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Dissipation in Few-Photon Waveguide Transport [Invited] 110 Photon. Res. / Vol. 1, No. 3 / October 2013 E. Rephaeli and S. Fan Dissipation in few-photon waveguide transport [Invited] Eden Rephaeli1,* and Shanhui Fan2 1Department of Applied Physics, Stanford University, Stanford, California 94305, USA 2Department of Electrical Engineering, Stanford University, Stanford, California 94305, USA *Corresponding author: [email protected] Received April 15, 2013; revised June 9, 2013; accepted June 9, 2013; posted June 10, 2013 (Doc. ID 188761); published August 30, 2013 We develop a formulation of few-photon Fock-space waveguide transport that includes dissipation in the form of reservoir coupling. We develop the formalism for the case of a two-level atom and then show that our formalism leads to a simple rule that allows one to obtain the dissipative description of a system from the nondissipative case. © 2013 Chinese Laser Press OCIS codes: (270.2500) Fluctuations, relaxations, and noise; (270.4180) Multiphoton processes; (270.5580) Quantum electrodynamics; (290.5850) Scattering, particles; (270.5585) Quantum information and processing; (270.0270) Quantum optics. http://dx.doi.org/10.1364/PRJ.1.000110 1. INTRODUCTION photons to interfere coherently, which can give rise to signifi- – The topic of dissipation in quantum optics is both important cant capabilities in controlling transport [6,7,11,16,19 23], and subtle. Dissipation is important because many quantum switching [17,24], and amplification [9,15,25] processes for optical systems are open; therefore tracking all of the out few-photon states. – flowing energy is often not possible, but accounting for it is A formulation of input output formalism has been recently often quite crucial. Dissipation is subtle because energy in developed that enables analytical calculation of few-photon quantum systems cannot simply disappear, and naively de- scattering matrices in these systems of waveguide coupled scribing dissipation with phenomenological damping factors to quantum multilevel atoms. In this paper we extend this in the system’s equations of motion is formally incorrect, lead- formalism to include dissipation. We model loss by introduc- ing to a non-Hermitian Hamiltonian, violation of commutation ing an auxiliary reservoir that transports energy away from relations [1], and time-reversal invariance [2]. the atom. – A correct treatment of dissipation entails coupling the sys- In applying the input output formalism to treat two-photon tem under study, which usually contains discrete energy lev- transport [8,18], a key step is to insert a complete set of basis els, to a reservoir that is modeled as having a continuum of states for the single-excitation Hilbert space. Here we show energy states [1]. When this is done, the presence of dissipa- that, to obtain the correct result for two-photon transport tion in the system is accompanied by a source of fluctuation in the presence of dissipation, such a basis set needs to from the reservoir, enabling two-way energy exchange. include the states of the reservoir. In quantum optics, the system–reservoir model is treated The paper is organized as follows. In Section 2 we present with many different approaches. In the Schrödinger picture, the Hamiltonian describing a two-level atom with reservoir this model has been treated in the master equation approach, coupling. In Section 3 we obtain the one and two-photon – where multiple-time averages are calculated via the quantum S-matrices of the system using the input output formalism, regression theorem [2]. Alternatively in the Schrödinger pic- showing that excluding the reservoir from the system ture one can use the stochastic approach of the Monte Carlo description leads to violation of outgoing particle exchange wave function [3], which involves evolving the wave function symmetry. Finally, in Section 4 we conclude with a general of the system with a non-Hermitian Hamiltonian term. In the recipe for the inclusion of dissipation in a general class of Heisenberg picture, the input–output formalism [4] is widely important and relevant systems. used. In typical use of the input–output formalism, one as- sumes that the input is a weak coherent state. The nonlinear 2. SYSTEM HAMILTONIAN response is then often treated approximately in the weak We consider a two-level atom side-coupled to a waveguide excitation limit [5]. The input–output formalism is attractive with right and left moving photons, as described by the because the external degrees of freedom, such as those of Hamiltonian the