Introduction to Circuit QED: Part II

Josephson junction qubits

Schrödinger cat states of photons

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

Remarkable Progress in Coherence

Progress = better designs & materials

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

dressed resonator dressed qubit

† † † †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

18

19

Coherent state is closest thing to a classicalsinusoidal RF signal

0( ) ( )

20

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: the experiment

Q

cavity

qubit

𝐶𝜋

P

M

Making a cat: the experiment

Q

cavity

qubit

𝐶𝜋

P

M

Making a cat: the experiment

Q

cavity

qubit

𝐶𝜋

P

M

Making a cat:

qubit acquires p phase per photon…

t after time:

Q

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

quantized light shift in ‘even’ cat

…

even cat state:

…

odd cat state:

quantized light shift in ‘odd’ cat

Parity

400 consecutive parity measurements

Time s4000 300200100

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

57

Time evolution of a coherent state in cavity for timeproduces an entangled ‘parity cat’ 2

T

P P

1ZZZZ 1ZZZZ

Measurement of cavity yields N-way parity of qubits

Magic identity: 1

1

2( )

Nzj

j

i

jj

NN

e i Z

z

j jZ