Point-coupling Hamiltonian for frequency-independent linear optical devices
Rahul Trivedi, Kevin Fischer, Sattwik Deb Mishra, Jelena Vuckovic
PPoint-coupling Hamiltonian for frequency-independent linear optical devices
Rahul Trivedi ξ , Kevin Fischer ξ , Sattwik Deb Mishra, and Jelena Vuˇckovi´c E. L. Ginzton Laboratory, Stanford University, Stanford, CA 94305, USA ξ Both the authors contributed equally to this work (Dated: September 24, 2019)We present the point-coupling Hamiltonian as a model for frequency-independent linear opticaldevices acting on propagating optical modes described as a continua of harmonic oscillators. Weformally integrate the Heisenberg equations of motion for this Hamiltonian, calculate its quantumscattering matrix, and show that an application of the quantum scattering matrix on an input stateis equivalent to applying the inverse of classical scattering matrix on the annihilation operators de-scribing the optical modes. We show how to construct the point-coupling Hamiltonian correspondingto a general linear optical device described by a classical scattering matrix, and provide examplesof Hamiltonians for some commonly used linear optical devices. Finally, in order to demonstratethe practical utility of the point-coupling Hamiltonian, we use it to rigorously formulate a matrix-product-state based simulation for time-delayed feedback systems wherein the feedback is providedby a linear optical device described by a scattering matrix as opposed to a hard boundary condition(e.g. a mirror with less than unity reflectivity).
I. INTRODUCTION
Linear-optical elements are key components in any quantum information processing system [1–3]. They can be usedfor applying phase shifts, interfering, and splitting quantum light propagating in waveguide modes or collimatedoptical beams. Consequently, they are traditionally described by scattering matrices [4, 5]. The response of the linearoptical element to an incoming state is often calculated by applying the inverse of the classical scattering matrix onthe annihilation operators [6–8]. The full time evolution of a quantum state incident on linear-optical elements hasalso been captured within the framework of quantum stochastic differential equations [9–11]. Several attempts havebeen made to calculate a Hamiltonian that captures the dynamics of linear optical elements [12–15]. In particular,Refs. [15] and [14] show the existence and explicit construction of an anti-hermitian matrix whose exponential repro-duces the classical scattering matrix of a linear-optical device. A Hamiltonian describing the quantum physics of thelinear-optical device can then be constructed from this matrix. However, this approach needs to introduce a fictitioustime corresponding to the duration for which the incident photons interact with the linear-optical device. While theintroduction of this fictitious time does not impact the relationship between the input and output states, it makes themodel unphysical if the full dynamics of the quantum state is desired. Alternatively, as shown in Ref. [16], a physicallyaccurate Hamiltonian for the linear-optical device can be written down if the normal modes of the linear-optical deviceare known.In this paper, we propose a Hamiltonian that can model an arbitrary frequency-independent linear optical deviceacting on propagating optical modes. The Hamiltonian assumes that the optical modes couple to each other ata single-point in space — we therefore call it the point-coupling Hamiltonian . For a given frequency-independentclassical scattering matrix implemented by the linear optical device, we provide a recipe to construct a point-couplingHamiltonian describing the device. We formally integrate the Heisenberg equations of motion for the point-couplingHamiltonian, and use the resulting solution