A loophole-free Bell test

A loophole-free Bell test Bas Hensen February 15, 2013 In 1964, John Bell showed that any local realistic theory disagrees with the predictions of quantum mechanics on measurements of distant objects. Local realism is the idea that objects have definite properties whether or not they are measured, and that measurements of these properties are not affected by events taking place sufficiently far away 1. If quantum mechanics is correct, it is inherently non-local. Whether quantum mechanics is correct, and we should indeed give up local realism, is one of the major open questions in modern physics. Both because of its fundamental implications to the understanding of our reality, but in particular because of a whole range of applications of quantum non-locality, some of which, such as quantum key distribution, are already exploited commercially Proposal: Fundamental proof of non-locality; A loophole-free Bell test Introduction: Bell s inequality, loopholes and previous experiments In 1935, Einstein Podolsky Rosen 3 described a famous thought-experiment: Two spin ½ particles are produced by a source, in the entangled state: 1 ( + A B A B) 2 Both particles leave the source in opposite directions, as in figure 1 below. y y x Source x Alice Bob L Figure 1: Schematic setup for a Bell test: A magic source prepares two spin systems in an (entangled) state, each flying in opposite directions. After the two systems are separated by a distance L, their spin is measured locally, by for instance a Stern Gerlach device. After a while, the particles spins are independently measured (for instance by a Stern Gerlach apparatus), in an arbitrary bases. According to Quantum Theory, the two separate spins do not have definite properties until they are measured, and measurements of one spin instantaneously affect measurements of the other spin, which can be very far away. Einstein, Podolsky, Rosen maintained that this could not be, and considered the predictions of quantum mechanics evidence that the theory was incomplete, and a more detailed description must lie underneath. It took until 1964 for John Bell to prove in a remarkably simple way that such an extension to quantum mechanics, based on local variables, is not possible. In particular, in the form of the experiment described above, he showed that any local realistic theory must satisfy the inequality 4 S : = C( ϕa, ϕb) + C( ϕa, ϕb' ) + C( ϕa', ϕb) C ( ϕa', ϕb' ) 2. (1) The correlation is defined by: C ( XY, ) = p ( XY, ) + p ( XY, ) p ( XY, ) p ( XY, ), (2) + + ++

where {+,-} is the result of the spin measurement of Alice, Bob, measuring along angles a,b,a,b in the x,y plane. According to quantum mechanics, the experiment above should resuld in the value S = 2 2, thus ruling out local realistic theories. The first experiment to demonstrate violation of Bell s inequality was conducted by Freidman and Clauser in 1972 5, using photons as entangled particles. All later experiments aimed to close two important loopholes in this first experiment: 1. The locality loophole 6 : The distance between Alice and Bob should be such that they can complete the measurement of their system before any signal regarding their measurement can travel between them. In particular, if the time it takes Alice and Bob to measure their system is T, then the distance L between Alice and Bob should be L > T c, where c is the speed of light. 2. The detection loophole 7 : In the proof of Bell s inequalities it is necessary that the added probability of all measurement outcomes of Alice and Bob sum to one: p+ ( XY, ) + p + ( XY, ) + p++ ( XY, ) + p ( XY, ) = 1. This sets a limit on the detection efficiency for Alice, Bob: whenever there is a particle in their detector, they should be able to assign a measurement value to it. If they only detect a subset of the particles, this allows the possibility that the subset of detected events agrees with quantum mechanics even though the entire set satisfies Bell's inequalities. Most experiments so far involved coincidence clicks of photons, which have low detection efficiencies. These experiments therefore had to assume that the detected events represent the entire ensemble; a fair-sampling hypothesis. Alain Aspect 8 was the first to provide experimental data aimed to close the locality loophole, and many later experiments confirmed his data 9. Recently, in 2001, Rowe et al. 10 were the first to close the detection loophole, by using trapped ions as entangled pair. However the ions were spaced only 5 um apart, so that the locality loophole remained open. A paper currently on the Arxiv (2012), describes an experiment by Giustina et al. 11, that achieves a Bell inequality violation using single photons with near unit efficiency single photon detectors, that doen not rely on the fail sampling hypothesis. A third, less often discussed loophole is the freedom of choice loophole: the chosen angles for the local manipulation could somehow be known to the source. To overcome this loophole the local manipulation angle should be chosen at random before any information regarding the choice can be transmitted between Alice and Bob. So far there has been no experiment that simultaneously closes both main loopholes, nor one that closes all three. Objective: A loophole free Bell test using entangled nitrogen-vacancy centre electronic spins. Intermediate goals, originating from the discussion below, are: 1. Improve single shot readout speed and fidelity. 2. Show violation of Bell s inequality (close the detection loophole) with the current setup. 3. (Down conversion of a single NV ZPL photon to telecom wavelength). 4. (Spin-spin entanglement with down converted photons). Why NV centres? Matter qubits provide the most straightforward way to eliminate the detection loophole 10 as a measurement outcome can be obtained with a probability approaching unity. The recently demonstrated long-distant spin-spin entanglement 2 demonstrated in our group makes use of time-bin spin-photon entanglement that is particularly robust to being transported over long distances in a fibre. This stands in contrast with spin photon entanglement between other photon degrees of freedom, as used in similar experiments with trapped atoms that used polarisation properties 12 or small frequency differences 13.

