The Real-Space Collapse of a Two Dimensional Polariton Gas

Photos of some of the authors -- From left to right: (top row) Lorenzo Dominici, Dario Ballarini, Milena De Giorgi; (bottom row) Blanca Silva Fernández, Fabrice Laussy, Daniele Sanvitto.

Authors:
Lorenzo Dominici¹, Mikhail Petrov², Michal Matuszewski³, Dario Ballarini¹, Milena De Giorgi¹, David Colas⁴, Emiliano Cancellieri^5,6, Blanca Silva Fernández^1,4, Alberto Bramati⁶, Giuseppe Gigli^1,7, Alexei Kavokin^2,8,9, Fabrice Laussy^4,10, Daniele Sanvitto¹.

Affiliation:
¹CNR NANOTEC—Istituto di Nanotecnologia, Lecce, Italy,
²Spin Optics Laboratory, Saint Petersburg State University, Russia,
³Institute of Physics, Polish Academy of Sciences, Warsaw, Poland,
⁴Física Teorica de la Materia Condensada, Universidad Autónoma de Madrid, Spain,
⁵Department of Physics and Astronomy, University of Sheffield, UK,
⁶Laboratoire Kastler Brossel, UPMC-Paris 6, ÉNS et CNRS, France,
⁷Università del Salento, Dipartimento di Matematica e Fisica “Ennio de Giorgi”,  Lecce, Italy,
⁸CNR-SPIN, Tor Vergata, Rome, Italy,
⁹Physics and Astronomy, University of Southampton, UK,
¹⁰Russian Quantum Center, Moscow Region, Skolkovo, Russia.

Can photons in vacuum interact?
The answer is not, since the vacuum is a linear medium where electromagnetic excitations and waves simply sum up, crossing themselves with no interaction. There exist a plenty of nonlinear media where the propagation features depend on the concentration of the waves or particles themselves. For example travelling photons in a nonlinear optical medium modify their structures during the propagation, attracting or repelling each other depending on the focusing or defocusing properties of the medium, and giving rise to self-sustained preserving profiles such as space and time solitons [1,2] or rapidly rising fronts such as shock waves [3,4].

One of the highest nonlinear effects can be shown by photonic microcavity (MC) embedding quantum wells (QWs), which are very thin (few tens of atomic distances) planar layers supporting electronic dipolar oscillations (excitons). What happens when a drop of photons, like a laser pulse, is trapped in a MC between two high reflectivity mirrors, and let to interact during this time with the electromagnetic oscillations of the QWs? If the two modes, photons and excitons, are tuned in energy each with the other, they cannot exist independently anymore and the result is the creation of a mixed, hybrid fluid of light and matter, which are known as the polaritons [5].

More specifically, we study the two-dimensional fluids of microcavity exciton polaritons, which can be enumerated among quantum or bosonic gases, and their hydrodynamics effects. Things become pretty nice since these polaritons behave partially as photons, in their light effective masses and fast speeds, and partially as excitons, with strong nonlinear interactions which can be exploited, for example, in all-optical transistors and logic gates [6]. Moreover, some photons continuously leak-out of the microcavity, bringing with them the information on the internal polariton fluid which can be on the one hand more straightforwardly studied with respect, for example, to atomic Bose-Einstein condensates, on the other hand making them out-of-equilibrium bosonic fluids.

Figure 1 (click on the image to view with higher resolution): Snapshots of the polariton fluid density and phase at significant instants. The amplitude and phase maps (the dashed circles depict the initial pump spot FWHM) have been taken at time frames of 0 ps, 2.8 ps and 10.4 ps, which correspond, respectively, to the pulse arrival, the ignition of the dynamical peak and its maximum centre density. The Figure has been extracted from Ref. [7].

In a recent study [7], we point out a very intriguing and unexpected effect, the dynamical concentration of the initial photonic pulse, upon its conversion into a polariton drop of high density. The accumulation of the field in a robust bright peak at the centre, as represented in Figure 1, is indeed surprising because it is at odds with the repulsive interactions of polaritons, which are expected to lead only to the expansion of the polariton cloud. The global phenomenology is spectacular because it is accompanied with the initial Rabi oscillations of the fluid [8,9] on a sub-picosecond scale, the formation of stable ring dark solitons [10,11], and the irradiation of planar ring waves on the external regions. Given the circular symmetry of the system, all these features can be represented in the time-space charts of Figure 2, where a central cross cut of the polariton cloud is represented during time.

Figure 2 (click on the image to view with higher resolution): Time-space charts of the polariton redistribution during time, for both the amplitude (a) and phase (b). The y-axis represents a central cross-cut of the circular-symmetry of the system and the x-axis represents time with a sample stepof 50 fs. Initially the polariton fluid oscillates with a Rabi period of about 800 fs (vertical stripes in the map), while the central density rapidly decays to zero before starting to rise as a bright peak. The two solid lines in both charts mark the phase disturbance delimiting the expanding region with large radial phase-gradient. The Figure has been extracted from Ref. [7].

From an application-oriented perspective we can devise features such as the enhancement ratio of the centre density with respect to the initial one (up to ten times in some experiments), the localization or shrinking factor of the original size (up to ten times as well), and the response speed (few picosecond rise time) and stability time (few tens of picosecond, well beyond the initial pulse length). These features can be tuned continuously with the intensity of the source laser pulse. Figure 3 reports the time dependence of the total population and of the relative centre density in one exemplificative case. The experiments have been reported in Nature Communications [7] and deserve, at least in a divulgative context, its own definition, which effect we like to refer to as the 'polariton backjet'. Indeed, its features are such to intuitively resemble the backjet of a water drop upon a liquid surface, while we devised the physics at the core as a collective polaron effect. This consists in the heating of the semiconductor lattice, resulting in the dynamical redshift of the exciton resonance. It is an interesting case of retarded nonlinearity inversion, leading to the self-sustained localization of the polariton condensate.

Figure 3. Total population and centre density versus time. Blue line are the experimental data of the area-integrated emission intensity, and the black line is a fit based on a model of coupled and damped oscillators. The red curve to be plotted on the right axis is the centre density versus time relative to that at the time of pulse arrival. The real enhancement factor obtained here in the centre density is 1.5, reached in a rise time of t = 10 ps. The Figure has been extracted from Ref. [7] Supplementary information.

