quantum coherence of josephson radio-frequency...

1

QUANTUM COHERENCE OF JOSEPHSON RADIO-FREQUENCY

CIRCUITS

INTERNATIONAL SCHOOL OF PHYSICS "ENRICO FERMI"QUANTUM COHERENCE IN SOLID STATE SYSTEMS

Michel Devoret, Applied Physics, Yale University, USAand Collège de France, Paris

OUTLINE

1. Motivation: Josephson circuits and quantum information

2. Quantum-mechanical superconducting LC oscillator

3. A non-dissipative, non-linear element: the Josephson junction

4. The Cooper pair box artificial atom circuit

5. Readout of the Cooper pair box qubit

6. Coherence of Josephson circuits

7. Quantum bus concept

BREAK

2

OUTLINE








• PHOTON

• NUCLEAR SPIN

• ION

• ATOM

• MOLECULE

• ATOMIC-SIZE DEFECT IN CRYSTAL

• QUANTUM DOT

• JOSEPHSON RF CIRCUIT

IMPLEMENTATIONS OF A SINGLE, MANIPULABLE, QUANTUM-MECHANICALLY

COHERENT SYSTEM

micro

MACRO

couplingwith envt.

(Leggett, '80)

3

CLASSICAL RADIO-FREQUENCY CIRCUITS

Hergé, Moulinsart

1940's : MHz 2000's : GHz

Communication

Computation

US Army Photo

Communications

Sony

WHAT IS A RADIO-FREQUENCY CIRCUIT? A RF CIRCUIT IS A NETWORK OF ELECTRICAL ELEMENTS

element

node p

loopbranch

np

node n

Vnp

Inp

TWO COLLECTIVE DYNAMICAL VARIABLES CHARACTERIZETHE STATE OF EACH DIPOLE ELEMENT AT EVERY INSTANT:

Voltage across the element:

Current through the element:

( )p

nnpV t E d= ⋅∫( ) nnp pj dI t σ= ⋅∫∫

4

EACH ELEMENT IS TAKEN FROM A FINITE SET OF ELEMENT TYPES

inductance:

capacitance:

V = L dI/dt

I = C dV/dt

resistance: V = R IIds = G(Igs,Vds,) VdsIgs = 0

transistor: g

s d

diode: I=G(V)

CONSTITUTIVE RELATIONS

EACH ELEMENT TYPE IS CHARACTERIZED BY A RELATION BETWEENVOLTAGE AND CURRENT

LINEAR NON-LINEAR

QUANTUM INFORMATION ECONOMY

suppose a function f { } ( ) { }0,1023 0,1023j k f j∈ → = ∈

Classically, need 1000 ×10-bit registers (10,000 bits) to storeinformation about this function and to work on it.

Quantum-mechanically, a 20-qubit register can suffice!

( )2 1

/ 20

12

N

Nj

j f j−

=

Ψ = ∑

Function encoded in a single quantum state of small register!(but a highly entangled one)

10 q-bits 10 q-bits

5

CAN WE BUILD A COMPLETE SET OF QUANTUM INFORMATION PROCESSING PRIMITIVES

OUT OF SIMPLE CIRCUITSTHAT WE WOULD CONNECT TO PERFORM ANY

DESIRED FUNCTION?

QUESTION:

TRANSM. LINES, WIRES

COUPLERS

CAPACITORS

GENERATORS

AMPLIFIERS

JOSEPHSON JUNCTIONS

FIBERS, BEAMS

BEAM-SPLITTERS

MIRRORS

LASERS

PHOTODETECTORS

ATOMS

DRAWBACKS OF CIRCUITS:

ADVANTAGES OF CIRCUITS: - PARALLEL FABRICATION METHODS- LEGO BLOCK CONSTRUCTION OF HAMILTONIAN- ARBITRARILY LARGE ATOM-FIELD COUPLING

ARTIFICIAL ATOMS PRONE TO VARIATIONS

QUANTUM OPTICS QUANTUM RF CIRCUITS

~

6

OUTLINE








SIMPLEST EXAMPLE: SUPERCONDUCTING LC OSCILLATOR CIRCUIT

A CIRCUIT BEHAVING QUANTUM-MECHANICALLYAT THE LEVEL OF CURRENTS AND VOLTAGES ?

