circuit quantum electrodynamics : beyond the linear...

Introduction Beyond linear dispersive Results Conclusion

Circuit quantum electrodynamics : beyond thelinear dispersive regime

Maxime Boissonneault1 Jay Gambetta2 Alexandre Blais1

1Departement de Physique et Regroupement Quebecois sur les materiaux de pointe, Universite de Sherbrooke

2 Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo

June 23th, 2008

Boissonneault, Gambetta and Blais, Phys. Rev. A77 060305 (R) (2008)

Maxime Boissonneault Universite de Sherbrooke


1 IntroductionAtom and cavityCavity QEDCharge qubit and coplanar resonatorCircuit QEDThe linear dispersive limitCircuit VS cavity QED

2 Beyond linear dispersiveUnderstanding the dispersive transformationThe dispersive limitDissipation in the systemDissipation in the transformed basis

3 ResultsReduction of the SNRMeasurement induced heat bathThe case of the transmon

4 ConclusionConclusion



Atom and cavity


Two-levels system Hamiltonian

H =ωa

2σz σz =

„

1 00 −1

«

ω01= ωa

En

erg

y

...


Atom and cavity


Cavity Hamiltonian

H =X

k

ωk

„

a†kak +

1

2

«

z

x

y

L


H =ωa

2σz σz =

„

1 00 −1

«

ω01= ωa

En

erg

y

...


Atom and cavity


Cavity Hamiltonian

H =X

k

ωk

„

a†kak +

1

2

«

Single-mode :

H = ωra†a

Re

sp

on

se

Input frequency, rf

κ = ωr/Q

ω2ω1ωr=


H =ωa

2σz σz =

„

1 00 −1

«

ω01= ωa

En

erg

y

...


Cavity QED

gg



Cavity QED

gg

Atom-cavity interaction

HI = − ~D · ~E ≈ g(a† + a)σx ≈ g(a†σ− + aσ+)

g(z) = −d0

r

ω

V ǫ0sinkz



Cavity QED

gg

Atom-cavity interaction

HI = − ~D · ~E ≈ g(a† + a)σx ≈ g(a†σ− + aσ+)

g(z) = −d0

r

ω

V ǫ0sinkz

Jaynes-Cummings Hamiltonian

H =ωa

2σz + ωra†a + g(a†σ− + aσ+)

Jaynes and Cummings, Proc. IEEE 51 89-109 (1963)Raimond, Brune and Haroche, Rev. Mod. Phys. 73 565–582 (2001)Mabuchi and Doherty, Science 298 1372-1377 (2002)



Charge qubit and coplanar resonator


Classical Hamiltonian

H = 4EC(n − ng)2 − EJ cos δ

EC =e2

2(Cg + CJ ), ng =

CgVg

2e

EJ =I0Φ0

2π

C J

E JVg

nCg

Vg

Cg

CJEJ

- - - - -




Quantum Hamiltonian

H =X

n

4EC(n − ng)2 |n〉〈n|

−X

n

EJ

2(|n〉〈n + 1| + h.c.)

Restricting to ng ∈ [0, 1] : H = ωaσz/2

Shnirman, Schon and Hermon, Phys. Rev. Lett. 79 2371–2374 (1997)Bouchiat et al., Physica Scripta T76 165-170 (1998)Nakamura, Pashkin and Tsai, Nature (London) 398 786 (1999)



EC =e2

2(Cg + CJ ), ng =

CgVg

2e

EJ =I0Φ0

2π

C J

E JVg

nCg

Vg

Cg

CJEJ

- - - - -




Quantum Hamiltonian

H =X

n

4EC(n − ng)2 |n〉〈n|

−X

n

EJ

2(|n〉〈n + 1| + h.c.)

Restricting to ng ∈ [0, 1] : H = ωaσz/2

Shnirman, Schon and Hermon, Phys. Rev. Lett. 79 2371–2374 (1997)Bouchiat et al., Physica Scripta T76 165-170 (1998)Nakamura, Pashkin and Tsai, Nature (London) 398 786 (1999)

En

erg

ie [A

rb. U

nits]

0 0.2 0.4 0.6 0.8 1

Gate charge, ng = CgVg/2e

EJ/4EC=0.1

EJ



EC =e2

2(Cg + CJ ), ng =

CgVg

2e

EJ =I0Φ0

2π

C J

E JVg

nCg

Vg

Cg

CJEJ

- - - - -





H =Φ2

2Lr+

1

2CrV 2

ωr =

s

1

LrCr

L r C r




Quantum Hamiltonian

V =

r

ωr

2Cr(a† + a), Φ = i

r

ωr

2Lr(a† − a)

H = ωr

„

a†a +1

2

«

Quantum Fluctuations in Electrical Circuits, M. H. Devoret, LesHouches Session LXIII, Quantum Fluctuations p. 351-386 (1995).


