Theory for investigating the dynamical Casimir effect in superconducting circuits

Transcription

1 Theory for investigating the dynamical Casimir effect in superconducting circuits Göran Johansson Chalmers University of Technology Gothenburg, Sweden International Workshop on Dynamical Casimir Effect Padova - Italy, 6-8th of June (2011)

2 Chalmers University of Technology, Gothenburg, Sweden Department of Microtechnology and Nanoscience (MC2)

3 Applied Quantum Physics Laboratory Theory Group at MC2 Hard Nano Mesoscopic Superconductivity Molecular Electronics Graphene Electronics Quantum Computation

4 Quantum Device Physics Laboratory Hard Nano MC2 Experimental Groups Per Delsing Chris Wilson Tauno Palomaki, IoChun Hoi, M. Sandberg, F. Persson,

5 Physical Review Letters 103, (2009), Physical Review A 82, (2010) Editors suggestion and highlighted at physics.aps.org (Theory for) Dynamical Casimir Effect in a Superconducting Coplanar Waveguide Robert Johansson, Göran Johansson, Chris Wilson and Franco Nori a RIKEN - Chalmers collaboration

6 Physical Review Letters 103, (2009), Physical Review A 82, (2010) Editors suggestion and highlighted at physics.aps.org Experimental progress in Chris Wilson s talk - in about 45 minutes (Theory for) Dynamical Casimir Effect in a Superconducting Coplanar Waveguide Robert Johansson, Göran Johansson, Chris Wilson and Franco Nori a RIKEN - Chalmers collaboration

7 Quantum Mechanics and Electrical Circuits An LC-oscillator in the microwave regime A QM harmonic oscillator: Quantized Amplitudes Zero point motion

8 Quantum Mechanics and Electrical Circuits f=2.4 GHz --> hf / k B = K An LC-oscillator in the microwave regime A QM harmonic oscillator: Quantized Amplitudes Zero point motion

9 Quantum Mechanics and Electrical Circuits f=2.4 GHz --> hf / k B = K Low temperatures needed! An LC-oscillator in the microwave regime ~3 MSEK or (300 K --> 6.3 THz) ~450 k$ A QM harmonic oscillator: Quantized Amplitudes Zero point motion

10 Quantum Mechanics and Electrical Circuits Low temperatures! Resistance/dissipation gives level broadening -> Minimize dissipation! An LC-oscillator in the microwave regime A QM harmonic oscillator: Quantized Amplitudes Zero point motion

11 Quantum Mechanics and Electrical Circuits Low temperatures! Resistance/dissipation gives level broadening -> Minimize dissipation! An LC-oscillator in the microwave regime In linear circuits, averages follow classical equations of motion, i.e. Kirchoff s rules. Quantum effects (photons) only in fluctuations (noise). -> Nonlinearity needed! A QM harmonic oscillator: Quantized Amplitudes Zero point motion

12 The Josephson Junction Tunnel junction between superconductors Current determined by phase difference of wave function on each side A nonlinear dissipationless (almost) inductor S I S ϕ 1 ϕ 2 Inductor: V = L I I = I 0 sin Φ Φ 0 I = 1 L Vdt = Φ L Φ 0 = h 2e

13 Electrical Circuits -> Classical Mechancis Simplest circuit: A current biased Josephson junction Josephson junction Kirchoff s rules <-> Dynamics of a fictitious phase particle with coordinate ϕj and mass CJ moving in a tilted washboard potential

14 Classical Mechanics -> Quantum Mechanics We derive a Lagrangian in terms of the phase that gives the Kirchoff s rules as equations of motion. Kinetic energy Potential energy Lagrangian Canonical momentum Hamiltonian Canonical quantization

15 Classical Mechanics -> Quantum Mechanics We derive a Lagrangian in terms of the phase that gives the Kirchoff s rules as equations of motion. Kinetic energy Potential energy Lagrangian Canonical momentum Hamiltonian Canonical quantization Classical -> Quantum

16 Quantum Mechanics and Electrical Circuits The quantum description is relevant. MQT experiment: Devoret, Martinis, Clarke, PRL (1985) Recently at Chalmers, MQT in HTc superconductor: Bauch et al, Science (2006) First superconducting qubit Today algorithms with 3 qubits. Also artificial atoms and circuit QED.

17 The SQUID - a Tunable Inductance The Superconducting Quantum Interference Device: a tunable Josephson junction ϕ ext = 2πΦ ext Φ 0 E J1 cos ϕ 1 + E J2 cos ϕ 2 L J = SQUID inductance: Φ0 2π A tunable dissipationless inductance Potential energy 2 1 2E J cos ϕ ext 2 Single valued phase, Ampère s law, ϕ ext + ϕ 1 ϕ 2 =0 (Fluxoid quantization) EJ1 2 + E2 J2 +2E ϕ1 + ϕ 2 J1E J2 cos ϕ ext cos (2E J cos ϕ ext 2 2 ) cos ϕ s

18 A Transmission Line with tunable boundary (SQUID) Quantum Network Analysis: Wallquist et al, PRB 2006 Yurke and Denker PRA 1984 Devoret, Les Houches 1997 =0

19 A Transmission Line with tunable boundary (SQUID) Quantum Network Analysis: Wallquist et al, PRB 2006 Yurke and Denker PRA 1984 Devoret, Les Houches 1997 SQUID Boundary Condition C 2 Φ(0,t) t 2 + 4π2 E J (t) Φ 2 0 Φ(0,t)+ 1 Φ(0,t) x=0 L 0 x =0

20 A Transmission Line with tunable boundary (SQUID) Quantum Network Analysis: Wallquist et al, PRB 2006 Yurke and Denker PRA 1984 Devoret, Les Houches 1997 SQUID Boundary Condition C 2 Φ(0,t) t 2 + 4π2 E J (t) Φ 2 0 Φ(0,t)+ 1 Φ(0,t) x=0 L 0 x =0 Specify in-field and calculate the out-field.

