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N-Photon Wave Packets Interacting with an Arbitrary Quantum System

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N-Photon Wave Packets Interacting with an Arbitrary Quantum System PHYSICAL REVIEW A 86, 013811 (2012) N-photon wave packets interacting with an arbitrary quantum system Ben Q. Baragiola,1 Robert L. Cook,1 Agata M. Branczyk,´ 2 and Joshua Combes1 1Center for Quantum Information and Control, University of New Mexico, Albuquerque, New Mexico 87131-0001, USA 2Department of Physics and Centre for Quantum Information and Quantum Control, University of Toronto, Toronto, Ontario M5S 1A7, Canada (Received 20 March 2012; published 9 July 2012) We present a theoretical framework that describes a wave packet of light prepared in a state of definite photon number interacting with an arbitrary quantum system (e.g., a quantum harmonic oscillator or a multilevel atom). Within this framework we derive master equations for the system as well as for output field quantities such as quadratures and photon flux. These results are then generalized to wave packets with arbitrary spectral distribution functions. Finally, we obtain master equations and output field quantities for systems interacting with wave packets in multiple spatial and/or polarization modes. DOI: 10.1103/PhysRevA.86.013811 PACS number(s): 42.50.Ct, 03.67.−a, 42.50.Lc, 03.65.Yz Nonclassical states of light are important resources for Quantum quantum metrology [1,2], secure communication [3], quantum wave packet envelope System networks [4–6], and quantum information processing [7,8]. Of particular interest for these applications are traveling wave packets prepared with a definite number of photons t in a continuous temporal mode, known as continuous-mode portion of the wave packet that Fock states [9–12]. As the generation of such states becomes has interacted with the system technologically feasible [13–24] a theoretical description of the light-matter interaction [25] becomes essential (see Fig. 1). FIG. 1. (Color online) Schematic depiction of a traveling wave Previously, aspects of continuous-mode single-photon packet interacting with an arbitrary quantum system. The temporal states interacting with a two-level atom have been examined. wave packet is described by a slowly varying envelope ξ(t)which Others have investigated master equations [26], two-time modulates fast oscillations at the carrier frequency. We consider the correlation functions [26,27], properties of scattered light case where the wave packet is prepared in a nonclassical state of [27–38], and optimal pulse shaping for excitation [38–42]. definite photon number. The results in these studies were produced with a variety of methods which have not been applied to many systems other extended in Sec. III to continuous-mode “N-photon states,” than two-level atoms or Fock states where N 1(however, where the spectral density function is not factorizable. Then, see [43]). in Sec. IV we apply our formalism to the study of a two-level One way to approach such problems is through the input- atom interacting with wave packets prepared in N-photon output formalism of Gardiner and Collett [44–48]. A central Fock states. This application is intended to serve as an result of input-output theory is the Heisenberg-Langevin instructive example that reproduces and extends results in equation of motion driven by quantum noise that originates previous studies [39–41]. In Sec. V, we present the second from the continuum of harmonic oscillator field modes [46,49]. main result: master equations and output field quantities for a The application of input-output theory to open quantum system interacting with Fock-state wave packets in two modes systems has historically been restricted to Gaussian fields (e.g., spatial or polarization). This sets the stage for the study of [45,46,50]—vacuum, coherent, thermal, and squeezed—with many canonical problems in quantum optics. As a two-mode several notable exceptions [26,51–54]. example, we examine the scattering of Fock states from a In this article we present a unifying method, based on input- two-level atom in Sec. VI. Finally, we conclude in Sec. VII output theory, for describing the interaction between a quan- with discussion and possible applications. tum system and a continuous-mode Fock state. Consequently, our formalism encapsulates and extends previous results. I. MODEL AND METHODS Specifically, our method allows one to derive the master A description of a system interacting with a traveling equations and output field quantities for an arbitrary quantum wave packet naturally calls for a formulation in the time system interacting with any combination of continuous-mode domain. The input-output theory developed in the quantum N-photon Fock states. optics community provides such a description [45–48,50–52]. This article is organized as follows. In Sec. I we introduce Often input-ouput