Introduction to Circuit QED: Part II Josephson junction qubits Schrödinger cat states of photons
Dec 29, 2015
Josephson plasma oscillation of
~3-4 Cooper pairs
Non-linearelectromagnetic oscillator
01 ~ 5 10GHz E
nerg
y
0
101
1201 12 Superconductor
Superconductor (Al)
Insulating barrier1 nm
2
The Josephson tunnel junction is the only known non-linear non-dissipative
circuit element
2† † †
J0
ˆ 2 ˆ2
cos2
H E b b b bQ
bbC
Transmon Qubit
01 ~ 5 10GHz
Ene
rgy
0
101
1201 12 Anharmonicity allows us
to approximately treat oscillator as a two-level ‘spin’.
1
0
1
21
2
Examples of coherentsuperpositions
~ mm
Transmon Qubit
Josephsontunnel
junction
Superconductivity gaps out single-particle excitations
1110 mobile electrons
Quantized energy level spectrum is simpler than hydrogen
1Q TQuality factor exceeds that of hydrogen
01 ~ 5 10GHz
Ene
rgy
0
101
1201 12
spin
50 mm
~ mm
Transmon Qubit in 3D Cavity
Josephsonjunction
† † † †0
† †
( )
2
R Qa b g a b ab
V
H a
b b
b
bb
rmsd Eg
h
spin
50 mm
~ mm
Transmon Qubit in 3D Cavity
Josephsonjunction
rmsd Eg
h
g 100 MHz
72 1 mm 10 Debye!!d e
Huge dipole moment: strong coupling
† † † †0
† †
( )
2
R Qa b g a b ab
V
H a
b b
b
bb
Diagonalize quadratic Hamiltonian to obtain normal modes
cos sin
sin cos
tan 2 2 ; Q R
a A B
b A B
g
† †0 R QH A A BB
† † † †0
† †
( )
2
R Qa b g a b ab
V
H a
b b
b
bb
† †0 R QH A A BB
Now express quartic term in normal modes:
4 † †
2 2 † †
4 † †
2
cos
sin cos4
sin
{
}
V
B BB
A AB B
A AA
B
A
(large) Dressed qubit anharmonicity
(medium) Qubit-Cavity cross-Kerr
(small) Cavity self-Kerr
4 † †
2 2 † †
4 † †
2
cos
sin cos4
sin
{
}
V
B BB
A AB B
A AA
B
A
(large) Dressed qubit anharmonicity
(medium) Qubit-Cavity cross-Kerr
(small) Cavity self-Kerr
Qubit-Cavity cross-Kerr:Frequency of cavity depends on excitation number of the qubit
† ˆA A n † † † †
2
ˆ2
sin
2
42
R QB B B B BH BB n B
Transmon Qubit
01 ~ 5 10GHz
Ene
rgy
0
101
1201 12 Anharmonicity allows us
to approximately treat oscillator as a two-level ‘spin’.
1
0
1
21
2
Examples of coherentsuperpositions
Qubit-Cavity cross-Kerr:Frequency of cavity depends on excitation number of the qubit
† ˆA A n † † † †
2
ˆ2
sin
2
42
R QB B B B BH BB n B
Qubit-Cavity cross-Kerr for two lowest levels of dressed transmon.
ˆ2QZ Z
RH n
‘Dispersive’ coupling
†
†
ˆ
10,1
2
Z
A A
B
n
B
R R
Qubit-Cavity cross-Kerr for two lowest levels of dressed transmon.
ˆ2QZ Z
RH n
‘Dispersive’ coupling
†
†
ˆ
10,1
2
Z
A A
B
n
B
Can read out qubit state by measuring cavity resonance frequency
cavi
ty r
espo
nse
R R
Can read out qubit state by measuring cavity resonance frequency
cavi
ty r
espo
nse
cavity circulator quantum limited amplifier
xreflection phase
X
Y
in
outi i
a b
a e b e
State of qubit is entangled with the ‘meter’ (microwave phase)Then ‘meter’ is read with amplifier.
cavity circulator quantum limited amplifier
xreflection phase
EXPERIMENTRob Schoelkopf, Michel Devoret Luigi Frunzio
M. HatridgeShyam ShankarG. KirchmairBrian VlastakisAndrei Petrenko
Departments of Physics and Applied Physics, Yale University Circuit QED:
Taming the World’s Largest Schrödinger Catand
Measuring Photon Number Parity(without measuring photon number!)
