View metadata, citation and similar papers at core.ac.uk brought to you by CORE weekprovided ending by Sussex Research Online PRL 110, 080502 (2013) PHYSICAL REVIEW LETTERS 22 FEBRUARY 2013

Mesoscopic Entanglement Induced by Spontaneous Emission in Solid-State

Alejandro Gonza´lez-Tudela1 and Diego Porras2 1Departamento de Fı´sica Teo´rica de la Materia Condensada, Universidad Autonoma de Madrid, 28040 Madrid, Spain 2Departamento de Fı´sica Teo´rica I, Universidad Complutense, 28040 Madrid, Spain (Received 21 September 2012; revised manuscript received 29 November 2012; published 20 February 2013) Implementations of solid-state quantum optics provide us with devices where qubits are placed at fixed positions in photonic or plasmonic one-dimensional waveguides. We show that solely by controlling the position of the qubits and with the help of a coherent driving, collective spontaneous decay may be engineered to yield an entangled mesoscopic steady state. Our scheme relies on the realization of pure superradiant Dicke models by a destructive interference that cancels dipole-dipole interactions in one dimension.

DOI: 10.1103/PhysRevLett.110.080502 PACS numbers: 03.67.Bg, 03.67.Ac, 37.10.Ty, 37.10.Vz

The study of atoms coupled to the electromagnetic (EM) distance. Adding a classical drive to the pure Dicke model field confined in cavities or optical waveguides has played we obtain a dissipative system with a phase diagram of a central role in the fields of quantum optics and atomic steady states showing mesoscopic squeezing and physics. In recent years the basics of that physical system entanglement. This model has been theoretically investi- have been realized with artificially designed atoms in gated in the past [25–27], but experimental realizations are solid-state setups. We may include here quantum dots scarce. Finally we upgrade our scheme to a set of N four- and nitrogen vacancy (NV) centers deterministically level emitters [28,29] and show that a judicious choice of coupled to photonic cavities [1–3], and plasmonic [4,5] couplings to the waveguide and dispersion relations may or photonic [6–10] waveguides, as well as circuit QED lead to a variety of many-body dissipative models which setups where superconducting qubits are coupled to show entangled steady states. microwave cavities [11,12]. Even though the physics of We start by modeling N two-level systems (2LSs), atomic and solid-state quantum optical systems is similar, fjginjeingn¼1...N, placed atpositions xn and coupled to a the latter show a crucial advantage: on a solid substrate, one-dimensional field [see Fig. 1(a)] with annihi- emitters may be placed permanently at fixed positions at lation operators aq, described by the Hamiltonian H ¼ separations of the order of relevant wavelengths [13]. H0 þ HI. The free term is H0 ¼ Hqb þ Hfield, with (@ ¼ 1) An important application of those systems is quantum XN X !0 information processing in solid-state devices [14], where ¼ z ¼ y Hqb 2 n;Hfield !qaq aq; (1) artificial atoms acting as qubits are placed within the EM n¼1 q field confined in a microcavity. Typically, the realization of those ideas requires unitary qubit-field evolutions induced where !0 is the qubit energy [see Fig. 1(b)] and !q is the by collective couplings to a single mode in a cavity. An field dispersion relation. We define the alternative pathway is to tailor the interaction with the environment to induce quantum correlations between qubits with dissipation [15,16]. This approach has been proven to be advantageous to generate entanglement between ensembles of atoms [17]. In this direction one- dimensional guided modes have been recently pointed out as a useful tool to create two-qubit entanglement [18–20] and many-qubit entanglement in cascaded quantum networks [21]. In this Letter, we show that by placing a set of qubits in a one-dimensional waveguide (see Fig. 1) the continuum of EM field modes induces a controllable dissipative coupling between the qubits. The possibility of deterministically positioning the artificial atoms or qubits allows us to FIG. 1 (color online). Panel (a) Experimental scheme of the engineer the paradigm for quantum optical collective ef- system: ensemble of equally spaced qubits placed in the vicinity fects, i.e., the Dicke model of [22] in its pure of a one-dimensional waveguide. Panel (b) Two-level system form. The observation of the latter in optical systems is configuration with resonant excitation. Panel (c): Four-level hindered due to dephasing caused by dipole-dipole inter- system configuration with two additional , where we im- actions [23,24]. In our scheme those interactions can be pose the condition: !L;1 !0 ¼ !L;2 þ !0 ¼ !a and define switched off by an appropriate choice of the interqubit !1 ¼ !L;1, !2 ¼ !L;2 !0.