to calculate its quantum scattering matrix [17–19]. It is shown that anapplication of the quantum scattering matrix on an incoming quantum state is equivalent to applying the inverseof the classical scattering matrix on the annihilation operators of the optical modes in the incoming quantum state,thereby reproducing the commonly used procedure for analyzing the quantum physics of linear-optical devices. Wealso ‘diagonalize’ the point-coupling Hamiltonian to calculate its normal modes — this provides a connection betweenthe Hamiltonian presented in this paper and the quantization approach described in Ref. [16]Finally, in order to demonstrate the utility of the point-coupling Hamiltonian proposed in this paper, we use itto rigorously derive and implement a matrix-product-state (MPS) [20, 21] based update to simulate time-delayedfeedback systems with linear optical devices providing feedback [22, 23]. Time-delayed feedback systems typicallyhave a low-dimensional quantum system, such as a two-level system, coupling to a well-defined optical mode and theemission from the quantum system is used to re-excite the quantum system via a feedback path. The feedback pathcan be constructed using a linear-optical device such as a mirror, or using another quantum system. Time-delayedfeedback systems possibly provide a platform for generation of highly entangled states for quantum computation[24, 25], quantum simulation [26] as well as implementation of quantum memories using long-lived bound states[27]. MPS based simulations of such systems were proposed in Ref. [22], although the formulation relied on hard a r X i v : . [ qu a n t - ph ] S e p boundary conditions e.g. it assumed that the feedback to the quantum system was provided by an ideal mirror.Using the point-coupling Hamiltonian, we outline a method to extend the MPS based simulation of this system toaccount for the situation wherein the mirror is described by a scattering matrix, and thus enable an analysis of theimpact of non-ideality in the mirror on the dynamics of the feedback-system. We expect the point-coupling Hamil-tonian proposed in this paper to be of utility in analyzing quantum systems with complicated linear optical devicesthat can only be described by a full scattering matrix, and cannot be well-approximated by hard boundary conditions.This paper is organized as follows — Section II introduces the point-coupling Hamiltonian and analyzes it in theHeisenberg picture. Using the solution of the Heisenberg equations of motion, we explicitly calculate the quantumscattering matrix for the linear-optical device. It is shown that an application of the quantum scattering matrix onan incoming state is equivalent to applying the inverse of the classical scattering matrix on the annihilation operatorsin the state. We also provide the point-coupling Hamiltonians corresponding to some commonly used linear opticaldevices such as phase shifters, beam splitters and optical circulators. Finally, we diagonalize the point-couplingHamiltonian and show that the classical modes corresponding to the annihilation operators that diagonalize theHamiltonian can be interpreted as the normal modes of the linear-optical device. In section III, we use the proposedpoint-coupling Hamiltonian to analyze feedback into a two-level system from a partially transmitting mirror usingMPS update. II. POINT-COUPLING HAMILTONIANA. Dynamics in the Heisenberg picture
