arXiv:1005.3435v1 [quant-ph] 19 May 2010 Experimental violation of a Bell���s inequality in time with weak measurement Agustin Palacios-Laloy1, Fran��ois Mallet1, Fran��ois Nguyen1, Patrice Bertet1*, Denis Vion1, Daniel Esteve1, and Alexander Korotkov2 1Quantronics Group, Service de Physique de l�����tat Condens�� (CNRS URA 2464), DSM/IRAMIS/SPEC, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, France and 2Deparment of Electrical Engineering, University of California, Riverside, CA 92521-0204, USA Abstract The violation of J. Bell���s inequality with two entangled and spatially separated quantum two- level systems (TLS) is often considered as the most prominent demonstration that nature does not obey ���local realism���. Under different but related assumptions of ���macrorealism���, plausible for macroscopic systems, Leggett and Garg derived a similar inequality for a single degree of freedom undergoing coherent oscillations and being measured at successive times. Such a ���Bell���s inequality in time���, which should be violated by a quantum TLS, is tested here. In this work, the TLS is a superconducting quantum circuit whose Rabi oscillations are continuously driven while it is continuously and weakly measured. The time correlations present at the detector output agree with quantum-mechanical predictions and violate the inequality by 5 standard deviations. PACS numbers: 74.50,03.65,82.25 1

Introduction The violation of J. Bell���s inequality [1, 2] is the most prominent example of a situation where the predictions of quantum mechanics are incompatible with a large class of classical theories. In the early 1980s, Aspect and coworkers [3] brought an experimental proof of this violation using pairs of spatially-separated polarization-entangled photons. By demon- strating an excess of correlations between the polarizations measured on the two photons of a pair, they ruled out descriptions of nature satisfying the very general conditions known as local realism. This striking finding also contributed to transform the so-called quantum weirdness into a useful resource for information processing. Shortly after, quantum cryptog- raphy protocols and quantum algorithms exploiting entanglement were indeed proposed [4]. Following a reasoning similar to that of Bell, Leggett and Garg derived in 1985 an inequality that can be seen as a ���Bell���s inequality in time���, which applies to any single macroscopic system measured at successive times [5] and fullfiling the assumptions of macrorealism: (A1) the system is always in one of its macroscopically distinguishable states, and (A2) this state can be measured in a non invasive way, i.e. without perturbing the subsequent dynamics of the system. Quantum mechanics however contradicts both assumptions, which can lead to an excess of correlations between subsequent measurements and to a violation of this inequal- ity. Ruskov and coworkers [6] then adapted the inequality to the situation where a two-level systems (TLS) is continuously and weakly monitored during its coherent oscillations. Us- ing such a weak monitoring, we report here an experimental test of a Bell���s inequality in time (see also the recent works [7, 8]), yielding results in excellent agreement with simple quantum-mechanical predictions and in contradiction with a large class of macrorealistic models. Bell���s inequalities in space and in time We start by briefly recalling the experimental protocol of the usual CHSH test [2] of Bell���s inequalities (see Fig. 1a). It consists in identically preparing many times a pair of quantum TLS in a maximally entangled state such as |�����angbracketright = (|������angbracketright ��� |������angbracketright)/ ��� 2. Each member of the pair is then distributed to two observers A and B, who perform projective measurements of the TLS spin ��i A,B = ��1 along one of two directions ai (i = 1, 2) for A and 2

g86A A B A g110 g112 g16 g112 g110B A g84 g84 g84 a1 a2 b2 b1 g86B yens g11g84 g12 2? g54 g100 yens - - x x g11 g12 2? g54 g84 g100 x V yens yens - + B x z t g86 t t+g87 t+2g87 yt + - + g11g87g12 1? g100 fLG x x x yt yt Figure 1: Comparison between two thought experiments which test the usual CHSH Bell���s inequality and the Bell���s inequality in time. (A) CHSH inequality: two maximally entangled spins ��A and ��B are sent to two spatially separated observers A and B. Each of the observers measures with pick-up coils his spin along one of two possible directions (a1 and a2 for A, and b1 and b2 for B ) the four directions make angles �� as depicted. By repeating this experiment on a statistical ensemble, a linear combination �� of the four possible correlators between measurements on a spin pair is computed. Local realism requires ���2 ��� �� ��� 2, while quantum mechanics predicts �� = 2 ��� 2 for �� = 45��. (B) Bell���s inequality in time with weak measurement: a single spin �� undergoing coherent oscillations at frequency ��R is continuously measured with a pick-up coil coupled to it so weakly that the time for a complete projective measurement would be much longer than the period of oscillations TR = 2��/��R. From the noisy time trace recorded in the steady state, one computes a linear combination fLG of the three time-averaged-correlators between the readout outcomes at three times separated by ��. Macrorealism requires fLG ��� 1 for any ��, while quantum mechanics predicts fLG = 1.5 at �� = TR/6. 3

