Circuit quantum electrodynamics : beyond the linear dispersive - - PowerPoint PPT Presentation

Introduction Beyond linear dispersive Results Conclusion

Circuit quantum electrodynamics : beyond the linear dispersive regime

Maxime Boissonneault1 Jay Gambetta2 Alexandre Blais1

1D´

epartement de Physique et Regroupement Qu´ eb´ ecois sur les mat´ eriaux de pointe, Universit´ e de Sherbrooke

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

June 23th, 2008

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

Maxime Boissonneault Universit´ e de Sherbrooke

Introduction Beyond linear dispersive Results Conclusion

1

Introduction Atom and cavity Cavity QED Charge qubit and coplanar resonator Circuit QED The linear dispersive limit Circuit VS cavity QED

2

Beyond linear dispersive Understanding the dispersive transformation The dispersive limit Dissipation in the system Dissipation in the transformed basis

3

Results Reduction of the SNR Measurement induced heat bath The case of the transmon

4

Conclusion Conclusion

Maxime Boissonneault Universit´ e de Sherbrooke

Introduction Beyond linear dispersive Results Conclusion Atom and cavity Maxime Boissonneault Universit´ e de Sherbrooke

Two-levels system Hamiltonian

H = ωa 2 σz σz = „1 −1 «

ω01= ωa

Energy

...

Introduction Beyond linear dispersive Results Conclusion Atom and cavity Maxime Boissonneault Universit´ e de Sherbrooke

Cavity Hamiltonian

H = X

k

ωk „ a†

kak + 1

2 « z x y L

Two-levels system Hamiltonian

H = ωa 2 σz σz = „1 −1 «

ω01= ωa

Energy

...

Introduction Beyond linear dispersive Results Conclusion Atom and cavity Maxime Boissonneault Universit´ e de Sherbrooke

Cavity Hamiltonian

H = X

k

ωk „ a†

kak + 1

2 « z x y L

Two-levels system Hamiltonian

H = ωa 2 σz σz = „1 −1 «

ω01= ωa

Energy

...

Introduction Beyond linear dispersive Results Conclusion Atom and cavity Maxime Boissonneault Universit´ e de Sherbrooke

Cavity Hamiltonian

H = X

k

ωk „ a†

kak + 1

2 « Single-mode : H = ωra†a

Response Input frequency,

rf

κ = ωr/Q

ω2 ω1 ωr = Two-levels system Hamiltonian

H = ωa 2 σz σz = „1 −1 «

ω01= ωa

Energy

...

Introduction Beyond linear dispersive Results Conclusion Cavity QED

g

Maxime Boissonneault Universit´ e de Sherbrooke

Introduction Beyond linear dispersive Results Conclusion Cavity QED

g Atom-cavity interaction

HI = − D · E ≈ g(a† + a)σx ≈ g(a†σ− + aσ+) g(z) = −d0 r ω V ǫ0 sin kz

Maxime Boissonneault Universit´ e de Sherbrooke

Introduction Beyond linear dispersive Results Conclusion Cavity QED

g Atom-cavity interaction

HI = − D · E ≈ g(a† + a)σx ≈ g(a†σ− + aσ+) g(z) = −d0 r ω V ǫ0 sin kz

Jaynes-Cummings Hamiltonian

H = ωa 2 σz + 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 Universit´ e de Sherbrooke

Introduction Beyond linear dispersive Results Conclusion Charge qubit and coplanar resonator Maxime Boissonneault Universit´ e 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 J Vg

n

Cg

Vg

Cg CJ E

J

• - - - -

Introduction Beyond linear dispersive Results Conclusion Charge qubit and coplanar resonator Maxime Boissonneault Universit´ e 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, Sch¨

• n 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 J Vg

n

Cg

Vg

Cg CJ E

J

• - - - -

Introduction Beyond linear dispersive Results Conclusion Charge qubit and coplanar resonator Maxime Boissonneault Universit´ e 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, Sch¨

• n 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)

Energie [Arb. Units] 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 J Vg

n

Cg

Vg

Cg CJ E

J

• - - - -

Introduction Beyond linear dispersive Results Conclusion Charge qubit and coplanar resonator Maxime Boissonneault Universit´ e de Sherbrooke

