:
naturally, this limits our ability to study the dynamics of the
system only to those
states that have negligible projections on the states over the
barrier energy V0.
Before blindly employing our calculus machinery to find ψ(x), we
make some logical
assumptions based on elementary quantum mechanics: we know that
any wavefunc-
tion with energy lower than V0 will have oscillatory nature
inside the two wells, while
it will be exponentially suppressed inside the barrier. Of
course, ψ(x) must also be
0 wherever the potential is infinite. Therefore we can
rightfully set:
ψ(x) := A ∗
φ1(x) := sin(kx) for 0 ≤ x < l,
φ2(x) := (Beλx + Ce−λx) for l ≤ x < l + b,
φ3(x) := D sin(k(l + b+ r − x)) for l + b ≤ x < l + b+ r,
0 elsewhere.
(2.3)
with k :=√E and λ :=
√V0 − E. Coefficient A is to be calculated via normaliza-
tion, while B, C and D will be determined by setting continuity
conditions for the
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2.2 ∼ Symmetric double-well potential 8
wavefunction and its derivative with respect to the spatial
coordinate x:
φ1(l) = φ2(l),
φ2(l + b) = φ3(l + b),
φ′1(l) = φ′2(l),
φ′2(l + b) = φ′3(l + b).
(2.4)
The reader will surely have spotted that we are trying to solve
a system of four
equations in only three variables B, C and D, seen as functions
of E (or rather k
and λ). Therefore, coefficients B(E), C(E) and D(E) can be
uniquely determined
by choosing a subsystem of only three of the equations, while
the fourth one will
act as a quantization condition for the spectrum E. Such
considerations lead to the
solution:
B(E) =e−λl
2
(sin(kl) +
k
λcos(kl)
), (2.5)
C(E) =e+λl
2
(sin(kl)− k
λcos(kl)
), (2.6)
D(E) =1
2 sin(kr)
(eλb(sin(kl) +
k
λcos(kl)) + e−λb(sin(kl)− k
λcos(kl))
), (2.7)
while the quantization condition for the momenta k is given by
the solutions of the
equation:
f(E) := e2λb −sin(kl)− kλ cos(kl)sin(kl) + kλ cos(kl)
∗sin(kr)− kλ cos(kr)sin(kr) + kλ cos(kr)
= 0. (2.8)
Finally, by imposing the normalization condition∫|ψ(x)|2dx = 1
we obtain:
A(E) =( l
2− sin(2kl)
4k+D2
(r2− sin(2kr)
4k
)+ 2bBC+
+1
2λ
(B2e2λl(e2λb − 1) + C2e−2λl(e−2λb − 1)
))− 12.
(2.9)
2.2 Symmetric double-well potential
As we have just demonstrated, the eigenvalues of the Hamiltonian
in equation 2.2
can be obtained by finding the roots of equation 2.8. However,
such task cannot be
accomplished analytically and one must employ some sort of
numerical algorithm:
unfortunately this can be somewhat difficult in the case of a
symmetric double-well
potential (l = r), as the eigenstates come in couples of
quasi-degenerate states, one
spatially symmetric and one anti-symmetric with respect to the
center of the barrier
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2.2 ∼ Symmetric double-well potential 9
(x = l + b/2), only separated by a small splitting in energy; a
not so careful choice
of the precision of the algorithm may either cause the loss of
some of the eigenval-
ues if the scanning is too rough, while a very meticulous
scanning greatly inflates
computational times.
A safer path is to impose the spatial symmetry (or
anti-symmetry) a priori alongside
the continuity conditions for ψ(x), so that this way we obtain
two different quanti-
zation conditions for the momenta, one for the even states and
one for the odd ones.
This is achieved by adding equation
ψ(x) = ±ψ(l + b+ r − x) (2.10)
to system 2.4, choosing + for the even states and − for the odd
ones.
Naturally, the previous definitions of A(E), B(E), C(E) and D(E)
still hold true
by setting l = r, while as expected we obtain two uncoupled
equations to determine
the eigenvalues of the Hamiltonian:
feven(E) :=λ
ktanh(λa)− tan(kc) sin(ka) + cos(ka)
sin(ka)− tan(kc) cos(ka)= 0, (2.11)
fodd(E) :=k
λtanh(λa)− sin(ka)− tan(kc) cos(ka)
tan(kc) sin(ka) + cos(ka)= 0, (2.12)
where for the sake of readability we have set a = b2 and c = a+
l.
Figure 2.2: this plots verifies that the quantization conditions
for even (2.11) and odd (2.12) states
yield the same eigenvalues as 2.8 in the case of a symmetric
potential; the parameters used in this
example are l = r = 5, b = 0.5 and V0 = 5.
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2.2 ∼ Symmetric double-well potential 10
Figure 2.3: lowest 8 eigenstates of a symmetric double-well
potential with parameters l = r = 2,
b = 0.2, V0 = 500; in total, such potential parameters allow for
28 bound eigenstates with energy
lower than the central barrier. The reader can notice the
aforementioned fact that states come
in doublets of spatially symmetric (left column) and
anti-symmetric (right column) wavefunctions,
with the states inside each doublet having the same number of
nodes inside each well. In red:
schematic representation of the double-well potential.
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2.3 ∼ Dynamics of the symmetric potential 11
2.3 Dynamics of the symmetric potential
To evaluate the behaviour of the system we calculate its
evolution in time: we expect
that even in the case of many non-interacting particles we
should obtain the same
results, provided analogous starting conditions.
Following the results from previous sections, given a set of
potential parameters
we can calculate the single-body spectrum Ei and the
corresponding eigenfunctions
ψi(x). We are free to choose any initial state Ψ0(x); it then
follows that the state
at any given time t must be:
Ψ(x, t) = 〈x|Ψ(t)〉 = 〈x|e−iĤt|Ψ0〉 = 〈x|e−iĤt(∑
n
〈ψn|Ψ0〉)|ψn〉 :=
=∑n
cne−iEnt 〈x|ψn〉 =
∑n
cne−iEntψn(x).
(2.13)
Naturally, the evolved state 2.13 in itself contains all the
information about the dy-
namics of the system. However, to make sense of such information
we must calculate
some more elementary observables, as for example the probability
of occupation of
the left and right halves of the system:
PL(t) :=
∫ l+b/20
|Ψ(x, t)|2dx; (2.14)
PR(t) :=
∫ l+b+rl+b/2
|Ψ(x, t)|2dx. (2.15)
We may start from any arbitrary initial state |Ψ0〉. However in
the case of a single
particle there are only two sensible choices: we can either
start with the particle
in the left or in the right well; we choose to start from the
left well by convention.
It is clear that such concept, while being perfectly intuitive,
makes no sense from
a mathematical point of view: in technical terms, we must choose
a starting state
where PL(0) ≈ 1 � PR(0) ≈ 0.1 A very simple choice that
satisfies such condition
is |Ψ0〉 = |∞L〉, with |∞L〉 being the ground state of the left
potential well with
V0 →∞. The reader may easily verify that this implies:
Ψ0(x) := 〈x|∞L〉 =
√
2l sin
(πxl
)for 0 ≤ x < l,
0 elsewhere.
(2.16)
1This is unfortunately easier said than done in practical terms:
as previously noted, to make it
possible to perform any numerical computation one must truncate
the Hilbert space, so that we
lose its completeness and therefore we lose the ability to
expand any arbitrary initial state onto a
given basis. One must be cautious to choose an initial state
that has negligible projections over the
states excluded by the truncation.
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2.3 ∼ Dynamics of the symmetric potential 12
As we have already underlined, in the two observable quantities
2.14 and 2.15 we
are not actually looking at the occupation of the sole two
wells, but more precisely
at the occupation of the two halves of the system with respect
to the center of the
barrier: this has been done for the sake of convenience as then
the two probabilities
are complementary and sum up to 1. In any case, results will not
be much different
than the more natural choice of only calculating the occupation
of the two wells
as, first and foremost, the eigenfunctions are strongly
suppressed inside the barrier;
moreover, we will usually choose the size of the barrier b to be
negligible in compar-
ison to the size of the two wells l and r.
Naturally, any numerical result must be partnered to some
analytical approximation
for comparison; we provide such in the form of a tight-binding
model where we ne-
glect the presence of any single-particle eigenstates besides
the lowest two in energy.
This hypothesis obviously discards any contribution given by
higher states of the
spectrum to the dynamics, but allows us to analytically compute
the left and right
well occupations in a simple and readable form. One must be sure
to check that the
higher states in the spectrum do not play an important role in
the dynamics before
making any comparison: in the case of a single particle, this is
simply guaranteed,
following 2.13, by choosing an initial state where c0, c1 � cn,
∀n ≥ 2 because coeffi-
cients cn do not evolve in time. Further details on the validity
of the tight-binding
model will be delayed until the following section.
Our tight-binding basis is restricted to only the lowest two
eigenstates of the total
Hamiltonian, |ψ0〉 and |ψ1〉. Following our previous findings, |
〈x|ψ0〉 |2 is spatially
symmetric around the center of the barrier, while | 〈x|ψ1〉 |2 is
antisymmetric, there-
fore for the sake of clarity in the next calculations we make
this property explicit by
calling |ψ0〉 := |s〉 (with s standing for symmetric) and |ψ1〉 :=
|a〉 (with a standing
for antisymmetric). We can exploit such symmetry to define the
localized wavefunc-
tions for each potential well in a very simple – although
approximate – form:
|L〉 := 1√2
(|s〉+ |a〉
), (2.17)
|R〉 := 1√2
(|s〉 − |a〉
), (2.18)
with |L〉 being localized inside the left well and |R〉 in the
right one. Unfortunately,
the contributions of |a〉 and |s〉 to |L〉 do not cancel out
exactly inside the right well,
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2.3 ∼ Dynamics of the symmetric potential 13
leaving a small oscillating probability amplitude; the same can
be said for |R〉 in the
left well. Therefore we cannot consider |L〉 and |R〉 as an exact
basis for the left and
right sites; once again, further discussion on this matter is
delayed until the next
section.
