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Mesoscopic Entanglement Induced by Spontaneous Emission in Solid-State Quantum Optics

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Mesoscopic Entanglement Induced by Spontaneous Emission in Solid-State Quantum Optics 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 Quantum Optics 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 spin 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 photon 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 Pauli matrices 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 superradiance [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 lasers, where we im- actions [23,24]. In our scheme those interactions can be pose the condition: !L;1 À !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 À !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, À 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 phase transition in the steady state of the model at a critical pumping rate ¼ NÀ=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 Àiqx 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, NÀ=2. Since LD;p can be obtained À À 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 ¼jÉcihÉcjþO ðÀÞ, where jÉci¼ master equation of the form d=dt ¼ LðÞ [30], with iÀSx 2 2 the superoperator e jN= ; ÀN= i is a spin coherent state. (ii) The mixed X state phase, NÀ=2. Here we get an infinite tempera- À þ þ À L L ðÞ¼ Jn;mðn m À 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 SÀ ! 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 À 1 X Sec. A of the Supplemental Material [31]). Special care L À 2 þ ð À 2 Þ þ H c D;p 2 SxSx Sx 2 S S 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 À which has the infinite temperature state s ¼ as the ¼ iqð!0ÞjxnÀxmj 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 À 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 NðÁSxÞ 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 À 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 À for increasing number of qubits N in Fig. 2. We find a range À þ þ À L DðÞ¼ ðS S À 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 À À þ þ Pwith 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 jÉ0i¼ states benefits from a collective enhancement scaling as njein ¼jN=2;N=2i the system evolves within the sector ÀN, so that the higher the number of qubits, the more J ¼ N=2.
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