Circuit quantum electrodynamics : beyond the linear dispersive regime

Circuit quantum electrodynamics : beyond the linear dispersive regime 1 Jay Gambetta 2 Alexandre Blais 1 1 Département de Physique et Regroupement Québécois sur les matériaux de pointe, 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)

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

Atom and cavity... Energy ω 01 = ω a Two-levels system Hamiltonian H = ωa σz σz = 2 1 0 0 1 «

Atom and cavity x y z Energy... L Two-levels system Hamiltonian H = ωa σz σz = 2 ω 01 = ω a 1 0 0 1 «Cavity Hamiltonian H = X k ω k a k a k + 1 «2

Atom and cavity ω r = ω 1 ω 2 Energy Response κ = ω r/q... Input frequency, rf Two-levels system Hamiltonian H = ωa σz σz = 2 ω 01 = ω a 1 0 0 1 «Cavity Hamiltonian H = X k Single-mode : ω k a k a k + 1 «2 H = ω ra a

Cavity QED g

Cavity QED g Atom-cavity interaction H I = D E g(a + a)σ x g(a σ + aσ + ) g(z) = d 0 r ω V ǫ 0 sinkz

Cavity QED g Atom-cavity interaction H I = D E g(a + a)σ x g(a σ + aσ + ) Jaynes-Cummings Hamiltonian g(z) = d 0 r ω V ǫ 0 sinkz 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)

Charge qubit and coplanar resonator Classical Hamiltonian Vg ----- C g E J C J Vg C g n E J C J E C = H = 4E C (n n g) 2 E J cos δ E J = I 0Φ 0 2π e 2 CgVg, ng = 2(C g + C J ) 2e

Charge qubit and coplanar resonator Classical Hamiltonian Vg ----- C g E J C J Vg C g n E J C J E C = H = 4E C (n n g) 2 E J cos δ E J = I 0Φ 0 2π e 2 CgVg, ng = 2(C g + C J ) 2e Quantum Hamiltonian H = X n 4E C (n n g) 2 n n X n E J ( n n + 1 + h.c.) 2 Restricting to n g [0,1] : H = ω aσ z/2 Shnirman, Schön 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)

Charge qubit and coplanar resonator Classical Hamiltonian Vg ----- C g E J C J Vg C g n E J C J E C = H = 4E C (n n g) 2 E J cos δ E J = I 0Φ 0 2π e 2 CgVg, ng = 2(C g + C J ) 2e Energie [Arb. Units] EJ/4EC=0.1 EJ 0 0.2 0.4 0.6 0.8 1 Gate charge, n g = C g V g /2e Quantum Hamiltonian H = X n X n 4E C (n n g) 2 n n E J ( n n + 1 + h.c.) 2 Restricting to n g [0,1] : H = ω aσ z/2 Shnirman, Schön 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)

Charge qubit and coplanar resonator

Charge qubit and coplanar resonator L r C r Classical Hamiltonian H = Φ2 2L r + 1 2 CrV 2 ω r = s 1 L rc r

Charge qubit and coplanar resonator L r C r Classical Hamiltonian H = Φ2 2L r + 1 2 CrV 2 ω r = s 1 L rc r Quantum Hamiltonian r ωr V = (a + a), 2C r H = ω r a a + 1 «2 r ωr Φ = i (a a) 2L r Quantum Fluctuations in Electrical Circuits, M. H. Devoret, Les Houches Session LXIII, Quantum Fluctuations p. 351-386 (1995).

Circuit QED Measurement output C g E J CJ Qubit control and readout ~ 10 GHz Atom: super conducting charge qubit Cavity: super conducting 1D transmission line resonator

Circuit QED Measurement output C g E J CJ Qubit control and readout ~ 10 GHz Atom: super conducting charge qubit Cavity: super conducting 1D transmission line resonator Parameters g : Qubit-cavity interaction ω a : Qubit frequency ω r : Resonator frequency = ω a ω r : Detuning H = ωa 2 σz + ωra a + g(a σ + aσ + ) 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)

Circuit QED C g E J CJ Parameters g : Qubit-cavity interaction ω a : Qubit frequency ω r : Resonator frequency = ω a ω r : Detuning H = ωa 2 σz + ωra a + g(a σ + aσ + ) 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)

The linear dispersive limit Jaynes-Cummings H = ω ra a+ω a σ z 2 +g(a σ +aσ + ) Small parameter λ = g/ Measurement output Qubit control and readout ~ 10 GHz Atom: super conducting charge qubit Cavity: super conducting 1D transmission line resonator

The linear dispersive limit Jaynes-Cummings H = ω ra a+ω a σ z 2 +g(a σ +aσ + ) Small parameter λ = g/ Measurement output Linear dispersive H D = (ω a + χ ) σz 2 + (ωr + χσz )a a Lamb shift (χ = gλ = g 2 / ) Stark shift or cavity pull Valid if n n crit., where n crit. = 1/4λ 2. Qubit control and readout ~ 10 GHz Atom: super conducting charge qubit Cavity: super conducting 1D transmission line resonator 2CP T ransmi ssio n (arb. units) κ Δ ~ 2π 1GHz 2χ κ Phase -2χ κ ωr CP ωr + CP ω a

