PHYSICAL REVIEW LETTERS week ending VOLUME 92, NUMBER 16 23 APRIL 2004 Proposal for Energy-Time Entanglement of Quasiparticles in a Solid-State Device Valerio Scarani,1 Nicolas Gisin,1 and Sandu Popescu2 1Group of Applied Physics, University of Geneva, 20, rue de l’Ecole-de-Me´decine, CH-1211 Geneva 4, Switzerland 2H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, United Kingdom (Received 29 July 2003; revised manuscript received 1 December 2003; published 23 April 2004) We present a proposal for the experimental observation of energy-time entanglement of quasiparticles in mesoscopic physics. This type of entanglement arises whenever correlated particles are produced at the same time and this time is uncertain in the sense of quantum uncertainty, as has been largely used in photonics. We discuss its feasibility for electron-hole pairs. In particular, we argue that junctions between materials in which electrons and holes, respectively, propagate ballistically and behave as ‘‘entanglers’’ for energy-time entanglement when irradiated with a continuous laser. DOI: 10.1103/PhysRevLett.92.167901 PACS numbers: 03.67.Mn, 03.65.Ud, 73.23.Ad Entanglement lies at the heart of quantum mechanics, photon pairs, Franson proposed, in 1989, a very conve- whose astonishing features mainly come from it [1]. nient interferometer [9], sketched in Fig. 1. Each particle Interest in entanglement has grown, since it was recog- is sent through unbalanced interferometers, with the same nized as a ressource needed to perform tasks that would difference between the long (L) and the short (S)arm: classically be impossible [2]. Although entangled states La ÿ Sa Lb ÿ Sb L.IfL is larger than the a;b arise in every subfield of quantum physics (e.g., the ei- single-particle coherence lengths ‘c , no single-particle genstates of total momentum), it is generally an experi- interferences appear. However, the coherence length ‘c of a;b mental challenge to achieve control over entanglement. the pair is usually much larger than ‘c . Then, let After the spectacular results of photonics [3], entangle- a;b ment has recently been demonstrated in other physical ‘c > L>‘c : (1) systems [4]. There is a growing list of proposals aimed at the observation of entanglement in solid-state physics, In this case, the alternatives ‘‘both particles have taken using quantum dots, Josephson junctions, and other de- the long arm’’ (LL) and ‘‘both particles have taken the vices. In this Letter we focus on quasiparticles in meso- short arm’’ (SS) are indistinguishable and exhibit inter- scopic devices. ference fringes; while the two other alternatives, LS and Coherent transport of quasiparticles in semiconductors SL, are distinguishable because one particle clearly ar- has been widely demonstrated, so one can envisage to rives to its detector before the other. Thus, in the runs demonstrate entanglement. A few years ago, Burkard and in which both particles are detected at the same time, co-workers noticed that electron-electron entanglement an interference pattern is observed that is due to the en- in spin could be detected by measuring correlation in tanglement in energy time. the current noise [5]. In Refs. [6], the scheme was com- The photonic setup.—We start by reviewing briefly the pleted with a proposal for an ‘‘entangler,’’ that is, for a photonic setup, which has already been the object of source of spin-entangled electrons: a Cooper pair from a successful experiments [3,10]. A nonlinear crystal is superconducting material. Recently, these ideas have been pumped by a cw laser with coherence time c.This extended to entanglement in spatial degrees of freedom, coherence time is defined as usual: the state of the laser for two electrons generated as Cooper pairs [7] and for light is a coherent beam with a fluctuating phase t, electron-hole pairs in edge states [8]. such that t ÿ t is a stationary Gaussian process 2 In the present Letter we take a different approach, of mean value hi0 and of variance h i2=c.A which works in principle for any kind of particles pro- nonlinear, purely quantum-mechanical process (para- duced in pairs. The basic idea is the following: two metric down conversion) takes place, that produces a field particles a and b are produced at the same, but uncertain, time. This quantum uncertainty is within the coherence α β time of the source. The latter is typically a photon from L L D+ Source D+ a laser beam, called the pump photon, whose well- S S determined energy is shared between the two produced D D− particles. Hence the energy and the time of creation of − * each particle are uncertain, but the sum of the energies FIG. 1. The Franson interferometer, drawn for photonics. and the difference of the times are well-determined. This Gray segments are 50-50 couplers; and are dephasing form of entanglement is known as energy-time entangle- elements (small delays). The difference between the two arms ment. To observe a signature of this entanglement for on each side, L L ÿ S, must satisfy requirement (1). 