Schmidt Decomposition Made Universal to Unveil the Entanglement of Identical Particles

From left to right: Stefania Sciara, Rosario Lo Franco, Giuseppe Compagno

Authors: Stefania Sciara^1,2, Rosario Lo Franco^2,3, Giuseppe Compagno²

Affiliation:
¹INRS-EMT, Varennes, Québec J3X 1S2, Canada,
²Dipartimento di Fisica e Chimica, Università di Palermo, Italy,
³Dipartimento di Energia, Ingegneria dell'Informazione e Modelli Matematici, Università di Palermo,  Italy.

The Schmidt decomposition is an important mathematical tool which has been already utilized during the early stages of quantum theory by Schrödinger in the context of quantum measurements [1-3]. This tool allows to determine the set of measurements on one part of the system such that the measurement outcome on the other part is determined, in the sense that to each outcome of the first measurement it corresponds a unique outcome for the second measurement. Schmidt decomposition has been shown to be at the heart of quantum information theory, quantifying entanglement in bipartite systems. It has been also widely employed in the context of Einstein-Podolski-Rosen (EPR) paradox, Bell non-locality and black-hole physics [2-4].

Every element of this decomposition consists unavoidably of two independent subsystems. Application of the Schmidt decomposition to identical particles is therefore hindered by the fact that overlapping particles can never be considered independent. In fact, despite its wide utilization in systems of distinguishable particles, the Schmidt decomposition has remained debated for identical particles [5,6]. For instance, it is well known that for distinguishable particles this tool assesses the entanglement of the system by the von Neumann entropy of the reduced density matrix, whose eigenvalues are the squares of the Schmidt coefficients appearing in the decomposition [3]. Differently, in the case of identical particles, it has been claimed that the relationship between the Schmidt coefficients and the eigenvalues of the reduced density matrix breaks down [6]. In strict connection with this issue, the partial trace operation to get the reduced state has not been considered suitable for quantifying the entanglement of pure states of identical particles [6-8].

We recall that, in Nature, particles are of different types, all particles of each type (electrons, protons, photons and so on) being identical. In the quantum world, the identity of particles gives rise to a new characteristic with respect to the classical world, that is the indistinguishability among particles of the same type [9,10]. This exclusive quantum trait leads to fundamental properties of matter such that particles can be of two classes, named bosons and fermions. Moreover, at variance with the case of distinguishable particles, when identical particles have wave functions that spatially overlap, they can never be taken as independent of each other [11,12].

The latter behavior is an essential requisite in determining features like quantum correlations (e.g., entanglement) among the particles themselves and in the theory of measurement [1]. Nevertheless, identical particles constitute the building blocks of quantum information and computation theory, being present in Bose-Einstein condensates [13,14], quantum dots [15,16], superconducting circuits [17] and optical setups [18,19]. It is thus important to have trustable methods and tools to characterize the quantum features of composite systems of identical particles under these general conditions.

A first step towards this goal has been provided by a recent non-standard approach [12] which deals with systems of identical particles within a particle-based description (that is, in terms of particle states) without resorting to the usual practice to assign fictitious labels to the particles [9,10], which render the latter distinguishable removing their indistinguishability. In this way, the ambiguity arising from the introduction of these labels in evaluating quantum correlations in identical particle systems is avoided. This method has, in fact, provided a way to calculate partial trace and von Neumann entropy for identical particles. Using this new approach, in a recent work [20] we have been able to demonstrate that the Schmidt decomposition is universal, meaning that it is also obtainable for an arbitrary state of indistinguishable particles (bosons or fermions) under general conditions of spatial overlap. Thanks to this achievement, the amount of entanglement present in identical particle systems in pure states can be evaluated by the von Neumann entropy of the reduced density matrix, as occurs in the case of distinguishable particles.

We have used the Schmidt decomposition to analyze some paradigmatic states of two identical particles to find the suitable measurement basis for unveiling their entanglement properties. These systems are illustrated in Figure 1 and represent simple yet effective examples which make it emerge the effect of particle identity. The first one is the well-known condition of two identical qubits with opposite pseudospins in spatially separated locations, denoted with left (L) and right (R) (see Fig. 1a).

Figure 1. (a) Two identical qubits in two spatially separated places with opposite pseudospins. (b) Two identical qubits in the same spatial mode with arbitrary pseudospins. (c) Two identical qutrits (three-level quantum systems) in the same spatial mode. The shaded ellipses indicate that the particles are entangled. Figure from Ref. [20].

Indeed, we have found that the Schmidt decomposition and the corresponding von Neumann entropy supply the results which are physically expected in this situation, such as zero entanglement for a product (separable) state of the two particles and maximal entanglement for a Bell state. We have also explicitly shown that nonlocal measurements induce entanglement in a state of distant identical particles. This means that, in principle, identical particles cannot be considered completely independent even when they are spatially separated. Application of the Schmidt decomposition to a system of two boson qubits in the same site (see Fig. 1b) has evidenced that the entanglement increases as the two internal states tend to be orthogonal, as displayed in the plot of Fig. 2. This finding is a generalization of a previous result obtained only for two identical qubits in the same spatial mode with orthogonal internal states (or pseudospins, that is θ = π in the plot of Fig. 2) [12].

Figure 2. Entanglement between the pseudospins of two identical particles in the same site, quantified by the von Neumann entropy, as a function of the angle θ between the directions of the two pseudospins (internal states). Figure from Ref. [20].