H =Φ2

2Lr+

1

2CrV 2

ωr =

s

1

LrCr

L r C r


Circuit QED


C g

C J

E J

Atom: super conducting

charge qubitCavity: super conducting 1D

transmission line resonator

Qubit control

and readout

~ 10 GHz

Measurement

output


Circuit QED


Blais et al., Phys. Rev. A 69 062320 (2004)

Wallraff et al., Nature 431 162 (2004)

Wallraff et al., Phys. Rev. Lett. 95 060501(2005)

Leek et al., Science 318 1889 (2007)

Schuster et al., Nature 445 515 (2007)

Houck et al., Nature 449 328 (2007)

Majer et al., Nature 449 443 (2007)

Parametersg : Qubit-cavity interaction

ωa : Qubit frequency

ωr : Resonator frequency

∆ = ωa − ωr : Detuning

H =ωa


C g

C J

E J




Qubit control

and readout

~ 10 GHz

Measurement

output


Circuit QED



Wallraff et al., Nature 431 162 (2004)

Wallraff et al., Phys. Rev. Lett. 95 060501(2005)

Leek et al., Science 318 1889 (2007)

Schuster et al., Nature 445 515 (2007)

Houck et al., Nature 449 328 (2007)

Majer et al., Nature 449 443 (2007)

Parametersg : Qubit-cavity interaction

ωa : Qubit frequency

ωr : Resonator frequency

∆ = ωa − ωr : Detuning

H =ωa


C g

C J

E J


The linear dispersive limit

Jaynes-Cummings

H = ωra†a+ωaσz

2+g(a†σ−+aσ+)

Small parameter λ = g/∆




Qubit control

and readout

~ 10 GHz

Measurement

output




Jaynes-Cummings

H = ωra†a+ωaσz

2+g(a†σ−+aσ+)





Qubit control

and readout

~ 10 GHz

Measurement

output

Linear dispersive

HD = (ωa + χ )σz

2+ (ωr + χσz )a†a

Lamb shift (χ = gλ = g2/∆)

Stark shift or cavity pull

Valid if n ≪ ncrit., where ncrit. = 1/4λ2 .


o is

s i m

sn

arT

n(a

rb. u

nits)

ωa

Ph

ase

Δ ~ 2π 1GHz

ωr − CP ωr + CP

2CP

2χ κ

-2χ κ

κ



Jaynes-Cummings

H = ωra†a+ωaσz

2+g(a†σ−+aσ+)





Qubit control

and readout

~ 10 GHz

Measurement

output

Linear dispersive

HD = (ωa + χ )σz

2+ (ωr + χσz )a†a



Valid if n ≪ ncrit., where ncrit. = 1/4λ2 .


Rabi π-pulseWallraff et al., Phys. Rev. Lett. 95 060501 (2005)

Averaged 50000 times.SNR for single-shot is 0.1.

o is

s i m

sn

arT

n(a

rb. u

nits)

ωa

Ph

ase

Δ ~ 2π 1GHz

ωr − CP ωr + CP

2CP

2χ κ

-2χ κ

κ


Circuit VS cavity QED

Symbol Optical cavity Microwave cavity Circuitωr/2π or ωa/2π 350 THz 51 GHz 10 GHz

g/π 220 MHz 47 kHz 100 MHzg/ωr 3 × 10−7 10−7 5 × 10−3

Hood et al., Science 287 1447 (2000)Raimond, Brune and Haroche, Rev. Mod. Phys. 73 565–582 (2001)






g/π 220 MHz 47 kHz 100 MHz

g/ωr 3 × 10−7 10−7 5 × 10−3



MotivationCircuit QED is harder than cavity QED on the dispersive limit (ncrit. is smaller)






g/ωr 3 × 10−7 10−7 5 × 10−3




The SNR is low, we want to measure harder... how does higher order termsaffect measurement ?






g/ωr 3 × 10−7 10−7 5 × 10−3




The SNR is low, we want to measure harder... how does higher order termsaffect measurement ?

