|
1.INTRODUCTIONIn 1935, Einstein, Podolsky and Rosen (EPR) reasoned that the quantum mechanical wave function can not completely describe physical reality.1 This argument can be easily understood using Bohm’s Gedankenexperiment:2,3 Consider a spin-0 molecule, that gets split up into two spin atoms and distributed to two space-like separated measurement stations, “Alice” and “Bob”. Whenever Alice measures the spin of her atom, she knows that – due to angular momentum conservation – the spin that Bob measures has to add up to 0 with her measurement outcome, whether she measures spin . In other words, her measurement outcome makes it possible for her to predict with certainty Bob’s outcome. Alice’ and Bob’s result will always exhibit this strong correlation whenever they have chosen to measure in the same measurement basis. Since there is no interaction between the two systems anymore, it might be reasonable to believe that the measurement outcomes have been predetermined, but the quantum mechanical description does not carry any information about the outcomes. This is the essence of the argument by EPR that the quantum state as a description of physical reality can not be complete. 1 Bell’s theorem of 1964 shows that local realism, the worldview under which physical influences are limited by the speed of light and measurement outcomes are defined prior to and independent of measurement, is inconsistent with the predictions of quantum mechanics.4 Specifically, correlations between measurement results from distant entangled systems would be smaller under the assumption of local realism than predicted by quantum mechanics. This is expressed in Bell’s inequalities. Since quantum mechanics predicts a violation of the inequality for the results of certain measurements on entangled particles, Bell’s inequality can be used here to rule out philosophical standpoints based on experimental results. Indeed, violations have been measured employing versions of Bell’s inequalities.5–7 Do these experimental violations invalidate local realism? That is not the only possibility. The experiments violating Bell’s inequality required extra assumptions, and therefore opened loopholes that in principle still permit that the measured data can be explained using a local realist model. 2.LOOPHOLESThe locality loophole (or communication loophole) is left open if the setting choice or the measurement result of one side could be communicated at the speed of light in vacuum or slower to the other side in time to influence the measurement result there. In order to close this loophole, it is necessary to space-like separate each local measurement from the distant setting choice as well as from the distant measurement. This can be guaranteed by independently choosing the measurement settings on both sides so quickly that no physical signal can pass information about the chosen setting or the measurement result to the other side in time to influence the measurement. The freedom-of-choice loophole regards the possibility of influences on the setting choices from any combination of hidden variables and/or other factors within the backward light-cone of the setting choice. Here, hidden variables represent “any number of hypothetical additional complementary variables needed to complete quantum mechanics in the way envisaged by EPR.”