0031-9007=13=110(8)=080502(5) 080502-1 Ó 2013 American Physical Society week ending PRL 110, 080502 (2013) PHYSICAL REVIEW LETTERS 22 FEBRUARY 2013

z ¼jei hejjgi hgj, þ ¼jei hgj, ¼jgi hej. The n n n n n n n L ð Þ¼L ð Þ ½ photon polarization is neglected to focus on the most D;p D i 2 Sx; : (6) relevant physics of our work. We consider a dipolar cou- Competition between the collective decay and the pumping pling of the form X leads to a nonequilibrium in the steady state of the model at a critical pumping rate ¼ N=2 HI ¼ ½nEðxnÞþH:c: (2) c n [25], manifested in a kink in the population inversion P observable hS i [see Fig. 2(a)]. Let us first give a brief iqx y iqx z with EðxÞ¼ qgqðaqe þ aq e Þ, and gq a dipolar description of the two limiting cases. (i) Coherent steady qubit-field coupling. We define as the reduced density state regime, N=2. Since LD;p can be obtained matrix for the qubits. In the weak coupling limit, the from LD by the substitution S ! S þ i=ð2Þ, one evolution of can be described by a Markovian 2 can easily show that s ¼jcihcjþO ðÞ, where jci¼ master equation of the form d=dt ¼ LðÞ [30], with iSx 2 2 the superoperator e jN= ; N= i is a spin . (ii) The mixed X state phase, N=2. Here we get an infinite tempera- þ þ L L ðÞ¼ Jn;mðn m mn ÞþH:c: (3) ture state. To show this, it is convenient to write D in the n;m interaction picture with respect to Sx=2. This allows us to make the replacement S ! S þð1=2Þ½cosðtÞS þ A detailed derivation follows the description in dimensions x y sinðtÞS . Averaging over time, leads to higher than one presented in previous works [24] (see z 1 X Sec. A of the Supplemental Material [31]). Special care L 2 þ ð 2 Þ þ H c D;p 2 SxSx Sx 2 SS S : :; must be paid to the counter rotating terms in Eq. (3), which ¼y;z have to be included to get the following result for the collective decay rates (7) 1 which has the infinite state s ¼ as the ¼ iqð!0Þjxnxmj Jn;m 2 e : (4) steady state. For calculations in the intermediate regime we use the full solution in the jJ; Mi basis. We define ¼ ð!0Þ, with the function ð!Þ¼ To quantify the entanglement we use the spin squeezing 2 as a figure of merit, gqð!ÞDð!Þ=, where qð!Þ is the resonant wave vector at !, and we have defined the EM density of states, Dð!Þ¼ 2 2 NðSxÞ ð2=LÞjdqð!Þ=d!j, with L the quantization length. A ¼ : (8) hS i2 þhS i2 crucial observation for this work is that the couplings y z Jn;m ideally do not decay with distance, a situation that is The latter is both an entanglement witness and it is also unique to one-dimensional waveguides. In free space, on linked to applications in quantum metrology [35–37]. 3 the contrary, collective couplings decay like 1=r or 1=r , Symmetric multiqubit states with <1 can be shown to depending on the relative dipole orientation [24]. be entangled [36] with pairwise entanglement between any 2 Homogeneous couplings [18,19,32–34] Jn;m ¼ = can pair of qubits [38]. Note that the above mentioned phases 2 2 be obtained from Eq. (4) by the choice xn ¼ n0, with (i) and (ii) lead to 1= ¼ 1 and 1= ¼ 0, respectively. 0 ¼ 2=q0, and n 2 Z. This condition cancels dipole- Another theoretical tool to be used is the purity, defined by dipole interactions and we get the pure Dicke superradiant P ðÞ¼Trð2Þ, although we note that witnesses entan- decay described by glement also for mixed states. Both magnitudes are plotted for increasing number of qubits N in Fig. 2. We find a range þ þ L DðÞ¼ ðS S S S ÞþH:c:; (5) 2 of pumping fields ( c) that induce a pure entangled P P s. This result leads to the controllable generation of þ þ withP S ¼ nn , S ¼ nn . We also define S ¼ entangled states in mesoscopic samples of artificial atoms. 