It is convenient to calculate the Hamiltonian matrix terms in
the {|L〉 , |R〉} basis.
As |s〉 and |a〉 are by definition eigenstates of the Hamiltonian,
we have:
Ĥ |s〉 = Es |s〉 , (2.19)
Ĥ |a〉 = Ea |a〉 , (2.20)
thus:
〈L|Ĥ|L〉 = 〈R|Ĥ|R〉 = 12
(Es + Ea) := E0, (2.21)
〈L|Ĥ|R〉 = 〈R|Ĥ|L〉 = 12
(Es − Ea) := Ω0. (2.22)
To obtain analogous dynamics as obtained in the numerical
calculations, we choose
our initial state to be |L〉, which guarantees that the particle
at the initial time is
mostly2 in the left well. so that we can calculate the
occupation probabilities for
the two wells in time by approximating them as the projections
of the evolved state
onto the |L〉 and |R〉 states:
PL(t) ≈∣∣∣ 〈L|e−iĤt|L〉 ∣∣∣2 = ∣∣∣1
2
(e−iEst + e−iEat
)∣∣∣2 ==∣∣∣e−iEa+Es2 t
2
(e−i
Ea−Es2
t + e+iEa−Es
2t)∣∣∣2 =
= cos2(Ω0t),
(2.23)
PR(t) ≈∣∣∣ 〈R|e−iĤt|L〉 ∣∣∣2 = ∣∣∣1
2
(e−iEst − e−iEat
)∣∣∣2 == (...) = sin2(Ω0t).
(2.24)
2It is not confined in the left well due to the aforementioned
fact that |L〉 has a small but non-zero
probability amplitude outside of the left well.
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2.4 ∼ Validity of the tight-binding approximation 14
Figure 2.4: numerical results for the left and right well
occupation probabilities 2.14 and 2.15 at
different times for a single particle starting from state 2.16
and potential parameters l = r = 2,
b = 0.2, V0 = 500; the black dotted lines represent the
tight-binding predictions 2.23 and 2.24,
which in this case overlap the numerical results. Ω0 ≈ 0.00238
as defined in 2.22.
2.4 Validity of the tight-binding approximation
The goodness of the tight-binding approximation relies on the
fact that the |L〉 and
|R〉 states, as defined 2.17 in and 2.18, approximate reasonably
well the “site” basis3
for the double-well potential. Now, there is no clear-cut
definition of such basis;
however, one reasonable choice would be to consider the ground
state of each of the
wells with a very large barrier size, i.e. b → ∞ (which is
equivalent to filling the
second well up to V0). This way, most of the contribution to the
probability is given
by the region inside the considered well, because outside of it
the wavefunction is
either zero or exponentially suppressed. For example, to obtain
such state for the
left well we must find the solutions of the Schrödinger
equation(− ∂
2
∂x2+ VL(x)
)ψL(x) = EψL(x), (2.25)
with:
VL(x) :=
∞ for x < 0,
0 for 0 ≤ x ≤ l,
V0 elsewhere.
(2.26)
3The term basis is used here quite liberally; our scope is to
find a set of (two) states for which
we would be allowed to say that the particle resides inside one
of the wells (almost) exclusively,
while keeping the treatment restricted to as few eigenstates as
possible as to allow for analytical
predictions to be calculated in a simple way.
-
2.4 ∼ Validity of the tight-binding approximation 15
(a) (b)
(c)
Figure 2.5: (a): first two eigenfunctions of the double-well
potential with parameters l = r = 2,
b = 0.2, V0 = 500; (b): näıve site basis as defined in 2.17 and
2.18 for the same potential parameters;
(c): zoom of plot (b) to highlight the contribution of each
näıve site basis state in the opposite
potential well.
Without dwelling into details, we infer that any wavefunction
with energy smaller
than the barrier height V0 must have oscillatory nature inside
the well, while it must
be exponentially suppressed inside the barrier:
ψL(x) := N ∗
sin(kLx) for 0 ≤ x < l,
sin(kLl)e−λL(x−l) for x ≥ l,
0 elsewhere.
(2.27)
where kL :=√EL and λL :=
√V0 − EL, with EL to be determined numerically via
the level quantization condition given by wavefunction
continuity:
tan(kLl) = −kLλL. (2.28)
Via normalization we also get:
N =( l
2− 1
2kLsin(kLl) cos(kLl) +
1
2λLsin2(kLl)
)− 12. (2.29)
Naturally, by symmetry the same holds true for the right
potential well, with the
isolated state being ψR(x) := ψL(l + b+ r − x).
-
2.4 ∼ Validity of the tight-binding approximation 16
(a) (b)
(c) (d)
(e) (f)
Figure 2.6: left column: 〈x|L〉 and ψL(x) defined respectively as
in 2.17 and 2.27 for different
sizes of the barrier b; right column:∣∣∣| 〈x|L〉 |2 − |ψL(x)|2∣∣∣
for the corresponding barrier size on the
left (b = 0.5 for (a) and (b), b = 1 for (c) and (d), b = 2.5
for (e) and (f)). The other potential
parameters are kept fixed at l = r = 5 and V0 = 10.
-
2.4 ∼ Validity of the tight-binding approximation 17
(a) (b)
(c) (d)
(e) (f)
Figure 2.7: left column: 〈x|L〉 and ψL(x) defined respectively as
in 2.17 and 2.27 for different
heights of the barrier V0; right column:∣∣∣| 〈x|L〉 |2 −
|ψL(x)|2∣∣∣ for the corresponding barrier height
on the left (V0 = 0.5 for (a) and (b), V0 = 1 for (c) and (d),
V0 = 25 for (e) and (f)). The other
potential parameters are kept fixed at l = r = 5 and b = 1.
Note: barrier height in the plots is not
to scale.
-
2.4 ∼ Validity of the tight-binding approximation 18
As the reader may appreciate from figures 2.6 and 2.7, our set
of states {|L〉 , |R〉}
approximates the localized states {|ψL〉 , |ψR〉} only when the
two potential wells
are separated by a barrier of sufficiently large width (b)
and/or height (V0): this is
particularly highlighted by the figures on the right-hand side,
which represent the
difference in the probability distributions, which are of
increasingly smaller order
of magnitude as the barrier gets stronger. Another interesting
feature is that for
strong barriers 〈x|L〉 and ψL(x) are well localized inside the
left-well region.
Our findings can be further confirmed by comparing the Rabi
oscillation frequency
obtained for our tight-binding approximation, which accordingly
to 2.14 and 2.15
is Ω0, with the model proposed by Landau and Lifshitz (page
175-176 of [7]): as
we have verified by solving the Schrödinger equation in the
previous sections, the
presence of a finite potential barrier splits the single-well4
energy levels into doublets
of even and odd states, separated by a small splitting energy.5
If, for example, we
take a single-well normalized wavefunction φ0(x) with
corresponding energy E0, the
presence of the barrier will split such level into a doublet
φ1(x) and φ2(x), with
respective energies E1 and E2. As noted, if we take our
coordinate system to be
centred around the middle of the barrier, such wavefunctions can
be approximated
as the symmetric and anti-symmetric combinations of φ0(x) and
φ0(−x):
φ1(x) :=1√2
(φ0(x) + φ0(−x)
),
φ2(x) :=1√2
(φ0(x)− φ0(−x)
).
(2.30)
According to Schrödinger’s equation we have:
∂2
∂x2φ0(x) +
2m
~2(E0 − U(x)
)φ0(x) = 0,
∂2
∂x2φ1(x) +
2m
~2(E1 − U(x)
)φ1(x) = 0,
(2.31)
with U(x) being a generic double-well potential. By multiplying
each equation in
2.30 respectively by φ1(x) and φ0(x), this implies:φ1(x)φ
′′0(x) +
2m~2
(E0 − U(x)
)φ1(x)φ0(x) = 0
φ0(x)φ′′1(x) +
2m~2
(E1 − U(x)
)φ0(x)φ1(x) = 0
(2.32)
4We are here referring to each one of the two-wells from the
double-well potential taken sepa-
rately; in our treatment it is analogous to 2.26.5In general,
the splitting energy is not constant along the spectrum and is
heavily dependent on
the shape of the potential barrier.
-
2.4 ∼ Validity of the tight-binding approximation 19
where short-hand apostrophe notation has been used for the
derivative over x. Com-
puting the difference of the two equations and integrating over
x between 0 and ∞
one obtains: (φ1(x)φ
′0(x)
)∣∣∣∞0−(φ0(x)φ
′1(x)
)∣∣∣∞0
+
+2m
~2(E0 − E1)
∫ ∞0
φ0(x)φ1(x)dx = 0.(2.33)
Finally, if we remember that φ1(0) =√
2φ0(0), φ′1(0) = 0 and we approximate
6∫ ∞0
φ0(x)φ1(x)dx ≈1√2
∫ ∞0
φ20(x)dx =1√2, (2.34)
where the last equality holds true because φ0(x) is normalized
and lives inside a
single well, we finally get:
E1 − E0 = −~2
m
(φ0(0)φ
′0(0)
). (2.35)
The same process can be repeated for φ2(x) and subtracting the
two results we
finally get an expression for the splitting energy:
∆ := E2 − E1 =2~2
m
(φ0(0)φ
′0(0)
). (2.36)
The power of this simple models relies on the fact that it makes
no assumptions
about the nature of the double-well potential; our treatment
instead is only valid
for square double-wells.
