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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
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
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)
Blais et al., Phys. Rev. A 69 062320 (2004)
Maxime Boissonneault Universite de Sherbrooke
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
g/π 220 MHz 47 kHz 100 MHz
g/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)
Blais et al., Phys. Rev. A 69 062320 (2004)
MotivationCircuit QED is harder than cavity QED on the dispersive limit (ncrit. is smaller)
Maxime Boissonneault Universite de Sherbrooke
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
g/π 220 MHz 47 kHz 100 MHz
g/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)
Blais et al., Phys. Rev. A 69 062320 (2004)
MotivationCircuit QED is harder than cavity QED on the dispersive limit (ncrit. is smaller)
The SNR is low, we want to measure harder... how does higher order termsaffect measurement ?
Maxime Boissonneault Universite de Sherbrooke
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
g/π 220 MHz 47 kHz 100 MHz
g/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)
Blais et al., Phys. Rev. A 69 062320 (2004)
MotivationCircuit QED is harder than cavity QED on the dispersive limit (ncrit. is smaller)
The SNR is low, we want to measure harder... how does higher order termsaffect measurement ?
Must consider higher order corrections in perturbation theory
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
Understanding the dispersive transformation
J-C : block diagonal
H = ωra†a + ωaσz/2 + g(a†σ− + aσ+)
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
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∞
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
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∞
Z
X
Hn
θn
θn = arctan(2g√
n/∆)
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
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∞
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
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
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∞
Diagonalization
Rotation around Y axis
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
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
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∞
Diagonalization
Rotation around Y axis
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
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
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
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
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
Result at order λ
HD = (ωa + χ )σz
2+(ωr + χσz )a†a
Lamb shift (χ = gλ = g2/∆)
Stark shift or cavity pull
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
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
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
Result at order λ
HD = (ωa + χ )σz
2+(ωr + χσz )a†a
Lamb shift (χ = gλ = g2/∆)
Stark shift or cavity pull
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
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
Dissipation in the system
Maxime Boissonneault Universite de Sherbrooke
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
Introduction Beyond linear dispersive Results Conclusion
Dissipation in the system
Maxime Boissonneault Universite de Sherbrooke
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)]σxdωE
ne
rgie
[A
rb. U
nits]
0 0.2 0.4 0.6 0.8 1
Gate charge, ng = CgVg/2e
EJ/4EC=0.1
δng
γ1
γϕ
κ
Introduction Beyond linear dispersive Results Conclusion
Dissipation in the system
Maxime Boissonneault Universite de Sherbrooke
Parametersκ : Rate of photon loss
γ1 : Transverse decay rate
γϕ : Pure dephasing 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
DephasingHϕ = ηf(t)σz
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
δng
γ1
γϕ
κ
Introduction Beyond linear dispersive Results Conclusion
Dissipation in the transformed basis
Z
X
γϕ
Hn
θn
γ1
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
Dissipation in the transformed basis
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´
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
Dissipation in the transformed basis
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´
MethodTransform the system-bath hamiltonian
Trace out heat bath and cavity degrees of freedom(Gambetta et al., Phys. Rev. A 77 012112 (2008))
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
Dissipation in the transformed basis
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´
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
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
Reduction of the SNR
Parameters for the SNRNumber of measurement photons
SNR ∼ Num. phot.
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
Reduction of the SNR
Parameters for the SNRNumber of measurement photons
Output rate : κ
SNR ∼ κ Num. phot.
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
Reduction of the SNR
Parameters for the SNRNumber of measurement photons
Output rate : κ
Fraction of photons detected : η
SNR ∼ κ×η×Num. phot.
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
Reduction of the SNR
Parameters for the SNRNumber of measurement photons
Output rate : κ
Fraction of photons detected : η
Info per photon : cavity pull
SNR ∼ κ×η×Num. phot.×Info per phot.
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
Reduction of the SNR
Parameters for the SNRNumber of measurement photons
Output rate : κ
Fraction of photons detected : η
Info per photon : cavity pull
Mixing rate :γ↓+γ↑ = γ1
ˆ
1 − 2λ2`
n + 12
´˜
+γκ+4λ2γϕn
SNR ∼ κ×η×Num. phot.×Info per phot.Mixing rate
Maxime Boissonneault Universite de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion
Reduction of the SNR
Parameters for the SNRNumber of measurement photons
Output rate : κ
Fraction of photons detected : η
Info per photon : cavity pull
Mixing rate :γ↓+γ↑ = γ1
ˆ
1 − 2λ2`
n + 12
´˜
+γκ+4λ2γϕn
SNR ∼ κ×η×Num. phot.×Info per phot.Mixing rate
ConclusionSNR levels off with non-linear effects !
Explains low experimental SNR
Applies to all dispersive homodynemeasurement
Maxime Boissonneault Universite de Sherbrooke
∆/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
Introduction Beyond linear dispersive Results Conclusion
Reduction of the SNR
Parameters for the SNRNumber of measurement photons
Output rate : κ
Fraction of photons detected : η
Info per photon : cavity pull
Mixing rate :γ↓+γ↑ = γ1
ˆ
1 − 2λ2`
n + 12
´˜
+γκ+4λ2γϕn
SNR ∼ κ×η×Num. phot.×Info per phot.Mixing rate
ConclusionSNR levels off with non-linear effects !
Explains low experimental SNR
Applies to all dispersive homodynemeasurement
Maxime Boissonneault Universite de Sherbrooke
∆/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
Introduction Beyond linear dispersive Results Conclusion
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)
Maxime Boissonneault Universite de Sherbrooke
-101
-1
0 5 10 15
〈σz〉
Time [µs]
Introduction Beyond linear dispersive Results Conclusion
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)
Maxime Boissonneault Universite de Sherbrooke
Increasing
power
-101
-101
-1
0 5 10 15
〈σz〉
Time [µs]
〈σz〉
Introduction Beyond linear dispersive Results Conclusion
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)
Maxime Boissonneault Universite de Sherbrooke
-101
-101
-101
0 5 10 15
〈σz〉
Time [µs]
〈σz〉
〈σz〉
Increasing
power
Introduction Beyond linear dispersive Results Conclusion
The case of the transmon
Maxime Boissonneault Universite de Sherbrooke
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
Gate charge, ng = CgVg/2e
EJ/4EC=0.1
EJ
Vg
Cg
CJEJ
- - - - -
Introduction Beyond linear dispersive Results Conclusion
The case of the transmon
Maxime Boissonneault Universite de Sherbrooke
Koch et al., Phys. Rev. A 76 042319 (2007)
C S
Vg
C g
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
Vg
Cg
CJEJ
- - - - -
Introduction Beyond linear dispersive Results Conclusion
The case of the transmon
Maxime Boissonneault Universite de Sherbrooke
Koch et al., Phys. Rev. A 76 042319 (2007)
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
Gate charge, ng = CgVg/2e
EJ/4EC=0.1
EJ
Vg
Cg
CJEJ
- - - - -
Introduction Beyond linear dispersive Results Conclusion
The case of the transmon
Maxime Boissonneault Universite de Sherbrooke
Koch et al., Phys. Rev. A 76 042319 (2007)
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
Gate charge, ng = CgVg/2e
EJ/4EC=0.1
EJ
Vg
Cg
CJEJ
- - - - -
Introduction Beyond linear dispersive Results Conclusion
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
Maxime Boissonneault Universite de Sherbrooke