Article

Proposal for Energy-Time Entanglement of in a -State Device

SCARANI, Valerio, GISIN, Nicolas, POPESCU, Sandu

Abstract

We present a proposal for the experimental observation of energy-time entanglement of quasiparticles in mesoscopic . This type of entanglement arises whenever correlated 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 - pairs. In particular, we argue that junctions between materials in which and holes, respectively, propagate ballistically and behave as "entanglers" for energy-time entanglement when irradiated with a continuous laser.

Reference

SCARANI, Valerio, GISIN, Nicolas, POPESCU, Sandu. Proposal for Energy-Time Entanglement of Quasiparticles in a Solid-State Device. Physical Review Letters, 2004, vol. 92, no. 16

DOI : 10.1103/PhysRevLett.92.167901 PMID : 15169260

Available at: http://archive-ouverte.unige.ch/unige:36730

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1 / 1 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 , pairs, Franson proposed, in 1989, a very conve- whose astonishing features mainly come from it [1]. nient interferometer [9], sketched in Fig. 1. Each 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 Sa Lb 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 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 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 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 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 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 et=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 et=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 eit j1 ;0i j1 ;0i ; (2) 1 t a t b V 2 , since one has 2 t= 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 0 h1t; 0j1t0 ; 0ia;b decreases rapidly as a function of jt 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 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 j1t; 0ia ! j1t; 0ia ij0; 1tia ie j0; 1ttia travels in a semiconductor structure, tailored for single- i 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 it i p-n junction can be an entangler: the open questions are j i’ dt e ‰j1t;1tie j1tt;1tti (i) whether will make the motion of the quasipar- i i e j1t;1ttie j1tt;1tiŠ: ticles diffusive, thus erasing quantum coherence, and (4) (ii) how to describe the interfaces between the bulk and the 2D gases. Fortunately, there is an elegant way of The two first terms correspond to the cases where the two bypassing both obstacles: by creating a junction between photons arrive at the same time in the detectors (paths SS a 2DEG and a 2DHG, the source is just the interface, and LL in Fig. 1); because of the invariance through the interferometer can be engineered in the same mate- translation in time, these two cases are indistinguishable, rials and the whole motion can be ballistic. The first and interfere. We can rewrite these two first terms as Z 2DEG-2DHG junction has been fabricated very recently j ~ i’ dt ‰eit ei‰ ttŠŠj1 ;1i: (5) in AlGaAs/GaAs heterostructures [17] according to the t t scheme that we reproduce in Fig. 2. The full understand- The third term is the case where photon b is delayed by t ing of the physics of such junctions and the optimization with respect to photon a(path SL), and the fourth term is of the parameters will need further work [18]. However, the opposite case (path LS). These last two cases are in there is no fundamental objection to considering that a principle distinguishable from the others, so they do not 2DEG-2DHG irradiated by a laser behave as an entangler 167901-2 167901-2 PHYSICAL REVIEW LETTERS week ending VOLUME 92, NUMBER 16 23 APRIL 2004