.8 In order to address this loophole, it is necessary to make specific assumptions about the origin of these hidden variables and generate the setting choices independently from past events and space-like separated from the hidden variables. We make the assumption that the hidden variables are created not before the emission event of the entangled photon pair. The fair-sampling loophole is about the idea, that a small sub-ensemble of all entangled particles could in principle be non-representative for the entire ensemble of entangled particles.9 For example, it is imaginable that the detected sub-ensemble could violate Bell’s inequality while the entire ensemble does not. It is possible to close this loophole by detecting the entangled particles with a sufficiently high efficiency. The coincidence-time loophole10,11 exploits the assumption that the timing statistics is the same for all detector clicks. This particularly applies to experiments in which the identification of pairs is done via a moving coincidence window. One way to avoid this loophole is to make a pulsed experiment with locally defined time slots. The memory loophole12 corresponds to the assumption that experimental trials are identical and independent (iid). In principle the outcomes of a specific trial could depend on all previous settings and outcomes since these are not space-like separated anymore. Exploiting this loophole, the statistical significance of a violation can be altered. This loophole can be closed by avoiding the iid assumption in the data analysis. Many experimental Bell tests have been performed6,7,13–25 closing individual loopholes. For example, Aspect et al.’s 1982 experiment7 first employed rapid switching in the measurement settings; Weihs et al.13 improved this with fast random switching; Scheidl et al.18 addressed the freedom-of-choice and locality loopholes in 2010 while Handsteiner et al.25 improved on that; Rowe et al.14 were first to close the fair-sampling loophole in 2001 and were followed by several experiments in a variety of systems.15,17,20–22 It has recently become possible to address all aforementioned loopholes in a single experiment.22–24,26 In this paper, we report the violation of a Bell inequality while closing all aforementioned loopholes in a single experiment with high statistical significance. Our experiment therefore strongly supports the claim that nature cannot be described within the framework of local realism. 3.SETUPIn Fig. 1(a), the experimental setup is illustrated. The source of polarization-entangled photon pairs made use of spontaneous parametric down-conversion (SPDC) in a periodically poled nonlinear crystal (ppKTP). The polarization entanglement was facilitated using a Sagnac configuration27, 28 and optimized focusing parameters for high heralding efficiency.29, 30 With single-mode fibers, the photons were distributed to the two measurement stations, “Alice” and “Bob” [Fig. 1(c)] to perform polarization measurements on them. While the photons were on their way towards the measurement station, a random number generator31,32 (RNG) made a choice between two linear polarization angles which were implemented by an electro-optical modulator (EOM) that acted as a polarization rotator in front of a polarizing beam splitter. The horizontal output of that beam splitter was connected to a transition-edge sensor (TES) single photon detector.33 The signal from the TES was amplified by several cryogenic34 and room-temperature amplifiers, digitized and recorded locally on a hard drive together with the time stamp and result of the basis choice. 3.1Closing the Freedom-of-choice and Locality LoopholesIn order to close the freedom-of-choice and locality loopholes, a very specific space-time arrangement was necessary as discussed above in sec. 2. In the space-time diagram of the experiment [Fig. 2], three events are of particular importance:
All relevant delays were characterized using an oscilloscope and a fast photodiode relative to a 1 MHz clock which was also used to control the pump laser and EOM. This clock was phase stable to a 10 MHz master oscillator which kept the time tagging devices, digitizer cards and random number generators synchronized. 3.2Closing the fair-sampling LoopholeThe closure of the fair sampling loophole can be observed in the measured data. It is the cleanest way to use an inequality that can be derived without the fair-sampling assumption. This applies to both the Clauser-Horne5 and Eberhard35 inequality which can be violated at a system heralding efficiency of larger than 2/3. We used a CH-Eberhard36 inequality which makes use of only one detector per side and considers the outcomes “+” for a detection event and “0” for no detection.36,37 For each trial, Alice choses between a1 and a2 and Bob choses between b1 and b2. For example, p+0(a1, b2) is the probability that Alice detected a photon and chose the angle a1 and Bob has no detection event and chose the angle b2. Both of them write down their outcomes “+” or “0” for each trial and compare their data after the experiment to estimate the probabilities and evaluate the inequality. This inequality is maximally violated by non-maximally entangled states of the form: The optimal parameter r was found using numerical simulations based on a quantum mechanical model38 and depends on the system efficiency, the visibility and the background rate. We used a parameter of r ≈ −2.9 and measured at the angles a1 = 94.4°, a2 = 62.4°, b1 = −6.5°, b2 = 25.5° for approximately 3510 seconds. 3.3Closing the Coincidence-time and Memory LoopholesThe coincidence-time loophole was avoided by using a pulsed experiment with locally defined time slots. Therefore, the identification of coincident photons does not rely on any method that opens the coincidence-time loophole. The assumption that the experimental trials are independent and identical was avoided in order to close the memory loophole.12 The statistical significance was computed assuming full experimental memory.23, 36 4.RESULTSWe characterized the system using a maximally entangled state (r = −1 in Eq. 2) and found a visibility of > 99% in both the diagonal and the HV-basis. The total system efficiency (i.e. the ratio of twofold coincidence events per single counts) was approximately 78.2% in the Alice arm and 76.2% in the Bob arm. Approximately 3500 pairs were created per second in the source. We determined a J value of 7.27 · 10−6. A p-value of 3.74 · 10−31 was computed under full experimental memory12,39,40 while taking into account the finite predictability of the random number generators.36 This is the purely statistical probability of our observed violation to be the result of statistical fluctuations under local realism. Given the very small probability, it should be mentioned that the confidence in this experiment is limited in general by other sources of errors including systematic and human mistakes rather than by the statistical significance. 5.SUMMARYWe demonstrated a strong violation of local realism with high statistical significance. We space-like separated the emission event of the down-conversion from the setting choice and the setting choice from the measurement using state-of-the-art random number generators. We closed the fair-sampling loophole with a very high system heralding efficiency. We also closed the coincidence-time loophole by using locally defined time slots and the memory loophole by adequate statistical analysis. Our experiment provides strong support for the viewpoint that local realism is untenable. The freedom-of-choice loophole was closed up to a reasonable point in time: The production of the entangled photon pair. However, this is just a few hundred nanoseconds before the measurement. What if the hidden variables have been created a long time before the experiment? It is possible to use setting choices that have been produced and space-like separated for billions of years by using light from different quasars on opposite sides of the night sky.41 Handsteiner et al.25 used basis settings derived from the light of Milky Way stars to push back the time