Violation of a Bell’s inequality in time with weak measurement SPEC CEA-Saclay IRFU, CEA, Jan A.Korotkov University of California, Riverside A. Palacios-Laloy F. Nguyen F. Mallet F. Ong P. Bertet D. Vion D. Esteve P. Senat P. Orfila Quantum world Macrorealism

AA ens x x a1a1 a2a2 b2b2 b1b1 x x BB The usual CHSH Bell’s inequality Classical hypothesis = local realism : the measured quantities exist locally already before measurement J.F. Clauser et al., Phys. Rev. Lett. 23 (1969)

AA ens x x a1a1 a2a2 b2b2 b1b1 x x BB The usual CHSH Bell’s inequality Quantum mechanically: Classical hypothesis = local realism : the measured quantities exist locally already before measurement J.F. Clauser et al., Phys. Rev. Lett. 23 (1969) A. Aspect et al., Phys. Rev. Lett. 49 (1982) Violation with polarized photons Violation with superconducting phase qubits: M. Ansmann, Nature 461, (24 Sept.2009)

z  A Bell’s inequality in time Classical assumptions = realism + non invasive measurement = macrorealism A. Leggett & A. Garg., Phys. Rev. Lett. 54 (1985)

z  A Bell’s inequality in time Classical assumptions = local realism + non invasive measurement = macrorealism A. Leggett & A. Garg., Phys. Rev. Lett. 54 (1985) This work (superconducting qubits) + arXiv: , M. E. Goggin et al.arXiv: M. E. Goggin Quantum mechanically: Excess of correlations is due to projective measurement

z  A Bell’s inequality in time with continuous weak measurement Quantum mechanically: t t V t+  t+2  t x x x Macrorealism:and R. Ruskov & A. Korotkov., Phys. Rev. Lett. 96 (2006) Can we test this inequality with an electrical circuit? same excess of correlations due to partial continuous measurement

resonator (  c ) = classical CW detector Cooper pair box = TLS (spin) A.Blais et al., Phys. Rev. A 69, (2004) A. Walraff et al., Nature 431, 162 (2004) J. Koch et al., Phys. Rev. A 76, (2007) YES WE CAN: continuous driving and monitoring of a transmon

resonator (  c ) = classical CW detector Cooper pair box = TLS (spin) A.Blais et al., Phys. Rev. A 69, (2004) A. Walraff et al., Nature 431, 162 (2004) J. Koch et al., Phys. Rev. A 76, (2007) YES WE CAN: continuous driving and monitoring of a transmon

The transmon circuit in the dispersive regime (reflection) cavity pull (readout) AC Stark shift Lamb shift

Readout from cavity pull L.O c frequency  /  c  

AC-stark back-action, dephasing and measurement time L.O c useful dephasing Ideally, discriminate, in

AC-stark back-action, dephasing and measurement time L.O c useful dephasing Other decoherence channels for the CPB:

5mm Implementation Nb antinode Si0 2 /Si chip CPW ~ 6 GHz 80  m C c ~100fF Q~200 2m2m Two J junctions Nb Si0 2 CPW 50  m Extra C 80  m CPB

18mK 4K 300K 600mK 10dB GHz 30dB 20dB DC-8 GHz 50  4-8 GHz G=40dB T N =2.5K VcVc G=56dB LO Fast Digitizer I Q 20dB dB 50MHz MW meas MW drive A(t)‏  (t)‏ COIL Experimental setup

Picture lab Experimental setup

Cavity, Cooper pair box, and coupling characterization MW q MW meas /c/c  

01 12 c Cavity, Cooper pair box, and coupling characterization  /  0 =  /2  MW q MW meas 01 = 5.3GHz  =1.69 rad  = 500MHz n crit = 31 working point c = GHz Q=193

|0> |1> T 1 = 280 ns T   = 365ns T 2 = 221ns relaxation dephasing CPB’s coherence characterization at zero measuring field

Measurement induced decoherence 1) Calibration of photon number from ac-Stark shift MW q MW meas spectro |0> see also D.I. Schuster et al, PRL 64 (2005)