Introduction Beyond linear dispersive Results Conclusion Charge qubit and coplanar resonator Maxime Boissonneault Universit´ e 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 Universit´ e 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, Les Houches 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 Universit´ e de Sherbrooke

C g C J E J

Atom: super conducting charge qubit Cavity: super conducting 1D transmission line resonator Qubit control and readout ~ 10 GHz Measurement

• utput

Introduction Beyond linear dispersive Results Conclusion Circuit QED Maxime Boissonneault Universit´ e 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)

Parameters

g : Qubit-cavity interaction ωa : Qubit frequency ωr : Resonator frequency ∆ = ωa − ωr : Detuning H = ωa 2 σz + ωra†a + g(a†σ− + aσ+)

C g C J E J

Atom: super conducting charge qubit Cavity: super conducting 1D transmission line resonator Qubit control and readout ~ 10 GHz Measurement

• utput

Introduction Beyond linear dispersive Results Conclusion Circuit QED Maxime Boissonneault Universit´ e 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)

Parameters

g : Qubit-cavity interaction ωa : Qubit frequency ωr : Resonator frequency ∆ = ωa − ωr : Detuning H = ωa 2 σz + 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+ωa σz

2 +g(a†σ−+aσ+)

Small parameter λ = g/∆

Atom: super conducting charge qubit Cavity: super conducting 1D transmission line resonator Qubit control and readout ~ 10 GHz Measurement

• utput

Maxime Boissonneault Universit´ e de Sherbrooke

Introduction Beyond linear dispersive Results Conclusion The linear dispersive limit

Jaynes-Cummings

H = ωra†a+ωa σz

2 +g(a†σ−+aσ+)

Small parameter λ = g/∆

Atom: super conducting charge qubit Cavity: super conducting 1D transmission line resonator Qubit control and readout ~ 10 GHz Measurement

• utput

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 Universit´ e de Sherbrooke

• i

s s i m s n a r T n (arb. units) ωa Phase

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+ωa σz

2 +g(a†σ−+aσ+)

Small parameter λ = g/∆

Atom: super conducting charge qubit Cavity: super conducting 1D transmission line resonator Qubit control and readout ~ 10 GHz Measurement

• utput

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 Universit´ e de Sherbrooke

• i

s s i m s n a r T n (arb. units) ωa Phase

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+ωa σz

2 +g(a†σ−+aσ+)

Small parameter λ = g/∆

Atom: super conducting charge qubit Cavity: super conducting 1D transmission line resonator Qubit control and readout ~ 10 GHz Measurement

• utput

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 Universit´ e de Sherbrooke

Rabi π-pulse

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

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

• i

s s i m s n a r T n (arb. units) ωa Phase

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 Circuit ωr/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) Maxime Boissonneault Universit´ e de Sherbrooke

Introduction Beyond linear dispersive Results Conclusion Circuit VS cavity QED

Symbol Optical cavity Microwave cavity Circuit ωr/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)

Motivation

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

Maxime Boissonneault Universit´ e de Sherbrooke

Introduction Beyond linear dispersive Results Conclusion Circuit VS cavity QED

Symbol Optical cavity Microwave cavity Circuit ωr/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)

Motivation

Circuit 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 terms affect measurement ?

Maxime Boissonneault Universit´ e de Sherbrooke

Introduction Beyond linear dispersive Results Conclusion Circuit VS cavity QED

Symbol Optical cavity Microwave cavity Circuit ωr/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)

Motivation

Circuit 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 terms affect measurement ? Must consider higher order corrections in perturbation theory

Maxime Boissonneault Universit´ e 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 Universit´ e 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 = ∆

2 σn z + g√nσn x

Total Hamiltonian H = H0 ⊕ H1 ⊕ H2 · · · ⊕ H∞

Maxime Boissonneault Universit´ e 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 = ∆

2 σn z + g√nσn x

Total Hamiltonian H = H0 ⊕ H1 ⊕ H2 · · · ⊕ H∞

Z X Hn θ

n

θn = arctan(2g√n/∆)

Maxime Boissonneault Universit´ e 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 = ∆

2 σn z + g√nσn x

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 θ

n

θn = arctan(2g√n/∆)

H Z Z X

D

X

D D

θ

n n Maxime Boissonneault Universit´ e 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 = ∆

2 σn z + g√nσn x

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 θ

n

θn = arctan(2g√n/∆)