when the hidden variables could have been created by ~ 600 years. Further steps could be to use photons from quasars that are space-like separated since the period of cosmic inflation and also close the fair-sampling loophole at the same time. AcknowledgmentsWe thank the Vienna Hofburg and especially Reinhold Sahl for the use of their basement. We acknowledge Max Tillmann, Johannes Steurer, Sven Ramelow, Scott Glancy and Witlef Wieczorek for helpful discussions and technical assistance. M. G. acknowledges support by the program CoQuS of the FWF (Austrian Science Fund). J. K. thanks Lucas Clemente for help with implementing data analysis code and acknowledges support from the EU Integrated Project SIQS. C. A., W. A., V. P., and M. W. M. acknowledge the European Research Council Project AQUMET, European Union Project QUIC (Grant Agreement No. 641122), Spanish MINECO under the Severo Ochoa programme (Grant No. SEV-2015-0522) and Projects MAGO (Grant No. FIS2011-23520) and XPLICA (Grant No. FIS2014-62181-EXP), Catalan AGAUR 2014 SGR Grants No. 1295 and No. 1623, the European Regional Development Fund (FEDER) Grant No. TEC2013-46168-R, and by Fundació Privada CELLEX. This work was also supported by the NIST Quantum Information Science Initiative. This project was supported by the Austrian Academy of Sciences (ÖAW), the European Research Council (SIQS Grant No. 600645 EU-FP7-ICT), and the Austrian Science Fund (FWF) with SFB F40 (FOQUS). REFERENCESEinstein, A., Podolsky, B., and Rosen, N.,
“Can quantum-mechanical description of physical reality be considered complete?,”
Phys. Rev., 47
(10), 777
(1935). https://doi.org/10.1103/PhysRev.47.777 Google Scholar
Bohm, D. and Aharonov, Y.,
“Discussion of Experimental Proof for the Paradox of Einstein, Rosen, and Podolsky,”
Phys. Rev., 108 1070
–1076
(1957). https://doi.org/10.1103/PhysRev.108.1070 Google Scholar
Bohm, D., Quantum theory, Courier Corporation(1951). Google Scholar
Bell, J. S., Physics, 1
(3), 195
–200
(1964). https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195 Google Scholar
Clauser, J. F. and Horne, M. A.,
“Experimental consequences of objective local theories,”
Phys. Rev. D, 10 526
–535
(1974). https://doi.org/10.1103/PhysRevD.10.526 Google Scholar
Freedman, S. J. and Clauser, J. F.,
“Experimental Test of Local Hidden-Variable Theories,”
Phys. Rev. Lett., 28 938
–941
(1972). https://doi.org/10.1103/PhysRevLett.28.938 Google Scholar
Aspect, A., Dalibard, J., and Roger, G.,
“Experimental Test of Bell’s Inequalities Using Time-Varying Analyzers,”
Phys. Rev. Lett., 49 1804
–1807
(1982). https://doi.org/10.1103/PhysRevLett.49.1804 Google Scholar
Bell, J. S., Speakable and Unspeakable in Quantum Mechanics, 232
–248 Cambridge University Press, Cambridge
(2004). https://doi.org/10.1017/CBO9780511815676 Google Scholar
Pearle, P. M.,
“Hidden-Variable Example Based upon Data Rejection,”
Phys. Rev. D, 2 1418
–1425
(1970). https://doi.org/10.1103/PhysRevD.2.1418 Google Scholar
Larsson, J.-Å. and Gill, R. D.,
“Bell’s inequality and the coincidence-time loophole,”
Europhys. Lett., 67 707
–713
(2004). https://doi.org/10.1209/epl/i2004-10124-7 Google Scholar
Larsson, J.-Å., Giustina, M., Kofler, J., Wittmann, B., Ursin, R., and Ramelow, S.,
“Bell-inequality violation with entangled photons, free of the coincidence-time loophole,”
Phys. Rev. A, 90 032107
(2014). https://doi.org/10.1103/PhysRevA.90.032107 Google Scholar
Barrett, J., Collins, D., Hardy, L., Kent, A., and Popescu, S.,
“Quantum nonlocality, Bell inequalities, and the memory loophole,”
Phys. Rev. A, 66 042111
(2002). https://doi.org/10.1103/PhysRevA.66.042111 Google Scholar
Weihs, G., Jennewein, T., Simon, C., Weinfurter, H., and Zeilinger, A.,
“Violation of Bell’s Inequality under Strict Einstein Locality Conditions,”
Phys. Rev. Lett., 81 5039
–5043
