Introduction Beyond linear dispersive Results Conclusion Circuit quantum electrodynamics : beyond the linear dispersive regime Maxime Boissonneault 1 Jay Gambetta 2 Alexandre Blais 1 1 D´ epartement de Physique et Regroupement Qu´ eb´ ecois sur les mat´ eriaux de pointe, Universit´ e de Sherbrooke 2 Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo June 23 th , 2008 Boissonneault, Gambetta and Blais, Phys. Rev. A 77 060305 (R) (2008) Maxime Boissonneault Universit´ e de Sherbrooke
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Introduction Beyond linear dispersive Results Conclusion
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
Introduction Beyond linear dispersive Results Conclusion
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
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
Atom and cavity
Maxime Boissonneault Universite de Sherbrooke
Two-levels system Hamiltonian
H =ωa
2σz σz =
„
1 00 −1
«
ω01= ωa
En
erg
y
...
Introduction Beyond linear dispersive Results Conclusion
Atom and cavity
Maxime Boissonneault Universite de Sherbrooke
Cavity Hamiltonian
H =X
k
ωk
„
a†kak +
1
2
«
z
x
y
L
Two-levels system Hamiltonian
H =ωa
2σz σz =
„
1 00 −1
«
ω01= ωa
En
erg
y
...
Introduction Beyond linear dispersive Results Conclusion
Atom and cavity
Maxime Boissonneault Universite de Sherbrooke
Cavity Hamiltonian
H =X
k
ωk
„
a†kak +
1
2
«
z
x
y
L
Two-levels system Hamiltonian
H =ωa
2σz σz =
„
1 00 −1
«
ω01= ωa
En
erg
y
...
Introduction Beyond linear dispersive Results Conclusion
Atom and cavity
Maxime Boissonneault Universite de Sherbrooke
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=
Two-levels system Hamiltonian
H =ωa
2σz σz =
„
1 00 −1
«
ω01= ωa
En
erg
y
...
Introduction Beyond linear dispersive Results Conclusion
Cavity QED
gg
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
Cavity QED
gg
Atom-cavity interaction
HI = − ~D · ~E ≈ g(a† + a)σx ≈ g(a†σ− + aσ+)
g(z) = −d0
r
ω
V ǫ0sinkz
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
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)
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
Charge qubit and coplanar resonator
Maxime Boissonneault Universite de Sherbrooke
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
- - - - -
Introduction Beyond linear dispersive Results Conclusion
Charge qubit and coplanar resonator
Maxime Boissonneault Universite de Sherbrooke
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)
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
- - - - -
Introduction Beyond linear dispersive Results Conclusion
Charge qubit and coplanar resonator
Maxime Boissonneault Universite de Sherbrooke
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
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
- - - - -
Introduction Beyond linear dispersive Results Conclusion
Charge qubit and coplanar resonator
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
Charge qubit and coplanar resonator
Maxime Boissonneault Universite de Sherbrooke
Classical Hamiltonian
H =Φ2
2Lr+
1
2CrV 2
ωr =
s
1
LrCr
L r C r
Introduction Beyond linear dispersive Results Conclusion
Charge qubit and coplanar resonator
Maxime Boissonneault Universite de Sherbrooke
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).
Classical Hamiltonian
H =Φ2
2Lr+
1
2CrV 2
ωr =
s
1
LrCr
L r C r
Introduction Beyond linear dispersive Results Conclusion
Circuit QED
Maxime Boissonneault Universite de Sherbrooke
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
Introduction Beyond linear dispersive Results Conclusion
Circuit QED
Maxime Boissonneault Universite de Sherbrooke
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
2σz + ωra†a + g(a†σ− + aσ+)
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
Introduction Beyond linear dispersive Results Conclusion
Circuit QED
Maxime Boissonneault Universite de Sherbrooke
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
2σz + ωra†a + g(a†σ− + aσ+)
C g
C J
E J
Introduction Beyond linear dispersive Results Conclusion
The linear dispersive limit
Jaynes-Cummings
H = ωra†a+ωaσz
2+g(a†σ−+aσ+)
Small parameter λ = g/∆
Atom: super conducting
charge qubitCavity: super conducting 1D
transmission line resonator
Qubit control
and readout
~ 10 GHz
Measurement
output
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
The linear dispersive limit
Jaynes-Cummings
H = ωra†a+ωaσz
2+g(a†σ−+aσ+)
Small parameter λ = g/∆
Atom: super conducting
charge qubitCavity: super conducting 1D
transmission line resonator
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 .
Maxime Boissonneault Universite de Sherbrooke
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
Introduction Beyond linear dispersive Results Conclusion
The linear dispersive limit
Jaynes-Cummings
H = ωra†a+ωaσz
2+g(a†σ−+aσ+)
Small parameter λ = g/∆
Atom: super conducting
charge qubitCavity: super conducting 1D
transmission line resonator
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 .
Maxime Boissonneault Universite de Sherbrooke
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
Introduction Beyond linear dispersive Results Conclusion
The linear dispersive limit
Jaynes-Cummings
H = ωra†a+ωaσz
2+g(a†σ−+aσ+)
Small parameter λ = g/∆
Atom: super conducting
charge qubitCavity: super conducting 1D
transmission line resonator
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 .
Maxime Boissonneault Universite de Sherbrooke
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
Introduction Beyond linear dispersive Results Conclusion
Circuit VS cavity QED
Symbol Optical cavity Microwave cavity Circuitωr/2π or ωa/2π 350 THz 51 GHz 10 GHz