e;h e;h ’ c : (8) We focus on the electron; the analog holds for the hole. We argue that (8) can be fulfilled if the electron is injected into the 2DEG close enough to the [refer to Fig. 2(c)]. The electron is injected into the 2DEG with an energy E F , where F is the Fermi energy of the 2DEG, typically some 0.1 eV [20]. Since the main relaxation mechanism will be e-e inelastic 2 scattering, the phase-relaxation time is ’ h F= [21]. For instance, if the electron is injected into the 3 2DEG with  1=100F 10 eV, then we obtain e ’ ’ 100 ps, in good agreement with the observed values e;h FIG. 2. Qualitative description of the 2DEG-2DHG junct- of L’ , that are typically some 10 m (see [15], and ion. (a) The design. (b),(b0) The gap along the vertical axis. references therein). The single-particle coherence time (c) Schematic description of the junction and of its behavior e can be estimated by c h= E, where E is the un- when irradiated with a laser spot (gray area). certainty in the electron kinetic energy. In Fig. 2(c), one can see clearly that this uncertainty is determined by 0 to generate electron-hole pairs entangled in energy time. the relation between the steepness  d=dxx 0 of the built-in potential and the width w of the laser Furthermore, this goal may be a strong motivation to 0 boost technical improvements. Finally, note that p-n spot. In particular, E & max e w, this being e junctions have also been fabricated in another material the largest value of . From the expressions of ’ and e exhibiting ballistic transport, namely, carbon nanotubes c, one derives immediately that condition (8) holds if and only if max F, and this can in principle be [19]. These can also be candidates as entanglers. 0 The interferometer.—From now on, we shall use for achieved by decreasing  (junction engineering) or w numerical estimates typical values for electrons, ex- (laser focusing). tracted from Ref. [20]. One must not forget that holes Condition (8) being possible, the calculation of the often have smaller mobilities in these structures, so the two-particle interference pattern goes along the same figures may not apply to one-half of the interferometer, lines as for the optical implementation [22], adding the but the principles of the analysis do apply. In the forth- presence of different environments. Following the ap- e e h coming discussion, we shift when convenient from proach of Stern et al. [23], we write jLi, jSi, jLi,and h ‘‘lengths’’ ‘ to times , the link being provided by ‘ jSi, the four environments associated with the paths, where L and S stand for long and short, and e and h stand vF where vF is the Fermi velocity in a 2DEG or 2DHG, 7 for electron and hole. The evolution of the electron state, typically vF 3 10 cm=s. In the optical experiment, we introduced the requirement (1) on L for the Franson replacing (3), is then interferometer to show two-particle interferences. In the e j1t; 0ie !j1t; 0ie ij0; 1tiej i present proposal, it is trivial to have the coherence length S i i e of the pair ‘c exceed all the other meaningful lengths, ie j0; 1ttie e j1tt; 0iejLi; (9) since ‘ is determined by the coherence time of the pump c and a similar evolution for the hole. In the interfering laser, and cw lasers easily have a coherence time of tens of terms j ~ i, the term into brackets in Eq. (5) is replaced nanoseconds. However, here we must meet an additional by constraint due to the role of the environment.While photons essentially do not couple to the environment, i’t e h i‰ ’ttŠ e h e jSijSie jLijLi; (10) electrons and holes propagating in semiconductors inter- act strongly, especially with the other quasiparticles. and finally the visibility is reduced with respect to the Because of this coupling, some which-path information analogous optical visibility as is transferred out of the system under study, whose coher- e e h h Ve-h Vopt jh jLij jh jLij : (11) ence is thus decreased. So the size of each interferometer S S must not be too big: for both electrons and holes, L S The phase-relaxation length that we introduced above is should be much smaller than L’, a phase-relaxation related to the expressions in (11), in the simplest case [23], x x x LxSx=L’ length that characterizes the coupling with the environ- through jhSjLij e for both x e; h. ment. Since (at least in principle) the S path can be made The detectors.—A convenient detection scheme arbitrarily short, we have L S L, and we can sum- should use the observables of mesoscopic physics, that e;h e;h e;h marize our present requirements as L’ > L >‘c . are current-correlation measurements [24]. We can con- Confident in the precision of semiconductor manufac- sider that the detector for the electron, respectively the turing techniques, we admit that this requirement holds hole, is a reservoir biased with a voltage V, e;h e;h if and only if L’ ‘c , that is, if respectively V. If the temperature is small enough 167901-3 167901-3 PHYSICAL REVIEW LETTERS week ending VOLUME 92, NUMBER 16 23 APRIL 2004