A recent violation of Bell s inequalities in our group 14 using two entangled nuclear spin qubits demonstrates that the readout is good enough to close the detection loophole. Readout of the NV centre can be done faster than 10 us, allowing separation of the setups over moderate distances less than three kilometres. (Current state of the art single shot readout for trapped atoms is 5 10 times slower 15, although a proposed method by state-selective ionisation is aimed to achieve readout times less than 1 us 16.) Practical Approach o Setup: The setup is essentially the same as used in Bernien et al. 2, where one of the two cryostats has to be relocated to a remote lab, a distance L away. This then automatically requires some additional changes: o The excitation pulses will be generated locally by two independent lasers. o One or both of the fibre arms of the beam splitter needs to be lengthened to match the distance between the setups. Although the asymmetric version shown in figure 2a is easier to implement in practice, due to large asymmetric losses in the fibres, it might be beneficial to use the symmetric version in (b). See also the feasibility section below. Figure 2: Sketch of two possible configurations for the Bell test setup: In (a), setup A, beam-splitter and detectors are located in a single lab, while setup B is located a distance L away. In (b), both setups A and B are in separate labs, and a third lab contains the beam-splitter and detectors. o Measurement sequence: To perform the Bell test, first an entangled state is created between the two remote NV centre electronic spins. This entanglement generation may be probabilistic, with an, in principle, arbitrarily low success probability. This forms the Magic box in figure 3. Once a successful entanglement event has been heralded, a Bell test measurement can be started: A random rotation setting must be chosen for both systems, the rotation must be performed, and finally the system must be read out. The required distance L between the NV s is set by the time T meas required to perform a single shot readout, and the time T rotate required to perform a random rotation of the measurement basis. We then require L > ( T meas + T rotate) c.

Figure 3: Measurement sequence for an implementation of the Bell test (adapted from Rowe et al. 10 ). Shown is a magic box that prepares two systems by some (not necessarily known) method. The two systems are separated by a distance L, and a randomly chosen, local manipulation is applied to each, characterised by some angle φ. Each system is then measured locally. The time between the decision of the manipulation and the end of the measurement has to be such that no information regarding the manipulation can reach the other system, i.e. the entire manipulation/measurement sequence of one setup is space-like separated from the other systems sequence. o Random rotation: To select a random rotation setting, a commercially available random number generator can be used 17. These generators can generate a (quantum-) random bit within 100 nanoseconds. The bit can then be used perform a pre-programmed rotation of the NV centre. This can be done using, for instance, the event-jump function of the Tektronics Arbitrary Waveform Generator (~200-500 ns delay), or by directly switching between two different I/Q amplitudes using a fast switch (~30-100 ns delay). Both options are readily available, using currently available devices. o Remote setup placement: The remote location of one of the setups requires an additional physical laboratory. Depending on the attainable readout duration, the distance L, and therefore the distance to the remote lab, needs to be between about 1 and 3 km. A possible location might be realisable inside the Reactor Institute Delft, a distance of 1.3 km from the existing laboratory(see figure 4). If the configuration is chosen with similar arm lengths for A,B, as in figure 2(b), an additional measurement location is required, about halfway between the two laboratories. This location only needs minimum resources: it would comprise only the beam splitter, two APD s, and an electrical-to-fiberoptic converter, to send back the detector clicks along secondary fibres.

Figure 4: Example locations for the two setups: One is at the current location in the B-wing of the physics building located at Lorentzweg 1, the other in the laboratory space of the Reactor Institute Delft located at Mekelweg 15. The direct distance is 1.3 km. Feasibility o Required entanglement and readout fidelities: To violate Bell s inequalities, both the degree of entanglement between the two systems, as well as the readout fidelity should be above a certain threshold. In Bernien et al. 2 a fidelity overlap with the maximally entangled anti-symmetric Bell state was estimated to be around F Ψ - 73%. For perfect readout, this results in correlations above the classical threshold of S = 2. However, in Bernien et al. 2 the readout fidelity was prohibitively low. In figure 5, the dependence of S on entanglement visibility V = C[0,0] (corresponding to the overlap with a maximally entangled state), and on readout fidelity of the m s = 0 state is show. Here it is assumed that the readout fidelity of m s = -1 is unity (in Bernien et al. 2 they are above 99.9%), and the readout of m s = 0 is identical for both systems. As indicated by the red dot, improvement of either the entanglement fidelity and/or the readout fidelity is necessary.