The results have been obtained on a very high-quality QW-MC sample (quality factor of 14000) and upon implementing a state-of-the-art real-time digital holography setup. This latter is based on the coherence characteristics of the resonant polariton fluid and the possibility of retrieving its amplitude and phase distribution during ultrafast times upon the interference of the device emission with the laser pulse itself. Indeed this allowed also to prepare other interesting experiments dedicated to peculiar phenomena, such as the Rabi oscillations and their coherent [8] or polarization control [9] and the integer and half-integer quantum vortices [12] which can be excited on the polariton fluid. For most of these cases we could retrieve the complex wavefunction (which is given by an amplitude and phase) of the polariton fluid, with time steps of 0.1 or 0.5 ps and space steps as small as 0.16 micrometers. Fundamentally it is like making a movie on the micrometer scale with a 1.000.000.000.000 slow-motion ratio, as in the following video:



The fabrication and use of high quality microcavity polariton devices coupled to the most advanced characterization technique is opening a deep insight on fundamental properties of the coupling between light and matter and into exotic phenomena linked to condensation, topological states and many-body coherent and nonlinear fluids. Applications can be expected on the front of new polariton lasers, sub-resolution pixels, optical storage and clocks, data elaboration and multiplexing, sensitive gyroscopes, polarization and angular momentum shaping for optical tweezers and advanced structured femtochemistry.

References:
[1] S. Barland, M. Giudici, G. Tissoni, J. R. Tredicce, M. Brambilla, L. Lugiato, F. Prati, S. Barbay, R. Kuszelewicz, T. Ackemann, W. J. Firth, G.-L. Oppo, "Solitons in semiconductor microcavities", Nature Photonics, 6, 204–204 (2012). Abstract.
[2] Stephane Barland, Jorge R. Tredicce, Massimo Brambilla, Luigi A. Lugiato, Salvador Balle, Massimo Giudici, Tommaso Maggipinto, Lorenzo Spinelli, Giovanna Tissoni, Thomas Knödl, Michael Miller, Roland Jäger, "Cavity solitons as pixels in semiconductor microcavities", Nature, 419, 699–702 (2002)  Abstract.
[3] Wenjie Wan, Shu Jia, Jason W. Fleischer, "Dispersive superfluid-like shock waves in nonlinear optics", Nature Physics, 3, 46–51 (2006). Abstract.
[4] N. Ghofraniha, S. Gentilini, V. Folli, E. DelRe, C. Conti, "Shock waves in disordered media", Physical Review Letters, 109, 243902 (2012). Abstract.
[5] Daniele Sanvitto, Stéphane Kéna-Cohen, "The road towards polaritonic devices", Nature Materials (2016). Abstract.
[6] D. Ballarini, M. De Giorgi, E. Cancellieri, R. Houdré, E. Giacobino, R. Cingolani, A. Bramati, G. Gigli, D. Sanvitto, "All-optical polariton transistor", Nature Communications, 4, 1778 (2013). Abstract.
[7] L. Dominici, M. Petrov, M. Matuszewski, D. Ballarini, M. De Giorgi, D. Colas, E. Cancellieri, B. Silva Fernández, A. Bramati, G. Gigli, A. Kavokin, F. Laussy,  D. Sanvitto, "Real-space collapse of a polariton condensate", Nature Communications, 6, 8993 (2015). Abstract.
[8] L. Dominici, D. Colas, S. Donati, J. P. Restrepo Cuartas, M. De Giorgi, D. Ballarini, G. Guirales, J. C. López Carreño, A. Bramati, G. Gigli, E. del Valle, F. P. Laussy, D. Sanvitto, "Ultrafast Control and Rabi Oscillations of Polaritons", Physical Review Letters, 113, 226401 (2014). Abstract.
[9] David Colas, Lorenzo Dominici, Stefano Donati, Anastasiia A Pervishko, Timothy CH Liew, Ivan A Shelykh, Dario Ballarini, Milena de Giorgi, Alberto Bramati, Giuseppe Gigli, Elena del Valle, Fabrice P Laussy, Alexey V Kavokin, Daniele Sanvitto "Polarization shaping of Poincaré beams by polariton oscillations", Light: Science & Applications, 4, e350 (2015). Abstract.
[10] Yuri S. Kivshar, Xiaoping Yang, "Ring dark solitons", Physical Review E, 50, R40–R43 (1994). Abstract.
[11] A S Rodrigues, P G Kevrekidis, R Carretero-González, J Cuevas-Maraver, D J Frantzeskakis, F Palmero, "From nodeless clouds and vortices to gray ring solitons and symmetry-broken states in two-dimensional polariton condensates", Journal of Physics: Condensed Matter, 26, 155801 (2014). Abstract.
[12] Lorenzo Dominici, Galbadrakh Dagvadorj, Jonathan M. Fellows, Dario Ballarini, Milena De Giorgi, Francesca M. Marchetti, Bruno Piccirillo, Lorenzo Marrucci, Alberto Bramati, Giuseppe Gigli, Marzena H. Szymańska, Daniele Sanvitto, "Vortex and half-vortex dynamics in a nonlinear spinor quantum fluid", Science Advances, 1, e1500807 (2015). Abstract.

Quantum Tunneling of Water in Ultra-Confinement

From Left to Right: (top row) Alexander I. Kolesnikov, George F. Reiter, Narayani Choudhury, Timothy R. Prisk, Eugene Mamontov; (bottom row) Andrey Podlesnyak, George Ehlers,  David J. Wesolowski, Lawrence M. Anovitz.

Authors: Alexander I. Kolesnikov¹, George F. Reiter², Narayani Choudhury³, Timothy R. Prisk⁴, Eugene Mamontov¹, Andrey Podlesnyak⁵, George Ehlers⁵, Andrew G. Seel⁶, David J. Wesolowski⁴, Lawrence M. Anovitz⁴

Affiliation:
¹Chemical and Engineering Materials Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA,
²Physics Department, University of Houston, Texas, USA,
³Math and Science Division, Lake Washington Institute of Technology, Kirkland, Washington, USA; School of Science, Technology, Engineering and Math, University of Washington, Bothell, Washington, USA,
⁴Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA,
⁵Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA,
⁶ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, United Kingdom.

The quantum-mechanical behavior of light atoms plays an important role in shaping the physical and chemical properties of hydrogen-bonded liquids, such as water [1,2]. Tunneling is a classic quantum effect in which a particle moves through a potential barrier despite classically lacking sufficient energy to transverse it. The tunneling of hydrogen atoms in condensed matter systems has been observed for translational motions through metals, anomalous proton diffusion in water phases, and in the rotation of methyl and ammonia groups, and Gorshunov et al. inferred on the basis of terahertz spectroscopy measurements that water molecules inside the mineral beryl may undergo rotational tunneling [3, 4].