(kT jeVj), no electron (hole) can be injected from the [5] G. Burkard, D. Loss, and E.V. Sukhorukov, Phys. Rev. B reservoirs into the semiconductor. The situation becomes 61, R16 303 (2000). then analog to those studied in Refs. [5–8]: entanglement [6] G. B. Lesovik, T. Martin, and G. Blatter, Eur. Phys. J. B can be detected by measuring the zero-frequency current 24, 287 (2001); N. M. Chtchelkatchev, G. Blatter, G. B. cross correlator. Since this detection scheme is not time- Lesovik, and T. Martin, Phys. Rev. B 66, 161320(R) (2002). resolved, it leads to (7) with the correction (11). We want [7] P. Samuelsson, E.V. Sukhorukov, and M. Bu¨ttiker, Phys. to conclude by addressing the question of time-resolved Rev. Lett. 91, 157002 (2003). detection, that is, the hope of observing the analog of (6). [8] C.W.J. Beenakker, C. Emary, M. Kindermann, and J. L. Typical for energy-time entanglement is the presence of van Velsen, Phys. Rev. Lett. 91, 147901 (2003). two meaningful time scales at the detection. The first, [9] J. D. Franson, Phys. Rev. Lett. 62, 2205–2208 (1989); standard one, comes from the fact that we want at most a M. O. Scully and M. S. Zubairy, Quantum Optics single electron-hole pair to be produced per coherence (Cambridge University Press, Cambridge, U.K., 1997), time of the pump c. This time scale determines the p. 597. [10] J. Brendel, E. Mohler, and W. Martienssen, Phys. Rev. current: taking c ’ 1ns, large enough to ensure c e;h Lett., 66, 1142 (1991); W. Tittel, J. Brendel, H. Zbinden, ’ ’ 100 ps, one expects a current at detection of some 100 pA. The second time scale determines whether one and N. Gisin, Phys. Rev. Lett. 81, 3563 (1998). [11] In our formalism, this qualitative reasoning amounts to can discriminate the interfering cases from the noninter- set h1t;0j1tt;0ia;b 0, that is, to neglect the single- fering ones. Explicitly, a time resolution for the single- a;b particle coherence times c before t. electron (-hole) measurement of meas < L=vF 10 ps [12] For the calculation of both (6) and (7), we use 1 i‰ttŠ jj= is needed to enter the regime V>2 (6). Both time reso- E‰e Še c , where E‰Š is the expectation lution and sensitivity are in principle achievable using value of the random variable. single-electron as detectors [25]. But very fast [13] Since there is only one particle per mode in a Franson measurements introduce unwanted excitations that may interferometer, the bosonic or fermionic nature of the e;h hide the signal; moreover, meas >c must hold in order particle does not introduce any difference. 1 [14] Y. Ji, Y. Chung, D. Sprinzak, M. Heiblum, D. Mahalu, to detect the quasiparticles. Anyway, V 2 would al- ready be a fair demonstration of entanglement, because and H. Shtrikman, Nature (London) 422, 415 (2003). the origin of this reduced visibility in a Franson setup is [15] I. Zailer et al., Phys. Rev. B 49, 5101 (1994). [16] G. Seelig and M. Bu¨ttiker, Phys. Rev. B 64, 245313 well understood. (2001). In conclusion, we have argued that energy-time entan- [17] B. Kaestner, J. Wunderlich, D. G. Hasko, and D. A. glement of quasiparticles can be observed. The task is Williams, Microelectron. J. 34, 423 (2003). challenging, but the goal seems within reach considering [18] For instance, in the device of Ref. [17], the 2DEG and present-day technology. 2DEG-2DHG junctions are 2DHG are separated vertically by 90 nm; this means that promising candidates as entanglers for energy-time en- the electron and the hole, before being coupled in the 2D tangled electron-hole pairs. gases, must travel some tens on nanometers in the in- We t h a n k M. B u¨ttiker, P. Samuelsson, E.V. Sukhoru- trinsic GaAs, where the phase coherence time is shorter. kov, and B. Kaestner for useful comments. V.S. acknowl- Also, it would be useful to decrease the steepness of the edges financial support from the Swiss NCCR ‘‘Quantum built-in potential (see main text, below). Photonics,’’ and fruitful discussions with members of the [19] G. Zhou, J. Kong, E. Yenilmez, and H. Dai, Science 290, 1552 (2000). network. [20] C.W.J. Beenakker and H. van Houten, in Solid State Physics (Academic Press, New York, 1991), Vol. 44, Table I; S. Datta, Electronic Transport in Mesoscopic Conductors (Cambridge University Press, Cambridge, [1] Quantum Theory and Measurement,editedbyJ.A. U.K., 1995), Chap. 1. Wheeler and W. H. Zurek, Princeton Series in Physics [21] This is valid for inelastic scattering of identical quasi- (Princeton University Press, Princeton, New Jersey, particles, e.g., R. D. Mattuck, A Guide to Feynman Dia- 1983). grams in the Many-Body Problem (Dover, New York, [2] Introduction to Quantum Computation and Information, 1967), Sec. 8.3; A. A. Abrikosov, Introduction to the edited by H. K. 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