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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 !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