The crystal structure of beryl, shown in Figure 1, contains hexagonally shaped nanochannels just wide enough to contain single water molecules. In a recently published paper [5], we presented evidence from inelastic neutron scattering experiments and ab initio computational modeling that these water molecules do, in fact, undergo rotational tunneling at low temperatures. In their quantum-mechanical ground state, the hydrogen atoms are delocalized among the six symmetrically-equivalent positions about the channels so that the water molecule on average assumes a double-top like shape.

Figure 1: The crystal structure of beryl

The first set of inelastic neutron scattering experiments was performed using the CNCS and SEQUOIA spectrometers located at Oak Ridge National Laboratory's Spallation Neutron Source. A number of transitions are observed in the energy spectrum that can only be attributed to quantum-mechanical tunneling. Alternative origins for these transitions, such as vibrational modes or crystal field effects of magnetic impurities, are inconsistent with the temperature and wavevector dependence of the energy spectrum. However, they are consistent with an effective one-dimensional orientational potential obtained from Density Functional Theory and Path Integral Molecular Dynamics calculations.

To confirm these results we performed neutron Compton scattering of experiments on beryl single-crystals using the VESUVIO spectrometer at the Rutherford Appleton Laboratory. In this technique, a high-energy incident neutron delivers an impulsive blow to a single atom in the sample, transferring a sufficiently large amount of kinetic energy to the target atom that it recoils freely from the impact. The momentum distribution n(p) of the hydrogen atoms may then be inferred from the observed dynamic structure factor S(Q, E) in this high-energy limit, providing a direct probe of the momentum-space wavefunction of the water hydrogens in beryl.

Figure 2: the measured momentum distribution n(p) in neutron Compton scattering experiments.

The tunneling behavior of the water protons is revealed in our neutron Compton scattering experiments by the measured momentum distribution n(p), illustrated as a color contour plot in Figure 2. The variation of n(p) with angle is due to vibrations of the O—H covalent bond. If it is true that water molecules undergo rotational tunneling between the six available orientations, then n(p) will include oscillations or interference fringes as a function of angle. On the other hand, if the water molecules are incoherently and randomly arranged among the possible positions, then no such interference fringes will be observed. As marked by the yellow line in Figure 2, the interference fringes were clearly observed in our experiment! The water molecule is, therefore, in a coherent superposition of states over the six available orientational positions.

Taken together, these results show that water molecules confined in the channels in the beryl structure undergo rotational tunneling, one of the hallmark features of quantum mechanics.

References:
[1] Michele Ceriotti, Wei Fang, Peter G. Kusalik, Ross H. McKenzie, Angelos Michaelides, Miguel A. Morales, Thomas E. Markland, "Nuclear Quantum Effects in Water and Aqueous Systems: Experiment, Theory, and Current Challenges", Chemical Reviews, 116, 7529 (2016). Abstract.
[2] Xin-Zheng Li, Brent Walker, Angelos Michaelides, "Quantum nature of the hydrogen bond", Proceedings of the national Academy of Sciences of the United States of America, 108, 6369 (2011). Abstract.
[3] Boris P. Gorshunov, Elena S. Zhukova, Victor I. Torgashev, Vladimir V. Lebedev, Gil’man S. Shakurov, Reinhard K. Kremer, Efim V. Pestrjakov, Victor G. Thomas, Dimitry A. Fursenko, Martin Dressel, "Quantum Behavior of Water Molecules Confined to Nanocavities in Gemstones", The Journal of Physical Chemistry Letters, 4, 2015 (2013). Abstract.
[4] Boris P. Gorshunov, Elena S. Zhukova, Victor I. Torgashev, Elizaveta A. Motovilova, Vladimir V. Lebedev, Anatoly S. Prokhorov, Gil’man S. Shakurov, Reinhard K. Kremer, Vladimir V. Uskov, Efim V. Pestrjakov, Victor G. Thomas, Dimitri A. Fursenko, Christelle Kadlec, Filip Kadlec, Martin Dressel, "THz–IR spectroscopy of single H₂O molecules confined in nanocage of beryl crystal lattice", Phase Transitions, 87, 966 (2014). Abstract.
[5] Alexander I. Kolesnikov, George F. Reiter, Narayani Choudhury, Timothy R. Prisk, Eugene Mamontov, Andrey Podlesnyak, George Ehlers, Andrew G. Seel, David J. Wesolowski, Lawrence M. Anovitz, "Quantum Tunneling of Water in Beryl: A New State of the Water Molecule", Physical Review Letters, 116, 167802 (2016). 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.

Demonstrating Quantum Advantage with the Simplest Quantum System -- Qubit

From left to right: Xiao Yuan, Ke Liu, Xiongfeng Ma, Luyan Sun.

Authors: Xiao Yuan, Ke Liu, Luyan Sun, Xiongfeng Ma

Affiliation:
Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China.

Along the search for quantum algorithms, the sophisticated quantum speed-up generally arises with delicately designed quantum circuits by manipulating quantum states that contain intricate multipartite correlations. While the essence of quantum correlation originates from coherent superposition of different states, it is natural to expect the essence of quantum advantage to also originate from coherence. This raises a fundamental question: Can quantum advantage be obtained even with the simplest quantum state system, qubit, i.e., superposition of a two level system. The question was answered affirmatively in our recent work published in Physical Review Letters on June 29th [1].

In our everyday life, a classical coin is called p-coin if it outputs a head and a tail with probability p and 1-p respectively. Given an unknown p-coin, a simple yet interesting problem is to construct an f (o)-coin, where f (p) is a given function of p and f(p)∈[0,1]. For example, when f (p)=1/2, a rather simple but heuristic strategy is given by Von Neumann [2]. Flip the p-coin (p ≠ 0) twice. If the outcomes are the same, start over; otherwise, output the second coin value as the 1/2-coin output. In general, such construction processing is called a Bernoulli factory. Solved by Keane and O’Brien [3], it says that not all functions can be constructed classically. Generally speaking, a necessary condition for f(p) being constructible is that f(p) ≠ 0 or 1 when p ∈ (0,1). A simple example is f (p) = 4p (1-p), where we have f (1/2)=1.

Figure 1: Classical and quantum coin.

As shown in Fig.1, a p-coin corresponds to a machine that outputs identically mixed qubit states, ρ_c = p|0⟩⟨0| + (1-p)|1⟩⟨1|, where p∈[0,1]. In general, such unknown p can also be encoded in a quantum way, |p⟩ = √p |0⟩ + √(1-p) |1⟩, which is called by a quoin. As classical coins can always be constructed via a quoin, a natural question is whether the set of quantum constructible functions (via a quantum Bernoulli factory) is strictly larger than the classical set.