167901-1 0031-9007=04=92(16)=167901(4)$22.50 2004 The American Physical Society 167901-1 PHYSICAL REVIEW LETTERS week ending VOLUME 92, NUMBER 16 23 APRIL 2004 in two initially empty modes a and b, whose wave vectors contribute to any interference [11]. It is a different matter and polarizations are determined by energy and momen- of course to distinguish them in practice.So,a priori we tum conservation. If the intensity of the pump is weak have to consider two possible outcomes of the experi- enough, the field in a and b consists essentially of a large ment: if one can select only the interfering cases, the vacuum component (that we neglect) plus a two-photon detection rate is [12] component. In each mode a or b, we shall write j1;0i R~ jj ~ jj2 / 1 eÿt=c cos ; (6) (respectively, j0;1i) for one photon propagating along a horizontal (respectively, vertical) direction in Fig. 1. The recalling that the visibility V is defined by R / 1 V cos state of the down-converted field can be written as a for a sinusoidal fringe, we find V eÿt=c 1. If one superposition of two-photon fields produced at any time t: cannot select only the interfering cases, the visibility of p Z the observed interference fringes will be reduced down to j i A dt ei t j1 ;0i j1 ;0i ; (2) 1 t a t b V 2 , since one has 2 ÿt= where A is proportional to the power of the laser and the R jj jj / 2 e c cos : (7) efficiency of the down-conversion process. The states In optics, for typical coherence times and jitters of the de- j1t; 0ia;b can be seen as an overcomplete set; the overlap tectors, one can select only those cases where the photons 0 h1t; 0j1t0 ; 0ia;b decreases rapidly as a function of jt ÿ t j= arrive at the same time; that is how V 1 has been a;b a;b c , where c are the single-photon coherence times. As reached and the Bell inequality could be violated [10]. we discussed, in our experiment this time is much shorter The proposal: overview.—We can now turn to the than the other times involved (t, c). main goal of this paper: a proposal for the production The state (2) can be seen as a continuous version of the and detection of energy-time entangled quasiparticles maximally entangled state of two d-dimensional quan- in mesoscopic physics [13]. Specifically, we consider tum systems, indexed by the parameter t. Franson’s setup electron-hole pairs produced in semiconductor junctions is a way of partially detecting this entanglement, by illuminated by a laser. A low intensity cw laser with projection onto a two-dimensional subspace and postse- coherence time c illuminates a junction, producing lection. The evolution of mode a in the unbalanced inter- electron-hole pairs. When the electron and the hole do ferometer (the beam splitters are 50-50 couplers) is not recombine, they will be accelerated out of the junction i in opposite directions. Once produced, each quasiparticle j1t; 0ia ! j1t; 0ia ij0; 1tia ie j0; 1ttia travels in a semiconductor structure, tailored for single- i ÿ e j1tt; 0ia; (3) mode coherent transport of the electron [14] or the hole 1 [15]: typically, a two-dimensional electron or hole gas, where we have omitted a global factor 2 and have rede- fined the origin of time to take into account the propaga- noted, respectively, as 2DEG and 2DHG. The unbalanced tion from the source. The evolution of mode b is identical, Mach-Zehnder interferometer is engineered in the semi- with a phase instead of . conductor, in the form of an asymmetric loop, where the The two-photon state at the detection stage is obtained phase between the two arms can be varied using the by replacing the evolved state into (2): it is a sum of 16 Aharonov-Bohm effect [16]. The two paths are then basic kets. We focus on a pair of detectors, say, the two recombined and split again, each ending in a detector. detectors labeled in Fig. 1. This means that we project The rest of this Letter is devoted to a detailed analysis of the three parts of the setup: entangler (preparation), onto the four kets of the form j1; 0iaj1; 0ib, which we write for conciseness j1; 1i: interferometer (evolution), and detectors (measurement). Z The entangler.—It is not clear whether a standard bulk i t i p-n junction can be an entangler: the open questions are j i’ dt e j1t;1tie j1tt;1tti (i) whether doping will make the motion of the quasipar- i i e j1t;1ttie j1tt;1ti: ticles diffusive, thus erasing quantum coherence, and (4) (ii) how to describe the interfaces between the bulk and the 2D gases.