Must consider higher order corrections in perturbation theory



Understanding the dispersive transformation

J-C : block diagonal

H = ωra†a + ωaσz/2 + g(a†σ− + aσ+)






1x1 block : H0 = −ωaI/2

2x2 blocks : Hn = ∆

2σn

z + g√

nσnx

Total Hamiltonian

H = H0 ⊕ H1 ⊕ H2 · · · ⊕ H∞








2σn

z + g√

nσnx

Total Hamiltonian

H = H0 ⊕ H1 ⊕ H2 · · · ⊕ H∞

Z

X

Hn

θn

θn = arctan(2g√

n/∆)








2σn

z + g√

nσnx

Total Hamiltonian

H = H0 ⊕ H1 ⊕ H2 · · · ⊕ H∞

Diagonalization

Rotation around Y axis

In all subspaces En

E0 = {|g, 0〉}En = {|g, n〉 , |e, n − 1〉} ≡ {|gn〉 , |en〉}

Z

X

Hn

θn

θn = arctan(2g√

n/∆)

HZ

Z

X

D

XD

D

θn

n








2σn

z + g√

nσnx

Total Hamiltonian

H = H0 ⊕ H1 ⊕ H2 · · · ⊕ H∞

Diagonalization


In all subspaces En

E0 = {|g, 0〉}En = {|g, n〉 , |e, n − 1〉} ≡ {|gn〉 , |en〉}The qubit is now part photon and

vice-versa

Z

X

Hn

θn

θn = arctan(2g√

n/∆)

HZ

Z

X

D

XD

D

θn

n








2σn

z + g√

nσnx

Total Hamiltonian

H = H0 ⊕ H1 ⊕ H2 · · · ⊕ H∞

Diagonalization


In all subspaces En

E0 = {|g, 0〉}En = {|g, n〉 , |e, n − 1〉} ≡ {|gn〉 , |en〉}The qubit is now part photon and

vice-versa

Z

X

Hn

θn

θn = arctan(2g√

n/∆) ≈2g

√n/∆

HZ

Z

X

D

XD

D

θn

n



The dispersive limit

Dispersive limitJaynes-Cummings hamiltonian

H = ωra†a + ωaσz

2+ g(a†σ− + aσ+)

Exact transformation : D

Small parameter λ = g/∆4λ2n ≪ 1






2+ g(a†σ− + aσ+)



Result at order λ

HD = (ωa + χ )σz

2+(ωr + χσz )a†a



o is

s i m

sn

arT

n(a

rb. u

nits)

ωa

Ph

ase

Δ ~ 2π 1GHz

ωr − CP ωr + CP

2CP

2χ κ

-2χ κ

κ






2+ g(a†σ− + aσ+)



Result at order λ

HD = (ωa + χ )σz

2+(ωr + χσz )a†a



Result at order λ2

HD = (ωa + χ′ )σz

2+ [ωr + (χ′ − ζa†a)σz ]a†a

χ′ = χ(1 − λ2) , ζ = λ2χ

The cavity pull decrease : 〈CP 〉 = χ′ − ζ˙

a†a¸

o is

s i m

sn

arT

n(a

rb. u

nits)

ωa

Ph

ase

Δ ~ 2π 1GHz

ωr − CP ωr + CP

2CP

2χ κ

-2χ κ

κ



Dissipation in the system


Parametersκ : Rate of photon loss

γ1 : Transverse decay rate

Model for dissipationCoupling to a bath

Hκ =R ∞0

p

gκ(ω)[b†κ(ω) + bκ(ω)][a† + a]dω

Hγ =R ∞0

p

gγ(ω)[b†γ(ω) + bγ(ω)]σxdω

γ1

γϕ

κ







Hκ =R ∞0

p


Hγ =R ∞0

p

gγ(ω)[b†γ(ω) + bγ(ω)]σxdωE

ne

rgie

[A

rb. U

nits]

0 0.2 0.4 0.6 0.8 1


EJ/4EC=0.1

δng

γ1

γϕ

κ






γϕ : Pure dephasing rate


Hκ =R ∞0

p


Hγ =R ∞0

p

gγ(ω)[b†γ(ω) + bγ(ω)]σxdω

DephasingHϕ = ηf(t)σz

En

erg

ie [A

rb. U

nits]

0 0.2 0.4 0.6 0.8 1


EJ/4EC=0.1

δng

γ1

γϕ

κ


Dissipation in the transformed basis

Z

X

γϕ

Hn

θn

γ1




Z

X

γϕ

Hn

θn

γ1

HZ

Z

X

γϕeff

γϕeff

γ↓

γ↓ γ↑,

D

XD

D

θn

n

Transformation of system-bath hamiltonian

aD→ a + λσ− + O

`

λ2´

σ−D→ σ− + λaσz + O

`

λ2´

σzD→ σz − 2λ(a†σ− + aσ+) + O

`

λ2´




Z

X

γϕ

Hn

θn

γ1

HZ

Z

X

γϕeff

γϕeff

γ↓

γ↓ γ↑,

D

XD

D

θn

n



`

λ2´


`

λ2´

σzD→ σz − 2λ(a†σ− + aσ+) + O

`

λ2´

MethodTransform the system-bath hamiltonian

Trace out heat bath and cavity degrees of freedom(Gambetta et al., Phys. Rev. A 77 012112 (2008))