Remarkably, Dale et al. [4] have theoretically proved the necessary and sufficient conditions for quantum Bernoulli factory. Especially for the function f(p) = 4 p (1-p), they proposed a method to construct it by simultaneously measuring two p-quoins. Essentially, entanglement is not necessary for constructing quantum Bernoulli factory. Therefore, we focus on the function f(p) = 4p (1-p) and show the quantum advantage in both theory and experiment with the simplest quantum system.

In practice, we cannot realize exact f(p)-coin due to imperfections, which may cause the realized function classically constructible. However, the number of classical coins N required to construct f(p) generally scales poorly to the inverse of the deviation. Thus, we need to implement high-fidelity state preparation and measurement to reduce the deviation as small as possible in order to faithfully demonstrate the quantum advantage. Superconducting quantum systems have made tremendous progress in the last decade, including a realization of long coherence times, showing great stability with fast and precise qubit manipulations, and demonstrating high-fidelity quantum non-demolition (QND) qubit measurement. Thus, it serves as a perfect candidate for our test.

Figure 2: Experimental setup. (a) Optical image of a transmon qubit located in a trench, which dispersively couples to two 3D Al cavities. (b) Optical image of the single-junction transmon qubit. (c) Scanning electron microscope image of the Josephson junction. (d) Schematic of the device with the main parameters.

The experiment setup is shown in Fig. 2. The necessary high fidelity (~99.6%) and QND qubit detection can be realized with the help of a near-quantum-limited Josephson parametric amplifier [4,5]. A randomized benchmark calibration shows that the single-qubit gate fidelity is about 99.8%, allowing for a highly precise qubit manipulation. Therefore, with the high fidelity state preparation, manipulation and measurement, we are able to achieve f_exp (1/2)=0.965. For a special model of the experiment data, we show that more than 10⁵ classical coins are needed for simulating this model, while the average number of quoins for our protocol is about 20.

Our experimental verification sheds light on a fundamental question about what the essential resource for quantum information processing is, which may stimulate the search for more protocols that show quantum advantages without multipartite correlations. Considering the conversion from coherence to multipartite correlation, investigating the power of coherence may also be helpful in understanding the power of multipartite correlation and universal quantum computation.

References:
[1] Xiao Yuan, Ke Liu, Yuan Xu, Weiting Wang, Yuwei Ma, Fang Zhang, Zhaopeng Yan, R. Vijay, Luyan Sun, Xiongfeng Ma, "Experimental Quantum Randomness Processing Using Superconducting Qubits", Physical Review Letters, 117, 010502 (2016). Abstract.
[2] J. Von Neumann, "Various Techniques used in connection with random digits", Journal of Research of the National Bureau of Standards -- Applied Mathematics Series, 12, 36 (1951). PDF File.
[3] M. S. Keane, George L. O’Brien, "A Bernoulli factory", ACM Transactions on Modeling and Computer Simulation, 4, 213 (1994). Abstract.
[4] Howard Dale, David Jennings, Terry Rudolph, Nature Communications, 6, 8203 (2015). Abstract.
[5] M. Hatridge, R. Vijay, D.H. Slichter, John Clarke, I. Siddiqi, "Dispersive magnetometry with a quantum limited SQUID parametric amplifier", Physical Review B, 83, 134501 (2011). Abstract.

Observation of Genuine One-Way Einstein-Podolsky-Rosen Steering

From Left to Right: Sabine Wollmann, Howard Wiseman, Geoff Pryde.

Authors:
Sabine Wollmann¹, Nathan Walk^1,2, Adam J. Bennet¹, Howard M. Wiseman¹, Geoff J. Pryde¹

Affiliations:
¹Centre for Quantum Computation and Communication Technology and Centre for Quantum Dynamics, Griffith University, Brisbane, Queensland, Australia.
²Department of Computer Science, University of Oxford, United Kingdom.

Quantum entanglement, a nonlocal phenomenon, is a key resource for foundational quantum information and communication tasks, such as teleportation, entanglement swapping and quantum key distribution. The idea of this widely investigated feature was first discussed by Albert Einstein, Boris Podolsky and Nathan Rosen (EPR) in 1935 [1]. In their famous thought experiment they consider a maximally-entangled state shared between two observers, Alice and Bob. Alice makes a measurement on her system and controls Bob’s measurement outcomes by her choice of measurement setting. They concluded from this counterintuitive effect, which Einstein later called ‘spooky action at a distance’, that quantum theory must be incomplete and an underlying hidden variable model must exist. It took another 29 years until it was proven by Bell that there exist predictions of quantum mechanics for which no possible local hidden variable model could account for [2].

Figure 1: Illustration of one-way EPR steering. Alice and Bob share a state which only allows Alice to demonstrate steering.

It was not until recently that the class of nonlocality described by EPR, being intermediate to entanglement witness tests and Bell inequality violations, was formalized by Wiseman et al. [3]. While the previous classes are a symmetric feature - in the sense that, the effects persist under exchange of the parties - this does not necessarily hold for EPR steering. The asymmetry arises because one of the parties is trusted (i.e. their measurements are assumed to be faithfully described by quantum mechanics) and the other is not. In this distinctive class of nonlocality we usually consider a shared state between the two parties Alice and Bob. The question which arises is whether sharing an asymmetric state can result in one-way EPR steering, where Alice can steer Bob, for example, but not the other way around.

This foundational question was first experimentally addressed by Haendchen et al. in 2012, who demonstrated experimentally Gaussian one-way EPR steering with two-mode squeezed states [4]. However their experimental investigation was in a limited context: Gaussian measurements on Gaussian states. As we have shown [5] there exist explicit examples of supposedly one-way steerable Gaussian states actually being two-way steerable for a broader class of measurements. So one could ask, do states exist which are one-way steerable for arbitrary measurements? And the answer is yes. Two independent groups, Nicolas Brunner’s in Geneva and Howard Wiseman’s in Brisbane, proved the existence of such states. Brunner’s approach holds for arbitrary measurements with infinite settings, so called infinite-setting positive-operator-valued measures (POVM), with the cost of using an exotic family of states to demonstrate the effect over an extremely small parameter range, which is unsuitable for experimental observation [6]. David Evans and Howard Wiseman showed one-way steerability exists for projective measurements of more practical, singlet states with symmetric noise - so called Werner states – and loss [7].

Figure 2: Creation of a one-way steerable state (see text for details). One half of a Werner state ρ_W is sent directly to Alice, whose measurements are described by {M_a|k}, while the other is transmitted to Bob through a loss channel, which replaces a qubit with the vacuum state and is parametrized by probability p. Bob’s measurements are described by {M_b|j}. For differing values of p the final state is unsteerable by Bob for arbitrary projective measurements or arbitrary POVMs. For the same range of p values, Alice can explicitly demonstrate steering via a finite number of Pauli measurements on both sides. She does this by steering Bob’s measurement outcomes so that their shared correlations exceed the upper bound Cn allowed in an optimal local hidden state model.