bi for B, with these directions forming angles (a1, b1) = (b1, a2) = (a2, b2) ��� �� as shown in Fig. 1a. The two observers then combine all their measurements to compute the Bell sum ��(��) = ���K11 + K12 ��� K22 ��� K21 of the correlators Kij(��) = angbracketleft��i A��j Bangbracketright. The Bell���s theorem, based on a simple statistical argument, states that according to all local realistic theories ��� 2 ��� ��(��) ��� 2. (1) However, standard quantum mechanics predicts that this inequality is violated, with a max- imum violation ��(�� = ��/4) = 2 ��� 2. Many experimental tests, and in particular those performed by A. Aspect [3] have verified this violation [11, 12]. While quantum entanglement between two spatially separated TLS is at the heart of the previous violation, Leggett and Garg proposed a similar inequality [5] holding for a single degree of freedom ���1 ��� z(t) ��� 1 fulfilling the assumptions of macrorealism (z(t) defined at any time, and measurable with no perturbation). Using a simple arithmetic argument �� la Bell, they showed that z(t0)z(t1) + z(t1)z(t2) ��� z(t0)z(t2) ��� 1 (2) for all {ti}. Consequently, an observer measuring z on many identical systems, either at t0 and t1 = t0 + ��, or at t0 and t2 = t0 + 2��, or at t1 and t2 should find ensemble-averaged correlators Kij(t0, ��) = angbracketleftz(ti)z(tj)angbracketright (for i, j = 0, 1, 2, with i j) satisfying the Leggett- Garg���s inequality: fLG(t0, ��) ��� K01 + K12 ��� K02 ��� 1. (3) Quantum mechanics on the other hand predicts that, applied to the case of a quantum TLS undergoing coherent oscillations at frequency ��R , this inequality is violated for well- chosen values of ��, with maximum violation fLG(t0, �� = ��/3��R) = 1.5 independent of t0. Here, the delay �� between successive measurements plays the role of the angle �� between the measurement directions in the Bell���s inequality (1), justifying the nickname ���Bell���s inequality in time���. The excess of correlations predicted by quantum mechanics, compared to the macrorealistic case, can be interpreted as resulting from the projection of the TLS state on a ��z eigenstate induced by the first measurement. As shown in [6], the very same conclusions also hold if the TLS undergoing coherent oscillations is continuously and weakly monitored along ��z (see Fig. 1B) instead of being 4

projectively measured at well-defined times. The detector now delivers an output signal V (t) = (��V/2)z(t) + ��(t) proportional to z(t) with some additional noise ��(t). Macrorealism implies that the dynamics of the system at time t + �� is fully uncorrelated with the detector noise at time t so that angbracketleft��(t)z(t + ��)angbracketrightt = 0. The detector���s output correlation function K(��) = angbracketleftV (t)V (t + ��)angbracketrightt/(��V/2)2 is then simply equal to angbracketleftz(t)z(t + ��)angbracketrightt. By averaging inequality (2) over t0 in the steady-state, the Bell���s inequality in time (3) becomes fLG(��) ��� 2K(��) ��� K(2��) ��� 1, (4) and should be violated by a quantum TLS in the very same way as discussed above. Here the violation is however not due to a strong projection of the TLS wavefunction induced by measurements at well-defined times of its evolution, but rather to the continuous partial projection caused by the measurement during the TLS coherent evolution, which reinforces correlations between the detector output at successive times. Experimental setup Our experimental setup (see Fig. 2A and supplementary informationA) for probing in- equality (4) closely implements the proposal discussed above, making use of the possibilities offered by the so-called circuit quantum electrodynamics (circuit-QED) architecture [13, 14] where a superconducting artificial TLS is coupled to a superconducting coplanar waveguide resonator. The TLS consists here of the two lowest energy states g and e of a modified Cooper-pair box of the transmon type [17, 18]. These two states can be regarded as ���macro- scopically distinguishable��� because the dipole moment of the g ��� e transition is of the order of 104 atomic units. On the other hand, the only degree of freedom of this system is the phase difference between the superconducting order parameters on both sides of the Joseph- son junction forming the Cooper-pair box, conjugate to the number of Cooper pairs passed through the junction this phase is a collective variable whose degree of macroscopicity is still under debate [15, 16]. The TLS transition frequency is ��ge/2�� = 5.304 GHz, below the resonance frequency ��c/2�� = 5.796 GHz of the resonator to which it is capacitively coupled for its measurement. Two microwave sources Vd and Vm drive and measure the TLS at frequencies ��ge and ��c, respectively. In order to continuously monitor the induced Rabi oscillations up to a few 5