Adapting result 2.36 to our model implies rescaling the units
such that ~ = 2m = 1
and taking φ0(x) = ψL(x), according to 2.27, with a simple
translation of the x
coordinate (x → x + l + b/2) to account for the fact that our
coordinate system is
not centred around the middle of the barrier:
|∆| =∣∣∣4ψL(l + b
2
)ψ′L
(l +
b
2
)∣∣∣ = ∣∣∣4λLψ2L(l + b2)∣∣∣, (2.37)where the last equality
directly descends from definition 2.27. In our tight-binding
model, the Rabi oscillation frequency for one particle is equal
to half the splitting
energy, Ω0; therefore, according to 2.37 we should expect this
value to approximate
∆/2 (see figure 2.8: the two models are in good agreement for
sufficiently large
barrier height V0, while a large barrier width b seems to
equally affect both models
by making the splitting energy smaller).6This is an
approximation due to the fact that we are neglecting the
contribution of φ0(−x) for
x ≥ 0; this can be done due to the fact that such contribution
is exponentially suppressed.
-
2.4 ∼ Validity of the tight-binding approximation 20
(a) (b)
(c)
Figure 2.8: Rabi oscillation frequency for the tight-binding
model Ω0 and according to Landau’s
approximation for varying barrier width b and different barrier
heights (V0 = 0.5 for (a), 5 for (b)
and 10 for (c)); the remaining potential parameters are l = r =
5.
In our treatment we have given for granted that states |L〉 and
|R〉 represent our best
effort in creating a site localized basis when restricting the
double-well spectrum to
just the first two states; despite this is common practice in
literature, we would like
to attempt to give a demonstration to the fact that this is
indeed a sensible choice.
If we take {|ψ0〉 , |ψ1〉} as our basis, any state of the system
can be expressed in the
form:
|χ〉 := cos(ϑ) |ψ0〉+ eiϕ sin(ϑ) |ψ1〉 , (2.38)
where ϕ is an arbitrary phase angle; the state is normalized by
construction. If we
wish |χ〉 to be completely localized inside the left well, we
expect:∫ l0| 〈x|χ〉 |2dx = 1. (2.39)
One can easily verify that:
| 〈x|χ〉 |2 = cos2(ϑ)|ψ0(x)|2 + sin2(ϑ)|ψ1(x)|2+
+ sin(2ϑ) cos(ϕ)ψ0(x)ψ1(x),(2.40)
-
2.4 ∼ Validity of the tight-binding approximation 21
where the last term can be written in such simple form due to
the fact that ψ0(x)
and ψ1(x) are real-valued functions. Therefore, by defining:I1
:=
∫ l0 |ψ0(x)|
2dx
I2 :=∫ l
0 |ψ1(x)|2dx
I3 :=∫ l
0 ψ0(x)ψ1(x)dx
(2.41)
according to 2.39 we should have:
cos2(ϑ)I1 + sin2(ϑ)I2 + sin(2ϑ) cos(ϕ)I3 = 1. (2.42)
Now, as the eigenfunctions are normalized over [0, 2l + b],
which is the width of
the symmetric double-well, and taken into account the spacial
symmetry properties
of the wavefunctions, we expect that the integral of their
square modulus over half
their support, [0, l+b/2], is 1/2; therefore, their integral
over the sole left well should
surely be less than 1/2 as the integrand is always positive,
implying I1, I2 < 1/2.
Moreover, as we have observed, inside the left well we have
ψ1(x) ≈ ψ0(x) (and the
same holds true for the right well with the addition of a minus
sign in front of either
wavefunction), so we may as well say I3 < 1/2. In general,
one should then calculate
the three integrals in 2.41 and solve equation 2.42 to find the
appropriate values for
ϑ and ϕ. If we set ξ := max(I1, I2) < 1/2, equation 2.42
reduces to:
sin(2ϑ) cos(ϕ) =1− ξI3
>1
2I3> 1, (2.43)
which one can quite easily see is not satisfied by any choice of
ϑ and ϕ, therefore
shattering our hope of finding a fully localized state by only
combining |ψ0〉 and
|ψ1〉.
However, if we take sufficiently large potential parameters b
and/or V0, ψ0(x) and
ψ1(x) will be strongly suppressed inside the barrier, so that
the contribution to I1,
I2 and I3 in the region [b, b/2] should be negligible,7 so that
we have I1, I2, I3 / 1/2,
therefore we may light-heartedly assume I1 = I2 = I3 = 1/2,
hence equation 2.42
reduces to:
sin(2ϑ) cos(ϕ) = 1. (2.44)
The reader may easily verify that, given ϑ, ϕ ∈ ]− π, π], this
leads to the solutions:
ϑ =π
4, ϕ = 0 ∨ ϑ = −π
4, ϕ = π (2.45)
7As an example, for potential parameters l = r = 2, b = 0.2 and
V0 = 500 we have that the
three integrals are just below 1/2 by order 10−5.
-
2.5 ∼ Loss dynamics for asymmetric potentials 22
which indeed imply either |χ〉 = |L〉 or |χ〉 = |R〉,8 confirming
the assumption that
this represents our best effort in building a localized site
basis only employing the
first two states in the spectrum in the context of strong enough
potential barriers.
2.5 Loss dynamics for asymmetric potentials
Even though our scope in the next chapters will mostly concern
symmetric double-
well potentials, we would like to give a brief overlook to the
dynamics of a strongly
asymmetrical double-well at least for the single-particle case:
the following para-
graph has no ambition to be regarded as a complete and
self-standing discussion on
the topic.
Naturally, for l ≈ r we expect to observe similar dynamics as
seen for the symmetric
potential, so the particle will oscillate between the left and
the right well if initially
prepared inside either one of them. Instead, when the size of
one of the two wells
is significantly larger than the other, for example r � l, the
system will radically
change its behaviour and an exponential loss of occupation
probability for the ini-
tial well is observed. This can be understood in the context of
Wigner’s theory
of decaying systems if we consider the two wells as separate but
interacting. This
becomes clear when we are faced with the expression for the
density of states in a
square well; we know from elementary quantum mechanics that the
energy levels for
a one-dimensional square potential well of size a, taken ~ = 2m
= 1, are:
En =π2n2
a2, (2.46)
where n ∈ N is a label for each state. Therefore we can quite
simply differentiate:
dE =2π2n
a2dn, (2.47)
from which the density of states ρ(E) descends directly:
=⇒ ρ(E) := dndE
=a2
2nπ2=
a
2π√E. (2.48)
We obtain that the spectral density of states is directly
proportional to the size of
the quantum well a: if the size is large enough, the spectrum
may therefore be taken
as a continuum. As anticipated, if we initially prepare the
particle in the left well
8The fact that we also get |R〉 as a possible solution even
though we only imposed conditions
for the integrals inside the left well is a by-product of the
spatial symmetry of the potential.
-
2.5 ∼ Loss dynamics for asymmetric potentials 23
with l� r, the problem can be reformulated as that of a single
energy level from the
left well coupled to the continuum of states from the right
well, justifying the usage
of Wigner’s prediction for which we expect that the projections
of the evolved state
over the eigenstates of the Hamiltonian |cn(t)|2 := | 〈ψn|ψ(0)〉
|2 are distributed with
a Lorentzian shape of width γ in energy space, which leads to an
exponential decay
with rate γ in the time domain. Following [6], we may
approximate the energy
spectrum as homogeneous and instead look at the distribution of
the projections
|cn(t)|2 in the space of state labels n, which under this
approximation still follows a
Lorentzian shape (see figure 2.9 (a)), but with a different
width parameter Γ, which
will be equal to γ multiplied by the supposedly constant density
of states in the
spectrum:
Γ := γρ(E). (2.49)
This gives a first numerically computable expression for the
decay rate:
γ0 :=2πΓ
r
√Ei, (2.50)
where Ei is the energy of the initial state |ψ(0)〉 at which the
density of states is
numerically evaluated according to 2.48. Still following [6], we
can make a sightly
more sophisticated assumption; first we define the participation
ratio for a given
state:
PR(|ψ(0)〉
):=(∑
n
|cn|4)−1
, (2.51)
which gives a measure of how many eigenstates contribute to the
initial state. If one
assumes a Lorentzian distribution for the projections and
performs the summation
in 2.51, we get:
PR(|ψ(0)〉
)= πΓ. (2.52)
By analogy with equation 2.49, we obtain a more refined
expression for the decay
rate which does not assume a perfectly Lorentzian distribution
of the projections:
γ1 :=2PR(|ψ(0)〉)
r
√Ei. (2.53)
Finally, another expression can be found in literature for the
decay rate (see [1] for
further details on its derivation):
γ2 :=8α3E
3/2i
V 20 (1 + αl2)e−2αb, (2.54)
-
2.5 ∼ Loss dynamics for asymmetric potentials 24
where for readability’s sake α :=√V0 − Ei.
The three proposed expressions for the decay rate have been
compared with the
numerical result for a particular set of parameters in figure
2.9 (b) and are all in
good agreement provided a sufficiently large right well size r;
this is made evident
by the simulations shown in figure 2.9 (c), where one can
appreciate the onset of the
particle loss regime only for sufficiently high r (r > 1000
for the particular choice of
parameters, refer to the figure caption for further details).