The linear dispersive limit Jaynes-Cummings H = ω ra a+ω a σ z 2 +g(a σ +aσ + ) Small parameter λ = g/ Measurement output Linear dispersive H D = (ω a + χ ) σz 2 + (ωr + χσz )a a Lamb shift (χ = gλ = g 2 / ) Stark shift or cavity pull Qubit control and readout ~ 10 GHz Atom: super conducting charge qubit Cavity: super conducting 1D transmission line resonator Valid if n n crit., where n crit. = 1/4λ 2. Rabi π-pulse Wallraff et al., Phys. Rev. Lett. 95 060501 (2005) 2CP T ransmi ssio n (arb. units) κ ωr CP Δ ~ 2π 1GHz ωr + CP ω a 2χ κ Phase -2χ κ Averaged 50000 times. SNR for single-shot is 0.1.

Circuit VS cavity QED Symbol Optical cavity Microwave cavity Circuit ω r/2π or ω a/2π 350 THz 51 GHz 10 GHz g/π 220 MHz 47 khz 100 MHz g/ω r 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)

Circuit VS cavity QED Motivation Symbol Optical cavity Microwave cavity Circuit ω r/2π or ω a/2π 350 THz 51 GHz 10 GHz g/π 220 MHz 47 khz 100 MHz g/ω r 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) Circuit QED is harder than cavity QED on the dispersive limit (n crit. is smaller)

Understanding the dispersive transformation J-C : block diagonal H = ω ra a + ω aσ z/2 + g(a σ + aσ +)

Understanding the dispersive transformation J-C : block diagonal H = ω ra a + ω aσ z/2 + g(a σ + aσ +) 1x1 block : H 0 = ω ai/2 2x2 blocks : H n = 2 σn z + g nσ n x Total Hamiltonian H = H 0 H 1 H 2 H

Understanding the dispersive transformation J-C : block diagonal H = ω ra a + ω aσ z/2 + g(a σ + aσ +) 1x1 block : H 0 = ω ai/2 Z θ n H n 2x2 blocks : H n = 2 σn z + g nσ n x Total Hamiltonian H = H 0 H 1 H 2 H Diagonalization Rotation around Y axis In all subspaces E n E 0 = { g,0 } E n = { g, n, e, n 1 } { g n, e n } X θ n = arctan(2g n/ ) Z Z D θ n D H n D X X

The dispersive limit Dispersive limit Jaynes-Cummings hamiltonian H = ω ra a + ω a σ z 2 + g(a σ + aσ + ) Exact transformation : D Small parameter λ = g/ 4λ 2 n 1

The dispersive limit Dispersive limit Jaynes-Cummings hamiltonian H = ω ra a + ω a σ z 2 + g(a σ + aσ + ) Exact transformation : D Small parameter λ = g/ 4λ 2 n 1 Result at order λ H D = (ω a + χ ) σz 2 +(ωr + χσz )a a Lamb shift (χ = gλ = g 2 / ) Stark shift or cavity pull T ransmi ssio n (arb. units) κ ωr CP 2CP Δ ~ 2π 1GHz ωr + CP ω a 2χ κ Phase -2χ κ

The dispersive limit Dispersive limit Jaynes-Cummings hamiltonian H = ω ra a + ω a σ z 2 + g(a σ + aσ + ) Exact transformation : D Small parameter λ = g/ 4λ 2 n 1 Result at order λ 2 H D = (ω a + χ ) σz 2 + [ωr + (χ ζa a)σ z ]a a χ = χ(1 λ 2 ), ζ = λ 2 χ The cavity pull decrease : CP = χ ζ a a Result at order λ H D = (ω a + χ ) σz 2 +(ωr + χσz )a a Lamb shift (χ = gλ = g 2 / ) Stark shift or cavity pull T ransmi ssio n (arb. units) κ 2CP Δ ~ 2π 1GHz 2χ κ Phase -2χ κ ωr CP ωr + CP ω a

Dissipation in the system κ γ ϕ γ 1 Model for dissipation Coupling to a bath H κ = R p 0 gκ(ω)[b κ(ω) + b κ(ω)][a + a]dω H γ = R p 0 gγ(ω)[b γ(ω) + b γ(ω)]σ xdω Parameters κ : Rate of photon loss γ 1 : Transverse decay rate

Dissipation in the system κ γ ϕ Energie [Arb. Units] EJ/4EC=0.1 δn g γ 1 0 0.2 0.4 0.6 0.8 1 Gate charge, n g = C g V g /2e Model for dissipation Coupling to a bath H κ = R p 0 gκ(ω)[b κ(ω) + b κ(ω)][a + a]dω H γ = R p 0 gγ(ω)[b γ(ω) + b γ(ω)]σ xdω Parameters κ : Rate of photon loss γ 1 : Transverse decay rate