H Z Z X

D

X

D D

θ

n n Maxime Boissonneault Universit´ e 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 = ∆

2 σn z + g√nσn x

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 θ

n

θn = arctan(2g√n/∆) ≈ 2g√n/∆

H Z Z X

D

X

D D

θ

n n Maxime Boissonneault Universit´ e de Sherbrooke

Introduction Beyond linear dispersive Results Conclusion The dispersive limit

Dispersive limit

Jaynes-Cummings hamiltonian H = ωra†a + ωa σz

2 + g(a†σ− + aσ+)

Exact transformation : D Small parameter λ = g/∆ 4λ2¯ n ≪ 1

Maxime Boissonneault Universit´ e de Sherbrooke

Introduction Beyond linear dispersive Results Conclusion The dispersive limit

Dispersive limit

Jaynes-Cummings hamiltonian H = ωra†a + ωa σz

2 + g(a†σ− + aσ+)

Exact transformation : D Small parameter λ = g/∆ 4λ2¯ n ≪ 1

Result at order λ

HD = (ωa + χ ) σz

2 +(ωr + χσz )a†a

Lamb shift (χ = gλ = g2/∆) Stark shift or cavity pull

• i

s s i m s n a r T n (arb. units) ωa Phase

Δ ~ 2π 1GHz

ωr − CP ωr + CP

2CP

2χ κ

• 2χ κ

κ

Maxime Boissonneault Universit´ e de Sherbrooke

Introduction Beyond linear dispersive Results Conclusion The dispersive limit

Dispersive limit

Jaynes-Cummings hamiltonian H = ωra†a + ωa σz

2 + g(a†σ− + aσ+)

Exact transformation : D Small parameter λ = g/∆ 4λ2¯ n ≪ 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¸

• i

s s i m s n a r T n (arb. units) ωa Phase

Δ ~ 2π 1GHz

ωr − CP ωr + CP

2CP

2χ κ

• 2χ κ

κ

Maxime Boissonneault Universit´ e de Sherbrooke

Introduction Beyond linear dispersive Results Conclusion Dissipation in the system Maxime Boissonneault Universit´ e de Sherbrooke

Parameters

κ : Rate of photon loss γ1 : Transverse decay rate

Model for dissipation

Coupling to a bath Hκ = R ∞ p gκ(ω)[b†

κ(ω) + bκ(ω)][a† + a]dω

Hγ = R ∞ p gγ(ω)[b†

γ(ω) + bγ(ω)]σxdω

γ1 γ

ϕ

κ

Introduction Beyond linear dispersive Results Conclusion Dissipation in the system Maxime Boissonneault Universit´ e de Sherbrooke

Parameters

κ : Rate of photon loss γ1 : Transverse decay rate

Model for dissipation

Coupling to a bath Hκ = R ∞ p gκ(ω)[b†

κ(ω) + bκ(ω)][a† + a]dω

Hγ = R ∞ p gγ(ω)[b†

γ(ω) + bγ(ω)]σxdω

Energie [Arb. Units] 0.2 0.4 0.6 0.8 1

Gate charge, ng = CgVg/2e EJ/4EC=0.1

δng

γ1 γ

ϕ

κ

Introduction Beyond linear dispersive Results Conclusion Dissipation in the system Maxime Boissonneault Universit´ e de Sherbrooke

Parameters

κ : Rate of photon loss γ1 : Transverse decay rate γϕ : Pure dephasing rate

Model for dissipation

Coupling to a bath Hκ = R ∞ p gκ(ω)[b†

κ(ω) + bκ(ω)][a† + a]dω

Hγ = R ∞ p gγ(ω)[b†

γ(ω) + bγ(ω)]σxdω

Dephasing Hϕ = ηf(t)σz

Energie [Arb. Units] 0.2 0.4 0.6 0.8 1

Gate charge, ng = CgVg/2e EJ/4EC=0.1

δng

γ1 γ

ϕ

κ

Introduction Beyond linear dispersive Results Conclusion Dissipation in the transformed basis

Z X

γ

ϕ

Hn θ

n

γ

1

Maxime Boissonneault Universit´ e de Sherbrooke

Introduction Beyond linear dispersive Results Conclusion Dissipation in the transformed basis

