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Exciting a Bound State in the Continuum Through Multiphoton Scattering Plus Delayed Quantum Feedback

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Exciting a Bound State in the Continuum Through Multiphoton Scattering Plus Delayed Quantum Feedback PHYSICAL REVIEW LETTERS 122, 073601 (2019) Exciting a Bound State in the Continuum through Multiphoton Scattering Plus Delayed Quantum Feedback Giuseppe Calajó,1,* Yao-Lung L. Fang (方耀龍),2,3,4 Harold U. Baranger,2 and Francesco Ciccarello2,5,6 1Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, Stadionallee 2, 1020 Vienna, Austria 2Department of Physics, Duke University, P.O. Box 90305, Durham, North Carolina 27708-0305, USA 3Computational Science Initiative, Brookhaven National Laboratory, Upton, New York 11973, USA 4National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, New York 11973, USA 5Universit`a degli Studi di Palermo, Dipartimento di Fisica e Chimica, via Archirafi 36, I-90123 Palermo, Italia 6NEST, Istituto Nanoscienze-CNR, Piazza S. Silvestro 12, 56127 Pisa, Italy (Received 17 November 2018; published 22 February 2019) Excitation of a bound state in the continuum (BIC) through scattering is problematic since it is by definition uncoupled. Here, we consider a type of dressed BIC and show that it can be excited in a nonlinear system through multiphoton scattering and delayed quantum feedback. The system is a semi-infinite waveguide with linear dispersion coupled to a qubit, in which a single-photon, dressed BIC is known to exist. We show that this BIC can be populated via multiphoton scattering in the non-Markovian regime, where the photon delay time (due to the qubit-mirror distance) is comparable with the qubit’s decay. A similar process excites the BIC existing in an infinite waveguide coupled to two distant qubits, thus yielding stationary entanglement between the qubits. This shows, in particular, that single-photon trapping via multiphoton scattering can occur without band edge effects or cavities, the essential resource being instead the delayed quantum feedback provided by a single mirror or the emitters themselves. DOI: 10.1103/PhysRevLett.122.073601 Introduction.—Waveguide Quantum ElectroDynamics stationary entanglement [14,15,18–20]. As a hallmark, (QED) is a growing area of quantum optics investigating this approach for exciting BICs is most effective in the the coherent interaction between quantum emitters and the Markovian regime where the characteristic photonic time one-dimensional (1D) field of a waveguide [1–3]. In such delays, denoted by τ, are very short (e.g., the photon round- systems, a growing number of unique nonlinear and trip time between a qubit and mirror or between two qubits). interference phenomena are being unveiled, the occurrence Indeed, as the time delay grows, the qubit component of the of which typically relies on the 1D nature of such setups. BIC decreases in favor of the photonic component [11–13], Among these is the formation of a class of bound states making such decay-based schemes ineffective for large in the continuum (BIC), which are bound stationary states mirror-emitter or interemitter distances. This is a major that arise within a continuum of unbound states [4]. Topical limitation when entanglement creation is the goal [12]. questions are how to form and prepare such states so as to In order to generate such dressed BICs in the non- enable potential applications such as quantum memory, Markovian regime of significant time delays, one needs which requires light trapping at the few-photon level, of initial states that overlap the BIC’s photonic component, interest for quantum information processing [5–7].We which in practice calls for photon scattering.Asingle show that addressing these questions involves studying photon scattered off the emitters cannot excite a BIC since delayed quantum dynamics in the presence of nonlinearity. the entire dynamics occurs in a sector of the Hilbert space An interesting class of BICs occurs in waveguide QED in orthogonal to the BIC. For multiphoton scattering, how- the form of dressed states featuring one or more emitters, ever, this argument does not hold because of the intrinsic usually qubits, dressed with a single photon that is strictly qubit nonlinearity. Indeed, the role of two-photon scattering confined within a finite region [8–15]. The existence of such has been recognized previously [21,22] in the context of BICs relies