d ~ 0.5 mm

7

MICROFABRICATION L ~ 3nH, C ~ 1pF, ωr /2π ~ 5GHz



d ~ 0.5 mm

d << λ: LUMPED ELEMENT REGIME

MICROFABRICATION L ~ 3nH, C ~ 1pF, ωr /2π ~ 5GHz



INCOMPRESSIBLE ELECTRONIC FLUID SLOSHES BETWEEN PLATES.NO INTERNAL DEGREES OF FREEDOM.

d ~ 0.5 mm

d << λ: LUMPED ELEMENT REGIME

SUPERCONDUCTIVITY ONLY ONE COLLECTIVE VARIABLE

8

Rydberg atom SuperconductingLC oscillator

L C

DEGREE OF FREEDOM IN ATOM vs CIRCUIT:SEMI-CLASSICAL DESCRIPTION


L C

velocity of electron → voltage across capacitorforce on electron → current through inductor

DEGREE OF FREEDOM IN ATOM vs CIRCUIT:SEMI-CLASSICAL DESCRIPTION

2 Possible correspondences: velocity of electron → current through inductor force on electron → voltage across capacitor

charge dynamics

field dynamics

9


velocity of electron → voltage across capacitorforce on electron → current through inductor

DEGREE OF FREEDOM IN ATOM vs CIRCUIT

2 Possible correspondences: velocity of electron → current through inductor force on electron → voltage across capacitor

charge dynamics

field dynamics

φL C

semiclassical picture

+Qφ

-Q

FLUX AND CHARGE IN LC OSCILLATOR

LIφ = Q VC=

V

-I

X

Mk

electrical world mechanical world

I

f

position variable: φ X-Xeqmomentum variable: Q Pgeneralized force : Ι fgeneralized velocity: V V

1eqX X

kf=− P VM=

generalized mass : C Mgeneralized spring constant: 1/L k

10

+Qφ

-Q

HAMILTONIAN FORMALISM

V

X

Mk


I

f

( ) ( )22

,2 2

eqQH QC L

φ φφ

−= +

charge on capacitor

time-integralof voltage oncapacitor

conjugate variables

( ) ( )22

,2 2

eqk X XPH X PM

−= +

momentum of mass

positionof massw/r wall

conjugate variables

+Qφ

-Q

HAMILTON'S EQUATION OF MOTION

V

X

Mk


I

f

H QQ C

HQL

φ

δφφ

∂= =

∂∂

= − =∂

H PXP MHP k XX

δ

∂= =

∂∂

= − =∂

1r LC

ω = rkM

ω =

11

FROM CLASSICAL TO QUANTUM PHYSICS:CORRESPONDENCE PRINCIPLE

ˆ

ˆX X

P P

→

→

( ) ( )ˆ ˆ ˆ, ,H X P H X P→

position operator

momentum operator

hamiltonian operator

{ } ˆ ˆ, , /PB

A B A B A B A B iX P P X

∂ ∂ ∂ ∂ ⎡ ⎤− = → ⎣ ⎦∂ ∂ ∂ ∂Poisson bracket

commutator

{ } ˆ ˆ, 1 ,PB

X P X P i⎡ ⎤= → =⎣ ⎦position andmomentumoperators donot commute

+Qφ

-Q

FLUX AND CHARGE DO NOT COMMUTE

V

I

ˆ ˆ, iQφ⎡ ⎤⎣ =⎦

We have obtained this resultthru correspondance principleafter having recognized fluxand charge as canonical conjugatecoordinates.

"Cannot quantize the equations of the stock market!"A.J. Leggett

We can also obtain this result from quantum field theory thru the commutationrelations of the electric and magnetic field. Tedious.