For small enough r the
dynamics still follow the characteristic behaviour of Rabi
oscillations, although with
a sightly smaller amplitude compared to the symmetric case.
(a) (b)
(c)
Figure 2.9: (a): distribution of the |cn|2 in the space of state
labels for l = 51, b = 4, V0 =
0.1 and r = 4000; inset: zoom of the plot around the peak of the
distribution, in red a least
squares Lorentzian fit with center ∼ 55.668 and width ∼ 0.985;
(b): numerically computed left well
occupation 2.14 for the same potential parameters and analytical
prediction according to decay rates
2.50, 2.53 and 2.54; (c): numerically computed left well
occupation 2.14 for potential parameters
l = 51, b = 4, V0 = 0.1 and different values of r: the
transition from Rabi oscillations to exponential
decay regime is observed for r > 1000.
-
Chapter 3
Many-body dynamics
3.1 Interacting N-body double-well dynamics
Now that we have laid the foundations for our analysis of the
double-well potential
by studying the dynamics of a single particle, we may now start
to tackle the more
sophisticated task of populating our system with a multitude of
particles. In this
introductory section we will begin by looking at the problem
from a general point
of view and, subsequently, we will move on to analyse more
specific cases so that we
can provide both numerical simulations and analytical
predictions to be confirmed
or, sometimes more interestingly, disproved.
First and foremost, dealing with more than one particle
introduces a new and ex-
tremely radical feature in our system: the possible presence of
an interaction be-
tween said particles, which can introduce novel effects in the
dynamics of the system.
Mathematically, if for the moment we restrict our treatment to N
distinguishable
particles, this translates into the presence of a new term in
the total Hamiltonian:
Ĥ(x1, . . . , xN ) :=
N∑i=1
Ĥ(0)i (xi) +
1
2
N∑i=1
∑j 6=i
Û(xi, xj), (3.1)
where xi is the position of the i-th particle, Û is an
interaction potential that acts
on any possible combination of two particles, and finally Ĥ(0)
is the non-interacting
Hamiltonian that acts separately on each single particle (the
dynamics of which have
been studied in the previous chapter):
Ĥ(0)i (xi) := T̂ (xi) + V̂ (xi), (3.2)
25
-
3.1 ∼ Interacting N -body double-well dynamics 26
with T̂ being the kinetic term and V̂ the double-well
potential.
Now, the interaction potential Û may take any arbitrary form,
one common example
would be a Coulomb type ∼ 1/r electrostatic interaction, with r
being the distance
between the two involved particles. However, such a choice would
make it very
difficult to express any analytical prediction as the
interaction matrix terms could
only be numerically calculated so that they could not be easily
expressed in terms of
the elementary quantities that characterise our physical system.
Instead, following a
common convention in literature (for example see[10]), we employ
a δ-type “contact”
interaction:
Û(xi, xj) := Uδ(xi − xj), (3.3)
where U is a tunable interaction strength parameter, which can
take both a pos-
itive or negative sign to emulate respectively a repulsive or
attractive interaction.
Of course, such a choice discards any long-range contributions
to the interaction,
making it an extremely simplistic model. On the good side, in
the limit of strong
interactions U → ∞ such a potential mimics the effect of the
Pauli exclusion prin-
ciple as the particles then act as impenetrable balls, making it
feasible to map the
problem of N interacting bosons to the possibly simpler one of N
free fermions (see
for example [10] for an in-depth but still extremely
straight-forward treatment of
the problem).
Getting onto more practical issues, we will now need to find
some observables to
be calculated at different times to have a picture of the
dynamics of the system:
naturally we wish to extend the concepts of left and right well
occupations to the
many-body case. As in any quantum-mechanical problem, the first
step is to find
a convenient basis. Following the results from the previous
chapter, we know how
to calculate the single-body spectrum {Ei} and eigenbasis
{|ψi〉}, where the i in-
dex runs over the states in the spectrum.1 Therefore an easy
choice is to build
the distinguishable many-body basis as an Hartree tensor prouct
of the single-body
eigenstates:
|Ψa1,...,aN 〉 := |ψa1〉1 ⊗ |ψa2〉2 ⊗ . . .⊗ |ψaN 〉N , (3.4)
where the ai is the quantum number for each particle, labelled
by the subscript | · 〉i.
One can see that in the position representation this simply
translates as the product
1Naturally we take for granted that all particles are identical
(however not always undistinguish-
able) and therefore they all share the same single-body
spectrum.
-
3.1 ∼ Interacting N -body double-well dynamics 27
of the single-particle wavefunctions:
Ψa1,...,aN (x1, . . . , xN ) := 〈x1, . . . , xN |Ψa1,...,aN 〉
=N∏i=1
ψai(xi). (3.5)
If we define the cardinality of the single body spectrum {Ei} as
C, we find that our
many-body Hilbert space will have dimension CN .2
Now, if for a moment we forget the interaction, we can verify
that this Hartree
product basis is an eigenbasis for the total non-interacting
Hamiltonian, which in
this case is just a sum of single-particle operators:
Ĥ(U=0) |Ψa1,...,aN 〉 =N∑i=1
Ĥ(0)i
(|ψa1〉1 ⊗ |ψa2〉2 ⊗ . . .⊗ |ψaN 〉N
)=
=
N∑i=1
Eai |Ψa1,...,aN 〉 .
(3.6)
This is, unfortunately, generally untrue for the interacting
term which would oth-
erwise just act as a simple energy shift for the free many-body
levels. Instead, the
eigenstates must be found by diagonalizing the total Hamiltonian
for each consid-
ered value of the interaction strength U ; let us say that in
general such interacting
eigenbasis will be indicated as {|α〉}, on the other hand the
free eigenbasis 3.4 will
be indicated as {|k〉}.3
In general, we may choose a different starting state for each
one of the N particles,
so that the many-body initial state can be expressed as:
|Ψ0〉 :=N⊗i=1
|ψ(0)〉i , (3.7)
with |ψ(0)〉i being the starting state for the i-th particle. The
trick up our sleeve is
that we know how to express the free eigenbasis states in the
position representation
in a simple analytical form, so that it is convenient for us to
express everything in
this basis. Consequently we define:
|Ψ0〉 =∑k
〈k|Ψ0〉 |k〉 :=∑k
ck0 |k〉 . (3.8)
2Naturally, if we don’t take any approximation C = ∞; as before,
to have any hope to perform
numerical calculations we will have to truncate the spectrum.
Anyways, the number of basis states
grows exponentially with the number of particles, so that we may
have to take a lower cut-off than
the barrier energy to keep the calculations manageable.3Formally
it would be more correct to label each state in the basis with a
subscript, but for the
sake of readability we will leave them unlabeled.
-
3.1 ∼ Interacting N -body double-well dynamics 28
However, to apply the total time evolution operator to the
initial state we must
make an intermediate step in the interacting eigenbasis:
|Ψ(t)〉 := e−iĤt |Ψ0〉 =∑α
∑k
ck0 〈α|k〉 e−iEαt |α〉 :=
=∑α
cα0 e−iEαt |α〉 =
∑k
∑α
cα0 e−iEαt 〈k|α〉 |k〉 :=
=∑k
∑α
cα0 e−iEαtckα |k〉 :=
∑k
ck(t) |k〉 ,
(3.9)
where the ckα are the {|α〉} → {|k〉} change of basis matrix
terms.
Finally, we may now extend the hole occupation observables that
we used in the
single particle case (2.14 and 2.15) to the many-body case; we
define Pm(t) as the
probability of finding the first m particles in the left
potential well at time t:4
Pm(t) :=
∫Ldx1 · · ·
∫Ldxm
∫Rdxm+1 · · ·
∫RdxN |Ψ(x1, . . . , xN , t)|2 =
=
∫Ldx1 · · ·
∫Ldxm
∫Rdxm+1 · · ·
∫RdxN 〈Ψ(t)|x〉 〈x|Ψ(t)〉 =
=
∫Ldx1 · · ·
∫Ldxm
∫Rdxm+1 · · ·
· · ·∫RdxN
∑k
∑k′
c∗k′(t)ck(t) 〈k′|x1, . . . , xN 〉 〈x1, . . . , xN |k〉 ,
(3.10)
where the L and R subscripts respectively denote that the
integration has to be
carried over the left and the right potential wells. If, for
example, we take:
|k〉 := |ψa1〉1 ⊗ . . .⊗ |ψaN 〉N (3.11)
|k′〉 := |ψb1〉1 ⊗ . . .⊗ |ψbN 〉N , (3.12)
we get an easily computable expression for Pm(t):
=⇒ Pm(t) =∑
a1,...,aN
∑b1,...,bN
c∗b1,...,bN (t)ca1,...,aN (t)
∗m∏i=1
∫Lψ∗bi(xi)ψai(xi)dxi
N∏j=m+1
∫Rψ∗bj (xj)ψaj (xj)dxj .
(3.13)
Now, calculating the probability of finding the first m
particles in the left well does
not make much physical sense: this is a consequence of the fact
that this observable
is inherently more suitable for indistinguishable particles, so
that we would have to
consider all possible ways in which we can fill the left well
with m particles, while
4As for the single-body case, we actually extend the occupation
up to the centre of the potential
barrier to have complementary probabilities.
-
3.2 ∼ Generalization to N interacting bosons 29
we are working with distinguishable ones; we shall later move on
to analyze such
case, which is surely more interesting from a physical point of
view.