2 nn = ,( ¼ x, y, z), and the basis fjJ; Mig of eigen- The time scale needed to achieve the stationary entangled ~2 states of S , Sz. Assuming an initial state like j0i¼ states benefits from a collective enhancement scaling as njein ¼jN=2;N=2i the system evolves within the sector N, so that the higher the number of qubits, the more J ¼ N=2. We note that Dicke superradiant decay is efficient is the preparation of states. achieved in one dimension without the restriction that the We now upgrade our 2LS into a four-level system whole qubit ensemble is confined within a region of length (4LS) configuration [see Fig. 1(c)], and show that this 0, which is a requirement for other realizations, i.e., is a more advantageous situation. Our scheme can be atomic ensembles. realized in the solid-state context [28,29] and describes a In this Letter we focus on the qubit steady state, s, variety of possible configurations in which a set of low- which for a given Liouvillian fulfills LðsÞ¼0.To level states are coupled to excited states by lasers with achieve some controllability on s, we add a pump term different polarizations. Two ground states (jgin; jein) 0 0 which physically can be implemented by the interaction of are coupled to high energy states (jg in, je in). The qubits with a resonant field with Rabi frequency , qubit part of the free Hamiltonian becomes now 080502-2 week ending PRL 110, 080502 (2013) PHYSICAL REVIEW LETTERS 22 FEBRUARY 2013

spontaneous coherence buildup induced by Lsq. The relative importance of those contributions depends on the photon density of states at frequencies !a and !1;2. We consider two limiting cases. (i) Small photonic band- width. This is the most favorable configuration. We assume that the density of states in the waveguide is peaked around !a, with a bandwidth ! j!1 !aj, z j!2 !aj such that ð!1;2Þ0 and therefore Jn;m 0. For example, this can be the case of one-dimensional waveguides consisting of coupled cavities forming a one- dimensional photonic crystal. Defining qð!aÞ¼2=a FIG. 2 (color online). Numerical results for the coherently and choosing xn ¼ na we arrive at a spin-squeezed pumped Dicke model [Eq. (6)] for increasing N. Different colors version of the Dicke superradiant model represent different number of qubits from N ¼ 2 to N ¼ 200 as h i sq þ þ shown in the legend. (a) Population inversion Sz =N. (b) Purity L sq;DðÞ¼ ðD D D D þ H:c:Þ; (13) P (dashed) and spin-squeezing parameter 1=2 (solid). 2 where we have introduced the collective spin-squeezed P P 0 0 0 0 operators Dþ= ¼ D=þ. In Fig. 3(a) we present a Hqb ¼ nð!g0 jg inhg jþ!e0 je inhe jþ!gjginhgjÞ.Two n n calculation of the spin squeezing in the steady state as a weak nonresonant fields with amplitudes L;1ð2Þ and function of the squeezing parameter r. Remarkably, we frequencies !L;1ð2Þ, induceP transitions described by a 2 0 i!L;1t observe an enhancement of the maximum value of the Hamiltonian term HL ¼ n½ð L;1= Þje inheje þ 0 i!L 2t entanglement of several orders of magnitude compared to ðL 2=2Þjg i hgje ; þH:c:. We impose the condition ; n the case of an ensemble of 2LSs. (ii) A large photonic !L 1 !0 ¼ !L 2 þ !0 ¼ ! , such that the two decay ; ; a bandwidth. In the opposite limit we consider a broadband channels in red of Fig. 1(c) (into modes aq) correspond waveguide [18,32](j!L 1 !L 2j!) such that the to photon emission with the same energy ! . After an ; ; a density of states at the frequencies considered here is adiabatic elimination of the excited states (see Sec. II of comparable ð!1Þð!2Þ. In experiments with optical the Supplemental Material [31] for details) we get an sq z transitions, for example with quantum dots in optical or effective qubit-field interaction HIðtÞ¼HI ðtÞþHI ðtÞ, plasmonic waveguides, the condition !1, !2, !a !0 written in the interaction picture with respect to H0. is found, since the transition energies are in the eV and The first term reads X meV ranges for high energy and low energy transitions, sq y i! t