For every branch in the circuit:

12

+Qφ

-Q

φ

E

LC CIRCUIT AS QUANTUMHARMONIC OSCILLATOR

rωh

( )†

†

ˆ 1ˆ ˆ 2ˆ ˆ ˆ ˆ

ˆ ˆ;

2

2

r

r r r r

r r

r r

H a a

Q Qa i a iQ Q

L

Q C

ω

φ φφ φ

φ ω

ω

= +

= + = −

=

=

annihilation and creation operators

0

φ

φ

SUPPRESSION OF THERMAL EXCITATIONOF LC CIRCUIT

Br k Tω5 GHz 10mK

rωhI

E

Can place the circuitin its ground state

13

φ

φ

WAVEFUNCTIONS OF LC CIRCUIT

rωhI

φ

E

Ψ(φ)Ψ0

0Ψ1

In every energy eigenstate,(photon state)

current flows in opposite directions simultaneously!

2 rφ

φ

φ

EFFECT OF DAMPING

important:as little dissipation

as possibledissipation broadens energy levels

E

112 2n r

r

iE n

RC

ω

ω

⎡ ⎤⎛ ⎞= + +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦=

QQ

seemorelater

14

φ

φ

E

ALL TRANSITIONS ARE DEGENERATE

CANNOT STEER THE SYSTEM TO AN ARBITRARY STATEIF PERFECTLY LINEAR

rωh

Potential energy

Position coordinate

NEED NON-LINEARITY TO FULLYREVEAL QUANTUM MECHANICS

15

OUTLINE








JOSEPHSON JUNCTIONPROVIDES A NON-LINEAR INDUCTOR

1nm SI

S

superconductor-insulator-

superconductortunnel junction

CjLJ

16


1nm SI

S



CjLJ

Ι

( )' 't

V t dtφ−∞

= ∫


1nm SI

S



φ

ΙΙ = φ / LJ

( )0 0sin /I I φ φ=

CjLJ

Ι

( )' 't

V t dtφ−∞

= ∫

0 2eφ =

20 0

0J

J

LE Iφ φ

= =

17

COUPLING PARAMETERSOF THE JOSEPHSON JUNCTION "ATOM"

THE hamiltonian:(we mean it!)

( )ˆ ' 't

V t dtφ−∞

= ∫REST

OFCIRCUIT

extq

( )21 2cos2 ext

jJj

CH EQ q eφ

= − −

I

( )ˆ ' 't

Q I t dt−∞

= ∫

2

0

212 ˆ

ˆ 14e

erm

AH epπε

⎛ ⎞= − −⎜ ⎟⎝ ⎠

RESTOF

CIRCUIT

extq

THE hamiltonian:(we mean it!)

( )21 2cos2 ext

jJj

CH EQ q eφ

= − −

COUPLING PARAMETERSOF THE JOSEPHSON JUNCTION "ATOM"

Comparable with lowest ordermodel for hydrogen atom

18

TWO ENERGY SCALESTwo dimensionlessvariables:

ˆˆ

ˆˆ

2

2QN

e

e

φϕ =

=

ˆˆ, iNϕ⎡ ⎤ =⎣ ⎦

( )2

co8 ˆs2

ext

JCj

NE EH

Nϕ

−= −

2

2Cj

E eC

=

2x

exte tqNe

=

Hamiltonian becomes :

Coulomb charging energy for 1e

18JE = ΔNT

Josephson energy

gap

# condion channels

barrier transpcy

reduced offset charge

valid foropaque barrier

LOW-LYING EXCITATIONS OF ISOLATEDJUNCTION: CHARGE STATES

____

++++

-ne+ne

0 2 4-2-4n = N/2 = 6

2Δ

e tunnelingCP tunneling

4 CE

JE

19

MECHANICAL ANALOG OF JOSEPHSONCOUPLING ENERGY

rotatingmagnet

angular velocity:4 C

extE NΩ =h

compass needle

needleangular momentum:

Nh

torque on needle due to magnetvelocity of needle in magnet frame

current through junctionvoltage across junction

ϕ̂

CHARGE STATES vs PHOTON STATES

ϕ

N A1

A2n

θ

Hilbert spaces have different topologies

20

HARMONIC APPROXIMATION

( )2

co8 ˆs2

ext

JCj

NE EH

Nϕ

−= −

( )22

,2ˆ

82

e

Cj h J

xtNE

NEH ϕ−

= +

8 CP

JE Eω =Josephson plasma frequency

Spectrum independent of DC value of Next

TUNNEL JUNCTIONSIN REAL LIFE

100nm

EJ ~ 50K

EJ ~ 0.5K

ωp ~ 30-40GHz

credit L. Frunzio and D. Schuster

credit I. Siddiqi and F.Pierre

21

OUTLINE








3 TYPES OF BIASES

I0

C

U

2Cg2Cg

I0

C

Ib

I0

C ΦbL

charge flux current

22

EFFECTIVE POTENTIALOF 3 BIAS SCHEMES

-1 +1 phase bias

flux bias

charge bias

UC Berkeley, NIST, UCSB,U. Maryland, CRTBT Grenoble...

TU Delft, NEC, NTT, MITUC Berkeley, IBM, SUNYIPHT Jena ....

CEA Saclay, YaleNEC, Chalmers, JPL, ...

22

eh

ϕπ

φ=

2ehφ

2ehφ

EFFECTIVE POTENTIALOF 3 BIAS SCHEMES

-1 +1 phase bias

flux bias

charge bias

UC Berkeley, NIST, UCSB,U. Maryland, CRTBT Grenoble...

TU Delft, NEC, NTT, MITUC Berkeley, IBM, SUNYIPHT Jena ....

CEA Saclay, YaleNEC, Chalmers, JPL, ...

2ehφ

2ehφ

2ehφ

23

U

Φ

island

COOPER PAIR"BOX"

Bouchiat et al. 97Nakamura, Pashkin & Tsai 99Vion et al. 2002

2 control parameters CgU/2e and Φ/Φ0

U

ΦCg

ANHARMONICITY vs CHARGE SENSITIVITY

Cooper pair box soluble in terms of Mathieu functions (A. Cottet, PhD thesis, Orsay, 2002)

24

290 μm

ANHARMONICITY vs CHARGE SENSITIVITYIN THE LIMIT EC/EJ << 1 ("TRANSMON")

peak-to-peak charge modulation amplitude of level m:

anharmonicity:

TRANSMON: SHUNT JUNCTION WITH CAPACITANCE

Courtesy of J. Schreier and R. Schoelkopf

J. Koch et al. 07

( )12 01

12 01 / 2 8C

J

EE

ω ωω ω

−→

+

ω01

ω12ω02 2

Sufficient to control the junction as a two level system

SPECTROSCOPY OF A COOPER PAIR BOXIN TRANSMON REGIME

Anharmonicity: 01 12 455MHz CEω ω− =

ω01

ω02 2

ω12J. Schreier et al. 07

Slide courtesy of J. Schreier and R. Schoelkopf

T1 > 2μs

25

ATOM IN SEMI-CLASSICAL REGIME:CIRCULAR RYDBERG ATOM

p+

2 nr

nω

2

20

2

1 3

1

Ry

Ry2

2

e

n

n

n n

nn n

m r n

r a n

En

nerd

ω

ω → ±

→ ±

=

=

= −

=

=Coulombpotential

old Bohr theory essentially “exact”!

constraint

valid when: ( ) 1EE

ω∂∂

( )2

, ,n x y zΨ

Josephsonpotential energy

Junction phase ϕ

TRANSMON AS ANALOG OF CIRCULARRYDBERG ATOMS

2EJ

ω

Spectroscopic lines (high T):

+π−π

0

01ω12ω , 1 1, 2 , 1n n n n n nω ω ω+ + + +−

26

Josephsonpotential energy

Junction phase ϕ

TRANSMON AS ANALOG OF CIRCULARRYDBERG ATOMS

2EJ

ω

Spectroscopic lines (low T):

+π−π

OUTLINE








27

THE READOUT PROBLEM

QUBIT READOUTON

OFF0 1

10 or

THE READOUT PROBLEM

QUBIT READOUTON

OFF0 1

QUBIT READOUTON

OFF0 1 QUBIT READOUT

ON

OFF

1) SWITCH WITH LARGE ON-OFF RATIO2) FAITHFUL READOUT CIRCUITWANT:

0 1

10

28

DIVERSITY OF CIRCUIT STRATEGIES

YALE (circuit-QED)analogous to cavity QED expts by

Haroche, Raimond, Brune et al.