Bernien et al. Bell s inequality threshold Figure 5: Contour plot of the S value from equation (1), as a function of the entanglement fidelity (overlap with the Bell state), and a function of the readout fidelity of the m s = 0 state. In the plot it is assumed that the readout for m s = -1 has fidelity F -1 = 1, and no errors are made in the four different microwave rotations necessary to do the Bell test. The red line shows the classical/quantum boundary of S = 2. Also indicated is the position of the experimental values in Bernien et al. 2 o Improving the readout: The single shot readout has to be enhanced, both in speed and in readout fidelity. Currently, as in Bernien et al. 2 the best readout characteristics are a 90%, 99.99% readout fidelity for readout of the states m s = 0, -1 respectively, performed in a 12 us readout time. By increasing the readout laser power, the readout can be pushed to shorter time scales as illustrated in figure 6. Here, the optimal readout time can be reduced by a factor 5 by increasing the readout power a factor 25, while the readout fidelity is unchanged. By this method, a readout time of 3-4 us seems RO power x 25 feasible. Figure 6: Readout fidelity versus readout time, for two different powers of the excitation readout laser. It can be seen that increasing the readout power can decrease the optimal readout time, without decreasing the maximum readout fidelity of either two states. Readout powers used here are 20 nw (left), and 1 uw (right). Because of the finite few us risetime of amplitude modulation used, the actual readout time is likely 1-2 us faster than shown. Further improvements of the readout fidelity might be gained by exploring different strain regimes of the NV centre by DC Stark tuning 18,19. Finally, both the readout time and fidelity can be increased by improving the optical collection efficiency, for instance by embedding the NV centre in a cavity, as described in Proposal 2 below.

o Fibre losses and dephasing: The required few kilometre length of fibre to connect both remote setups will result in a reduced detection efficiency for the single photons in the entanglement scheme. This will directly result in a lower success rate for the entanglement generation. Additionally, false entanglement events caused by detector dark counts become more frequent, which in turn decreases the entanglement fidelity. Currently 2, the single photon detection probability is about 100 times the detector dark count. For standard commercially available single mode fibres, the attenuation is less than 12 db/km 20. As one photon needs to be received from each NV, it advantageous to symmetrise the setup as in figure 2(b), as otherwise the effect of MW errors on the setup without the long fibre are amplified relative to the far away setup. Figure 7: Simulation of known errors in the entanglement protocol, used Bernien et al. 2. The effect of additional losses in two fibres of length L/2 in each arm of the beam splitter as in figure 2(b) is shown. The distance measure corresponds to an example fibre 20 for 637 nm, with 12 db/km losses. However, there are numerous ways to increase the entanglement fidelity. By decreasing the detection time window after each laser pulse (t det in Bernien et al. 2 ), the relative contribution of detector dark counts can be reduced; the dark-count rate is independent of time, whereas the NV detection rate is an exponentially decaying function of time. Furthermore, being more restrictive on the allowed times between the two subsequent clicks in the entanglement protocol (δτ in Bernien et al. 2 ), will boost the fidelity of the entangled state. In Bernien et al. 2 not enough data was taken to set δτ <5 ns, but a clear rising trend was observed towards zero time delay. Note that these restrictions can be applied as post selection as these operations are contained in the Magic box. If longer stretches of optical fibre are necessary (because the readout time cannot be shortened sufficiently), at some point it becomes advantageous/necessary to convert the single photons emitted by the NV centre to a longer wavelength, where fibre losses are reduced. This can be achieved by single photon down-conversion, as recently shown for single photons from quantum dots, emitting around 800 nm 21,22. This process can be as efficient as 60%, and losses near telecom wavelength are less than 0.35 db/km. However the construction and operation of the down-conversion setup will be challenging, and will require a substantial investment. Finally, optical path length stability is required over the entire fibre length, on the timescale set by the inter-pulse delay of the entanglement protocol. In Bernien et al. 2, this time is 600 ns. Over fibre lengths up to a few kilometres, this stability is expected to be achieved without active stabilisation 23 25. In summary: A 3 us readout with 95 % m s = 0 readout fidelity seems feasible. Including an additional 1 us for the random basis rotation, we need a setup separation of 1.3 km, which can be accomplished within the TU Delft campus area. Assuming we need two fibres of 1 km to connect the setups, we will reduce our current entanglement fidelity by about 3 %. A reasonable increase of about 10 % in entanglement fidelity is then required to violate Bell s inequalities. This might already be possible by stricter entanglement event time-filtering.