Another interesting observable that can be evaluated is the
probability density dis-
tribution in space for each particle: as an example, we will now
show how it can
be calculated for the first particle by marginalizing the total
probability density
function |Ψ(x1, . . . , xN , t)|2:
ρ(x1, t) :=
∫dx2 . . . dxN |Ψ(x1, . . . , xN , t)|2 =
=∑k
∑k′
c∗k′(t)ck(t)
∫dx2 . . . dxN 〈k′|x1, . . . , xN 〉 〈x1, . . . , xN |k〉 =
=∑
a1,...,aN
∑b1,...,bN
c∗b1,...,bN (t)ca1,...,aN (t)ψ∗b1(x1)ψa1(x1) ∗
∗N∏i=2
dxiψ∗bi
(xi)ψai(xi),
(3.14)
where we have recovered definitions 3.11 and 3.12.
3.2 Generalization to N interacting bosons
Studying the dynamics of many interacting bosons can be
particularly enlightening
if, for example, we choose to prepare all N particles in the
same state at t = 0: in
fact, this is a very simplified model of an interacting
Bose-Einstein condensate in
a double-well trap and the dynamics can be studied exactly,
opposed to the usual
mean-field treatment.
On paper, the problem of switching from distinguishable
particles to bosons is a
simple task: we must ensure that in our formalism the particles
are indistinguish-
able and that the states of the system are totally symmetric
under any exchange of
two particles (see for example chapter 7 in [8] for an in-depth
analysis of the issue).
Usually, this is performed by switching to the so-called second
quantization formal-
ism; however, as we wish to keep as much as we can of the
results we have obtained
so far5 we shall use what therefore should be called first
quantization formalism and
ensure the bosonic particle exchange symmetry for the states. In
general, given a N
distinguishable (but otherwise identical) particle state
|Ψa1,...,aN 〉, any permutation5We shall not forget that the scope
of our overview of the dynamics is to perform numerical
simulations, so we – or rather the author – wish to adapt the
code developed so far for distinguishable
particles with minimum effort.
-
3.3 ∼ Non-interacting tight-binding dynamics 30
of the label set {a1, . . . , aN} constitutes a new and
distinguished state; the number
of such states is equal ton1! . . . nN !
N !(3.15)
where ni is the number of times ai appears in {a1, . . . , aN}.
Imposing bosonic
particle exchange symmetry, all the permutations of a
distinguishable particle state
contribute to a single bosonic state, defined as the sum over
the permutations of the
particle labels:
|Ψa1,...,aN 〉+ :=√n1! . . . nN !
N !
∑{P}
P̂ |Ψa1,...,aN 〉 , (3.16)
where the + subscript denotes bosonic symmetry (opposed to − for
fermionic anti-
symmetry), and the sum is intended to be carried over the set of
all permutations
of the distinguishable particle states generated by the particle
permutation operator
P̂ . This last equation constitutes our Rosetta Stone for
translating bosonic states
into distinguishable particle states that we already know how to
treat. Naturally,
one must also take into consideration that the dimension of the
Hilbert space will
in general be different if we deal with bosons of
distinguishable particles, due to the
aforementioned symmetrization of the states.
3.3 Non-interacting tight-binding dynamics
In our thirst for analytical results we should start by
generalizing the single-body
tight-binding model to the many-body case. Naturally, as per
nature of a single-
body Hamiltonian, no interaction terms can be included in the
framework of the
tight-binding model without developing a specific many-body
theory, so that we are
restricted to generalizing the results for the non-interacting
case; naturally, we are
also restricted to symmetric double-well potentials. We recall
that in this case the
total Hamiltonian is just a sum of single particle
operators:
Ĥ(x1, . . . , xn) =N∑i=1
Ĥ(0)i (xi), (3.17)
therefore also the time evolution operator can be factorized
into single-particle op-
erators:
Û(t) = e−iĤt = exp(− i
N∑i=1
Ĥ(0)i t)
=N∏i=1
e−iĤ(0)i t. (3.18)
-
3.3 ∼ Non-interacting tight-binding dynamics 31
Recalling the single-body tight-binding approximation, we
postulated that the sys-
tem can be fully described in the site basis {|L〉 , |R〉}, as
defined in 2.17 and 2.18.
Restricting our treatment to distinguishable particles for the
moment, we can gen-
eralize the tight-binding site basis to its many-body
equivalent:
{|α1〉1 ⊗ |α2〉2 ⊗ . . .⊗ |αN 〉N}, |αi〉i ∈ {|L〉i , |R〉i},
(3.19)
where the i subscript indicates that the state refers to the
i-th particle. We now
have all the ingredients needed to calculate site occupation
probabilities as we did
for the single-particle case; naturally we now have a plethora
of such probabilities,
2N to be exact, because each particle can either occupy the left
or the right potential
well.
Let us suppose that, in analogy to the single-particle case, at
time t = 0 we prepare
the system with all the particles located in the left well; any
of such 2N probabilities
can be calculated as:
Pw1,...,wn(t) :=∣∣∣(1 〈w1| ⊗ . . .⊗ N 〈wN |) N∏
i=1
e−iĤ(0)i t(|L〉1 ⊗ . . .⊗ |L〉N
)∣∣∣2, (3.20)with wi ∈ {L,R} expresses whether particle i
occupies the left or the right well. We
also used the fact that the initial state can be expressed
as:
|Ψ(0)〉 :=N⊗i=1
|L〉i . (3.21)
In the single particle case we have calculated:
PL(t) := |i 〈L|e−iĤ(0)i t|L〉i |
2 = cos2(Ω0t), (3.22)
PR(t) := |i 〈R|e−iĤ(0)i t|L〉i |
2 = sin2(Ω0t), (3.23)
following definitions 2.23 and 2.24. On the basis of these
results we can write:
Pw1,...,wn(t) = cos2NL(Ω0t) sin
2NR(Ω0t), (3.24)
with NL :=∑N
i=1 δwi,L being the number of particles inside the left well in
the
target state, with δ being the Kronecker delta (and analogously
for NR). Therefore,
following 3.13 we have:
P(U=0)NL
(t) ≈ cos2NL(Ω0t) sin2NR(Ω0t). (3.25)
Equation 3.24 could also have been obtained by simple
statistics; we have demon-
strated that the total Hamiltonian acts independently on each
particle, and so does
-
3.3 ∼ Non-interacting tight-binding dynamics 32
the time evolution operator. Therefore the N particles can be
regarded as a set of
N disjoint events: this implies that their conjoined probability
is just the product of
the probabilities of the single events, namely PL(t) and PR(t)
being the probability
of finding a particle in the left or right well at time t, in
accordance with 3.24.
The previous results can be extended also to the bosonic case;
we may either re-
peat the calculation in 3.20 but substituting the target state
with its symmetric
counterpart
|w1, . . . , wn〉+ :=√n1! . . . nN !
N !
∑{P}
P̂( N⊗i=1
|wi〉i)
(3.26)
in accordance with definition 3.16, or once again follow our
statistical analogy. The
main difference with bosonic particles resides in their
indistinguishability: this means
that, unlike in the distinguishable case, there is a multitude
of ways in which we
can populate the left well with NL bosons and the right well
with NR bosons.6
Therefore, we must multiply result 3.25 by the number of times
we can group NL
bosons from a pool of N particles:
P(U=0)NL
(t) ≈(N
NL
)cos2NL(Ω0t) sin
2NR(Ω0t). (3.27)
(a) (b) (c)
Figure 3.1: numerical simulations of |Ψ(x1, x2)|2 for 2 bosons
in the non-interacting case at the
start (a), at a quarter (b) and at half (c) of the oscillation
cycle. The potential parameters are
l = r = 2, b = 0.2, V0 = 500 and the starting state is 2.16 for
both particles; Ω0 ≈ 0.00238. The
lower left (resp. upper right) corner represents the region
where both particles occupy the left (resp.
right) well; in the other two regions they are separated.
6The reader may argue that this is obviously also true for
distinguishable particles; however we
have to recall that, according to definition 3.13, in 3.25 we
are looking at the probability of finding
the first NL particles in the left well, which can only be done
in one way.
-
3.3 ∼ Non-interacting tight-binding dynamics 33
(a) (b)
(c)
Figure 3.2: equation 3.27 for 2 (a), 3 (b) and 4 (c) bosons; the
results are in very good agreement
with the analytical results from the tight-binding
approximation, even though the simulations also
include the contributions of the higher states in the spectrum.
The potential parameters are l = r =
2, b = 0.2, V0 = 500 and the starting state was chosen to be
2.16 for each particle; Ω0 ≈ 0.00238.