SACLAY -- YALE (Quantronium) TU DELFT -- NEC -- NTT

NIST -- UCSB

QUBITCHIP

DISPERSIVE READOUT STRATEGY

10 or

rf signal in

01ω ω≠

29

QUBITCHIP

or

10 or

QUBIT STATEENCODED IN PHASE

OF OUTGOING SIGNAL,NO ENERGY DISSIPATED

ON-CHIP

rf signal in rf signal out


QUBITCHIP

10

or

or


DO WITH ENOUGH PHOTONS TO GET1 CLASSICAL BIT OF INFORMATION


Requirements: 1) low noise in detection2) enough phase shift

30

QUBITCHIP

10

or

or


DO WITH ENOUGH PHOTONS TO GET1 CLASSICAL BIT OF INFORMATION


Requirements: 1) low noise in detection2) enough phase shift

PLACE QUBIT IN MICROWAVE CAVITYTO FILTER OUT IRRELEVANT PART OF SPECTRUM

Si or Al2O3

Al or Nb

SUPERCONDUCTING MICROWAVE RESONATOR: ANALOG OF FABRY-PEROT CAVITY

can get Qup to 106

( in bulk:Q ~ 1010)

length ~1cm, but photon propagation length ~ 10km!

mirror = capacitors

COPLANAR WAVEGUIDE: 2D VERSION OF COAXIAL CABLE

31

Si or Al2O3

Al or Nb


~

can get Qup to 106

( in bulk:Q ~ 1010)

ADC

IN OUT

CHIP

length ~1cm, but photon propagation length ~ 10km!RFGEN.

Si or Al2O3

Al or Nb


~

can get Qup to 106

( in bulk:Q ~ 1010)

ADC

length ~ 1cm, but photon decay length ~ 10km!

IN OUT

1/2 photon: ~1nA~100nV

ν ~ 10GHz

CHIP

w~20μm

VERY SMALL MODE VOLUME

32

3mm

10mm

100μm

Nb

SUPERCONDUCTING CAVITY = FABRY-PEROT

qubit goes here

10µm

CPW ground

CPW ground

ISLAND

READOUT JUNCTION

"QUANTRONIUM" IN MICROWAVE CAVITY

33

10-20 mK2-20 GHz

HIGH FREQUENCIES, LOW TEMPERATURES

OFF-RESONANT NMR-TYPEPULSE SEQUENCE

FOR QUBIT MANIPULATION

QUANTRONIUM IN MICROWAVE CAVITY

34

|0>

|1>

READOUTPROBING

PULSE

QUBIT STATEENCODED IN PHASE

OF TRANSMITTED PULSE

or


|0>

|1>


~ ADCIN OUT

AWG

40ns40ns200ns

t t

Metcalfe et al.Phys. Rev. B6174516 (2007)

35

|0>

|1>


~ ADCIN OUT

AWG

40ns40ns200ns

t t

Metcalfe et al.Phys. Rev. B6174516 (2007)

latching effectdue to bifurcation

of cavity mode

CAVITY BIFURCATION : ANALOG OF OPTICALBISTABILITY

36

HYSTERESIS

0

1

Ω/2π = 4.2GHzT=15mK

1 ms sweep

Urf

MEASUREMENT OFREADOUT FIDELITY

Slope3 timestoo smallcomparedwith thy

Relaxationinduced byreadout

π pulse Urf

TOO MANY READOUT PHOTONS?