Two identical (boson) qutrits, that is three-level quantum systems, constitute a system of interest being promising candidates for quantum processors thanks to their good capacity to store quantum information [21,22]. We consider the qutrits in the same site, that is under the condition of complete spatial overlap, as depicted in Fig. 1c. Our method easily determines the two-qutrit entanglement for given combinations of their internal states and provides a physical interpretation. We remark that the entanglement found for the two qutrits is different from that obtained for the same system by an alternative approach, based on a so-called subalgebra technique [8]. The origin of this difference in the entanglement measure requires further investigation, for instance by comparing the two theoretical approaches or by experimental verifications.

We now briefly discuss the practical aspect concerning the possibility to exploit the entanglement of identical particles, as identified by the Schmidt decomposition we have introduced. In the context of quantum information processing with distinguishable particles, a well-established resource theory is based on local operations and classical communication (LOCC), where each particle is individually addressed [1-3]. On the other hand, a peculiar trait of indistinguishable particles is that it is not possible to operate on a given individual particle. This characteristic in general hinders the manipulation of composite systems of identical particles for desired tasks of quantum information and computation. Nevertheless, it is known that the entanglement quantified by the von Neumann entropy can be extracted from the state of identical particles and then utilized in a conditional way by LOCC [23].

In conclusion, the universal character of the Schmidt decomposition shown in our work allows a faithful treatment of the entanglement of composite systems of identical particles and opens the way to more general investigations of collective properties of such systems.

References:
[1] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, Karol Horodecki, “Quantum entanglement”, Review of Modern Physics, 81, 865 (2009). Abstract.
[2] John Preskill, “Lectures notes for physics 229: Quantum information and computation” (1998) Link.
[3] Michael A. Nielsen and Isaac L. Chuang, “Quantum Computation and Quantum Information” (Cambridge University Press, Cambridge, 2000).
[4] E.D. Belokolos, M.V. Teslyk, “Scalar field entanglement entropy of a Schwarzschild black hole from the Schmidt decomposition viewpoint”, Classical and Quantum Gravity, 26, 235008 (2009). Abstract.
[5] R. Paškauskas, L. You, “Quantum correlations in two-boson wave functions”, Physical Review A, 64, 042310 (2001). Abstract.
[6] Malte C Tichy, Florian Mintert, Andreas Buchleitner, “Essential entanglement for atomic and molecular physics”, Journal of Physics B: Atomic, Molecular and Optical Physics. 44, 192001 (2011). Abstract.
[7] GianCarlo Ghirardi, Luca Marinatto, Tullio Weber, “Entanglement and properties of composite quantum systems: a conceptual and mathematical analysis”, Journal of Statistical Physics 108, 49 (2002). Abstract.
[8] A. Balachandran, T. Govindarajan, Amilcar R. de Queiroz, A. Reyes-Lega, “Entanglement and particle identity: A unifying approach”, Physical Review Letters, 110, 080503 (2013). Abstract.
[9] Asher Peres, “Quantum Theory: Concepts and Methods” (Springer, Dordrecht, The Netherlands, 1995).
[10] Claude Cohen-Tannoudji, Bernard Diu, Franck Laloe, “Quantum mechanics, Vol. 2” (Wiley-VCH, Paris, France, 2005).
[11] John Schliemann, J. Ignacio Cirac, Marek Kuś, Maciej Lewenstein, Daniel Loss, “Quantum correlations in two-fermion systems”, Physical Review A, 64, 022303 (2001). Abstract.
[12] Rosario Lo Franco, Giuseppe Compagno, “Quantum entanglement of identical particles by standard information-theoretic notions”, Scientific  Reports, 6, 20603 (2016). Abstract.
[13] Immanuel Bloch, Jean Dalibard, Wilhelm Zwerger, “Many-body physics with ultracold gases”, Review of Modern Physics, 80, 885 (2008). Abstract.
[14] Marco Anderlini, Patricia J. Lee, Benjamin L. Brown, Jennifer Sebby-Strabley, William D. Phillips,  J. V. Porto, “Controlled exchange interaction between pairs of neutral atoms in an optical lattice”, Nature 448, 452 (2007). Abstract.
[15] Michael H. Kolodrubetz, Jason R. Petta, “Coherent holes in a semiconductor quantum dot”, Science 325, 42–43 (2009). Abstract.
[16] Frederico Martins, Filip K. Malinowski, Peter D. Nissen, Edwin Barnes, Saeed Fallahi, Geoffrey C. Gardner, Michael J. Manfra, Charles M. Marcus, Ferdinand Kuemmeth, “Noise suppression using symmetric exchange gates in spin qubits”, Physical Review Letters, 116, 116801 (2016). Abstract.
[17] R. Barends, L. Lamata, J. Kelly, L. García-Álvarez, A. G. Fowler, A Megrant, E Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Yu Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill,  P. J. J. O’Malley,  C. Quintana,  P. Roushan, A. Vainsencher, J. Wenner, E. Solano, John M. Martinis, “Digital quantum simulation of fermionic models with a superconducting circuit”, Nature Communications, 6, 7654 (2015). Abstract.
[18] Andrea Crespi, Linda Sansoni, Giuseppe Della Valle, Alessio Ciamei, Roberta Ramponi, Fabio Sciarrino, Paolo Mataloni, Stefano Longhi, Roberto Osellame, “Particle statistics affects quantum decay and Fano interference”, Physical Review Letters, 114, 090201 (2015). Abstract.
[19] Christian Reimer, Michael Kues, Piotr Roztocki, Benjamin Wetzel, Fabio Grazioso, Brent E. Little, Sai T. Chu, Tudor Johnston, Yaron Bromberg, Lucia Caspani, David J. Moss, Roberto Morandotti, “Generation of multiphoton entangled quantum states by means of integrated fre-quency combs”, Science 351, 1176 (2016). Abstract.
[20] Stefania Sciara, Rosario Lo Franco, Giuseppe Compagno, “Universality of Schmidt decomposition and particle identity”, Scientific Reports, 7, 44675 (2017). Abstract.
[21] B.P. Lanyon, T.J. Weinhold, N.K. Langford, J.L. O’Brien, K.J. Resch, A. Gilchrist, A.G. White, “Manipulating biphotonic qutrits”, Physical Review Letters, 100, 060504 (2008). Abstract.
[22] K. S. Kumar, A. Vepsalainen, S. Danilin,  G.S. Paraoanu, “Stimulated Raman adiabatic passage in a three-level superconducting circuit”, Nature Communications, 7, 10628 (2016). Abstract.
[23] N. Killoran, M. Cramer, M. B. Plenio, “Extracting entanglement from identical particles”, Physical Review Letters, 112, 150501 (2014). Abstract.