i! t HI ðtÞ¼ Eðxn;tÞðDn e a þ Dne a Þ; (9) respectively [28,29]. Thus, we can safely assume n qð!aÞqð!1Þqð!2Þ¼2=a, and consider that 2 2 2 where ¼ðL;1=21Þ ðL;2=22Þ (1ð2Þ ¼ !e0ðg0Þ quantum dots can be placed at the same relative optical þ path with respect to all frequencies. To give a more !L;1ð2Þ) is a normalization constant. Dn ¼ un þ vn is a jump resulting from the cross radiative decay, quantitative argument for this approximation we define 1 1 the group velocity of the modes of the waveguide with u ¼ L;1=21 and v ¼ L;2=22, fulfilling u2 v2 ¼ 1. The latter condition will be useful in the vgð!Þ¼j@q!qj, and consider the limit j!1 !ajvgð!aÞ; discussion below, and allows characterization by a single j!2 !ajvgð!bÞqð!aÞ, which corresponds to small parameter r, such that u ¼ cosh ðrÞ, v ¼ sinh ðrÞ. After wave vector differences. In the case of constant vg and eliminating the EM field degrees of freedom we arrive at optical transitions, this condition leads to differences of 3 the Liouvillian 10 in qð! Þ, qð!1 2Þ. We neglect for the moment those X a ; sq þ þ differences, which may lead to the inhomogeneous L sqðÞ¼ Jn;mðDn Dm DmDn ÞþH:c: (10) n;m broadening effects that are discussed later in this Letter. sq 2 Thus, the condition xn ¼ na leads to a collective iqð!aÞjxnxmj with Jn;m ¼ sqe and sq ¼ ð!aÞ. The dephasing term of the form second term in the effective qubit-field interaction describes the longitudinal decay processes L ð Þ¼ z ð z z z þ H c Þ X z;D 2 S S S : : ; (14) HzðtÞ¼ Eðx ;tÞðuz ei!1t þ vz ei!2t þ H:c:Þ; (11) I n n n ¼ z n where we have introduced the rate z Jn;n. In the large photonic bandwidth limit we get thus two competing and leads to X terms L ¼ Lsq;D þ Lz;D. Collective dephasing increases L z z z z z H c zðÞ¼ Jn;mðnm mnÞþ : : (12) with the squeezing parameter r, as depicted in solid black n;m in Fig. 3(b). The competition between the dephasing and z 2 2 iqð!1Þjx x j 2 iqð!2Þjx x j with Jn;m¼ ½ð!1Þu e n m þð!2Þv e n m squeezing mechanisms determines an optimal r to gener- where !1 ¼ !L;1 and !2 ¼ !L;2 !0. The term Lz ate maximal entanglement. The latter can be higher than induces a dephasing mechanism that competes with the the one generated by LD;p in the 2LS scheme considered 080502-3 week ending PRL 110, 080502 (2013) PHYSICAL REVIEW LETTERS 22 FEBRUARY 2013 above. The large bandwidth limit is a worst-case scenario Recent experimental results [28,29] show level schemes as typically the waveguide modes are peaked around a in quantum dots similar to those required in our work. certain energy chosen by fabrication. Thus, in the realistic (iii) Markovian approximation. We require N !0 in case the entanglement generation will be in a situation the 2LS scheme and N !L1;2;!a in the 4LS case, between the two limits. The purity of the system is also such that the cooperative decay rate is much smaller affected by the dephasing term; however, one can still find than the transition frequencies; the latter determine the a region that combines high purity and high values of photonic bath memory time [30]. This condition is well entanglement as shown in Fig. 3(b). satisfied in the case of optical transitions of quantum dots. Finally, we discuss the feasibility of our ideas, focusing (iv) Independent decay channels on each transition fre- on the following points. (i) One-dimensional waveguides. quency. This is required for the 4LS’s scheme in the We require a long propagation length and efficient cou- large bandwith limit, to obtain Eqs. (10) and (12), and is pling to the qubits. Coupling to guided modes of 85% to justified as long as N j!L;1 !aj, j!L;2 !aj. Since 89% has been reported for photonic [9,10] and plasmonic differences in the transition energies are of the order of [4,5] waveguides. Theoretical predictions of even higher millielectron volts, this condition imposes