theory is formulated for a one-dimensional the white-noise Langevin equations of motion, the mathe- (1D) electromagnetic field, although this is not a necessary matical description of quantum white noise, and the formal restriction [50]. (Such effective one-dimensional models are definition of continuous-mode Fock states. In Sec. II we typically thought about in the context of optical cavities [55] present the first main results: the method for deriving master or photonic waveguides [35,56–58].) In this formalism the equations for systems interacting with continuous-mode Fock rotating wave approximation, the weak-coupling limit (the states and related output field equations. This result is then Born approximation), and the Markov approximation are made 1050-2947/2012/86(1)/013811(18)013811-1 ©2012 American Physical Society BARAGIOLA, COOK, BRANCZYK,´ AND COMBES PHYSICAL REVIEW A 86, 013811 (2012) [59,60]. Strict enforcement of these approximations is known the quantum noise increments: as the quantum white-noise limit [61]. t+dt t+dt In Appendix A1we review the quantum white-noise limit; = † = † dBt ds b(s) and dBt ds b (s), (3) other introductory material can be found in Refs. [45,46, t t 59,62]. The main result is a quantum stochastic differential t+dt = † equation (QSDE) for the unitary time evolution operator that dt ds b (s)b(s), (4) governs the system-field dynamics. From this equation one can t derive QSDEs for system and field operators driven by white which drive the Heisenberg dynamics in Eq. (1). noise, also known as white-noise Langevin equations. These Under vacuum expectation, the calculus rules for manipu- equations of motion are what lie at the heart of the derivation lating QSDEs are summarized by the relations of Fock-state master equations. † = = dBt dBt dt, dBt dt dBt , The Langevin equations derived in the white-noise limit (5) are in Stratonovich form [9,46,63]. Stratonovich QSDEs obey = † = † dt dt dt ,dt dBt dBt . the standard rules of calculus, but expectations can be hard to calculate because the quantum noises do not commute with These composition rules are often referred to as the vacuum the operators to which they couple. Stratonovich QSDEs can Ito¯ table. be converted to an equivalent form known as the ItoQSDEs.¯ As a prelude to the derivation of the Fock-state master In Ito¯ form the quantum noises commute with the operators equations, we derive the vacuum master equation. First, we to which they couple, which facilitates taking expectations. take vacuum expectations of Eq. (1) using the following = ⊗ However, differentials must be calculated to second order [46]. notation (to be explained in Sec. II): E0,0[dX] Tr[(ρsys † To derive master equations we take expectations over field |00|) dX]. Consequently, we need the action of the quantum states and consequently work solely with ItoQSDEs.¯ noise increments on vacuum, dBt |0=0, (6) A. Derivation of the vacuum master equation from the Ito¯ Langevin equations dt |0=0. (7) Consider an arbitrary system operator in the interaction All of the quantum noise terms in Eq. (1) vanish under vacuum. = ⊗ picture, X(t), with the initial condition X(t0) X Ifield.The Then, using the cyclic property of the trace we obtain the time evolution of X is given by the Ito¯ Langevin equation [see vacuum master equation: Appendix (A3)] d =− + L = + L† + † + † † 0,0(t) i[H,0,0] [L]0,0, (8) dX (i[H,X] [L]X)dt [L ,X]SdBt S [X,L]dBt dt † + (S XS − X)dt , (1) where the Lindblad superoperator is defined as L = † − 1 † + † where the action of the superoperator is [L] LL 2 (L L L L), (9) † † 1 † † L = − + and the subscripts on 0,0 denote that Eq. (8) is a vacuum [L]X L XL 2 (L LX XL L). (2) master equation. The operators (S,L,H) act on the system Hilbert space. The quantum noise increments dB , dB †, and d are field t t t B. Continuous-mode Fock states operators, discussed in more detail shortly. The first two terms in Eq. (1) describe smooth evolution A continuous-mode single-photon state [9,10,12] can be from an external Hamiltonian on the system and from a interpreted as a single photon coherently superposed over Lindblad-type dissipator. The second two terms describe the many spectral modes [66,67] with weighting given by the influence of quantum noise through coupling of a system spectral density function (SDF) ξ˜(ω), operator L linearly to the field operators, for example, | = ˜ † | dipole-type coupling. The final term arises from coupling 1ξ dωξ(ω)b (ω) 0 . (10) of a system operator S to a quantity quadratic in the field operators, such as photon number. Such effective couplings We focus on quasimonochromatic wave packets, where the appear in optomechanical systems [64] and arise after adiabatic spectral spread is much smaller than the carrier frequency, elimination of the excited states in multilevel atoms [65], for ω ωc [68]. This holds for optical carriers, whose band- example.
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