(1998). https://doi.org/10.1103/PhysRevLett.81.5039 Google Scholar
Rowe, M. A., Kielpinski, D., Meyer, V., Sackett, C. A., Itano, W. M., Monroe, C., and Wineland, D. J.,
“Experimental violation of a Bell’s inequality with efficient detection,”
Nature, 409 791
–794
(2001). https://doi.org/10.1038/35057215 Google Scholar
Matsukevich, D. N., Maunz, P., Moehring, D. L., Olmschenk, S., and Monroe, C.,
“Bell Inequality Violation with Two Remote Atomic Qubits,”
Phys. Rev. Lett., 100 150404
(2008). https://doi.org/10.1103/PhysRevLett.100.150404 Google Scholar
Hofmann, J., Krug, M., Ortegel, N., Gerard, L., Weber, M., Rosenfeld, W., and Weinfurter, H.,
“Heralded Entanglement Between Widely Separated Atoms,”
Science, 337 72
–75
(2012). https://doi.org/10.1126/science.1221856 Google Scholar
Ansmann, M., Wang, H., Bialczak, R. C., Hofheinz, M., Lucero, E., Neeley, M., O’Connell, A. D., Sank, D., Weides, M., Wenner, J., Cleland, A. N., and Martinis, J. M.,
“Violation of Bell’s inequality in Josephson phase qubits,”
Nature, 461 504
–506
(2009). https://doi.org/10.1038/nature08363 Google Scholar
Scheidl, T., Ursin, R., Kofler, J., Ramelow, S., Ma, X.-S., Herbst, T., Ratschbacher, L., Fedrizzi, A., Langford, N. K., Jennewein, T., and Zeilinger, A.,
“Violation of local realism with freedom of choice,”
Proc. Natl. Acad. Sci., 107 19708
–19713
(2010). https://doi.org/10.1073/pnas.1002780107 Google Scholar
Agüero, M. B., Hnilo, A. A., and Kovalsky, M. G.,
“Time-resolved measurement of Bell inequalities and the coincidence loophole,”
Phys. Rev. A, 86 052121
(2012). https://doi.org/10.1103/PhysRevA.86.052121 Google Scholar
Giustina, M., Mech, A., Ramelow, S., Wittmann, B., Kofler, J., Beyer, J., Lita, A., Calkins, B., Gerrits, T., Nam, S. W., Ursin, R., and Zeilinger, A.,
“Bell violation using entangled photons without the fair-sampling assumption,”
Nature, 497 227
(2013). https://doi.org/10.1038/nature12012 Google Scholar
Christensen, B. G., McCusker, K. T., Altepeter, J. B., Calkins, B., Gerrits, T., Lita, A. E., Miller, A., Shalm, L. K., Zhang, Y., Nam, S. W., Brunner, N., Lim, C. C. W., Gisin, N., and Kwiat, P. G.,
“Detection-Loophole-Free Test of Quantum Nonlocality, and Applications,”
Phys. Rev. Lett., 111 130406
(2013). https://doi.org/10.1103/PhysRevLett.111.130406 Google Scholar
Hensen, B., Bernien, H., Dréau, A. E., Reiserer, A., Kalb, N., Blok, M. S., Ruitenberg, J., Vermeulen, R. F. L., Schouten, R. N., Abellán, C., Amaya, W., Pruneri, V., Mitchell, M. W., Markham, M., Twitchen, D. J., Elkouss, D., Wehner, S., Taminiau, T. H., and Hanson, R.,
“Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres,”
Nature, 526 682
–686
(2015). https://doi.org/10.1038/nature15759 Google Scholar
Giustina, M., Versteegh, M. A. M., Wengerowsky, S., Handsteiner, J., Hochrainer, A., Phelan, K., Steinlechner, F., Kofler, J., Larsson, J.-A., Abellán, C., Amaya, W., Pruneri, V., Mitchell, M. W., Beyer, J., Gerrits, T., Lita, A. E., Shalm, L. K., Nam, S. W., Scheidl, T., Ursin, R., Wittmann, B., and Zeilinger, A.,
“Significant-loophole-free test of bell’s theorem with entangled photons,”
Phys. Rev. Lett., 115 250401
(2015). https://doi.org/10.1103/PhysRevLett.115.250401 Google Scholar
Shalm, L. K., Meyer-Scott, E., Christensen, B. G., Bierhorst, P., Wayne, M. A., Stevens, M. J., Gerrits, T., Glancy, S., Hamel, D. R., Allman, M. S., Coakley, K. J., Dyer, S. D., Hodge, C., Lita, A. E., Verma, V. B., Lambrocco, C., Tortorici, E., Migdall, A. L., Zhang, Y., Kumor, D. R., Farr, W. H., Marsili, F., Shaw, M. D., Stern, J. A., Abellán, C., Amaya, W., Pruneri, V., Jennewein, T., Mitchell, M. W., Kwiat, P. G., Bienfang, J. C., Mirin, R. P., Knill, E., and Nam, S. W.,
“Strong loophole-free test of local realism,”
Phys. Rev. Lett., 115 250402
(2015). https://doi.org/10.1103/PhysRevLett.115.250402 Google Scholar