Indications of an Influence of Solar Neutrinos on Beta Decays

From left to right: Peter A. Sturrock, Ephraim Fischbach, Jeffrey D. Scargle

Authors: Peter A. Sturrock¹, Ephraim Fischbach², Jeffrey D. Scargle<sup>3

1</sup>Kavli Institute for Particle Astrophysics and Cosmology and the Center for Space Science and Astrophysics, Stanford University, California, USA,
²Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana, USA,
³NASA/Ames Research Center, Moffett Field, California, USA.

Eckhard D. Falkenberg, who found evidence of an annual oscillation in the beta-decay rate of tritium, was either the first or one of the first to propose that some beta-decay rates may be variable [1]. He suggested that the beta-decay process may be influenced by neutrinos, and attributed the annual variation to the varying Earth-Sun distance that leads to a corresponding variation in the flux of solar neutrinos as detected on Earth. Supporting evidence for the variability of beta-decay rates could be found in the results of an experiment carried out at the Brookhaven National Laboratory. Alburger et al. had measured the decay rates of ³⁶Cl and of ³²Si from 1982.13 to 1986.13 (later extended to 1989.93), and reported finding “small periodic annual deviations of the data points from an exponential decay … of uncertain origin” [2].

In 2006, Fischbach and Jenkins of Purdue University set up an experiment to track the decay rate of ⁵⁴Mn as part of a project to determine whether or not beta decays are strictly random. They found, as had Falkenberg, that the decay rate appeared to be variable. This led them to examine the publication of Alburger et al. [2] and an article by Siegert et al. [3] concerning measurements of the decay rates of ²²⁶Ra acquired at the Physikalisch Technische Bundesanstalt in Germany. Jenkins and Fischbach proposed, as had Falkenberg, that the beta decay process may be influenced by solar neutrinos, and that the annual variation may be due to the varying Earth-Sun distance [4]. They also found evidence of a notable variation in the decay rate at the time of a solar flare on December 13, 2006, which led them to suggest that the beta decay process may somehow influence or be influenced by solar activity [5]. Although these two suggestions (that an annual oscillation in beta decays is due to the varying Earth-Sun distance and that beta decays are somehow correlated with solar activity) have not been substantiated by subsequent investigations, their articles were effective in drawing attention to the possibility of decay-rate variability.

Contrary to what one might hope for, but in line with reality, their suggestions quickly led to several nay-saying articles [6,7,8], to which Fischbach and his colleagues responded [9,10,11]. More recently, Kossert and Nahle have claimed to have evidence that beta-decay rates are constant [12], but their claims have been refuted [13]. The latest such article is one by Pomme et al. [14], who have published data concerning 67 measurements of a variety of decay processes, examining the measurements only for evidence of an Earth-Sun-Distance effect (which is known to be an inadequate test of variability) and claim to establish that there is no such effect. Their data have not yet been subjected to an independent analysis.

Although the first evidence for variability was the discovery of annual oscillations in decay rates, this approach to the problem has the obvious defect that an annual variation may be caused by any one of several experimental or environmental influences, which led us to seek evidence for an influence of solar rotation. We have learned from helioseismology that the synodic rotation rate (as seen from Earth) of the radiative zone is in the range 12.5 to 12.8 year^-1, whereas the synodic rotation rate of the photosphere extends to 13.8 year^-1 [15]. Variation of the solar neutrino flux may be attributable to the RSFP (Resonant Spin Flavor Precession) process [16], which can more easily occur in the deep solar interior (where there can be much stronger magnetic field) than in the convection zone [17]. This scenario would not lead one to expect an association between decay-rate variations and flare-like solar activity, which takes place in the outermost layer of the convection zone and in the solar atmosphere.