Consider N propagating optical modes (which can physically be waveguide modes, or collimated optical beams)interacting with each other through a linear-optical device (Fig. 1). Note that optical modes that are identical butfor the direction of propagation are counted as separate modes. Labelling by a n ( ω ) the annihilation operator of the n th optical mode at frequency ω , we propose the following Hamiltonian for describing the dynamics of the system: H = N (cid:88) n =1 (cid:90) ∞−∞ ωa † n ( ω ) a n ( ω )d ω + N (cid:88) n =1 N (cid:88) m =1 V n,m a † n ( x n ) a m ( x m ) , (1)where a n ( x n ) is the position-domain annihilation operator for the n th optical mode at coordinate x n along itsdirection of propagation: a n ( x n ) = (cid:90) ∞−∞ a n ( ω ) e i ωx n d ω √ π . (2)Here the position x n is expressed in units of time such that the group velocity, assumed to be frequency independentand uniform across all the optical modes, is unity. The coordinate of the linear-optical device along the direction Linear optical device
FIG. 1. Schematic of a linear optical device acting on N optical modes. The frequency-domain and position-domain annihilationoperator fo the n th optical mode are denoted by a n ( ω ) and a n ( x n ) respectively where x n is the position of the point underconsideration in the coordinate system attached to the n th optical mode (note that we each optical mode to, in general, haveits own independent coordinate system with the linear optical device being at x n = x n in the coordinate system of the n th optical mode.). of propagation of the n th optical mode is x n . Note that we allow for the coordinate x n for different optical modesto be expressed in different coordinate systems — in general, each optical mode can have its own coordinate sys-tem with the x axis of the coordinate system being along its direction of propagation. For e.g. the x axis for aforward propagating mode will be in a direction opposite to the x axis for a backward propagating mode, or modeswaveguides oriented in physically different directions will have differently oriented coordinate systems. We note that a n ( ω ), and consequently a n ( x n ), satisfy the usual bosonic commutation relations: [ a n ( ω ) , a † m ( ω (cid:48) )] = δ n,m δ ( ω − ω (cid:48) ),[ a n ( ω ) , a m ( ω (cid:48) )] = 0, [ a n ( x n ) , a † m ( x (cid:48) m )] = δ n,m δ ( x n − x (cid:48) n ) and [ a n ( x n ) , a m ( x (cid:48) m )] = 0. Moreover, as a consequence of theHermiticity of H , the coupling coefficients V k,m satisfy V n,m = V ∗ m,n .The dynamics of this Hamiltonian can be easily analyzed using the Heisenberg’s equations of motion, which aregiven by: i d a n ( ω ; t )d t = ωa n ( ω ; t ) + N (cid:88) m =1 V n,m √ π a m ( x m ; t ) e − i ωx n ∀ n ∈ { , . . . N } . (3)As is shown in appendix A, these equations can easily be integrated to relate the position-domain annihilation operatorsat time t = t + τ and displaced by a distance x ∈ ( −∞ , ∞ ) from the linear optical device to the position-domainannihilation operators at time t = t : a ( x + x ; t + τ ) a ( x + x ; t + τ )... a N ( x N + x ; t + τ ) = (cid:20) I − i V (cid:18) I + i V (cid:19) − Θ(0 ≤ y ≤ τ ) (cid:21) a ( x + x − τ ; t ) a ( x + x − τ ; t )... a N ( x N + x − τ ; t ) , (4)where V is a N × N Hermitian matrix formed by V m,n as its elements, I is the identity matrix of size N andΘ( x ≤ x ≤ x ) is defined by: Θ( x ≤ x ≤ x ) = x ∈ ( x , x )0 if x ∈ ( −∞ , x ) ∪ ( x , ∞ ) if x ∈ { x , x } . (5)To intuitively interpret the result in Eq. 4, note that if y <
0, then the position-domain annihilation operator at t = t + τ is simply a propagated version of itself at t = t . This is a simply a consequence of the physical pointsdescribed by x < x > τ , since the scattered lightfrom the linear-optical device has not had sufficient time to propagate to the points in question from the location ofthe linear optical device (i.e. from x = 0). For 0 < x < τ , the optical mode annihilation operator at t = t + τ isa sum of a propagated version of itself and contributions from other optical modes as scattered by the linear opticaldevice, and Eq. 4 can be simplified to: a ( x + x ; t + τ ) a ( x + x ; t + τ )... a N ( x N + x ; t + τ ) = S a ( x + x − τ ; t ) a ( x + x − τ ; t )... a N ( x N + x − τ ; t ) , (6)where S = I − i V (cid:18) I + i V (cid:19) − = (cid:18) I − i V (cid:19)(cid:18) I + i V (cid:19) − . (7)Since S relates the optical mode annihilation operators before and after scattering from the linear optical-device hasoccurred, it can be interpreted as the classical scattering matrix corresponding to the linear optical-device. It canreadily be verified that a consequence of V being Hermitian is that the matrix S is unitary. Moreover, if S has thediagonalization S = U diag (cid:2) e i φ (cid:3) U † , it follows from Eq. 7 that: V = − U diag (cid:20) (cid:18) φ (cid:19)(cid:21) U † . (8)Therefore, given the classical matrix S of a linear-optical device, Eq. 8 allows us to construct the matrix V and byextension the point-coupling Hamiltonian that can model the device. As examples, we consider some commonly usedlinear optical devices (Fig. 2) and construct the point-coupling Hamiltonians that describe their dynamics:(a) Phase shifter : The classical scattering matrix S of the phase-shifter [Fig. 2(a)] is a single-element matrix: S = [ e i ϕ ]. From Eq. 8, we obtain V = [ − ϕ/ H phase-shifter = (cid:90) ∞−∞ ωa † ( ω ) a ( ω )d ω − (cid:18) ϕ (cid:19) a † ( x = 0) a ( x = 0) , (9)where a ( ω ) [ a ( x )] is the frequency-domain (position-domain) annihilation operator for the optical mode thatthe phase-shifter is acting on.(b) Beam-splitter : The beam-splitter [Fig. 2(b)] is described by the following classical scattering matrix: S = (cid:20) cos θ sin θ e i ϕ − sin θ e − i ϕ cos θ (cid:21) . (10)Again, using Eq. 8, we obtain V = (cid:20) θ/ e i ϕ −
2i tan( θ/ e − i ϕ (cid:21) , (11)from which we can construct the beam splitter Hamiltonian: H beam-splitter = (cid:88) k ∈{ , } (cid:90) ∞−∞ ωa † k ( ω ) a k ( ω )d ω + (cid:20)
2i tan (cid:18) θ (cid:19) e i ϕ a † ( x = 0) a ( x = 0) + h.c. (cid:21) , (12)where a , ( ω ) [ a , ( x , )] are the frequency-domain (position-domain) annihilation operators for the opticalmodes that the beam-splitter is acting on.(c) Optical circulator : The optical circulator [Fig. 2(c)] routes an excitation in optical mode 1 to optical mode 2,optical mode 2 to optical mode 3 and optical mode 3 to optical mode 1. It can thus be described by the following (a) (c)(b)
FIG. 2. Schematic figures showing the action of (a) phase-shifter with imparted phase ϕ acting on a optical mode denoted by a , (b) beam-splitter with parameters ( θ, ϕ ) acting on optical modes denoted by a and a and (c) an optical circulator designedacting on optical modes a , a and a . The relationship between the input and output signals of the linear-optical device arealso shown in the diagrams. × S = , (13)and therefore V = − −
2i 0 2i2i −
2i 0 , (14)from which we can construct the optical circulator Hamiltonian: H circulator = (cid:88) k ∈{ , , } (cid:90) ∞−∞ ωa † k ( ω ) a k ( ω )d ω + (cid:20) (cid:88) k ∈{ , , } a † k ( x k = 0) a k +1 ( x k +1 = 0) + h.c. (cid:21) , (15)where a , , ( ω ) [ a , , ( x , , )] are the frequency-domain (position-domain) annihilation operators for the opticalmodes that the optical circulator acts on, and a ( ω ) /a ( x ) is to be interpreted as a ( ω ) /a ( x ). B. Quantum scattering matrix of the point-coupling Hamiltonian