THEORYSMG, L. Glazman, Liang Jiang Simon NiggM. MirrahimiZ. Leghtas
Claudia deGrandiUri VoolHuaixui ZhengRichard BrierleyMatti Silveri
How do we create a cat?
‘Classical’ signal generators only displace the vacuum and create coherent states.
We need some non-linear coupling to the cavity via a qubit.
21
22
Quantum optics at the single photon level
0 1 2 30 1 2 3a a a a
• Photon state engineering
Goal: arbitrary photon Fock state superpositions
Use the coupling between the cavity (harmonic oscillator)and the two-level qubit (anharmonic oscillator) to achieve this goal.
Previous State of the Art for Complex Oscillator States
Haroche/Raimond, 2008 Rydberg (ENS)
Expt’l. Wigner tomography: Leibfried et al., 1996 ion traps (NIST – Wineland group)
Hofheinz et al., 2009 (UCSB – Martinis/Cleland)
Rydberg atom cavity QED Phase qubit circuit QED
~ 10 photons ~ 10 photons
Q
24
Quantum optics at the single-photon level
• Quantum engineer’s toolbox to make arbitrary states:
‘Dispersive’ Hamiltonian: qubit detuned from cavity -qubit can only virtually absorb/emit photons
q† †r damping2
z zH a a a a H
resonator qubit Dispersive coupling
r q
(DOUBLY QND)
Large dipole coupling of transmon qubit to cavity permits:
25
Dispersive Hamiltonian
q† †r damping2
z zH a a a a H
resonator qubit dispersivecoupling
rcavity frequency z
eg
r r
‘strong-dispersive’ limit
32 ~ 2 10
26
Strong-Dispersive Limit yields a powerful toolbox
eg
r r
Microwave pulse at this frequency excites cavityonly if qubit is in ground state
Microwave pulse at this frequency excites cavityonly if qubit is in excited state
gD Conditional displacement of cavity
Engineer’s tool #1:
Cavity frequency depends onqubit state
cavity
qubit
𝐶𝜋
P
M
Making a cat: the experiment
Q
(*fine print for the experts: this is the Husimi Q function not Wigner)
cavity
qubit
𝐶𝜋
P
M
Making a cat:
qubit acquires p phase per photon…
t after time:
Qubit fully entangled with cavity
‘cat is dead; poison bottle open’‘cat is alive; poison bottle closed’
Q
33
n Conditional flip of qubit if exactly n photons
Engineer’s tool #2:
q† †r damping2
z zH a a a a H
resonator qubit dispersivecoupling
Reinterpret dispersive term:- quantized light shift of qubit frequency
†q 2
2za a
Microwaves are particles!
…
2
- quantized light shift of qubit frequency (coherent microwave state)
†q 2
2za a
N.B. power broadened100X
New low-noise way to do axion dark matter detection?
35
†DISPERSIVE
zV a a
strong dispersive coupling I
2
Qubit Spectroscopy
Coherent state in the cavity
Conditional bit flip n
Strong Dispersive Coupling Gives Powerful Tool Set
36
Cavity conditioned bit flip
Qubit-conditioned cavity displacement gD
n
• multi-qubit geometric entangling phase gates (Paik et al.)• Schrödinger cats are now ‘easy’ (Kirchmair et al.)
experiment theoryG. Kirchmair M. MirrahimiB. Vlastakis Z. LeghtasA. Petrenko
Photon Schrödinger cats on demand
37
Combining conditional cavity displacements with conditionalqubit flips, one can disentangle the qubit from the photons
1
2g gD0gD
Qubit in ground state; cavity in photon cat state
1
2g e
Does it work in practice?
To prove the cat is not an incoherent mixture:
- measure photon number parity in the cat
- measure the Wigner function (phase space distribution of cat)
Vlastakis et al. Science 342, 607 (2013)
1
2g
Photon number
Rea
dout
sig
nal
0246810
Spectroscopy frequency (GHz)
Coherent state:
Mean photon number: 4
Even parity cat state:
Odd parity cat state:
Only photon numbers: 0, 2, 4, …
Only photon numbers: 1, 3, 5, …
Proving phase coherence via Parity
P̂
P̂
2
Wigner Function MeasurementVlastakis, Kirchmair, et al., Science (2013)
Density Matrix: ( , )
Wigner Function:
, ) ( , )2 2
( iQQ d eW
(( ˆ) ) ( )D DW P
Handy identity: ˆˆ ( 1) parityNP
(will explain paritymeasurement later)
Qi
0
0
4
4
-4
-4
Wigner Function MeasurementVlastakis, Kirchmair, et al., Science (2013)
1
2
Re
Im
Q
Rapid parity oscillationsWith small displacements
Interference fringes prove cat is coherent:
0
0
44
-4
-4
Deterministic Cat State Production
Data!