Handsteiner, J., Friedman, A. S., Rauch, D., Gallicchio, J., Liu, B., Hosp, H., Kofler, J., Bricher, D., Fink, M., Leung, C., Mark, A., Nguyen, H. T., Sanders, I., Steinlechner, F., Ursin, R., Wengerowsky, S., Guth, A. H., Kaiser, D. I., Scheidl, T., and Zeilinger, A.,
“Cosmic bell test: Measurement settings from milky way stars,”
Phys. Rev. Lett., 118 060401
(2017). https://doi.org/10.1103/PhysRevLett.118.060401 Google Scholar
Rosenfeld, W., Burchardt, D., Garthoff, R., Redeker, K., Ortegel, N., Rau, M., and Weinfurter, H.,
“Eventready bell test using entangled atoms simultaneously closing detection and locality loopholes,”
Physical Review Letters, 119
(1), 010402
(2017). https://doi.org/10.1103/PhysRevLett.119.010402 Google Scholar
Fedrizzi, A., Herbst, T., Poppe, A., Jennewein, T., and Zeilinger, A.,
“A wavelength-tunable fiber-coupled source of narrowband entangled photons.,”
Opt. Express, 15
(23), 15377
–15386
(2007). https://doi.org/10.1364/OE.15.015377 Google Scholar
Kim, T., Fiorentino, M., and Wong, F. N.,
“Phase-stable source of polarization-entangled photons using a polarization sagnac interferometer,”
Physical Review A, 73
(1), 012316
(2006). https://doi.org/10.1103/PhysRevA.73.012316 Google Scholar
Bennink, R. S.,
“Optimal collinear Gaussian beams for spontaneous parametric down-conversion,”
Phys. Rev. A, 81 053805
(2010). https://doi.org/10.1103/PhysRevA.81.053805 Google Scholar
Steinlechner, F., Sources of Photonic Entanglement for Applications in Space, Barcelona, Spain
(2015). Google Scholar
Abellán, C., Amaya, W., Jofre, M., Curty, M., Acín, A., Capmany, J., Pruneri, V., and Mitchell, M. W.,
“Ultra-fast quantum randomness generation by accelerated phase diffusion in a pulsed laser diode,”
Opt. Express, 22 1645
(2014). https://doi.org/10.1364/OE.22.001645 Google Scholar
Abellán, C., Amaya, W., Mitrani, D., Pruneri, V., and Mitchell, M. W.,
“Generation of fresh and pure random numbers for loophole-free bell tests,”
Physical review letters, 115
(25), 250403
(2015). https://doi.org/10.1103/PhysRevLett.115.250403 Google Scholar
Lita, A. E., Miller, A. J., and Nam, S. W.,
“Counting near-infrared single-photons with 95% efficiency,”
Opt. Express, 16
(5), 3032
(2008). https://doi.org/10.1364/OE.16.003032 Google Scholar
Drung, D., Assmann, C., Beyer, J., Kirste, A., Peters, M., Ruede, F., and Schurig, T.,
“Highly Sensitive and Easy-to-Use SQUID Sensors,”
IEEE Trans. Appl. Supercond., 17 699
–704
(2007). https://doi.org/10.1109/TASC.2007.897403 Google Scholar
Eberhard, P. H.,
“Background level and counter efficiencies required for a loophole-free Einstein-Podolsky-Rosen experiment,”
Phys. Rev. A, 47 R747
–R750
(1993). https://doi.org/10.1103/PhysRevA.47.R747 Google Scholar
Kofler, J., Giustina, M., Larsson, J.-Å., and Mitchell, M. W.,
“Requirements for a loophole-free photonic bell test using imperfect setting generators,”
Physical Review A, 93
(3), 032115
(2016). https://doi.org/10.1103/PhysRevA.93.032115 Google Scholar
Bierhorst, P.,
“A robust mathematical model for a loophole-free clauser–horne experiment,”
Journal of Physics A: Mathematical and Theoretical, 48
(19), 195302
(2015). https://doi.org/10.1088/1751-8113/48/19/195302 Google Scholar
Kofler, J., Ramelow, S., Giustina, M., and Zeilinger, A.,
“On ‘Bell violation using entangled photons without the fair-sampling assumption’,”
(2013). Google Scholar
Gill, R.,
in Proceedings of foundations of probability and physics-2,
(2003). Google Scholar
Moore, M. and van Eeden, C.,
“Mathematical statistics and applications: Festschrift for constance van eeden,”
in IMS,
(2003). Google Scholar
Gallicchio, J., Friedman, A. S., and Kaiser, D. I.,
“Testing Bell’s Inequality with Cosmic Photons: Closing the Setting-Independence Loophole,”
Phys. Rev. Lett., 112 110405
(2014). https://doi.org/10.1103/PhysRevLett.112.110405 Google Scholar
|