Fit to Bloch solution with MW q MW meas tt |0> 0 to 15 % disagreement with theory for  , photon ( R = 2.5 to 20 MHz, n=0-5) T 1 =306 ns diffusive Rabi? quantum Zeno Measurement induced decoherence 2) Rabi oscillations perturbed by the resonator field

as expected for AC-Stark shift induced dephasing BUT depends on Rabi frequency See also D. Schuster et al., Phys. Rev. Lett. 94 (2005) Photon noise induced dephasing: check experiment

Coherent transient Important remarks about ensemble averaged Rabi oscillations Incoherent steady-state ?? zero signal due to ensemble average - Same signal as that of a driven classical spin submitted to the same noise. - No proof of quantumness - No violation of Bell’s inequality in time here (A. Korotkov and D. Averin, PRB 64 (2001)) Do not ensemble average! Average autocorrelation function or spectrum Spectrum is more practical to subtract noise and make bandwidth corrections

Frequency spectrum measurement : listening to the Rabi tone MW q ON MW meas ON x N~ times

Rabi tone in the frequency spectrum MW q ON MW meas ON x N~ times amplifier white noise (T N =3K) See also Il’ichev PRL 91 (2003)

MW q ON MW meas ON x N~ times amplifier white noise (T N =3K) See also Il’ichev PRL 91 (2003) Rabi tone in the frequency spectrum

 V/2 50% 100% 0% Signal calibration in  z units

Weak to strong measurement transition on a continuously monitored driven qubit Rabi tone in the frequency spectrum

A. Korotkov phenomenological theory (no fitting parameter) Numerical integration of master equation + quantum regression theorem Rabi tone in the frequency spectrum

Spectral density of noise in units of  z MW q ON MW meas ON diffusive Rabi quantum Zeno cavity cutoff Does it violate the Bell’s inequality in time? indep. meas. Rabi tone in the frequency spectrum

Experimental violation of Leggett-Garg’s inequality? S v /(  V/2) 2 Frequency (MHz) =1 photon cavity cutoff corrected measured spectrum theory (no fitting param) inverse Fourier transform

Experimental violation of Leggett-Garg’s inequality by 3.3 sigmas theory (no fitting param) Total error bar = Maximum systematic error (5% due to  V/2 and Q) + statistical error on the measured points of the spectrum

- Understanding of dephasing of Rabi oscillations during measurement - First violation of the Garg-Leggett inequality (weak measurement version) - Rabi spectra : from diffusive Rabi to Quantum Zeno regime. Semi-quantitative understanding Conclusion Quantum world Macrorealism Perspective Quantum feedback need nearly quantum limited linear phase- preserving amplifiers R. Ruskov & A. Korotkov., Phys. Rev. B 66 (2002)

« Qubit team » : A. Palacios-Laloy F. Nguyen M. Kende F. Mallet F. Ong P. Bertet D. Vion D. Esteve P. Senat P. Orfila SPEC CEA-Saclay Thank you ! DV PB FO FN FM DE AP A.Korotkov University of California, Riverside

Introduction: Continuous Measurement of a Driven «Two Level Atom » drive measure info back-action electrical circuit ? Diffusive Rabi oscillations Quantum Zeno effect 1 See J. Gambetta et al., Phys. Rev. A 77, (2008) density matrix conditional to V(t): Quantum trajectory

MW q OFF MW meas CW Resonator characterization

Calibration of photon number from ac-Stark shift MW q MW meas spectro |g> see also D.I. Schuster et al, PRL 64 (2005)

By-product: Low frequency noise in transmon frequency  /  0 =0  /  0 ~0.5 (preliminary) =10 10 min meas

Quantum Zeno Effect Fits : Bloch with Fit with the Bloch solutions R =2.5MHz R =5MHz R =10MHz R =20MHz A. Palacios-Laloy et al., in preparation

Fit with the Bloch solutions as expected for AC-Stark shift induced dephasing BUT depends on Rabi frequency D. Schuster et al., Phys. Rev. Lett. 94, (2005)

Rabi oscillations damping due to measurement time Ithier et al., PRB 72, (2005) FILTERING OF SHOT NOISE

Quantitative understanding of transient Rabi oscillations dephasing due to measurement Rabi oscillations damping due to measurement