Dissipation in the system κ γ ϕ Energie [Arb. Units] EJ/4EC=0.1 δn g γ 1 0 0.2 0.4 0.6 0.8 1 Gate charge, n g = C g V g /2e Model for dissipation Coupling to a bath H κ = R p 0 gκ(ω)[b κ(ω) + b κ(ω)][a + a]dω H γ = R p 0 gγ(ω)[b γ(ω) + b γ(ω)]σ xdω Dephasing H ϕ = ηf(t)σ z Parameters κ : Rate of photon loss γ 1 : Transverse decay rate γ ϕ : Pure dephasing rate

Dissipation in the transformed basis Z H n θ n γ 1 γ ϕ X

Dissipation in the transformed basis Z θ n γ 1 γ ϕ H n X Transformation of system-bath hamiltonian a D a + λσ + O `λ2 D σ σ + λaσ z + O `λ2 D σ z σz 2λ(a σ + aσ + ) + O `λ2 Z D Z D H n θ n γ ϕeff γ γ ϕeff γ γ, X X D

Dissipation in the transformed basis Z D Z Z θ n γ 1 H γ ϕ D n H n X Transformation of system-bath hamiltonian a D a + λσ + O `λ2 D σ σ + λaσ z + O `λ2 D σ z 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)) θ n γ ϕeff γ γ ϕeff γ γ, X X D

Reduction of the SNR Parameters for the SNR Number of measurement photons SNR Num. phot.

Reduction of the SNR Parameters for the SNR Number of measurement photons Output rate : κ SNR κ Num. phot.

Reduction of the SNR Parameters for the SNR Number of measurement photons Output rate : κ Fraction of photons detected : η κ η Num. phot. SNR

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 κ η Num. phot. Info per phot. SNR

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 = γ 1 ˆ1 2λ 2 ` n + 1 2 +γκ+4λ 2 γ ϕ n SNR κ η Num. phot. Info per phot. Mixing rate SNR 15 10 5 Cooper-Pair Box Linear With corrections Conclusion SNR levels off with non-linear effects! Explains low experimental SNR Applies to all dispersive homodyne measurement 0 0.0 0.1 0.2 0.3 0.4 0.5 n/n crit. /2π = 1.7 GHz, g/2π = 170 MHz κ/2π = 34 MHz, γ 1 /2π = 0.1 MHz γ ϕ = 0.1 MHz, η = 1/80 n crit. = 1/4λ 2 = 25

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 = γ 1 ˆ1 2λ 2 ` n + 1 2 +γκ+4λ 2 γ ϕ n SNR Conclusion κ η Num. phot. Info per phot. Mixing rate SNR levels off with non-linear effects! Explains low experimental SNR Applies to all dispersive homodyne measurement SNR 15 10 5 Cooper-Pair Box Linear With corrections Transmon 0 0.0 0.1 0.2 0.3 0.4 0.5 n/n crit. /2π = 1.7 GHz, g/2π = 170 MHz κ/2π = 34 MHz, γ 1 /2π = 0.1 MHz γ ϕ = 0.1 MHz, η = 1/80 n crit. = 1/4λ 2 = 25

Measurement induced heat bath Mixing rates Downward rate : γ ( n) Upward rate : γ ( n) Heat bath with temperature T( n) = ( ω r/k B )/log(1 + 1/ n) σ z -1 1 0-1 0 5 10 15 Time [µs]

Measurement induced heat bath Mixing rates Downward rate : γ ( n) Upward rate : γ ( n) Heat bath with temperature T( n) = ( ω r/k B )/log(1 + 1/ n) σ z σ z -1 1 0-1 1 0-1 0 5 10 15 Time [µs] Increasing power

Measurement induced heat bath Mixing rates Downward rate : γ ( n) Upward rate : γ ( n) Heat bath with temperature T( n) = ( ω r/k B )/log(1 + 1/ n) σ z σ z σ z 1 0-1 1 0-1 1 0-1 0 5 10 15 Time [µs] Increasing power

The case of the transmon ----- C g Vg E J C J Energie [Arb. Units] EJ/4EC=0.1 EJ 0 0.2 0.4 0.6 0.8 1 Gate charge, n g = C gv g/2e Koch et al., Phys. Rev. A 76 042319 (2007)

The case of the transmon ----- C g C g Vg E J C J Vg C S Energie [Arb. Units] EJ/4EC=0.1 EJ 0 0.2 0.4 0.6 0.8 1 Gate charge, n g = C gv g/2e Koch et al., Phys. Rev. A 76 042319 (2007)

The case of the transmon ----- C g C g C g Vg E J C J Vg C S Vg C S Energie [Arb. Units] EJ/4EC=0.1 EJ 0 0.2 0.4 0.6 0.8 1 Gate charge, n g = C gv g/2e Koch et al., Phys. Rev. A 76 042319 (2007)

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