In our work, recently published in [5], we ask if we can extend the result in Ref. [6] to find a simple state which is steerable in one direction but cannot be steered in the other direction even for the case of arbitrary measurements and infinite settings. For that we consider a shared Werner state

between our observers Alice and Bob. This is a probabilistic mixture of a maximally entangled singlet state with a symmetric noise state parametrised by the mixing probability, or Werner parameter, µ. Using the theorem of Quintino et al. [5] allowed us to construct a state

where the probability p of a vacuum state represents adding asymmetric loss in Bob’s arm. This state is one-way steerable for arbitrary measurements, if we can fulfil the condition

for loss p .

Figure 3: In the experimental scheme, Alice and Bob are represented by black and green boxes, respectively. Both are in control of their line and their detectors. The party that is steering is additionally in control of the source. Entangled photon pairs at 820 nm were produced via SPDC in a Sagnac interferometer. Different measurement settings are realized by rotating half- and quarter-wave plates (HWP and QWP) relative to the polarizing beam splitters. A gradient neutral density (ND) filter is mounted in front of Bob’s line to control the fraction of photon qubits passing through. Long pass (LP) filters remove 410 nm pump photons copropagating with the 820 nm photons before the latter are coupled into single-mode fibers and detected by single photon counting modules and counting electronics.

In our experiment we generate an one-way steerable state for projective measurements with a fidelity of (99.6±0.01)% with a Werner state of µ = 0.991±0.003 and insert a filter into Bob’s line to generate the loss p = (87±3)%. To demonstrate that Alice remains able to steer Bob’s state, it is necessary to violate the EPR steering inequality. That means measuring a correlation function – the so called steering parameter S_n – which exceeds the classically allowed value. We observe that Alice’s steering parameter of S₁₆ = 0.970±0.004 is 7.3 standard deviations above the classical bound at an heralding efficiency of η = (17.11±0.07)%. The loss of information in Bob’s arm makes him unable to steer the other party. We observe a steering parameter of S₁₆ = 0.963±0.006. In this case, this S value would not have violated a steering inequality even with an infinite number of measurements.

The second one-way steerable regime which we investigate, does still allow Alice to steer Bob’s state but he remains unable to steer hers even by using POVMs. To demonstrate this case, we produce a state with a fidelity of (99.1±0.3)% with a Werner state of µ = 0.978±0.008 and applied a loss p =(99.5±0.3)%. Alice remains able to steer Bob with a steering parameter S₁₆ = 0.951±0.005, being 6.6 standard deviations above the classical bound, at an heralding efficiency of η = (17.17±0.04)%. Bob’s steering parameter S₁₆ = 0.951±0.006 does not violate the inequality and there is no kind of measurement he could choose, even in principle, to be able to steer Alice. We note that the shared state is not exactly a Werner state, but the extremely high fidelities imply, with low probability of error, that the state is only one-way steerable.

Thus, we observe genuine one-way EPR steering for the first time. We note that an independent demonstration was realised in Ref.[8]. While their result is restricted to two measurement settings, our experimental demonstration holds for an arbitrary number of measurements.

References:
[1] A. Einstein, B. Podolsky, N. Rosen, "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?", Physical Review, 47, 777 (1935). Abstract.
[2] John S. Bell, “On the Einstein Podolsky Rosen paradox”, Physics, 1, 195 (1964). Full Text.
[3] H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox”, Physical Review Letters, 98, 140402 (2007). Abstract.
[4] Vitus Händchen, Tobias Eberle, Sebastian Steinlechner, Aiko Samblowski, Torsten Franz, Reinhard F. Werner, and Roman Schnabel, “Observation of one-way Einstein-Podolsky-Rosen steering”, Nature Photonics, 6, 596 (2012). Abstract.
[5] Sabine Wollmann, Nathan Walk, Adam J. Bennet, Howard M. Wiseman, and Geoff J. Pryde, "Observation of Genuine One-Way Einstein-Podolsky-Rosen Steering", Physical Review Letters, 116, 160403 (2016). Abstract.
[6] Marco Túlio Quintino, Tamás Vértesi, Daniel Cavalcanti, Remigiusz Augusiak, Maciej Demianowicz, Antonio Acín, Nicolas Brunner, "Inequivalence of entanglement, steering, and Bell nonlocality for general measurements", Physical Review A, 92, 032107 (2015). Abstract.
[7] D. A. Evans, H. M. Wiseman, "Optimal measurements for tests of Einstein-Podolsky-Rosen steering with no detection loophole using two-qubit Werner states", Physical Review A, 90, 012114 (2014). Abstract.
[8] Kai Sun, Xiang-Jun Ye, Jin-Shi Xu, Xiao-Ye Xu, Jian-Shun Tang, Yu-Chun Wu, Jing-Ling Chen, Chuan-Feng Li, and Guang-Can Guo, “Experimental quantification of asymmetric Einstein-Podolsky-Rosen steering”, Physical Review Letters, 116, 160404 (2016). Abstract. 2Physics Article.

Demonstrating One-Way Einstein-Podolsky-Rosen Steering in Two Qubits

Some authors of the PRL paper (Reference 6) published on Thursday. From Left to Right: (top row) Kai Sun, Xiang-Jun Ye, Jin-Shi Xu, (bottom row) Jing-Ling Chen, Chuan-Feng Li, Guang-Can Guo.

Authors: Kai Sun¹, Xiang-Jun Ye¹, Jin-Shi Xu¹, Jing-Ling Chen², Chuan-Feng Li¹, Guang-Can Guo¹

Affiliation:
¹Key Laboratory of Quantum Information, CAS Centre for Excellence and Synergetic Innovation Centre in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, China.
²Chern Institute of Mathematics, Nankai University, Tianjin, China.

Asymmetric Einstein-Podolsky-Rosen (EPR) steering is an important “open question” proposed when EPR steering is reformulated in 2007 [1]. Suppose Alice and Bob share a pair of two-qubit states; it is easy to imagine that if Alice entangles with Bob, then Bob must also entangle with Alice. Such a symmetric feature holds for both entanglement and Bell nonlocality [2]. However, the situation is dramatically changed when one turns to a novel kind of quantum nonlocality, the EPR steering, which stands between entanglement and Bell nonlocality. It may happen that for some asymmetric bipartite quantum states, Alice can steer Bob but Bob cannot steer Alice. This distinguished feature would be useful for the one-way quantum tasks. The first experimental verification of one-way EPR steering was performed by using two entangled continuous variable systems in 2012 [3]. However, the experiments demonstrating one-way EPR steering [3,4] are restricted to Gaussian measurements, and for more general measurements, like projective measurements, there is no experiment realizing the asymmetric feature of EPR steering, even though the theoretical analysis has been proposed [5].