Power spectrum analysis of BNL data has in fact yielded evidence of oscillations in the frequency range 11 – 13 year^-1, supporting our conjecture that there may be a rotational influence on the solar neutrino flux [18,19]. However, these results raised the question of whether one could find corresponding evidence for solar rotation in measurements of the solar neutrino flux made by neutrino observatories such as Super-Kamiokande. An analysis of Super-Kamiokande data that takes account only of the mid-time of each bin, ignores the error estimates, and adopts an unrealistically wide search band, yields inconclusive results [20], but an analysis that takes account of the start and stop time of each measurement bin and of the upper and lower error estimates, and adopts an appropriate search band, yields strong evidence of an oscillation of frequency 9.43 year^-1 [21].

A crucial issue is whether or not solar influences are steady or time variable. Our investigations of GALLEX solar neutrino data indicated that the influence of rotation tends to be episodic [22]. This suggested that one should examine neutrino and beta-decay data by means of spectrograms rather than periodograms. These considerations led us to carry out a comparative analysis of spectrograms formed from the BNL data and from Super-Kamiokande data. The results have recently been published in Solar Physics [23].

Figure 1. Spectrogram formed from ³⁶Cl data for the frequency band 8 – 16 year^-1.

Figures 1 and 2, taken from that article, show spectrograms formed from BNL ³⁶Cl and ³²Si data, respectively. In order to focus on possible evidence of solar rotation, we show spectrograms only for the frequency range 8 to 16 year^-1. Figure 1 shows evidence of strong but transient oscillations at frequencies of approximately 11 year^-1 and 12.6 year^-1. Figure 2 also shows evidence of an oscillation at approximately 12.6 year^-1, but only slight evidence of an oscillation at approximately 11 year^-1. Evidence of these two oscillations has previously been found in power-spectrum analyses [18,19].

Figure 2. Spectrogram formed from ³²Si data for the frequency band 8 – 16 year^-1.

Figure 3 shows a spectrogram formed from Super-Kamiokande data, again for the frequency range 8 – 16 year^-1. This spectrogram shows evidence of a strong and steady oscillation at approximately 9.5 year^-1, as expected from our earlier power-spectrum analysis [21]. However, it also shows evidence of a transient oscillation with a frequency of approximately 12.6 year^-1, supporting the proposition that beta-decay variability may be attributed to an influence of solar neutrinos.

Figure 3. Spectrogram formed from Super-Kamiokande data for the frequency band of 8 – 16 year^-1.

The schedules of the relevant experiments were such that measurements leading to Figures 1 and 2 and measurements leading to Figure 3 were not acquired at the same time. It would clearly be desirable to compare beta-decay measurements and solar neutrino measurements that are acquired in the same time frame. The most extensive set of beta-decay measurements is the sequence currently (beginning in early 1992) being acquired by Steinitz and his colleagues at the Geological Survey of Israel (GSI) [24, 25]. The Borexino solar neutrino experiment began operation in 2007 and is still operational [26], so it may be possible at some time to compare beta-decay data with contemporaneous solar-neutrino data. It is however important to note that one may not find a perfect match between the two sets of data, even if beta decays are in fact influenced by neutrinos, since beta decays and neutrino detectors may respond to neutrinos of different energies and (since we have no theoretical understanding of beta-decay variability) conceivably of different flavors.