Z X

γ

ϕ

Hn θ

n

γ

1

H Z Z X

γϕeff γϕeff

γ↓ γ↓ γ↑

,

D

X

D D

θ

n n

Transformation of system-bath hamiltonian

a D → a + λσ− + O `λ2´ σ−

D

→ σ− + λaσz + O `λ2´ σz

D

→ σz − 2λ(a†σ− + aσ+) + O `λ2´

Maxime Boissonneault Universit´ e de Sherbrooke

Introduction Beyond linear dispersive Results Conclusion Dissipation in the transformed basis

Z X

γ

ϕ

Hn θ

n

γ

1

H Z Z X

γϕeff γϕeff

γ↓ γ↓ γ↑

,

D

X

D D

θ

n n

Transformation of system-bath hamiltonian

a D → a + λσ− + O `λ2´ σ−

D

→ σ− + λaσz + O `λ2´ σz

D

→ σz − 2λ(a†σ− + aσ+) + O `λ2´

Method

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

Maxime Boissonneault Universit´ e de Sherbrooke

Introduction Beyond linear dispersive Results Conclusion Dissipation in the transformed basis

Z X

γ

ϕ

Hn θ

n

γ

1

H Z Z X

γϕeff γϕeff

γ↓ γ↓ γ↑

,

D

X

D D

θ

n n

Transformation of system-bath hamiltonian

a D → a + λσ− + O `λ2´ σ−

D

→ σ− + λaσz + O `λ2´ σz

D

→ σz − 2λ(a†σ− + aσ+) + O `λ2´

Method

Transform 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 ˆ1 − 2λ2 `¯ n + 1

2

´˜ + γκ + 2λ2γϕ¯ n γ↑ = 2λ2γϕ¯ n γκ = λ2κ

Maxime Boissonneault Universit´ e de Sherbrooke

Introduction Beyond linear dispersive Results Conclusion Reduction of the SNR

Parameters for the SNR

Number of measurement photons SNR ∼

• Num. phot.

Maxime Boissonneault Universit´ e de Sherbrooke

Introduction Beyond linear dispersive Results Conclusion Reduction of the SNR

Parameters for the SNR

Number of measurement photons Output rate : κ SNR ∼ κ

• Num. phot.

Maxime Boissonneault Universit´ e de Sherbrooke

Introduction Beyond linear dispersive Results Conclusion Reduction of the SNR

Parameters for the SNR

Number of measurement photons Output rate : κ Fraction of photons detected : η SNR ∼ κ×η×Num. phot.

Maxime Boissonneault Universit´ e de Sherbrooke

Introduction Beyond linear dispersive Results Conclusion Reduction of the SNR

Parameters for the SNR

Number 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 Universit´ e de Sherbrooke

Introduction Beyond linear dispersive Results Conclusion Reduction of the SNR

Parameters for the SNR

Number of measurement photons Output rate : κ Fraction of photons detected : η Info per photon : cavity pull Mixing rate : γ↓+γ↑ = γ1 ˆ1 − 2λ2 `¯ n + 1

2

´˜+γκ+4λ2γϕ¯ n SNR ∼ κ×η×Num. phot.×Info per phot. Mixing rate

Maxime Boissonneault Universit´ e de Sherbrooke

Introduction Beyond linear dispersive Results Conclusion Reduction of the SNR

Parameters for the SNR

Number of measurement photons Output rate : κ Fraction of photons detected : η Info per photon : cavity pull Mixing rate : γ↓+γ↑ = γ1 ˆ1 − 2λ2 `¯ n + 1

2

´˜+γκ+4λ2γϕ¯ n SNR ∼ κ×η×Num. phot.×Info per phot. Mixing rate

Conclusion

SNR levels off with non-linear effects ! Explains low experimental SNR Applies to all dispersive homodyne measurement

Maxime Boissonneault Universit´ e de Sherbrooke ∆/2π = 1.7 GHz, g/2π = 170 MHz κ/2π = 34 MHz, γ1/2π = 0.1 MHz γϕ = 0.1 MHz, η = 1/80

• ncrit. = 1/4λ2 = 25

5 10 15 0.0 0.1 0.2 0.3 0.4 0.5

SNR n/ncrit. Linear Cooper-Pair Box With corrections

Introduction Beyond linear dispersive Results Conclusion Reduction of the SNR

Parameters for the SNR

Number of measurement photons Output rate : κ Fraction of photons detected : η Info per photon : cavity pull Mixing rate : γ↓+γ↑ = γ1 ˆ1 − 2λ2 `¯ n + 1