-
3.4 ∼ Hubbard model 34
3.4 Hubbard model
We will now review a model which is usually employed in
calculating the dynamics
of many particles in a lattice-style potential: the model can be
downsized to a
lattice of just two sites, corresponding to a double-well
potential. The model was
first introduced by John Hubbard in 1963 (see [5]) and is
extremely streamlined in
its form: we suppose to populate with a given number of
particles a lattice-style
potential made of wells, or lattice sites, each containing one
single bound state and
separated by potential barriers of finite height. The total
Hamiltonian of the system
is composed of three simple terms: the single-body Hamiltonian
for each particle,
accounting for the bound-state energy of the occupied lattice
sites; a single-body
hopping term between neighbour lattice sites, characterised by a
constant tunnelling
amplitude; finally, a very simple interaction term that adds a
constant contribution
to the total energy whenever a lattice site is multiply
occupied: we suppose that
particles cannot interact when they occupy different sites. In
the so-called second
quantization formalism the total Hamiltonian can be written
as:
Ĥ : =∑i,s
(E0 â
†i,sâi,s +
∑j
U
2â†i,sâi,sâ
†j,sâj,s
)+
+ Ω0∑i,s,s̄
(â†i,s̄âi,s + â
†i,sâi,s̄
),
(3.28)
where the i and j are labels for each particle, s is a label for
each lattice site, E0 is the
bound-state energy of the sites, U the constant energy
contribution given whenever
two particles occupy the same site (divided by two to account
for double-counting
in the summation) and Ω0 is the single-body tunnelling amplitude
between site s
and its neighbours s̄. Naturally, â†i,s and âi,s represent the
usual ladder operators
for particle i in the lattice site s, which act as follows:
â†i,s⊗j
|n(j)0 , n(j)1 , . . . , n
(j)s , . . .〉j =
=
√n
(i)s + 1 |n(i)0 , n
(i)1 , . . . , n
(i)s + 1, . . .〉i
⊗i 6=j|n(j)0 , n
(j)1 , . . . , n
(j)s , . . .〉j
(3.29)
âi,s⊗j
|n(j)0 , n(j)1 , . . . , n
(j)s , . . .〉j =
=
√n
(i)s |n(i)0 , n
(i)1 , . . . , n
(i)s + 1, . . .〉i
⊗i 6=j|n(j)0 , n
(j)1 , . . . , n
(j)s , . . .〉j
(3.30)
-
3.4 ∼ Hubbard model 35
with n(i)s ∈ {0, 1} being the occupation number for particle i
in site s,
∑s n
(i)s = 1.7
We will now have a brief look at the dynamics of two
distinguishable particles in a
double-well potential according to the Hubbard model: in
equation 3.28 this means
that we should take i, j ∈ {1, 2} and s ∈ {L,R} (referring
respectively to the left
and right potential wells). In the spirit of second quantization
we shall build a basis
for the Hilbert space in the site occupation number
representation; there are four
possible ways in which we can populate the double-well with two
particles8 (two
where both occupy the same well and two – due to the fact that
we are working
with distinguishable particles – where they are in separated
wells):
|ΨLL〉 := â†1,Lâ†2,L |0〉 , (3.31)
|ΨLR〉 := â†1,Lâ†2,R |0〉 , (3.32)
|ΨRL〉 := â†1,Râ†2,L |0〉 , (3.33)
|ΨRR〉 := â†1,Râ†2,R |0〉 , (3.34)
where |0〉 is the vacuum state where the system is not populated
by any particle.
We may verify the action of the Hamiltonian on the basis states
by computing the
matrix elements in the {|ΨLL〉 , |ΨLR〉 , |ΨRL〉 , |ΨRR〉}
basis:
Ĥ.=
2E0 + U Ω0 Ω0 0
Ω0 2E0 0 Ω0
Ω0 0 2E0 Ω0
0 Ω0 Ω0 2E0 + U
. (3.35)
The reader may observe from the Hamiltonian matrix
representation that there is
no non-zero matrix term linking states where both particles
change their site: this
means that the Hubbard Hamiltonian only allows for single-body
tunnelling events;
7This cumbersome and rather ugly looking notation is a
by-product of the fact that we are
trying to use second quantization not for its intended scope,
that is dealing with undistinguishable
particles; therefore we have to keep track of the occupation
numbers of each single particle for all
the lattice sites, instead of just labelling the states based on
how many bosons/fermions occupy
each site.8Once again, this is only true in the Hubbard model
approximation where each well has only
one bound state. We have verified from exact calculations that,
instead, the eigenfunctions of the
double-well potential are delocalized over both wells; the
difficulties of defining a bound state for
the wells have been analyzed in section 2.4.
-
3.4 ∼ Hubbard model 36
we shall discuss the implication of this fact in a deeper
fashion in the following sec-
tions.
Due to basis completeness, any state of the system at a given
time t may be decom-
posed over the basis states:
|ψ(t)〉 := bLL(t) |ΨLL〉+ bLR(t) |ΨLR〉+ bRL(t) |ΨRL〉+ bRR(t) |ΨRR〉
, (3.36)
where normalization imposes:
|bLL(t)|2 + |bLR(t)|2 + |bRL(t)|2 + |bRR(t)|2 = 1. (3.37)
Therefore, the evolution of the state of the system is dictated
by the Schrödinger
equation, which applied to 3.36 yields:
iḃLL(t) =(
2E0 + U)bLL(t) + Ω0
(bLR(t) + bRL(t)
)iḃLR(t) =
(2E0
)bLR(t) + Ω0
(bLL(t) + bRR(t)
)iḃRL(t) =
(2E0
)bRL(t) + Ω0
(bLL(t) + bRR(t)
)iḃRR(t) =
(2E0 + U
)bRR(t) + Ω0
(bLR(t) + bRL(t)
). (3.38)
The exact solutions of such system can be found analytically but
are extremely
cumbersome. Instead, we are looking for some simple analytical
result to have
a better intuitive understanding of the system’s behaviour:
following [3], we may
provide an approximate solution in the strongly-interacting
regime. First, we begin
by making the solution of the system more accessible by defining
new variables as a
combination of the probability amplitudes of the four basis
states:
x1(t) := bLL(t)− bRR(t)
x2(t) := bLR(t)− bRL(t)
y1(t) := bLL(t) + bRR(t)
y2(t) := bLR(t) + bRL(t)
. (3.39)
This way we obtain two decoupled equations that can be solved
independently from
the others:
iẋ1(t) =(
2E0 + U)x1(t) =⇒ x1(t) = Ae−i(2E0+U)t, (3.40)
iẋ2(t) =(
2E0
)x2(t) =⇒ x2(t) = De−i(2E0)t. (3.41)
-
3.4 ∼ Hubbard model 37
We are left with two coupled equations:iẏ1(t) =
(2E0 + U
)y1(t) + 2Ω0y2(t)
iẏ2(t) =(
2E0
)y2(t) + 2Ω0y1(t)
. (3.42)
To uncouple the two equations we diagonalize the system in its
matrix representa-
tion. The eigenvalues can be calculated to be:
λ1,2 = 2E0 +U
2±√U2 + 16Ω20. (3.43)
In order to obtain a simple analytical form for the solutions we
approximate the
eigenvalues with a second order Taylor series expansion in
function of ω := 2Ω20/U :
λ1,2 ≈ 2E0 +U
2± 1
2
(U + 4ω +O(ω2)
). (3.44)
Naturally, this holds true in the strongly-interacting regime
where U � Ω20. In this
approximation the eigenvectors are:z1(t) = Ω0y1(t) + ωy2(t)
z2(t) = ωy1(t)− Ω0y2(t), (3.45)
which lead to the solutions:
iż1(t) =(
2E0 + U + 2ω)z1(t) =⇒ z1(t) = Be−i(2E0+U+2ω)t, (3.46)
iż2(t) =(
2E0 − 2ω)z2(t) =⇒ z2(t) = Ce−i(2E0−2ω)t. (3.47)
We may now finally write the solution for the basis probability
amplitude coefficients:
bLL(t) = Ae−i(2E0+U)t +Be−i(2E0+U+2ω)t + Ce−i(2E0−2ω)t
bLR(t) = De−i2E0t + ωΩ0Be
−i(2E0+U+2ω)t + Ω0ω Ce−i(2E0−2ω)t
bRL(t) = −De−i2E0t + ωΩ0Be−i(2E0+U+2ω)t + Ω0ω Ce
−i(2E0−2ω)t
bRR(t) = −Ae−i(2E0+U)t +Be−i(2E0+U+2ω)t + Ce−i(2E0−2ω)t
. (3.48)
To evaluate the dynamics, we must now choose some initial
conditions: we decide
to place the two particles inside the left well at t = 0. This
roughly corresponds to
imposing bLL(0) = 1 and bLR(0) = bRL(0) = bRR(0) = 0 which, if
we take ω ≈ 0,
implies A = B = 1/2 and C = D = 0.
Finally, we can compute the occupation probabilities for each of
the basis states for
-
3.4 ∼ Hubbard model 38
the evolving system:
PLL(t) := |bLL(t)|2 =∣∣∣ e−i(2E0+U+ω)t2 (eiωt + eiωt)∣∣∣2 = cos2
(2Ω20U t)
PLR(t) := |bLR(t)|2 = (. . .) = 4Ω0U2 sin2(U2 t)
PRL(t) := |bRL(t)|2 = (. . .) = 4Ω0U2 sin2(U2 t)
PRR(t) := |bRR(t)|2 = (. . .) = sin2(
2Ω20U t)
. (3.49)
We stress once again that those results are only valid for U �
Ω20; moreover, the
applicability of this approximate solutions to the exact
dynamics of the double-well
must also take into account the intrinsic limitations of the
Hubbard model: in re-
ality U must be small enough not to involve the higher states of
the spectrum in
the dynamics, as the Hubbard model only considers a single
“band” that can be
populated by the particles.