reservoir, are integrated out, allowing one to focus on the Z dynamics of the system alone. † d † d Hwvgatom dx −ivgc x cR xivgc x cL x Recently, there has been great interest in describing the R dx L dx h i properties of photons propagating in a waveguide coupled p † † κδ x c xσ− σ c xc xσ− σ c x to quantum multilevel atoms either directly [6–16] or through R R L L the use of a microcavity [17,18]. The one-dimensional nature Ω σ : (1) of waveguide propagation allows incident and scattered 2 z 2327-9125/13/030110-05 © 2013 Chinese Laser Press E. Rephaeli and S. Fan Vol. 1, No. 3 / October 2013 / Photon. Res. 111 † † Here cR xcR x and cL xcL x create (annihilate) a right or a x − a −x κ a x R p L ; (3b) left moving photon with group velocity vg, respectively. is o 2 the coupling rate between the atom and waveguide. The form of the interaction here assumes the rotating-wave approxima- with the above operators, the even subspace Hamiltonian He tion. σ σ− raises (lowers) the state of the two-level atom and the odd subspace Hamiltonian Ho follow whose transition frequency is Ω. We note that the waveguide Z model in Hwvgatom can describe any single-mode dielectric † d † d or plasmonic waveguide in the optical regime [13] or a trans- H dx −iv a x a x − iv b x b x e g dx r dx mission line in the microwave regime [26]. Since the atomic linewidth is typically quite narrow, a linear approximation δ † σ σ δ † σ σ V xa x − a x V r xb x − b x of the waveguide dispersion relation, which we adopt here, is generally sufficient to describe the waveguide. We further 1 Ωσz; note that the two-level atom in Hwvgatom can describe either a 2 real atom or an artificial atom, such as a quantum dot [13]or Z superconducting qubit [15]. † d Ho −ivg dxao x ao x. To model dissipation we introduce reservoir–atom cou- dx pling as described by Hr and consider the composite Here H describes a one-way propagating chiral waveguide system H Hwvgatm Hr as shown schematically in Fig. 1, e where mode that interacts with the atom. Ho describes a one-way propagating waveguide mode without any interaction with Z the atom. Once the solutions for He are known, the solution † d p † Hr dx −ivrb x b x γδ xb xσ− σ b x : to H H can be straightforwardly obtained by using dx wvgatm r Eqs. (3a) and (3b). In what follows we consider only the even (2) Hamiltonian, since it contains all of the nontrivial physics of the system. Here b x and b† x are the reservoir photon operators The Hamiltonian He can be recast in k-spacep by and γ is the atom–reservoir coupling rate. We assume the res- ≡ 1∕ 2π Rdefining momentum-spacep photonR operators ak ervoir has a continuum of modes centered around the atomic dxa xe−ikx and b ≡ 1∕ 2π dxb xe−ikx, transition frequency Ω. We further assume that within the k r spectral range of interest the dispersion of these reservoir Z † † κ † H dk kv a a kv b b a σ− σ a modes is well described by a group velocity vr. The reservoir e g k k r k k 2π k k will initially be set to the vacuum state. As a result of interac- r tion with the atom, photons originating in the waveguide may γ † 1 b σ− σbk Ωσz: leak back into the waveguide, described in Eq. (1), or they 2π k 2 may leak into the reservoir as described in Eq. (2), resulting in loss. RWe note thatR the total number of excitations Nexc † † σ σ To proceed, we exploit the spatial inversion symmetry of H dkakak dkbkbk − commutes with He, Ho, allowing for the solution to proceed independently in each excitation and decompose it to even and odd subspaces H He Ho, number manifold. where He;Ho0. We transform from right- and left-moving waveguide photon operators to even and odd photon opera- tors. We omit from here on the subscripts of the even mode operators: 3. INPUT–OUTPUT FORMALISM In the absence of dissipation, the one- and two-photon scat- a xa −x tering matrices of this system were solved in [10], to which a x R p L ; (3a) 2 we refer the reader for a detailed derivation of the formalism we use. Our focus here is on the addition of dissipation to the formalism. By defining input and output field operators for the waveguide, Z 1 −ik t−t0 a tp dka t0e ; in 2π k Z 1 −ik t−t1 a tp dka t1e ; out 2π k and reservoir, Z 1 −ik t−t0 Fig. 1. Schematic of sample system considered. A waveguide side b tp dkb t0e ; κ in 2π k coupled with rate to a two-level atom with transition frequency Z Ω. The atom is additionally coupled with rate γ to a reservoir. The 1 −ik t−t1 initial state in the waveguide is either a one- or a two-photon Fock bout tp dkbk t1e ; state, whereas the auxiliary waveguide is initially in the vacuum state.
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