Figure 1: Illustration of one-way EPR steering. In one direction (red), EPR steering is realized and this direction is safe for quantum information. In the other direction (blue), steering task fails and this direction is not safe.

Recently, we for the first time quantified the steerability and demonstrated one-way EPR steering in the simplest entangled system (two qubits) using two-setting projective measurements [6]. The asymmetric two-qubit states in the form of ρ_AB = η |Ψ(θ)⟩⟨Ψ(θ)| + (1-η) |Φ(θ)⟩⟨Φ(θ)|, where 0 ≤ η≤ 1, |Ψ(θ)⟩ = cos ⁡θ |0_A 0_B⟩ + sin⁡θ |1_A 1_B⟩, |Φ(θ)⟩ = cos⁡θ |1_A 0_B⟩ + sin⁡θ |0_A 1_B⟩, are prepared in this experiment (see Figure 2(a) ). Based on the steering robustness [7], an intuitive criterion R called as “steering radius” is defined to quantify the steerability (see Figure 2 (c) ). The different values of R on two sides clearly illustrate the asymmetric feature of EPR steering. Furthermore, the one-way steering is demonstrated when R > 1 on one side and R < 1 on the other side (see Figure 2 (b)).

Figure 2:  (click on the figure to view with higher resolution)  Experimental results for asymmetric EPR steering. (a) The distribution of the experimental states. The right column shows the entangled states we prepared, and the left column is a magnification of the corresponding region in the right column. The states located in the yellow (grey) regions are predicted to realize one-way (two-way) steering theoretically in the case of two-setting measurements. The blue points and red squares represent the states realizing one-way and two-way EPR steering, respectively. The black triangles represent the states for which EPR steering task fails for both observers. (b) The values of R for the states are labeled in the left column in (a). The red squares represent the situation where Alice steers Bob's system, and the blue points represent the case where Bob steers Alice's system. (c) Geometric illustration of the strategy for local hidden states (black points) to construct the four normalized conditional states (red points) obtained from the maximally entangled state.

For the failing EPR steering process, the local hidden state model, which provides a direct and convinced contradiction between the nonlocal EPR steering and classical physics, is prepared experimentally to reconstruct the conditional states obtained in the steering process (see Figure 3).

Figure 3. (click on the figure to view with higher resolution) The experimental results of the normalized conditional states and local hidden states shown in the Bloch sphere. The theoretical and experimental results of the normalized conditional states are marked by the black and red points (hollow), respectively. The blue and green points represent the results of the four local hidden states in theory and experiment, respectively. The normalized conditional states constructed by the local hidden states are shown by the brown points. Spheres (a) and (c) are for the case in which Alice steers Bob's system, whereas (b) and (d) show the case in which Bob steers Alice's system. The parameters of the shared state in (a) and (b) are θ = 0.442 and η = 0.658; the parameters of the shared state in (c) and (d) are θ = 0.429 and η = 0.819. The spheres (a), (b) and (d) show that the local hidden state models exist, and the steering tasks fail. The sphere (c) Shows that no local hidden state model exists for the steering process with the constructed hidden states located beyond the Bloch sphere and R = 1.076.

The quantification of EPR steering provides an intuitional and fundamental way to understand the EPR steering and the asymmetric nonlocality. The demonstrated asymmetric EPR steering is significant within quantum foundations and quantum information, and shows the applications in the tasks of one-way quantum key distribution [8] and the quantum sub-channel discrimination [7], even within the frame of two-setting measurements.

References:
[1] H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox”, Physical Review Letters, 98, 140402 (2007). Abstract.
[2] John S. Bell, “On the Einstein Podolsky Rosen paradox”, Physics, 1, 195 (1964). Full Text.
[3] Vitus Händchen, Tobias Eberle, Sebastian Steinlechner, Aiko Samblowski, Torsten Franz, Reinhard F. Werner, and Roman Schnabel, “Observation of one-way Einstein-Podolsky-Rosen steering”, Nature Photonics, 6, 596 (2012). Abstract.
[4] Seiji Armstrong, Meng Wang, Run Yan Teh, Qihuang Gong, Qiongyi He, Jiri Janousek, Hans-Albert Bachor, Margaret D. Reid, and Ping Koy Lam, “Multipartite Einstein-Podolsky-Rosen steering and genuine tripartite entanglement with optical networks”, Nature Physics, 11, 167 (2015). Abstract.
[5] Joseph Bowles, Tamás Vértesi, Marco Túlio Quintino, and Nicolas Brunner, “One-way Einstein-Podolsky-Rosen steering”, Physical Review Letters, 112, 200402 (2014). Abstract.
[6] Kai Sun, Xiang-Jun Ye, Jin-Shi Xu, Xiao-Ye Xu, Jian-Shun Tang, Yu-Chun Wu, Jing-Ling Chen, Chuan-Feng Li, and Guang-Can Guo, “Experimental quantification of asymmetric Einstein-Podolsky-Rosen steering”, Physical Review Letters, 116, 160404 (2016). Abstract.
[7] Marco Piani, John Watrous, “Necessary and sufficient quantum information characterization of Einstein-Podolsky-Rosen steering”, Physical Review Letters, 114, 060404 (2015). Abstract.
[8] Cyril Branciard, Eric G. Cavalcanti, Stephen P. Walborn, Valerio Scarani, and Howard M. Wiseman, “One-sided device-independent quantum key distribution: security, feasibility, and the connection with steering”, Physical Review A, 85, 010301 (2012). Abstract.

Efficient Long-Distance Heat Transport by Microwave Photons

Research team behind the original discovery (Reference [11] ) from left to right: Tuomo Tanttu, Joonas Goveenius, Mikko Möttönen, Matti Partanen, and Miika Mäkelä (Missing from the figure: Kuan Yen Tan and Russell Lake). Photo Credit: Vilja Pursiainen/Kaskas Media.

Authors: Matti Partanen and Mikko Möttönen

Affiliation: QCD Labs, Department of Applied Physics, Aalto University, Finland.

Link to the Quantum Computing and Devices (QCD) Group >>

Quantum computers are predicted to vastly speed up the computation for certain problems of great practical interest [1]. One of the most promising architectures for quantum computing is based on superconducting quantum bits [2], or qubits, which are the key ingredients in circuit quantum electrodynamics [3]. In such systems, the control of heat at the quantum level is extremely important, and remote cooling may turn out to be a viable option.