A brief look at system 3.49 reveals that the probability of
finding the two particles
separated is strongly suppressed by the presence of the
interaction: this is striking,
considering that the model is insensitive to the sign of U ,
meaning that the dynamics
of an attractive or repulsive interaction are the same. Instead,
the particles tend
to oscillate together between the two wells but a much lower
frequency compared
to the non-interacting Rabi oscillations as 2Ω20/U � Ω0 in the
strongly interacting
regime. This behaviour can be simply explained in terms of
energy conservation:
when two particles are prepared together in the same well, their
total mean energy
is 2E0 + U ; instead, when they occupy different wells, their
total mean energy is
only 2E0 due to the fact that in the Hubbard regime the range of
the interaction is
confined inside each lattice site. This means that, as the
interaction grows stronger
(may U be either positive or negative), the two mentioned states
are increasingly
detuned so that the probability of the two particles jumping
from being together in
the same well to being separated is suppressed, in accordance
with the approximate
results.
As an addendum, we now want to provide an example on how the
Hubbard model
can be generalized to N bosons; the total Hamiltonian reads:
Ĥ :=∑s
(E0 n̂s +
U
2n̂s(n̂s − 1)
)+ Ω0
∑s,s̄
(â†s̄âs + â
†sâs̄
), (3.50)
-
3.4 ∼ Hubbard model 39
where we have defined n̂s := â†sâs as the number operator that
counts the number
of bosons inside the s-th lattice site. The reader may notice
that the operators
have lost the index pertaining to the particle label, in
compliance with the fact that
bosons have to be treated as identical and indistinguishable
particles and therefore
they all share the same ladder operators. The interaction terms
has slightly changed
its form, but not its function: it counts the number of particle
pairs inside each well
and appends an U contribution to the energy for each one.
As an example, we now solve the same problem of two particles
inside the symmetric
double-well potential. Differently from the distinguishable
particle case, we must
keep in mind that all states must obey bosonic particle exchange
symmetry rules,
according to definition 3.16; therefore we can define a basis
for bosonic particles by
a symmetrization of the distinguishable particle basis:
|20〉 := |ΨLL〉 , (3.51)
|11〉 := 1√2
(|ΨLR〉+ |ΨRL〉
), (3.52)
|02〉 := |ΨRR〉 , (3.53)
where we have switched to the site occupation number
representation, in great sec-
ond quantization style. In this basis the Hamiltonian matrix
reads:
Ĥ.=
2E0 + U
√2Ω0 0
√2Ω0 2E0
√2Ω0
0√
2Ω0 2E0 + U
. (3.54)
The state of the system may be expanded on this basis as:
|ψ(t)〉 := b20(t) |20〉+ b11(t) |11〉+ b02(t) |22〉 . (3.55)
Applying the Schrödinger equation to state |ψ(t)〉 we
get:iḃ20(t) =
(2E0 + U
)b20(t) +
√2Ω0b11(t)
iḃ11(t) =(
2E0
)b11(t) +
√2Ω0
(b20(t) + b02(t)
)iḃ02(t) =
(2E0
)b02(t) +
√2Ω0b11(t).
(3.56)
We will not go into the details of the calculations, which are
similar to the distin-
guishable case. The solution of the system in the strong
interaction limit U � 2Ω20
-
3.4 ∼ Hubbard model 40
gives the probability amplitudes:b20(t) = Ae
−i(2E0+U)t +Be−i(2E0+U+2ω)t + Ce−i(2E0−2ω)t
b11(t) =√
2ωΩ0
Be−i(2E0+U+2ω)t +√
2Ω0ω Ce
−i(2E0−2ω)t
b02(t) = −Ae−i(2E0+U)t +Be−i(2E0+U+2ω)t + Ce−i(2E0−2ω)t
, (3.57)
where A, B and C are constant to be determined via initial
conditions. We choose
once again to prepare the system with both particles in the left
well at t = 0, which
yields A = B = 1/2 and C = 0. The state occupation probabilities
can therefore be
calculated: P20(t) := |b20|2 = cos2
(2Ω20U t)
= PLL(t)
P02(t) := |b02|2 = sin2(
2Ω20U t)
= PRR(t)
P11(t) := |b11|2 =8Ω20U2
sin2(U2 t)
= PLR(t) + PRL(t)
. (3.58)
The reader may see that the dynamics are very similar to the
distinguishable parti-
cle case; once again the effect of the interaction is to slow
down the transfer of the
particles between the two wells and suppressing the amplitude of
the state where
the two bosons are separated, independently of the sign of the
interaction (in the
limits of the approximation U � 2Ω20). This can be appreciated
in figure 3.3: as
long as we choose U � Ω20, but still not too large to excite the
upper states in the
spectrum, the approximate Hubbard model solutions mimic quite
well the behaviour
of the exact dynamics.
How high we can push the interaction strength shall be estimated
from the single-
body spectrum: in the exact model, the non-interacting many-body
eigenenergies
are∑
n nnEn with nn being the occupation number of the n-th
eigenstate with
energy En. As we shall see in section 3.5, the single-band
Hubbard model approxi-
mation translates in the exact model to truncating the
single-body spectrum to the
first two states, {E0, E1}, meaning that the uppermost many-body
eigenvalue shall
be �a := NE1. The next highest eigenvalue, excluded from the
single-band model, is
therefore �b := (N − 1)E1 +E2: therefore, the single-band
approximation holds true
until the interaction is strong enough to drive the transition
�a → �b, which roughly
corresponds to U ≈ (�b − �a) = E2 − E1 := ∆E. Referring to the
parameters used
in figure 3.3 (which are reported in the caption), ∆E ≈ 7.
-
3.4 ∼ Hubbard model 41
(a) (b)
(c) (d)
(e) (f)
(g) (h)
Figure 3.3: numerical simulations for left well occupation
probability 3.13 for 2 bosons (left
column) and 1 boson (right column) in the N = 2 system for
various strengths of the interaction
(see plot title): in red the repulsive case (U = +|U |) and in
blue the attractive one (U = −|U |);
the dashed black line represent the approximate solution from
the Hubbard model (3.58). The
potential parameters are l = r = 2, b = 0.2, V0 = 500 and the
starting state was chosen to be
2.16 for each particle; Ω0 ≈ 0.00238. The gap between the first
and the second doublet in the
single-body spectrum is ∆E ≈ 7.
-
3.5 ∼ Extended Hubbard model 42
Finally, we would like to spend some words on a comparison
between the Hubbard
model and the exact dynamics for the double-well potential. In
both models, the
interaction strength can be tuned by changing the parameter U ;
however, the pa-
rameter is fundamentally different in the two cases: while in
the exact dynamics
it is just a multiplicative constant in front of the interaction
matrix elements (see
definitions 3.1 and 3.3), in the Hubbard model it is the energy
shift to the state
where two particles occupy the same lattice site due to the
interaction. We shall
therefore refer to the two quantities with different labels, so
we choose to indicate
the Hubbard energy shift as Ueff as it is an effective
contribution given by the inter-
action to the spectrum. If we want to compare the two models we
have to assure
that Ueff is equal to the contribution given by the δ-type
potential for a given U
whenever two-particles reside in the same well; as we have
mentioned before, there
is no clear-cut definition of such localized states, so we make
the arbitrary, but sen-
sible, choice of putting each particle in the ground state of a
single potential well in
the limit V0 →∞. Recalling definition 2.16 we have:
Ueff =( N⊗i=1
i 〈∞L|)Û( N⊗i=1
i |∞L〉)
=
= UN∑i=1
N∑j=i
∫dxi
∫dxj | 〈xi|∞L〉 |2| 〈xj |∞L〉 |2δ(xi − xj) =
= UN(N − 1)
2
∫dxi| 〈xi|∞L〉 |4 =
=N(N − 1)
2
4U
l2
∫ l0
sin4(πxl
)dx =
N(N − 1)2
3U
2l.
(3.59)
This is consistent with the fact that the factor N(N − 1)/2
counts the number of
distinguished particle pairs inside the left well, each
contributing a factor 3U/2l.
3.5 Extended Hubbard model
We wish now to sightly improve the Hubbard model by getting rid
of some of the
more simplistic approximations, in particular in regard to the
interaction. We will
still keep the model as a single-band theory: as we have seen,
the symmetric double-
well single-body spectrum is made of doublets of symmetric and
antisymmetric
states; we will therefore only consider the contribution to the
dynamics given by
the first “band” (that is the first doublet of states) for each
particle, in a similar
way as we have done in the tight-binding approximation. This
will allow us to
-
3.5 ∼ Extended Hubbard model 43
define an approximate site basis as discussed in section 2.4 for
a symmetric double-
well potential.
The model can be easily built from the N -body total Hamiltonian
3.1 and restricting
the spectrum of each particle to {E0, E1}, the first two
eigenvalues of the single-body
double-well Hamiltonian Ĥ(0)i ; no further approximations will
be taken. This implies
that the model should give accurate results, provided that the
higher states in the
spectrum do not provide significant contribution to the
dynamics: in comparison to
the single-body case, now this is not only guaranteed by
choosing an initial state with
small projections on the upper states, but also that the
interaction strength U should
be small enough not to excite the upper states, as discussed in
the previous section.
We may offer a graphical justification to this fact by looking
at figure 3.4, where the
probability density distribution |Ψ(x1, x2, t)|2 for two
interacting bosons has been
calculated at various points of the system’s evolution. For weak
interactions (first
row), the probability distribution inside each well is described
by a single “lobe”;
instead, as the interaction gets stronger (second and third
rows), the probability is
either enhanced or suppressed for x1 = x2 (depending on whether
the interaction is
attractive or repulsive), consistently with what we may näıvely
expect from a δ(x1−
x2) interaction, so that the distribution has a more peculiar
shape. Remembering
definition 3.4 and the single-body wavefunctions (see figure
2.3), one may see that
this peculiar shape requires the many-body state Ψ(x1, x2, t) to
have non-negligible
projections over the states that present several nodes in their
spatial distribution,
which are the more energetic ones.