Expt’l Wigner function
Vlastakis, Kirchmair, et al., Science (2013)
0
0
44
-4
-4
Deterministic Cat State Production
0.8
0.4
0.0
-0.4
-0.8-2 0 2 -2 0 2 -2 0 2 -2 0 2
18.7 photons 32.0 photons 38.5 photons 111 photons
determined by fringe frequency
Data!
Expt’l Wigner function
111 photons
Most macroscopic superposition ever created?
Vlastakis, Kirchmair, et al., Science (in press 2013)
0
0
44
-4
-4
Deterministic Photon Cat Production
0.8
0.4
0.0
-0.4
-0.8-2 0 2 -2 0 2 -2 0 2 -2 0 2
18.7 photons 32.0 photons 38.5 photons 111 photons
determined by fringe frequency
Three-component cat: Four-component cat:
111 photons
Vlastakis, Kirchmair, et al., Science (2013)
Zurek ‘compass’ state for sub-Heisenberg metrology
45
- use quantized light shift of qubit frequency
†q 2
2za a
Measuring Photon Number Parity
ˆ ˆ22 2ez z
i nt i ne
ˆ 0,2,4,...nˆ 1,3,5,...n x
z
“qubit is in |+x>”
“qubit is in |-x>”
Fidelity of produced cats:
Cat = Coherent State Projected onto Parity!L. Sun et al., Nature (July 2014)
No time to talk about:
-Continuous QND monitoring of -photon number parity-multi-qubit parity (stabilizers)
-Bell inequality violation between a qubit and a macroscopic cat-Quantum error correction using cat state encoding
Stabilizer Quantum Error Correction Toolbox for Superconducting Qubits(topological Kitaev toric code) Simon Nigg and SMGPhys. Rev. Lett. 110, 243604 (2013)
Dynamically protected cat-qubits: a new paradigm for universal quantum computation Zaki Leghtas et al. New J. Phys. 16, 045014 (2014)
Four-component cat:
Z Z
Z
Z
XX
X
X
Circuit QED Team Members 2013
KevinChou
ChrisAxline
BrianVlastakis
JacobBlumoff
LuyanSun
LuigiFrunzio
ReinierHeeres
SteveGirvin
AndreiPetrenko
Funding:
ChenWang
Eric Holland
TeresaBrecht
NissimOfek
PhillipReinhold
MichelDevoret
MattReagor
YvonneGao
LeonidGlazman
Z. Leghtas M. Mirrahimi
GerhardKirchmair
LiangJiang
Tracking photon jumps with repeated quantumnon-demolition parity measurements
L. Sun et al., Nature (July 2014)
evenodd evenodd odd even
400 consecutive parity measurements
1.0 1.0
2.0 2.0 ODD CAT
ODD CAT EVEN CAT
EVEN CAT
Number of parity jumps Number of parity jumps
Pro
bab
ility
(%
)P
roba
bili
ty (
%)
Pro
bab
ility
(%
)P
roba
bili
ty (
%)
0 2 4 6 8 100 2 4 6 8 10
0 2 4 6 8 10Number of parity jumps
0 2 4 6 8 10Number of parity jumps
12 12
1212
51
arXiv:1212.4000Stabilizer Quantum Error Correction Toolbox for superconducting qubitsSimon Nigg and SMG
Map multi-qubit parity onto cavity stateusing new toolbox
Z Z
Z
Z
XX
X
X
Kitaev Toric/Surface Topological QEC Code
1ZZZZ 1XXXX
Stabilizers
qj† †r 2
z
j j
zj j jH a a a a
j j (fine tuning)
52
arXiv:1212.4000Stabilizer Quantum Error Correction Toolbox for superconducting qubitsSimon Nigg and SMG
qj† †r 2
z zj
j jjH a a a a
j j
(fine tuning)
Magic identity: 1
1
2( )
Nzj
j
i
jj
NN
e i Z
z
j jZ
56
arXiv:1212.4000Stabilizer Quantum Error Correction Toolbox for superconducting qubitsSimon Nigg and SMG
Z Z
Z
Z
XX
X
X
Kitaev Toric/Surface Topological QEC Code
1ZZZZ 1XXXX
Stabilizers
N-way stabilizer measurements/pumping for QEC