In one dimension, heat transport may be described by individual heat conduction channels -- each corresponding to a certain quantized profile of the heat carriers in the transverse direction. Importantly, the maximum heat power flowing in a single channel between bodies at given temperatures is fundamentally limited by quantum mechanics [4,5]. This quantum limit has previously been observed for phonons [6], sub-wavelength photons [7,8], and electrons [9]. Among these, the longest distance of roughly 50 μm [7,8] was recorded in the photonic channel [10]. Such short distance may be undesirable in cooling quantum devices which are sensitive to spurious dissipation.

In our recent work [11], we observe quantum-limited heat conduction by microwave photons flying in a superconducting transmission line of length 20 cm and 1 m. Thus we were able to extend the maximum distance 10,000 fold compared with the previous experiments.

Figure 1: (click on the figure to view with higher resolution) Sample structure and measurement scheme. The electron temperature of the right resistor is controlled with an external voltage while the temperatures of both resistors are measured. Microwave photons transport heat through the spiraling transmission line.

Our sample is shown in Figure 1. The heat is transferred between two normal-metal resistors functioning as black-body radiators to the transmission line [10,12]. To be able to fabricate the whole sample on a single relatively small chip, the transmission line has a double spiral structure. We have measured such spiraling transmission lines without resistors and confirmed that photons travel along the line; they do not jump through vacuum from one end to the other. Thus for heat transport, the distance should be measured along the line.

We measure the electron temperatures of both normal-metal resistors while we change the temperature of one of them [13]. The obtained temperature data agrees well with our thermal model, according to which the heat conduction is very close to the quantum limit.

In contrast to subwavelength distances employed in References [7,8], we need to match the resistance of the normal-metal parts to the characteristic impedance of the transmission line to reach the quantum limit. Furthermore, the transmission line itself has to be so weakly dissipative that almost no photons are absorbed even over distances of about a meter. However, we managed to develop nanofabrication techniques which enabled us to satisfy these conditions well. In fact, the losses in the transmission line are so weak they allow a further increment of the distance by several orders of magnitude.

We consider that long-distance heat transport through transmission lines may be a useful tool for certain future applications in the quickly developing field of quantum technology. If the coupling of a quantum device to a low-temperature transmission line can be well controlled in situ, the device may be accurately initialized without disturbing its coherence properties when the coupling is turned off [14]. Furthermore, the implementation of such in-situ-tunable environments opens an interesting avenue for the study of the detailed dynamics of open quantum systems and quantum fluctuation relations [15].

Acknowledgements: We thank M. Meschke, J. P. Pekola, D. S. Golubev, J. Kokkala, M. Kaivola and J. C. Cuevas for useful discussions, and L. Grönberg, E. Mykkänen, and A. Kemppinen for technical assistance. We acknowledge the provision of facilities and technical support by Aalto University at Micronova Nanofabrication Centre. We also acknowledge funding by the European Research Council under Starting Independent Researcher Grant No. 278117 (SINGLEOUT), the Academy of Finland through its Centres of Excellence Program (project nos 251748 and 284621) and grants (nos 138903, 135794, 265675, 272806 and 276528), the Emil Aaltonen Foundation, the Jenny and Antti Wihuri Foundation, and the Finnish Cultural Foundation.

References:
[1] T.D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, J.L. O'Brien, “Quantum computers”, Nature, 464, 45 (2010). Abstract.
[2] J. Kelly, R. Barends, 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, A.N. Cleland, John M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit”, Nature, 519, 66 (2015). Abstract.
[3] A. Wallraff, D.I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S.M. Girvin, R.J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum Electrodynamics”, Nature, 431, 162 (2004). Abstract.
[4] J.B. Pendry, “Quantum limits to the flow of information and entropy”, Journal of Physics A: Mathematical and General, 16, 2161 (1983). Abstract.
[5] Luis G. C. Rego, George Kirczenow, “Fractional exclusion statistics and the universal quantum of thermal conductance: A unifying approach”, Physical Review B, 59, 13080 (1999). Abstract.
[6] K. Schwab, E.A. Henriksen, J.M.Worlock, M.L. Roukes, “Measurement of the quantum of thermal conductance”, Nature, 404, 974 (2000). Abstract.
[7] Matthias Meschke, Wiebke Guichard, Jukka P. Pekola, “Single-mode heat conduction by photons”, Nature, 444, 187 (2006). Abstract.
[8] Andrey V. Timofeev, Meri Helle, Matthias Meschke, Mikko Möttönen, Jukka P. Pekola, “Electronic refrigeration at the quantum limit”, Physical Review Letters, 102, 200801 (2009). Abstract.
[9] S. Jezouin, F.D. Parmentier, A. Anthore, U. Gennser, A. Cavanna, Y. Jin, and F. Pierre, “Quantum limit of heat flow across a single electronic channel”, Science, 342, 601 (2013). Abstract.
[10] D.R. Schmidt, R.J. Schoelkopf, A.N. Cleland, “Photon-mediated thermal relaxation of electrons in nanostructures”, Physical Review Letters, 93, 045901 (2004). Abstract.
[11] Matti Partanen, Kuan Yen Tan, Joonas Govenius, Russell E. Lake, Miika K. Mäkelä, Tuomo Tanttu, Mikko Möttönen, “Quantum-limited heat conduction over macroscopic distances”, Nature Physics, Advance online publication, DOI:10.1038/nphys3642 (2016). Abstract.
[12] L.M.A. Pascal, H. Courtois, F.W.J. Hekking, “Circuit approach to photonic heat transport”, Physical Review B, 83, 125113 (2011). Abstract.
[13] Francesco Giazotto, Tero T. Heikkilä, Arttu Luukanen, Alexander M. Savin, Jukka P. Pekola, “Opportunities for mesoscopics in thermometry and refrigeration: Physics and applications”, Reviews of Modern Physics, 78, 217 (2006). Abstract.
[14] P. J. Jones, J.A.M. Huhtamäki, J. Salmilehto, K.Y. Tan, M. Möttönen, “Tunable electromagnetic environment for superconducting quantum bits”, Scientific Reports, 3, 1987 (2013). Abstract.
[15] Jukka P. Pekola, “Towards quantum thermodynamics in electronic circuits”, Nature Physics, 11, 118 (2015). Abstract.