a restriction on efficiencies have been pointed out [32,39], although at the the achievable rates for entanglement generation in our expense of reducing the field propagation length. In addi- scheme. (v) Homogeneous couplings. So far we have tion. the precise location of the qubits is also required, neglected inhomogeneities in the couplings and qubit ener- which is possible for solid-state emitters using, i.e., litho- gies, which take the steady state out of the jJ; Mi basis. graphic methods which nowadays have a precision larger This may be a severe restriction in quantum optical solid- than 50 nm [13]. (ii) Lambda transitions in solid-state state devices. Although inhomogeneous broadening in qubits. We assume a degree of addressability of electronic solid-state setups is still of the order of millielectron volts levels similar to the one achieved in atomic physics, espe- for quantum dots [13] and microelectron volts for nitrogen cially the 4LS scheme. Applications in quantum infor- vacancy centers [41], the feasibility of our proposal will mation processing [14,40] typically require controlling benefit from current experimental efforts in the field. optical transitions for spin pumping and initialization. We have carried out calculations to check the effect of experimental imperfections with a focus on an inhomoge- 100 neous random distribution ofP qubit energies, !j, z 50 described by a term Hinh ¼ j !jj (with !j 2 1.2 ½; ). Exact calculations are very demanding; how- 20 1.0 ever, for a limited number of qubits N ¼ 2,3,4,weare 10 0.8 0.01 0.1 1 10 100 1000 able to show that Hinh induces a dephasing time td, such 5 that for t>td, spin squeezing is totally degraded in the 2LS scheme, or strongly decreases below its maximum 2 value in the 4LS case (see Ref. [31] for details). In the 1 10 4LS scheme, one can use a bosonic approximation in the master equation (n bn, with bn a bosonic annihilation 1 operator) and render the problem solvable in a low occu- þ 0 0.1 pation limit hn ni . This method has allowed us to study the scaling of the spin squeezing for large N. Our 0.01 main result is that, under the effect of Hinh, the system 0.1 0.2 0.5 1.0 2.0 reaches the steady-state spin-squeezing values, and after a time td, entanglement degrades down to a residual value. FIG. 3 (color online). Entanglement witness for the 4LS The robustness of the 4LS scheme increases for large N, scheme in the small and large bandwidth limits. since td grows with N. In the inset of Fig. 3(a) we show (a) Entanglement witness (1=2, solid) as a function of the results for small values of r, which are particularly well squeezing parameter, r, for increasing number of qubits (N ¼ described by the bosonization method. We confirm the 2, 10, 50, 100, 200) in the small photonic bandwidth limit, where same scaling with larger values of r with higher degrees z 0 P 1 2 Jm;n . In all the cases, the purity of the system is . of 1= (see Ref. [31] for details). Our conclusion is that 2 Inset: Dynamics of the entanglement witness (1= ) for the the 4LS is advantageous with respect to the 2LS, since it ensembles of qubits with a fixed random dispersion of qubit allows us to generate higher spin-squeezing values with d ¼ 10 energies, , for the different number of qubits depicted increasing with N. in the main panel. (b) Entanglement witness (1=2, solid) and purity (P , dashed) as a function of the parameter r for increasing In conclusion, we have proved that one-dimensional number of qubits (N ¼ 2, 10, 50, 100, 200) in the large plasmonic [4,5] or photonic [6–10] waveguides can be bandwidth limit. The evolution of the collective dephasing used to correlate a large number of qubits by collective mechanism, z, with the squeezing parameter is also plotted in radiative decay. 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