21 Fixed EJ - The effective length of the SQUID

22 Fixed EJ - The effective length of the SQUID

23 A Transmission Line with tunable boundary (SQUID) A photon trombone, Applied Physics Letters (2008) Experiment: Sandberg, Wilson, Persson, Duty, Delsing Theory: Johansson, Shumeiko

24 A Transmission Line with tunable boundary (SQUID) A photon trombone, Applied Physics Letters (2008) Experiment: Sandberg, Wilson, Persson, Duty, Delsing Theory: Johansson, Shumeiko

25 A Transmission Line with tunable boundary (SQUID) Tune 330 MHz in 10 ns! Effective velocity: A photon trombone, Applied Physics Letters (2008) Experiment: Sandberg, Wilson, Persson, Duty, Delsing Theory: Johansson, Shumeiko ~5%! (More details in Chris Wilson s talk)

26 Harmonic Drive Small Amplitude Drive

27 Harmonic Drive Small Amplitude Drive Identical to single-sided single mirror spectrum

28 The dynamical Casimir effect Single Oscillating Mirror Moore (1970), Fulling-Davies (1975) Recent review: Dodonov (2010)

29 The dynamical Casimir effect Single Oscillating Mirror Moore (1970), Fulling-Davies (1975) Recent review: Dodonov (2010) Lambrecht, Jaekel, Reynaud, PRL (1996) Output power Photon spectrum ω/ω

30 The dynamical Casimir effect Single Oscillating Mirror Moore (1970), Fulling-Davies (1975) Recent review: Dodonov (2010) Description of experimental confirmation in 20 minutes Lambrecht, Jaekel, Reynaud, PRL (1996) Output power Photon spectrum ω/ω

31 The effect of finite temperature

32 The effect of finite temperature ~10 5 photons per second in 100 MHz bandwidth

33 The effect of finite temperature ~10 5 photons per second in 100 MHz bandwidth Power comparable to 2mK noise temperature Experimentally detectable

34 The dynamical Casimir effect in a coplanar waveguide Case Frequency (Hz) Amplitude (m) Maximum velocity (m/s) Photon production rate (# photons / s) moving a mirror by hand ~2e-18 nanomechanical oscillator 1e+9 1e-9 2 ~2e-9 SQUID in coplanar waveguide 18e+9 ~1e-4 ~4e6 ~2e5 Photon production rate: Lambrecht et al., PRL

35 Early version of the Experiment a proper update in 20 minutes Output 5.5 mm meandering waveguide Arsalan Pourkabirian (PhD student), Chris Wilson, Per Delsing Flux drive

36 Early version of the Experiment a proper update in 20 minutes Output 5.5 mm meandering waveguide Arsalan Pourkabirian (PhD student), Chris Wilson, Per Delsing Flux drive SQUID Point like boundary condition

37 Effect of weak resonances (It is not easy to make a transmission line completely flat over 10 GHz) J. R. Johansson, G. Johansson, C. M. Wilson, F. Nori, PRA 82, (2010).

38 Effect of weak resonances (It is not easy to make a transmission line completely flat over 10 GHz) J. R. Johansson, G. Johansson, C. M. Wilson, F. Nori, PRA 82, (2010).

39 Pairs of Photons - Parametric down-conversion Photons are created in pairs ω 1 + ω 2 = ω d n out (ω 1 )=n out (ω 2 ) Q(ω 1 )Q(ω 2 ) J. R. Johansson, G. Johansson, C. M. Wilson, F. Nori, PRA 82, (2010).

40 Two weak resonances symmetric in frequency Photons are created in pairs ω 1 + ω 2 = ω d n out (ω 1 )=n out (ω 2 ) Q(ω 1 )Q(ω 2 ) J. R. Johansson, G. Johansson, C. M. Wilson, F. Nori, PRA 82, (2010).

41 Photon Correlations a(ω +)a(ω ) = i v eff c ω+ ω ω ± = ω d 2 ± ω

42 Two-mode Squeezing Ω = ω d 2 Frequency of Local Oscillator θ Phase of Local Oscillator (compared to drive) Walls & Milburn, Quantum Optics J. R. Johansson, G. Johansson, C. M. Wilson, F. Nori, PRA 82, (2010).

43 Two-mode squeezing with an amplifier Measure simultaneously both quadrature voltages at both frequencies: I ± ω ± (a(ω + )+a(ω ) ) Q ± i ω ± (a(ω + ) a(ω ) ) Two-mode squeezing: 2 ω ± = ω d 2 ± ω I + I Q + Q I Q I 2 + Q 2 Symmetries: I + I = Q + Q I + Q = I Q + No single-mode squeezing: I 2 ± = Q 2 ± Caves & Schumaker PRA (1985).

44 Photon Correlations - Bunching J. R. Johansson, G. Johansson, C. M. Wilson, F. Nori, PRA 82, (2010).

45 Conclusions and outlook SQUID = Tunable length mirror Effective velocities close to the speed of light Boundary condition identical to dynamical Casimir effect Look for truly broadband radiation! Look for broadband two-mode squeezing correlations, symmetric around half the drive frequency! DCE : Physical Review Letters 103, (2009) Long paper, correlations, squeezing, PRA 82, (2010) Photon Trombone : Sandberg et al, Applied Physics Letters 92, (2008)

46

47 The dynamical Casimir effect