on the quantum feedback provided by a mirror or exciting normal bound states (i.e., outside the continuum) the qubits themselves (since a qubit behaves as a perfect that occur in cavity arrays coupled to qubits [23–26]. mirror under 1D single-photon resonant scattering [16,17]). We show that dressed BICs in waveguide-QED setups A natural way to populate these states is to excite the can be excited via multiphoton scattering in two paradig- emitters and then let them decay: the system evolves towards matic setups: a qubit coupled to a semi-infinite waveguide the BIC with an amplitude equal to the overlap between the (see Fig. 1) and a pair of distant qubits coupled to an infinite BIC and the initial state. This results in incomplete decay of waveguide [see Fig. 5(a)]. A perfectly sinusoidal photon the emitter(s) and, in the case of two or more qubits, wave function and stationary excitation of the emitters 0031-9007=19=122(7)=073601(7) 073601-1 © 2019 American Physical Society PHYSICAL REVIEW LETTERS 122, 073601 (2019) qubit with Γ 2V2=v the qubit’s decay rate (without mirror). ¼ mirror incoming scattered The qubit’sexcited-statepopulation(referredtosimplyas BIC wave packet wave packet “population” henceforth) is given by 2 1 εb ; 3 j j ¼ 1 1 Γτ ð Þ þ 2 where τ 2a=v is the delay time. Equations (2) and (3) fully FIG. 1. One-qubit setup: a semi-infinite waveguide, whose end ¼ lies at x 0 and acts as a perfect mirror, is coupled to a qubit at specify the BIC. The photonic wave function has shape ¼ † † † † x a. When a resonant standing wave can fit between the qubit [we set x aˆ x 0 with aˆ x aˆ R x aˆ L x ] and¼ the mirror (k a mπ), an incoming two-photon wave packet j i¼ ð Þj i ð Þ¼ ð Þþ ð Þ 0 ¼ is not necessarily fully scattered off the qubit: a fraction remains x ϕb ∝sin k0x for 0 ≤x ≤a; 4 trapped in the form of a dressed single-photon BIC. h j i ð Þ ð Þ while it vanishes at x ∉ 0;a: the BIC is formed strictly ½ represents a clear signature of single-photon trapping. This between the qubit and the mirror, where the field profile is a pure sinusoid. When the BIC exists (i.e., for k a mπ)the provides a solvable example of non-Markovian quantum 0 ¼ dynamics in a nonlinear system, a scenario of interest in qubit does not fully decay in vacuum [11,39,40]: since the overlap of the initial state e; 0 with the BIC is ε , ε 2 is many areas of contemporary physics [27–33]. j i b j bj Model and BIC.—Consider first a qubit coupled to the also the probability of generating the BIC via vacuum decay. 1D field of a semi-infinite waveguide [Fig. 1(a)] having This probability decreases monotonically with delay time a linear dispersion ω v k (with v the photon group [Eq. (3)], showing that vacuum decay is most effective when velocity and k the wave¼ vector).j j The qubit’s ground and Γτ is small. excited states g and e , respectively, are separated in BIC generation scheme.—Bound state [Eq. (2)] cannot j i j i be generated, however, via single-photon scattering, which energy by ω0 vk0 (we set ℏ 1 throughout). The end of the waveguide¼ at x 0 is effectively¼ a perfect mirror, while involves only the unbound states ϕk that are all orthogo- ¼ nal to ϕ : during a transientfj timeig the photon may be the qubit is placed at a distance a from the mirror. The j bi Hamiltonian under the rotating wave approximation (RWA) absorbed by the qubit, but it is eventually fully released. reads [16,34 37] We thus send a two-photon wave packet such that the initial – ∞ L L joint state is Ψ 0 A 0 dx dy φ1 x φ2 y 1 ↔ 2 † † j ð Þi¼ ½ ð Þ ð Þþ L ∞ aˆ L x aˆ L y g 0 , where A is for normalization, φi x is † d † d ð Þ ð Þj ij i RR ð Þ Hˆ ω σˆ †σˆ − iv dx aˆ x aˆ x − aˆ x aˆ x the wave function of a single left-propagating photon, and ¼ 0 Rð Þdx Rð Þ Lð Þdx Lð Þ Z0 the qubit is not excited. The ensuing dynamics in the two- ∞ † † excitation sector (N 2) is given by V dx aˆ L x aˆ R x σˆ H:c: δ x − a ; 1 ¼ þ 0 ½ð ð Þþ ð ÞÞ þ ð Þ ð Þ Z ∞ † † Ψ t dx ψ η x; t aˆ η x σˆ j ð Þi¼ 0 ð Þ ð Þ with σˆ g e , aˆ R L x the bosonic field operator anni- η R;L Z ¼j ih j ð Þð Þ X¼ hilating a right-going (left-going) photon at position x, and 1 ∞ dx dy χ x; y; t V the atom-photon coupling. Due to the RWA, the total ηη0 þ p2 0 ð Þ ˆ † † η;η0 R;L ZZ number of excitations N σˆ σˆ η R;L dx aˆ η x aˆ η x X¼ ¼ þ ¼ ð Þ ð Þ is conserved. † † ffiffiffi R × aˆ η x aˆ η y g 0 ; 5 In the single-excitation subspaceP (N 1), the spectrum ð Þ 0 ð Þ j ij i ð Þ of Eq.
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