References:
[1] Eckhard Dieter Falkenberg, "Radioactive Decay Caused by Neutrinos?", Apeiron, 8, 32 (2001). Full Article.
[2] D.E. Alburger, G. Harbottle, E.F. Norton, "Half Life of ³²Si", Earth and Planetary Science Letters, 78, 168 (1986). Abstract.
[3] Helmut Siegert, Heinrich Schrader, Ulrich Schötzig, "Half-life measurements of Europium radionuclides and the long-term stability of detectors", Applied Radiation and Isotopes, 49, 1397 (1998). Abstract.
[4] Jere H. Jenkins, Ephraim Fischbach, John B. Buncher, John T. Gruenwald, Dennis E. Krause, Joshua J. Mattes, "Evidence of correlations between nuclear decay rates and Earth–Sun distance", Astroparticle Physics, 32, 42 (2009). Abstract.
[5] Jere H. Jenkins, Ephraim Fischbach, "Perturbation of nuclear decay rates during the solar flare of 2006 December 13", Astroparticle Physics, 31, 407 (2009). Abstract.
[6] Peter S. Cooper, "Searching for modifications to the exponential radioactive decay law with the Cassini spacecraft", Astroparticle Physics, 31, 267 (2009). Abstract.
[7] Eric B. Norman, Edgardo Browne, Howard A. Shugart, Tenzing H. Joshi, Richard B. Firestone, "Evidence against correlations between nuclear decay rates and Earth–Sun distance", Astroparticle Physics, 31, 135 (2009). Abstract.
[8] T.M. Semkowa, D.K. Hainesa, S.E. Beacha, B.J. Kilpatricka, A.J. Khana, K. O'Brienb, "Oscillations in radioactive exponential decay", Physics Letters B, 675, 415 (2009). Abstract.
[9] D.E. Krause, B.A. Rogers, E. Fischbach, J.B. Buncher, A. Ging, J.H. Jenkins, J.M. Longuski, N. Strange, P.A. Sturrock, "Searches for solar-influenced radioactive decay anomalies using spacecraft RTGs", Astroparticle Physics, 36, 51 (2012). Abstract.
[10] D. O’Keefe, B.L. Morreale, R.H. LeeJohn, B. Buncher, J.H. Jenkins, Ephraim Fischbach, T. Gruenwald, D. Javorsek II, P.A. Sturrock, "Spectral content of ²²Na/⁴⁴Ti decay data: implications for a solar influence", Astrophysics and Space Science, 344, 297 (2013). Abstract.
[11] Jere H. Jenkins, Daniel W. Mundy, Ephraim Fischbach, "Analysis of environmental influences in nuclear half-life measurements exhibiting time-dependent decay rates", Nuclear Instruments and Methods in Physics Research Section A. 620, 332 (2010). Abstract.
[12] Karsten Kossert, Ole J. Nähle, "Disproof of solar influence on the decay rates of ⁹⁰Sr/⁹⁰Y", Astroparticle Physics, 69, 18 (2015). Abstract.
[13] P.A. Sturrock, G. Steinitz, E. Fischbach, A. Parkhomov, J.D. Scargle, "Analysis of beta-decay data acquired at the Physikalisch-Technische Bundesanstalt: Evidence of a solar influence", Astroparticle Physics, 84, 8 (2016). Abstract.
[14] S. Pomméa, H. Stroh, J. Paepen, R. Van Ammel, M. Marouli, T. Altzitzoglou, M. Hult, K. Kossert, O. Nähle, H. Schrader, F. Juget, C. Bailat, Y. Nedjadi, F. Bochud, T. Buchillier, C. Michotte, S. Courte, M.W. van Rooy, M.J. van Staden, J. Lubbe, B.R.S. Simpson, A. Fazio, P. De Felice, T.W. Jackson, W.M. Van Wyngaardt, M.I. Reinhard, J. Golya, S. Bourke, T. Roy, R. Galea, J.D. Keightley, K.M. Ferreira, S.M. Collins, A. Ceccatelli, M. Unterweger, R. Fitzgerald, D.E. Bergeron, L. Pibida, L. Verheyen, M. Bruggeman, B. Vodenik, M. Korun, V. Chisté, M.-N. Amiot, "Evidence against solar influence on nuclear decay constants", Physics Letters B, 761, 281 (2016). Abstract.
[15] J. Schou, R. Howe, S. Basu, J. Christensen-Dalsgaard, T. Corbard, F. Hill, R. Komm, R. M. Larsen, M. C. Rabello-Soares, M. J. Thompson, "A Comparison of Solar p-Mode Parameters from the Michelson Doppler Imager and the Global Oscillation Network Group: Splitting Coefficients and Rotation Inversions", Astrophysical Journal, 567, 1234 (2002). Abstract.
[16] E. Kh. Akhmedov, "Resonant amplification of neutrino spin rotation in matter and the solar-neutrino problem", Physics Letters B, 213, 64 (1988). Abstract.
[17] João Pulido, C R Das, Marco Picariello, "Remaining inconsistencies with solar neutrinos: Can spin flavour precession provide a clue?", Journal of Physics: Conference series, 203, 012086 (2009). Abstract.
[18] P.A. Sturrock, J.B. Buncher, E. Fischbach, J.T. Gruenwald, D. Javorsek II, J.H. Jenkins, R.H. Lee, J.J. Mattes, J.R. Newport, "Power spectrum analysis of BNL decay rate data", Astroparticle Physics, 34, 121 (2010). Abstract.
[19] D. Javorsek II, P.A. Sturrock, R.N. Lasenby, A.N. Lasenby, J.B. Buncher, E. Fischbach, J.T. Gruenwald, A.W. Hoft, T.J. Horan, J.H. Jenkins, J.L. Kerford, f, R.H. Lee, A. Longman, J.J. Mattes, B.L. Morreale, D.B. Morris, R.N. Mudry, J.R. Newport, D. O’Keefe, M.A. Petrelli, M.A. Silver, C.A. Stewart, B. Terry, "Power spectrum analyses of nuclear decay rates", Astroparticle Physics, 34, 173 (2010). Abstract.
[20] J. Yoo et al. (Super-Kamiokande Collaboration), "Search for periodic modulations of the solar neutrino flux in Super-Kamiokande-I", Physical Review D, 68, 092002 (2003). Abstract.
[21] P.A. Sturrock, J.D. Scargle, "Power-Spectrum Analysis of Super-Kamiokande Solar Neutrino Data, Taking into Account Asymmetry in the Error Estimates", Solar Physics, 237, 1 (2006). Abstract.
[22] P.A. Sturrock, "Time–Frequency Analysis of GALLEX and GNO Solar Neutrino Data", Solar Physics, 252, 1 (2008). Abstract.
[23] P.A. Sturrock, E. Fischbach, J.D. Scargle, "Comparative Analyses of Brookhaven National Laboratory Nuclear Decay Measurements and Super-Kamiokande Solar Neutrino Measurements: Neutrinos and Neutrino-Induced Beta-Decays as Probes of the Deep Solar Interior", Solar Physics, 291, 3467 (2016). Abstract.
[24] G. Steinitz, O. Piatibratova, P. Kotlarsky, "Sub-daily periodic radon signals in a confined radon system", Journal of Environmental Radioactivity, 134, 128 (2014). Abstract.
[25] G. Steinitz, P. Kotlarsky, O. Piatibratova, "Observations of the relationship between directionality and decay rate of radon in a confined experiment", European Physical Journal, 224, 731 (2015). Abstract.
[26] S. Davini, G. Bellini, J. Benziger, D. Bick, G. Bonfini, D. Bravo, B. Caccianiga, F. Calaprice, A. Caminata, P. Cavalcante, A. Chepurnov, D. D'Angelo, A. Derbin, A. Etenko, K. Fomenko, D. Franco, C. Galbiati, C. Ghiano, A. Goretti, M. Gromov, Aldo Ianni, Andrea Ianni, V. Kobychev, D. Korablev, G. Korga, D. Kryn, M. Laubenstein, T. Lewke, E. Litvinovich, F. Lombardi, P. Lombardi, L. Ludhova, G. Lukyanchenko, I. Machulin, S. Manecki, W. Maneschg, S. Marcocci E. Meroni, M. Misiaszek, P. Mosteiro, V. Muratova, L. Oberauer, M. Obolensky, F. Ortica, K. Otis, M. Pallavicini, L. Papp, A. Pocar, G. Ranucci, A. Razeto, A. Re, A. Romani, N. Rossi, C. Salvo, S. Schönert, H. Simgen, M. Skorokhvatov, O. Smirnov, A. Sotnikov, S. Sukhotin, Y. Suvorov, R. Tartaglia, G. Testera, D. Vignaud, R. B. Vogelaar, J. Winter, M. Wojcik, M. Wurm, O. Zaimidoroga, S. Zavatarelli, G. Zuzel, "New results of the Borexino experiment: pp solar neutrino detection", Il Nuovo Cimento C, 38, 120 (2015). Abstract.