2

´˜+γκ+4λ2γϕ¯ n SNR ∼ κ×η×Num. phot.×Info per phot. Mixing rate

Conclusion

SNR levels off with non-linear effects ! Explains low experimental SNR Applies to all dispersive homodyne measurement

Maxime Boissonneault Universit´ e de Sherbrooke ∆/2π = 1.7 GHz, g/2π = 170 MHz κ/2π = 34 MHz, γ1/2π = 0.1 MHz γϕ = 0.1 MHz, η = 1/80

• ncrit. = 1/4λ2 = 25

Transmon

5 10 15 0.0 0.1 0.2 0.3 0.4 0.5

SNR n/ncrit. Linear Cooper-Pair Box With corrections

Introduction Beyond linear dispersive Results Conclusion Measurement induced heat bath

Mixing rates

Downward rate : γ↓(¯ n) Upward rate : γ↑(¯ n) Heat bath with temperature T(¯ n) = (ωr/kB)/ log(1 + 1/¯ n)

Maxime Boissonneault Universit´ e de Sherbrooke

• 1

1

• 1

5 10 15

σz Time [µs]

Introduction Beyond linear dispersive Results Conclusion Measurement induced heat bath

Mixing rates

Downward rate : γ↓(¯ n) Upward rate : γ↑(¯ n) Heat bath with temperature T(¯ n) = (ωr/kB)/ log(1 + 1/¯ n)

Maxime Boissonneault Universit´ e de Sherbrooke

Increasing power

• 1

1

• 1

1

• 1

5 10 15

σz Time [µs] σz

Introduction Beyond linear dispersive Results Conclusion Measurement induced heat bath

Mixing rates

Downward rate : γ↓(¯ n) Upward rate : γ↑(¯ n) Heat bath with temperature T(¯ n) = (ωr/kB)/ log(1 + 1/¯ n)

Maxime Boissonneault Universit´ e de Sherbrooke

• 1

1

• 1

1

• 1

1 5 10 15

σz Time [µs] σz σz

Increasing power

Introduction Beyond linear dispersive Results Conclusion The case of the transmon Maxime Boissonneault Universit´ e de Sherbrooke Koch et al., Phys. Rev. A 76 042319 (2007) Energie [Arb. Units] 0.2 0.4 0.6 0.8 1

Gate charge, ng = CgVg/2e EJ/4EC=0.1 EJ

Vg

Cg CJ E

J

• - - - -

Introduction Beyond linear dispersive Results Conclusion The case of the transmon Maxime Boissonneault Universit´ e de Sherbrooke Koch et al., Phys. Rev. A 76 042319 (2007)

C S

Vg

C g

Energie [Arb. Units] 0.2 0.4 0.6 0.8 1

Gate charge, ng = CgVg/2e EJ/4EC=0.1 EJ

Vg

Cg CJ E

J

• - - - -

Introduction Beyond linear dispersive Results Conclusion The case of the transmon Maxime Boissonneault Universit´ e de Sherbrooke Koch et al., Phys. Rev. A 76 042319 (2007)

C S

Vg

C g

C S

Vg

C g

Energie [Arb. Units] 0.2 0.4 0.6 0.8 1

Gate charge, ng = CgVg/2e EJ/4EC=0.1 EJ

Vg

Cg CJ E

J

• - - - -

Introduction Beyond linear dispersive Results Conclusion The case of the transmon Maxime Boissonneault Universit´ e de Sherbrooke Koch et al., Phys. Rev. A 76 042319 (2007)

C S

Vg

C g

C S

Vg

C g

Energie [Arb. Units] 0.2 0.4 0.6 0.8 1

Gate charge, ng = CgVg/2e EJ/4EC=0.1 EJ

Vg

Cg CJ E

J

• - - - -

Introduction Beyond linear dispersive Results Conclusion Conclusion

Main results

Simple model describing the physics of measurement Side-effect of measuring harder : you heat your qubit (even with photons that can’t be directly absorbed) Side-effect of measuring harder : each photon you add carries less information than the previous one Measuring harder = bigger SNR

Coming soon

The 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 Universit´ e de Sherbrooke