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
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 Energy ... ω 01 = ω a Two-levels system Hamiltonian H = ω a „ 1 « 0 2 σ z σ z = 0 − 1 Maxime Boissonneault Universit´ e de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion Atom and cavity x z y Energy ... L ω 01 = ω a Cavity Hamiltonian „ k a k + 1 « X a † H = ω k 2 Two-levels system Hamiltonian k H = ω a „ 1 « 0 2 σ z σ z = 0 − 1 Maxime Boissonneault Universit´ e de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion Atom and cavity x z y Energy ... L ω 01 = ω a Cavity Hamiltonian „ k a k + 1 « X a † H = ω k 2 Two-levels system Hamiltonian k H = ω a „ 1 « 0 2 σ z σ z = 0 − 1 Maxime Boissonneault Universit´ e de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion Atom and cavity ω r = ω 1 ω 2 Response κ = ω r / Q Energy ... Input frequency, rf ω 01 = ω a Cavity Hamiltonian „ k a k + 1 « X a † H = ω k 2 Two-levels system Hamiltonian k Single-mode : H = ω a „ 1 « 0 2 σ z σ z = 0 − 1 H = ω r a † a Maxime Boissonneault Universit´ e de Sherbrooke
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 E ≈ g ( a † + a ) σ x ≈ g ( a † σ − + aσ + ) H I = − � D · � r ω g ( z ) = − d 0 sin kz V ǫ 0 Maxime Boissonneault Universit´ e de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion Cavity QED g Atom-cavity interaction E ≈ g ( a † + a ) σ x ≈ g ( a † σ − + aσ + ) H I = − � D · � r ω g ( z ) = − d 0 sin kz V ǫ 0 Jaynes-Cummings Hamiltonian H = ω a 2 σ z + ω r a † 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 Classical Hamiltonian H = 4 E C ( n − n g ) 2 − E J cos δ - - - - - C g C g n e 2 n g = C g V g V g E C = 2( C g + C J ) , E J E C J 2 e J V g C J E J = I 0 Φ 0 2 π Maxime Boissonneault Universit´ e de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion Charge qubit and coplanar resonator Classical Hamiltonian H = 4 E C ( n − n g ) 2 − E J cos δ - - - - - C g C g n e 2 n g = C g V g V g E C = 2( C g + C J ) , E J E C J 2 e J V g C J E J = I 0 Φ 0 2 π Quantum Hamiltonian 4 E C ( n − n g ) 2 | n � � n | X H = n E J X − 2 ( | n � � n + 1 | + h . c . ) n Restricting to n g ∈ [0 , 1] : H = ω a σ z / 2 Shnirman, Sch¨ on 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) Maxime Boissonneault Universit´ e de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion Charge qubit and coplanar resonator Classical Hamiltonian H = 4 E C ( n − n g ) 2 − E J cos δ - - - - - C g C g n e 2 n g = C g V g V g E C = 2( C g + C J ) , E J E C J 2 e J V g C J E J = I 0 Φ 0 2 π Quantum Hamiltonian 4 E C ( n − n g ) 2 | n � � n | X H = Energie [Arb. Units] E J /4E C =0.1 n E J X − 2 ( | n � � n + 1 | + h . c . ) n Restricting to n g ∈ [0 , 1] : H = ω a σ z / 2 E J Shnirman, Sch¨ on and Hermon, Phys. Rev. Lett. 79 2371–2374 (1997) Bouchiat et al ., Physica Scripta T76 165-170 (1998) 0 0.2 0.4 0.6 0.8 1 Nakamura, Pashkin and Tsai, Nature (London) 398 786 (1999) Gate charge, n g = C g V g /2e Maxime Boissonneault Universit´ e de Sherbrooke