Experimental Simulation of the Exchange of Majorana Zero Modes

From left to right: (top row) Jin-Shi Xu, Kai Sun, Yong-Jian Han; (bottom row) Chuan-Feng Li, Jiannis K. Pachos, and Guang-Can Guo.

Authors: Jin-Shi Xu^1,2, Kai Sun^1,2, Yong-Jian Han^1,2, Chuan-Feng Li^1,2, Jiannis K. Pachos³, Guang-Can Guo^1,2

Affiliation:
¹Key Laboratory of Quantum Information, Department of Optics and Optical Engineering, University of Science and Technology of China, China,
²Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, China,
³School of Physics and Astronomy, University of Leeds, UK.

The exchange character of identical particles plays an important role in physics. For bosons, such an exchange leaves their quantum state the same, while a single exchange between two fermions gives a minus sign multiplying their wave function. A single exchange between two Abelian anyons gives rise to a phase factor that can be different than 1 or -1, that corresponds to bosons or fermions, respectively. More exotic exchanging character are possible, namely non-Abelian anyons. These particles have their quantum state change more dramatically, when an exchange between them takes place, to a possibly different state. Such non-Abelian anyons are the Majorana fermions that were first proposed by the physicist Majorana [1].

Majorana zero modes (MZMs) are quasiparticle excitations of topological phases of matter that have the same exchange character of the Majorana fermions, that is, they are non-Abelian anyons. When two MZMs are exchanged, the process cannot be described only by a global phase multiplying their wave function. Instead, their internal states, corresponding to degenerate ground states of the topological system, are transformed by a unitary operator. In addition, states encoded in the ground-state space of systems with MZMs are naturally immune to local errors. The degeneracy of the ground state, driven by the corresponding Hamiltonian, cannot be lifted by any local perturbation. Therefore, the encoded quantum information is topologically protected. These unusual characteristics imply that MZMs may potentially provide novel and powerful methods for quantum information processing [2]. Rapid theoretical developments have greatly reduced the technological requirements and made it possible to experimentally observe MZMs. However, until now, only a few positive signatures of the formation of MZMs have been reported in solid-state systems. The demonstration of the essential characteristic of non-Abelian exchange and the property of topological protection of MZMs is a considerable challenge.

Recently, we use a photonic quantum simulator to experimentally investigate the exchange of MZMs supported in the 1D Kitaev Chain Model (KCM) [3]. The Fock space of the Majorana system is mapped to the space of the quantum simulator by employing two steps. First, we perform the mapping of the Majorana system to a spin-1/2 system via the Jordan-Wigner (JW) transformation. Then we perform the mapping of the spin system to the spatial modes of single photons. In this way, we are able to demonstrate the exchange of two MZMs in a three-site Kitaev chain encoded in the spatial modes of photons. We further demonstrate that quantum information encoded in the degenerate ground state is immune to local phase and flip noise errors.

We consider a three-fermion KCM which is the simplest model that supports isolated two MZMs. Six Majorana fermions are involved and the exchange of two isolated Majoranas can be realized by a set of projective measurement, which can be expressed as imaginary-time evolution (ITE) operators with a sufficiently large evolution time. These processes depend on the corresponding Hamiltonians. Figure 1 shows the exchanging process.

Figure 1: The exchange of Majorana zero modes. The spheres with numbers at their centers represent the Majorana fermions at the corresponding sites. A pair of Majorana fermions bounded by an enclosing ring represents a normal fermion. The wavy lines between different sites represent the interactions between them. The interactions illustrated in a, b, c and d represent different Hamiltonians, respectively. The figure is adapted from Reference [3].

We transformed the KCM to a spin model through the JW transformation. Although these two models have some different physics, they share the same spectra in the ferromagnetic region and their corresponding quantum evolution are equivalent. The geometric phases obtained from the exchanging evolution are invariant under the mapping. As a result, the well-controlled spin system offers a good platform to determine the exchanging matrix and investigate the exchange behavior of MZMs.