Distinguishable particles
In this model, the Hartree product basis can be built as:
|Ψa1,...,aN 〉 :=N⊗i=1
|ψai〉i , (3.60)
with ai ∈ {0, 1} denoting the first two single-particle
eigenstates. As we have seen,
in this basis the non-interacting part of the Hamiltonian is
diagonal. Instead, the
-
3.5 ∼ Extended Hubbard model 44
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Figure 3.4: numerical simulations for the probability density
function |Ψ(x1, x2)|2 for 2 bosons at
the start (left column), at a quarter (central column) and at
half (right column) of the oscillation
cycle (see figure 3.3) for U = 0.1 ((a), (b), (c)), U = −3 ((d),
(e), (f)) and U = 3 ((g), (h), (i)). The
potential parameters are l = r = 2, b = 0.2, V0 = 500 and the
starting state was chosen to be 2.16 for
both particles; Ω0 ≈ 0.00238. The lower left (resp. upper right)
corner represents the region where
both particles occupy the left (resp. right) potential well; in
the other two regions the particles
are separated. Compared to the non-interacting case (figure
3.1), the probability of finding the
particles separated is strongly suppressed, in accordance with
the Hubbard model. We can see that
for strong attractive (resp. repulsive) interaction the
probability is enhanced (resp. suppressed) for
x1 = x2, consistently with what we expect from a δ(x1 − x2)
interaction. The formation of “lobes”
in the probability distribution is a symptom of the involvement
of higher states of the spectrum
in the dynamics, which is the breakdown point for both the
traditional and the extended Hubbard
model.
-
3.5 ∼ Extended Hubbard model 45
interaction matrix terms can be calculated as:
〈Ψb1,...,bN |Û |Ψa1,...,aN 〉 =( N⊗j=1
j 〈ψbj |)Û( N⊗i=1
|ψai〉i)
=
=
∫dx1 · · ·
∫dxN
N∏j=1
ψ∗bj (xj)N∏i=1
ψai(xi)U
2
N∑m=1
∑n6=m
δ(xm − xn).
(3.61)
Naturally this holds for distinguishable particles; for bosons
one can employ the
same results after performing state symmetrization.
To analyse the differences with the traditional Hubbard model,
we will now show a
practical example by calculating the Hamiltonian matrix terms
for two distinguish-
able particles. The Hartree product basis will be composed of
four states:
|Ψ00〉 := |ψ0〉1 ⊗ |ψ0〉2 , (3.62)
|Ψ01〉 := |ψ0〉1 ⊗ |ψ1〉2 , (3.63)
|Ψ10〉 := |ψ1〉1 ⊗ |ψ0〉2 , (3.64)
|Ψ11〉 := |ψ1〉1 ⊗ |ψ1〉2 . (3.65)
The non-interacting part of the total Hamiltonian is, as
mentioned, diagonal in this
basis and reads:
Ĥ(U=0).=
2E0 0 0 0
0 E0 + E1 0 0
0 0 E0 + E1 0
0 0 0 2E1
. (3.66)
Instead, the interaction matrix terms will have to be calculated
according to defini-
tion 3.61. As an example, we explicitly calculate the first
term:
〈Ψ00|Û |Ψ00〉 =
= U
∫dx1
∫dx2ψ
∗0(x1)ψ
∗0(x2)ψ0(x1)ψ0(x2)δ(x1 − x2) =
= U
∫|ψ0(x1)|4dx1.
(3.67)
For the sake of clean notation, we define the quantities:a
:=
∫|ψ0(x)|4dx
b :=∫|ψ1(x)|4dx
c :=∫|ψ0(x)|2|ψ1(x)|2dx
. (3.68)
-
3.5 ∼ Extended Hubbard model 46
To make writing the interaction matrix easier, we define a
fourth quantity:
d : =1
4
∫ (|ψ0(x)|2 − |ψ1(x)|2
)2dx =
=1
4
∫ ((ψ0(x) + ψ1(x))(ψ0(x)− ψ1(x))
)2dx =
=
∫ (〈x|L〉 〈x|R〉
)2dx,
(3.69)
where we used the fact that the wavefunctions are real valued
and definitions 2.17
and 2.18 for the |L〉 and |R〉 states. The reader may notice that
this is the interaction
matrix term that links the state |LL〉 := |L〉1⊗|L2〉 with the
state |RR〉 := |R〉1⊗|R2〉
(and vice-versa) and therefore corresponds to the amplitude for
the simultaneous co-
tunnelling process of the two particles.
The previous definitions allow us to write:
c =
∫|ψ0(x)|2|ψ1(x)|2dx =
=1
2
∫ (|ψ0(x)|4 + |ψ1(x)|4 −
(|ψ0(x)|2 − |ψ1(x)|2
)2)dx =
=a+ b
2− 2d,
(3.70)
so that the interaction matrix reads:
Û.= U
a 0 0 a+b2 − 2d
0 a+b2 − 2da+b
2 − 2d 0
0 a+b2 − 2da+b
2 − 2d 0
a+b2 − 2d 0 0 b
, (3.71)
where we have used the fact that:∫ψ30(x)ψ1(x)dx =
∫ψ0(x)ψ
31(x)dx = 0 (3.72)
due to the symmetry of the integrand function.
Now, to draw any comparison with the Hubbard model we must write
the Hamil-
tonian matrix in the site basis; we follow the usual {|L〉 , |R〉}
approximation that
we have developed for the single-body tight-binding model.
Therefore, following
-
3.5 ∼ Extended Hubbard model 47
definitions 2.17 and 2.18 we can build the basis:
|LL〉 : = |L〉1 ⊗ |L〉2 =1√2
(|ψ0〉1 + |ψ1〉1
)⊗ 1√
2
(|ψ0〉2 + |ψ1〉2
)=
=1
2
(|Ψ00〉+ |Ψ01〉+ |Ψ10〉+ |Ψ11〉
),
(3.73)
|LR〉 : =(. . .)
=1
2
(|Ψ00〉 − |Ψ01〉+ |Ψ10〉 − |Ψ11〉
), (3.74)
|RL〉 : =(. . .)
=1
2
(|Ψ00〉+ |Ψ01〉 − |Ψ10〉 − |Ψ11〉
), (3.75)
|RR〉 : =(. . .)
=1
2
(|Ψ00〉 − |Ψ01〉 − |Ψ10〉+ |Ψ11〉
). (3.76)
We can now calculate the {|Ψ00〉 , |Ψ01〉 , |Ψ10〉 , |Ψ11〉} → {|LL〉
, |LR〉 , |RL〉 , |RR〉}
change of basis matrix:
T̂ :.=
1
2
+1 +1 +1 +1
+1 −1 +1 −1
+1 +1 −1 −1
+1 −1 −1 +1
, (3.77)
so that for the Hamiltonian matrix terms can be rewritten in the
site basis:
Ĥ.=
2EL Ω0 Ω0 0
Ω0 2EL 0 Ω0
Ω0 0 2EL Ω0
0 Ω0 Ω0 2EL
+ U
a+ b− 3d a−b4a−b
4 d
a−b4 d d
a−b4
a−b4 d d
a−b4
d a−b4a−b
4 a+ b− 3d
, (3.78)
where we have separated the free and the interaction
contributions;
EL : =1
2(E0 + E1), (3.79)
Ω0 : =1
2(E0 − E1) (3.80)
in compliance with the single-body tight binding definitions. If
we compare 3.78 with
3.35, first and foremost we see that by calculating the exact
interaction terms we have
a tunnelling amplitude d whenever the Hamiltonian connects two
states where both
particles switch their site location: as we have noted before,
those terms represent
the two-body co-tunnelling amplitudes and constitute an
additional physical process
that was not accounted for in the Hubbard Hamiltonian. Moreover,
the interaction
matrix provides a correction to the single-body tunnelling
amplitudes Ω0:
Ω0 → Ω0 + Ua− b
4. (3.81)
-
3.5 ∼ Extended Hubbard model 48
This contribution is sensitive to the sign U , so it is
different whether the interaction
is attractive or repulsive.
Bosons
Analogously to what we have done for the traditional Hubbard
model, we will now
repeat the same calculations for two bosons to highlight any
difference in the dy-
namics. We must also add that for N ≥ 2 it makes more sense to
study only bosonic
dynamics, as we usually prepare the state so that at the initial
time all the particles
occupy the same state (and this can only be done for bosons);
instead, studying
the distinguishable particle case for N = 2 makes sense if, for
example, we load the
system with a doublet of opposite spin electrons.
We will only highlight the differences from the distinguishable
particle case, as oth-
erwise the calculations are similar; we must only obey the state
symmetrization rule
3.16. Therefore, the Hartree product basis is:
|Ψ00〉+ := |ψ0〉1 ⊗ |ψ0〉2 , (3.82)
|Ψ01〉+ :=1√2
(|ψ0〉1 ⊗ |ψ1〉2 + |ψ1〉1 ⊗ |ψ0〉2
), (3.83)
|Ψ11〉+ := |ψ1〉1 ⊗ |ψ1〉2 . (3.84)
In this basis, the Hamiltonian matrix reads:
Ĥ.=
2E0 0 0
0 E0 + E1 0
0 0 2E1
+ U
a 0 a+b2 − 2d
0 a+ b− 4d 0
a+b2 − 2d 0 b
, (3.85)
with a, b and d defined according to 3.68 and 3.69. Once again,
comparisons with
the näıve Hubbard mo