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 L r C r Classical Hamiltonian H = Φ 2 + 1 2 C r V 2 2 L r s 1 ω r = L r C r Maxime Boissonneault Universit´ e de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion Charge qubit and coplanar resonator L r C r Quantum Hamiltonian Classical Hamiltonian r ω r r ω r ( a † + a ) , ( a † − a ) H = Φ 2 + 1 V = Φ = i 2 C r V 2 2 C r 2 L r 2 L r s „ a † a + 1 « 1 H = ω r ω r = 2 L r C r Quantum Fluctuations in Electrical Circuits, M. H. Devoret, Les Houches Session LXIII, Quantum Fluctuations p. 351-386 (1995). Maxime Boissonneault Universit´ e de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion Circuit QED Measurement output C g E J C J Qubit control Atom: super conducting and readout charge qubit ~ 10 GHz Cavity: super conducting 1D transmission line resonator Maxime Boissonneault Universit´ e de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion Circuit QED Measurement output C g E J C J Qubit control Atom: super conducting and readout charge qubit ~ 10 GHz Cavity: super conducting 1D transmission line resonator Parameters Blais et al ., Phys. Rev. A 69 062320 (2004) g : Qubit-cavity interaction Wallraff et al ., Nature 431 162 (2004) ω a : Qubit frequency Wallraff et al ., Phys. Rev. Lett. 95 060501 (2005) ω r : Resonator frequency Leek et al ., Science 318 1889 (2007) ∆ = ω a − ω r : Detuning Schuster et al ., Nature 445 515 (2007) H = ω a Houck et al ., Nature 449 328 (2007) 2 σ z + ω r a † a + g ( a † σ − + aσ + ) Majer et al ., Nature 449 443 (2007) Maxime Boissonneault Universit´ e de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion Circuit QED C g E J C J Parameters Blais et al ., Phys. Rev. A 69 062320 (2004) g : Qubit-cavity interaction Wallraff et al ., Nature 431 162 (2004) ω a : Qubit frequency Wallraff et al ., Phys. Rev. Lett. 95 060501 (2005) ω r : Resonator frequency Leek et al ., Science 318 1889 (2007) ∆ = ω a − ω r : Detuning Schuster et al ., Nature 445 515 (2007) H = ω a Houck et al ., Nature 449 328 (2007) 2 σ z + ω r a † a + g ( a † σ − + aσ + ) Majer et al ., Nature 449 443 (2007) Maxime Boissonneault Universit´ e de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion The linear dispersive limit Jaynes-Cummings H = ω r a † a + ω a σ z 2 + g ( a † σ − + aσ + ) Small parameter λ = g/ ∆ Measurement output Qubit control Atom: super conducting and readout charge qubit ~ 10 GHz Cavity: super conducting 1D transmission line resonator Maxime Boissonneault Universit´ e de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion The linear dispersive limit Jaynes-Cummings Linear dispersive H D = ( ω a + χ ) σ z H = ω r a † a + ω a σ z 2 + g ( a † σ − + aσ + ) 2 + ( ω r + χσ z ) a † a Small parameter λ = g/ ∆ Lamb shift ( χ = gλ = g 2 / ∆ ) Measurement Stark shift or cavity pull output n ≪ n crit . , where n crit . = 1 / 4 λ 2 . Valid if ¯ Qubit control Atom: super conducting and readout charge qubit ~ 10 GHz Cavity: super conducting 1D transmission line resonator 2 CP n (arb. units) 2 χ κ Phase κ o i s Δ ~ 2 π 1GHz s i m s n a r -2 χ κ T ω r − CP ω r + CP ω a Maxime Boissonneault Universit´ e de Sherbrooke
Introduction Beyond linear dispersive Results Conclusion The linear dispersive limit Jaynes-Cummings Linear dispersive H D = ( ω a + χ ) σ z H = ω r a † a + ω a σ z 2 + g ( a † σ − + aσ + ) 2 + ( ω r + χσ z ) a † a Small parameter λ = g/ ∆ Lamb shift ( χ = gλ = g 2 / ∆ ) Measurement Stark shift or cavity pull output n crit . = 1 / 4 λ 2 . Valid if ¯ n ≪ n crit . , where Qubit control Atom: super conducting and readout charge qubit ~ 10 GHz Cavity: super conducting 1D transmission line resonator 2 CP n (arb. units) 2 χ κ Phase κ o i s Δ ~ 2 π 1GHz s i m s n a r -2 χ κ T ω r − CP ω r + CP ω a Maxime Boissonneault Universit´ e de Sherbrooke
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