Necessary technological advances: Create mobile version of one side of the entanglement setup. Create long-distance fibre infrastructure. (Single photon down-conversion with retention of phase).

References 1. Bell, J. S. Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy. (Cambridge University Press, 2004). 2. Bernien, H. et al. Heralded entanglement between solid-state qubits separated by 3 meters. arxiv:1212.6136 (2012). at <http://arxiv.org/abs/1212.6136> 3. Einstein, A., Podolsky, B. & Rosen, N. Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Phys. Rev. 47, 777 780 (1935). 4. Clauser, J. F., Horne, M. A., Shimony, A. & Holt, R. A. Proposed Experiment to Test Local Hidden- Variable Theories. Phys. Rev. Lett. 23, 880 884 (1969). 5. Freedman, S. J. & Clauser, J. F. Experimental Test of Local Hidden-Variable Theories. Phys. Rev. Lett. 28, 938 941 (1972). 6. Bohm, D. Quantum Theory. (Courier Dover Publications, 1951). 7. Garg, A. & Mermin, N. D. Detector inefficiencies in the Einstein-Podolsky-Rosen experiment. Phys. Rev. D 35, 3831 3835 (1987). 8. Aspect, A., Dalibard, J. & Roger, G. Experimental Test of Bell s Inequalities Using Time- Varying Analyzers. Phys. Rev. Lett. 49, 1804 1807 (1982). 9. Percival, I. C. Why do Bell experiments? arxiv:quant-ph/0008097 (2000). doi:10.1016/s0375-9601(00)00799-4 10. Rowe, M. A. et al. Experimental violation of a Bell s inequality with efficient detection. Nature 409, 791 794 (2001). 11. Giustina, M. et al. Bell violation with entangled photons, free of the fair-sampling assumption. arxiv:1212.0533 (2012). at <http://arxiv.org/abs/1212.0533> 12. Hofmann, J. et al. Heralded Entanglement Between Widely Separated Atoms. Science 337, 72 75 (2012). 13. Moehring, D. L. et al. Entanglement of single-atom quantum bits at a distance. Nature 449, 68 71 (2007). 14. Pfaff, W. et al. Demonstration of entanglement-by-measurement of solid-state qubits. Nature Physics (2012). doi:10.1038/nphys244415. Bochmann, J. Coherent dynamics and state detection of single atoms in a cavity. (2010). 16. Henkel, F. Photoionisation detection of single Rb-87 atoms using channel electron multipliers. (2011). 17. Quantis RNG - True Random Number Generator - Resource Center. at <http://www.idquantique.com/random-number-generators/resources/quantis-resource-center.html> 18. Bassett, L. C., Heremans, F. J., Yale, C. G., Buckley, B. B. & Awschalom, D. D. Electrical Tuning of Single Nitrogen Vacancy Center Optical Transitions Enhanced by Photoinduced Fields. 1104.3878 (2011). at <http://arxiv.org/abs/1104.3878> 19. Hensen, B. Measurement-based Quantum Computation with the Nitrogen-Vacancy centre in Diamond. (2011). 20. Thorlabs - PM-S630-HP 630-780 nm PM Fiber w/ Pure Silica Core, 4.2 µm MFD. at <http://www.thorlabs.de/thorproduct.cfm?partnumber=pm-s630-hp> 21. Zaske, S. et al. Visible-to-Telecom Quantum Frequency Conversion of Light from a Single Quantum Emitter. Phys. Rev. Lett. 109, 147404 (2012). 22. Greve, K. D. et al. Quantum-dot spin-photon entanglement via frequency downconversion to telecom wavelength. Nature 491, 421 425 (2012). 23. Ye, J. et al. Delivery of high-stability optical and microwave frequency standards over an optical fiber network. J. Opt. Soc. Am. B 20, 1459 1467 (2003). 24. Minář, J., De Riedmatten, H., Simon, C., Zbinden, H. & Gisin, N. Phase-noise measurements in longfiber interferometers for quantum-repeater applications. Phys. Rev. A 77, 052325 (2008). 25. Holman, K. W., Hudson, D. D., Ye, J. & Jones, D. J. Remote transfer of a high-stability and ultralowjitter timing signal. Opt. Lett. 30, 1225 1227 (2005).