Z

X

γϕ

Hn

θn

γ1

HZ

Z

X

γϕeff

γϕeff

γ↓

γ↓ γ↑,

D

XD

D

θn

n



`

λ2´


`

λ2´

σzD→ σz − 2λ(a†σ− + aσ+) + O

`

λ2´

MethodTransform the system-bath hamiltonian

Trace out heat bath and cavity degrees of freedom(Gambetta et al., Phys. Rev. A 77 012112 (2008))

New rates (assuming white noises)

γ↓ = γ1

ˆ

1 − 2λ2`

n + 12

´˜

+ γκ + 2λ2γϕn

γ↑ = 2λ2γϕn

γκ = λ2κ



Reduction of the SNR

Parameters for the SNRNumber of measurement photons

SNR ∼ Num. phot.





Output rate : κ

SNR ∼ κ Num. phot.





Output rate : κ

Fraction of photons detected : η

SNR ∼ κ×η×Num. phot.





Output rate : κ


Info per photon : cavity pull

SNR ∼ κ×η×Num. phot.×Info per phot.





Output rate : κ



Mixing rate :γ↓+γ↑ = γ1

ˆ

1 − 2λ2`

n + 12

´˜

+γκ+4λ2γϕn

SNR ∼ κ×η×Num. phot.×Info per phot.Mixing rate





Output rate : κ




ˆ

1 − 2λ2`

n + 12

´˜

+γκ+4λ2γϕn


ConclusionSNR levels off with non-linear effects !

Explains low experimental SNR

Applies to all dispersive homodynemeasurement


∆/2π = 1.7 GHz, g/2π = 170 MHzκ/2π = 34 MHz, γ1/2π = 0.1 MHzγϕ = 0.1 MHz, η = 1/80

ncrit. = 1/4λ2 = 25

0

5

10

15

0.0 0.1 0.2 0.3 0.4 0.5

SN

R

n/ncrit.

Linear

Cooper-Pair Box

Withcorrections




Output rate : κ




ˆ

1 − 2λ2`

n + 12

´˜

+γκ+4λ2γϕn


ConclusionSNR levels off with non-linear effects !

Explains low experimental SNR

Applies to all dispersive homodynemeasurement


∆/2π = 1.7 GHz, g/2π = 170 MHzκ/2π = 34 MHz, γ1/2π = 0.1 MHzγϕ = 0.1 MHz, η = 1/80

ncrit. = 1/4λ2 = 25

Transmon

0

5

10

15

0.0 0.1 0.2 0.3 0.4 0.5

SN

R

n/ncrit.

Linear

Cooper-Pair Box

Withcorrections


Measurement induced heat bath

Mixing ratesDownward rate : γ↓(n)

Upward rate : γ↑(n)

Heat bath with temperature T (n) = (~ωr/kB)/ log(1 + 1/n)


-101

-1

0 5 10 15

〈σz〉

Time [µs]







Increasing

power

-101

-101

-1

0 5 10 15

〈σz〉

Time [µs]

〈σz〉







-101

-101

-101

0 5 10 15

〈σz〉

Time [µs]

〈σz〉

〈σz〉

Increasing

power


The case of the transmon


Koch et al., Phys. Rev. A 76 042319 (2007)

En

erg

ie [A

rb. U

nits]

0 0.2 0.4 0.6 0.8 1


EJ/4EC=0.1

EJ

Vg

Cg

CJEJ

- - - - -





C S

Vg

C g

En

erg

ie [A

rb. U

nits]

0 0.2 0.4 0.6 0.8 1


EJ/4EC=0.1

EJ

Vg

Cg

CJEJ

- - - - -





C S

Vg

C g

C S

Vg

C g

En

erg

ie [A

rb. U

nits]

0 0.2 0.4 0.6 0.8 1


EJ/4EC=0.1

EJ

Vg

Cg

CJEJ

- - - - -


Conclusion

Main resultsSimple model describing the physics of measurement

Side-effect of measuring harder : you heat your qubit (even with photons thatcan’t be directly absorbed)

Side-effect of measuring harder : each photon you add carries less informationthan the previous one

Measuring harder 6= bigger SNR

Coming soonThe transmon (3 level system) (Koch et al., Phys. Rev. A 76 042319 (2007))

Taking advantage of the non-linearity

Comparison with experiments

More information : Boissonneault, Gambetta and Blais, Phys. Rev. A 77 060305 (R)(2008)

FQRNT


circuit quantum electrodynamics : beyond the linear...

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