In our experiment, the states of three spin-1/2 sites correspond to an eight-dimension Hilbert space, which are encoded in the optical spatial modes of a single photon. To complete the exchange, we implement the ITE by designing appropriate dissipative evolution. The ground state information of the corresponding Hamiltonian is preserved but the information of the other states is dissipated. We use beam-displacers to prepare the initial states and the dissipative evolution is accomplished by passing the photons through a polarization beam splitter. In our case, the optical modes with horizontal polarization are preserved which represent the ground states of the Hamiltonian. The optical modes with vertical polarization are discarded.

Figure 2 shows the experimental results of simulating the exchanging evolution. States encoded in the two-dimension degeneracy space are represented in Bloch spheres. The final states (Figure 2b) after the exchanging evolution are obtained by rotating the initial states (Figure 2a) counterclockwise along the X axis through an angle of π/2. We obtain the exchanging matrix through the quantum process tomography [4]. The real and imaginary parts of the exchanged operator are presented in Figures 2c and d. Compare with the theoretical operation, the fidelity is calculated to be 94.13±0.04%.

Figure 2: Experimental results on simulating the exchanging evolution. a. The six experimental initial states in the Bloch sphere. b. The corresponding experimental final states after the braiding evolution. The final states are shown to be rotated along the X axis by π/2 from the initial states. c. Real (Re) and d. Imaginary (Im) parts of the exchange operator with a fidelity of 94.13±0.04%. The figure is adapted from Reference [3].

Figures 3a and b show the real and imaginary parts of the flip-error protection operator with a fidelity of 97.91±0.03%. Figures 4c and d show the real and imaginary parts of the phase-error protection operator with a fidelity of 96.99±0.04%. The high fidelity reveals the protection from the local flip error and phase error of the information encoded in the ground state space of the Majorana zero modes. The total operation behaves as an identity, thus demonstrating immunity against noise.

Figure 3: Experimental results on simulating local noises immunity. a. Real (Re) and b. Imaginary (Im) parts of the flip-error protection operator, with a fidelity of 97.91±0.03%. c. Real (Re) and d. Imaginary (Im) parts of the phase-error protection operator, with a fidelity of 96.99±0.04%. The high fidelity reveals the protection from the local flip error and phase error of the information encoded in the ground state space of the Majorana zero modes. The figure is adapted from Reference [3].

In our experiment, the optical quantum simulator provides a versatile medium that can efficiently simulate the Kitaev chain model that supports MZMs at its endpoints. It also opens the way for the efficient realization and manipulation of MZMs in complex architectures. The gained know-how can be picked up by other technologies that offer scalability, like ion traps or optical lattices. This work achieves the realization of non-Abelian exchanging and may provide a novel way to investigate topological quantities of quantum systems.

References:
[1] Ettore Majorana, "Symmetrical theory of electrons and positrons", Nuovo Cimento 14, 171 (1937). Abstract.
[2] Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, Sankar Das Sarma, "Non-Abelian anyons and topological quantum computation". Review of Modern Physics, 80, 1083 (2008). Abstract.
[3] Jin-Shi Xu, Kai Sun, Yong-Jian Han, Chuan-Feng Li, Jiannis K. Pachos, Guang-Can Guo, "Simulating the exchange of Majorana zero modes with a photonic system". Nature Communications", 7, 13194 (2016). Abstract.
[4] J. L. O'Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, A. G. White, "Quantum process tomography of a Controlled-NOT gate". Physical Review Letters, 93, 080502 (2004). Abstract.

There Are Many Ways to Spin a Photon

Left to Right: Paul Eastham, Kyle Ballantine, John Donegan 

 Authors: Kyle E. Ballantine, John F. Donegan, Paul R. Eastham

 Affiliation: School of Physics and CRANN, Trinity College Dublin, Ireland

Can a boson, like a photon, have half-integer angular momentum? In three dimensions, no. The familiar quantum numbers l and m_l, for orbital angular momentum; s and m_s, for spin angular momentum; and j and m_j for the resulting total angular momentum, are all integers. However, a beam of light singles out a particular direction in space. The electric field, which must be perpendicular to this direction, is essentially a two-dimensional vector, specified over the plane perpendicular to the beam. Particles moving in two dimensions can have strange properties, including quantum numbers which are fractions of those expected in the general three-dimensional setting [1]. Given the restricted geometry of a beam of light, and the analogy with quantum mechanics in two-dimensions, it is intriguing to ask whether we could see similar effects there.

In our recent paper [2] we find that this is indeed the case: we show there is a physically reasonable form of angular momentum in a beam of light, which has an unexpected half-integer spectrum.

The study of light’s angular momentum [3] is an old one, going back to Poynting’s realization that circularly polarized light carries angular momentum because the electric field vector rotates. This spin angular momentum is one contribution to the total angular momentum carried by a light wave; the other is the orbital angular momentum, which arises from the spatial variation of the wave amplitude. We were led to the idea of the half-quantized angular momentum by the structure of beams generated by conical refraction, which is shown in Figure 1.

Figure 1: Cross section of conically refracted beam. The beam is a hollow cylinder, as can be seen from the intensity plotted in the gray scale. The direction of linear polarization at each point around the beam is shown by the red arrows; it takes a half-turn for one full turn around the beam. Figure adapted from [2].