A New Approach to Quantum Entanglement for Identical Particles

Rosario Lo Franco (right) and Giuseppe Compagno (left)

Authors: Rosario Lo Franco¹ and Giuseppe Compagno²

Affiliation:
¹Dipartimento di Energia, Ingegneria dell'Informazione e Modelli Matematici, Università di Palermo, Italy,
²Dipartimento di Fisica e Chimica, Università di Palermo, Italy.

Entanglement for distinguishable particles is well established from a conceptual point of view with standard tools capable to identify and quantify it [1, 2]. This is instead not the case for identical particles, bosons (e.g., photons, atoms) and fermions (e.g., electrons), where particle identity may give place to fictitious contributions to entanglement which has been the origin of a long-standing debate [2-4]. For all practical purposes, when two identical particles are spatially separated, as in experiments with photons in different optical modes or with strongly repelling trapped ions, no ambiguity is possible for which particle has a given property so they can be effectively treated as distinguishable objects [3]: in this case, no physical contribution to entanglement arises due to indistinguishability.

Figure 1: Asymmetric double-well configuration. One particle has a localized (orange) wave function A in the left well L, while one particle has a (blue) wave function B which overlaps with A, being nonzero in both the left well L and the right well R. This is a typical instance where one particle can tunnel from a site to the other one and indistinguishability counts.

This aspect comes from a natural requirement known as cluster decomposition principle stating that distant experiments are not influenced by each other [6]. Otherwise, quantum indistinguishability comes into play when the constituting particles are close enough to spatially overlap. This happens for all the applications of quantum information processing based, for instance, on quantum dot technology with electrons [7,8] or on Bose-Einstein condensates [9,10], where the particles have the possibility to tunnel from a location to the other (Fig. 1). Hence, correctly treating identical particle entanglement, besides its fundamental interest, is a central requirement in quantum information theory. Despite this, the analysis of identical particle entanglement has been suffering both conceptual and technical pitfalls [2-4].

The ordinary approach to deal with identical particles in quantum mechanics textbooks consists in assigning them unobservable labels which give rise to a new fictitious system of distinguishable particles [5]. In order that this new system behaves as the original (bosonic or fermionic) one, only symmetrized or anti-symmetrized states with respect to labels are allowed. The byproduct is that, according to the usual notion of non-separability employed in quantum information theory to define entanglement, such states entangled. Ordinary entanglement measures, such as the von Neumann entropy of the reduced state obtained by partial trace, fail then to be directly applied to these states. In particular, they witness entanglement even for independent separated particles which are clearly uncorrelated and also show contradictory results for bosons and fermions [3].

As a consequence, methods utilizing notions at variance with the ordinary ones adopted for distinguishable particles have been formulated to overcome this issue [3,4]. In any case, these alternative methods remain technically awkward and unsuited to quantify entanglement under general conditions of scalability and wave function overlap. The use of new notions to discuss quantum entanglement for identical particles looks surprising, not less than the introduction of unobservable labels which is in contrast with the quantum mechanical requirement that the state of any physical system is uni-vocally described by a complete set of commuting observables. So far, there has not been general agreement even whether the entanglement between two identical particles in the same site may exist [3, 11, 12]. The characterization of quantum entanglement for identical particles has thus remained problematic, jeopardizing the general understanding and exploitation of composite systems.

In our recent work [13], we provide a straightforward description of entanglement in systems of identical particles, based on simple physical concepts, which unambiguously answers the general question: when and at which degree the identity of quantum particles plays a physical role in determining the entanglement among the particles? This is achieved by introducing a novel approach for identical particles without resorting to fictitious labels, differently from the usual textbook practice. The core of this approach is that the state of several identical particles must be considered a whole entity while the transition probability amplitude between two of such states is expressible in terms of single-particle amplitudes by applying the basic quantum-mechanical superposition principle with no which-way information to alternative paths. Our approach enables the determination of entanglement for both bosons and fermions by the same notions usually adopted in entanglement theory for distinguishable particles, such as the von Neumann entropy of the reduced state. The latter is obtained through the partial trace defined by local single-particle measurements.

Figure 2: Panel A. Entanglement as a function of system parameters for a fixed degree of spatial overlap for bosons (blue dotted line) and fermions (orange dashed line), compared to the corresponding entanglement of nonidentical particles (red solid line). Panel B. Density plot of bosonic entanglement as a function of both relative phase in the system state and degree of spatial overlap.

We have analyzed a system of two identical qubits (two-level systems) with orthogonal internal states (opposite pseudospins). The qubits are supposed to have wave functions (spatial modes) which can overlap at an arbitrary extent. A simple system which realizes this condition is that of the asymmetric double-well configuration illustrated in Fig. 1. When the two particles partially overlap in a spatial region where local single-particle measurements can be done, entanglement depends on their overlap and an ordering emerges for different particle types, fermions or bosons (Fig. 2). Moreover, identical particles are found to be at least as entangled as non-identical ones placed in the same quantum state.

This result suggests that identical particles may be more efficient than distinguishable ones for entanglement-based quantum information tasks. The main findings of this analysis can be summarized as: (i) an absolute degree of entanglement for identical particles, independent of local measurements, can be assigned only when the particles are spatially separated or in the same site; (ii) the act of bringing identical particles into overlapping spatial modes creates an “entangling gate” whose effectiveness depends on the amount of overlap. Our results finally show that a natural creation of maximally entangled states is possible just by moving two identical particles with opposing pseudospin states into the same site, supplying theoretical support to recent observations in an experiment with ultracold atoms transported in an optical tweezer [14]. This gives a definitive positive answer whether identical particles in the same site can be entangled.

Our approach contributes, from a fundamental point of view, to clarify the relation between quantum entanglement and identity of particles. It remarkably allows the quantitative study of entanglement under completely general conditions of overlap and scalability, motivating studies on correlations other than entanglement [15] in the context of identical particle systems. Moreover, our study paves the way to interpret experiments which use quantum correlations in relevant scenarios where identical particles can overlap.

References:
[1] Luigi Amico, Rosario Fazio, Andreas Osterloh, Vlatko Vedral, “Entanglement in many-body systems”, Review of Modern Physics, 80, 517 (2008). Abstract.
[2] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, Karol Horodecki, “Quantum entanglement”, Review of Modern Physics, 81, 865 (2009). Abstract.
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[5] Claude Cohen-Tannoudji, Bernard Diu, Franck Laloë, “Quantum mechanics. Vol. 2” (Willey-VCH, Paris, France, 2005).
[6] Asher Peres, “Quantum Theory: Concepts and Methods” (Kluwer Academic,1995).
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[9] Immanuel Bloch, Jean Dalibard, Wilhelm Zwerger, “Many-body physics with ultracold gases”, Review of Modern Physics, 80, 885 (2008). Abstract.
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