37

JOSEPHSONPARAMETRIC

AMPLIFIERSIGNAL

IDLER

180ºhybrid

coupler

Design goals:TN < hf/kBGain > 20 dBBandwidth > 5 MHzDynamic range > 20 dB chip

Φ0/2Josephson ringmodulator

Bergeal et al. 08

OUTLINE








38

CAVITY QUANTRONIUMRABI OSCILLATIONS

M. Metcalfe et al.

0

1

νLarmor=14.5GHz

20%

40%

60%

acqu

isiti

on ti

me

(s)

RAMSEYFRINGES

T2 =500ns+-

200ns

0

1

39

Lopsided frequency fluctuations

Charge noise amplitude1/f2

3( ) , 1.9 10gNS eαω α

ω−= ∼

value OKEJ/EC = 3.6

(Karlsruhe)

method A of van Harlingen et al. 2004

DISTRIBUTION OF RAMSEY DATA

Need larger EJ/EC

Lopsided frequency fluctuations

Charge noise amplitude1/f2

3( ) , 1.9 10gNS eαω α

ω−=

value OK

DISTRIBUTION OF RAMSEY DATA

(Karlsruhe)

van Harlingen et al. 2004

40

DECOHERENCE TIME ( T2 )

Δt (ns)

Δt

Qφ = ωLarmorT2

'02 Vion et al. 50,000'05 Siddiqi et al. 30,000'05 Wallraff et al. 19,000'06 Metcalfe et al. 70,000'08 Houck et al. 100,000

1 param. fit→ T2 = 300ns

νLarmor=18.984GHz

Siddiqi et al, 06

phase

flux

charge

ΔECΔEJΔQoffbias

junction noises

SENSITIVITY OF BIAS SCHEMES TO NOISE (EXPD)

chargequbit

fluxqubit

phasequbit

- +-

CHARGED IMPURITYTUNNEL CHANNEL

ELECTRIC DIPOLES

41

OUTLINE








QUANTUM BUS CONCEPT FOR SOLID STATE QUBITS

Nature, 27 Sept. 2007

artificial atom

microwave transmission line resonator

coupling element

42

2g = 250 MHz!

cavityqubit

Msmt. of qubit-cavity avoided crossing vacuum Rabi splitting

STRONG QUBIT-CAVITY COUPLING

~ ADCIN OUT

~

CPB

RESOLVING PHOTONNUMBER IN CAVITY

( )( )!

nnnP n e

n−=

• Mean shifts

• Peaks are Poisson distributed

• Peaks get broader n∝

Houck et al., Nature 07

43

RESOLVING PHOTONNUMBER IN CAVITY

CONDITIONAL QUBITROTATION POSSIBLE!

Blais A. et al., Phys. Rev. A 75, 032329 2007

SCHEMATIC OF 2 COOPER PAIR BOXESIN A MICROWAVE CAVITY

44

SWAPPING INFORMATION BETWEEN QUBITS

Background due to

off-resonant driving by the

Stark tone

Nota Bene:(Using first generationtransmon with ‘short’

T2=100 ns.)

J. Majer, J. M. Chow et al, Nature, 449, 443 (2007)Similar results with phase qubits: Sillanpaa et al, Nature, 449, 438 (2007)

QUANTUM COMPUTATIONviolation of Bell's inequalities teleportationerror correctionalgorithms

MEASUREMENTS AND CONTROL OF QUANTUM ENGINEERED SYSTEMS

amplification at the quantum limitquantum noise squeezing (1-mode and 2-mode)quantum feedbacksingle photon detection for radioastronomycharge counting for metrology

PERSPECTIVES

Qφ ~ 107

NEWMATERIALS?

Qφ ~ 104

OK

45

Circuit Quantum Electrodynamics GroupsDepts. Applied Physics and Physics, Yale

P.I.'sGrads

Post-Docs

Res. Sc.

UndrgrdsCollab.

M. D.R. VIJAYC. RIGETTIM. METCALFEV. MANUCHARIANI. SIDDIQI (UCB)E. BOAKNIN (Mtrl)N. BERGEALP. HYAFIL

S. FISSETTED. ESTEVE(Saclay)

S. GIRVINT. YUL. BISHOP

J. KOCHJ. GAMBETTA

F. MARQUARDT(Munich)A. BLAIS(Sherbrooke)

A. CLERK (McGill)

R. SCHOELKOPFJ. CHOWB. TUREKJ. SCHREIERB. JOHNSONA. WALRAFF (ETH)H. MAJER (Vienna)A. HOUCKD. SCHUSTERL. FRUNZIOJ. SCHWEDEP. ZOLLER(Innsbruck)

quantum coherence of josephson radio-frequency...

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