This exotic form of refraction was discovered in our own institution, Trinity College Dublin, almost 200 years ago, by William Rowan Hamilton and Humphrey Lloyd. They showed that on passing through a “biaxial” crystal a ray of light became a hollow cylinder [4]. At each point around the cylinder the light is linearly polarised, meaning the electric field oscillates in a particular direction. However, if we take one full turn around the beam, the direction of linear polarisation takes only a half-turn. Conical refraction has introduced a topological defect into the beam [5]: a knot in the wave amplitude, which cannot be untied by smooth deformations of polarisation or phase. Similar transformations can be achieved using inhomogeneous polarizers called q-plates.

Any beam of light a beam can be decomposed into beams which have an exact value of some angular momentum. These are eigenstates of that angular momentum, defined by the property that when they are rotated they change only by a phase. For spin angular momentum, the relevant rotation is that of the electric field vectors, while for orbital angular momentum, it is a rotation of the amplitude. These rotations are both symmetries of Maxwell’s equations in the paraxial limit, so that they can be performed independently, or in any combination.

Thus the choice of basis for optical angular momentum, and the definition of the angular momentum operators, is not unique. If we consider beams which are rotationally symmetric under an equal rotation of the image and the polarisation, we get the conventional total angular momentum: the sum of orbital and spin quantum numbers, which is always an integer multiple of Planck’s constant, ħ. We showed that an equally valid choice is those beams which are symmetric when we rotate the image by one angle, and simultaneously rotate the polarisation by a half-integer multiple of that angle. The conically refracted beam is exactly of this form. The corresponding total angular momentum is a sum of the orbital contribution and one-half of the spin contribution, so that these beams have a total angular momentum which is shifted by ħ/2.

Figure 2: (A) Average angular momentum per photon as measured by interferometer. As the input beam is varied the average angular momentum goes from 1/2 to -1/2 in units of Planck's constant. (B) The quantum noise in the measured angular momentum. The minimum value corresponds exactly 1/2 of Planck’s constant being carried by each photon. (This Figure is reproduced from Ref.[2] ).

To measure this effect we built an interferometer, similar to the design used by Leach et al. [6]. The angular momentum eigenstates which make up any beam are, by definition, invariant under rotations up to a phase. When we rotate the beam, this phase means each component will interfere either constructively or destructively with the unrotated beam, so we can infer the amplitude of that component from the resulting intensity. In our experiment we rotated the amplitude and the polarisation by different amounts, which allowed us to measure the different types of angular momentum described above. The experimental results are shown in Fig 2(A). We use a quarter-wave plate (QWP) to vary the polarisation of our laser, and generate conically refracted beams with opposite handedness. As we move gradually between these beams the average of the relevant angular momentum varies between 1/2 and -1/2, in units of ħ.

Since photons with varying integer angular momentum could combine to give a fractional average, we wanted to show each photon carries exactly this amount. Rather than measuring single photons individually, we adopted a technique previously used to measure the charge of quasiparticles in the fractional quantum Hall effect [7]. This relies on the fact that in a current of particles there will be some inherent quantum noise, due to the discrete arrival of those particles, which is proportional to the size of the quantum of that current. We measured the quantum noise in the output angular momentum current of the interferometer described above. Fig 2(B) shows this noise, normalised in such a way that the minimum is the angular momentum carried by each photon, plus any excess classical noise still present. When the input beam is in either conical refraction state, this value dips well below one and approaches one half, demonstrating the half-integer angular momentum of each photon.

The possibility of exotic “fractional” quantum numbers [1] in two-dimensional quantum mechanics is known to occur in practice in electronic systems, and specifically in the quantum Hall effect. Our work is the first to show such behaviour for photons, and suggests that other aspects of this physics might be possible with light. Quantum optics gives the ability to transmit quantum information over large distances and process it at very high speeds. We have identified a new form of a familiar property, optical angular momentum, that may prove useful in such developments, and gives a new twist in our understanding of light.

References:
[1] Frank Wilczek, "Magnetic Flux, Angular Momentum, and Statistics", Physical Review Letters, 48, 1144 (1982). Abstract.
[2] Kyle E. Ballantine, John F. Donegan, Paul R. Eastham, "There are many ways to spin a photon: Half-quantization of a total optical angular momentum", Science Advances, 2, e1501748. Abstract.
[3] L. Allen, Stephen M. Barnett, Miles J. Padgett, "Optical Angular Momentum" (Institute of Physics Publishing, 2003).
[4] M. V. Berry, M. R. Jeffrey, "Conical diffraction: Hamilton's diabolical point at the heart of crystal optics", Progress in Optics, 50, 13 (2007). Abstract.
[5] J. F. Nye, "Lines of circular polarization in electromagnetic wave fields", Proceedings of the Royal Society A, 389, 279 (1983). Abstract.
[6] Jonathan Leach, Johannes Courtial, Kenneth Skeldon, Stephen M. Barnett, Sonja Franke-Arnold, Miles J. Padgett. "Interferometric Methods to Measure Orbital and Spin, or the Total Angular Momentum of a Single Photon", Physical Review Letters, 92, 013601 (2004). Abstract.
[7] C. L. Kane, Matthew P. A. Fisher, "Nonequilibrium noise and fractional charge in the quantum Hall effect", Physical Review Letters, 72, 724 (1994). Abstract.