While the previous subsection analyzed the point-coupling Hamiltonian in the Heisenberg picture, in this subsectionwe analyze its dynamics in the Schroedinger picture. In particular, we consider the problem of exciting the linear-optical device with an input state, and attempt to calculate the output state produced by the device.The key object relating the input state (assumed to be the asymptote [28] to the system at t → −∞ ) of thesystem to its output state (assumed to be the asymptote [28] to the state of the system at t → ∞ ) is the quantum[28]: S = lim t + →∞ t − →−∞ e i H t + e − i H ( t + − t − ) e − i H t − , (16)where H is the Hamiltonian of the optical modes without accounting for the linear-optical device: H = N (cid:88) k =1 (cid:90) ∞−∞ ωa † k ( ω ) a k ( ω )d ω. (17)Consider now the computation of the following K photon matrix element of the scattering matrix ( x ≡ { x , x . . . x K } , x (cid:48) ≡ { x (cid:48) , x (cid:48) . . . x (cid:48) K } , µ = { µ , µ . . . µ K } and µ (cid:48) = { µ (cid:48) , µ (cid:48) . . . µ (cid:48) K } ): S ( x , µ ; x (cid:48) , µ (cid:48) ) = (cid:104) vac | (cid:20) K (cid:89) i =1 a µ i ( x i ) (cid:21) S (cid:20) K (cid:89) i =1 a † µ (cid:48) i ( x (cid:48) i ) (cid:21) | vac (cid:105) . (18)Note that the relations exp( − i H t ) a k ( x k ) exp(i H t ) = a k ( x k + t ) and exp(i Ht ) a k ( x k ) exp( − i Ht ) = a k ( x k ; t ) togetherwith H | vac (cid:105) = H | vac (cid:105) = 0 immediately imply the following relationship between the scattering matrix element inEq. 18 and the Heisenberg picture optical mode position-domain annihilation operators: S ( x , µ ; x (cid:48) , µ (cid:48) ) = lim t + →∞ t − →−∞ (cid:104) vac | (cid:20) K (cid:89) i =1 a µ i ( x i + t + ; t + ) (cid:21)(cid:20) K (cid:89) i =1 a † µ (cid:48) i ( x (cid:48) i + t − ; t − ) (cid:21) | vac (cid:105) . (19)Using Eq. 4, and in the limit of t + → ∞ and t − → −∞ , it follows that : a µ i ( x i + t + ; t + ) = N (cid:88) ν i =1 S µ i ,ν i a ν i ( x i − x µ i + x ν i + t − ; t − ) , (20)where S i,j are the elements of the classical scattering matrix S . With this, the following explicit expression for S ( x , µ ; x (cid:48) , µ (cid:48) ) as given by Eq. 19 can be obtained: S ( x , µ ; x (cid:48) , µ (cid:48) ) = (cid:88) P K K (cid:89) l =1 S µ l , P K µ (cid:48) l δ (cid:0) ( x l − x µ l ) − ( P K x (cid:48) l − x P K µ (cid:48) l ) (cid:1) , (21)where P K is a K element permutation. It can immediately be noticed that the quantum scattering matrix elementsare completely determined in terms of the classical scattering matrix elements S i,j . From Eq. 21, we can alsoevaluate the frequency domain scattering matrix elements S ( ω , µ ; ω (cid:48) , µ (cid:48) ) ( ω = { ω , ω . . . ω K } , ω (cid:48) = { ω (cid:48) , ω (cid:48) . . . ω (cid:48) K } , µ = { µ , µ . . . µ K } and µ (cid:48) = { µ (cid:48) , µ (cid:48) . . . µ (cid:48) K } ): S ( ω , µ ; ω (cid:48) , µ (cid:48) ) = (cid:90) R K (cid:90) R K S ( x , µ ; x (cid:48) , µ (cid:48) ) (cid:20) K (cid:89) l =1 e i( ω (cid:48) l x (cid:48) l − ω l x l ) π (cid:21) d K x d K x (cid:48) = (cid:88) P K K (cid:89) l =1 S P K µ l ,µ (cid:48) l e − i ω l (cid:0) x P Kµl − x µ (cid:48) l (cid:1) δ ( P K ω l − ω (cid:48) l ) . (22)Note that the frequency domain scattering matrix doesn’t have any connected parts [18] i.e. scattering of a K photonwave-packet from the linear optical device conserves the individual input frequencies. This is a direct consequence ofthe ‘linearity’ of the optical device. To gain more insight into the form of the scattering matrix in Eqs. 21 and 22,consider the calculation of the output state corresponding to a K -photon input state | ψ in (cid:105) : | ψ in (cid:105) = (cid:88) µ (cid:90) R K ψ in ( ω , µ ) (cid:20) K (cid:89) l =1 a † µ l ( ω l ) (cid:21) | vac (cid:105) d K ω , (23)where the amplitude ψ in ( ω , µ ) can be chosen, without any loss of generality, to be symmetric with respect to asimultaneous permutation of the indices µ and ω : ψ in ( ω , µ ) = ψ in ( P K ω , P K µ ) ∀ K -element permutations P K . Theoutput state is then given by: | ψ out (cid:105) = (cid:88) µ (cid:90) R K ψ out ( ω , µ ) (cid:20) K (cid:89) l =1 a † µ l ( ω l ) (cid:21) | vac (cid:105) d K ω , (24)where ψ out ( ω , µ ) is given by: ψ out ( ω , µ ) = 1 N ! (cid:88) µ (cid:48) (cid:90) R K S ( ω , µ ; ω (cid:48) , µ (cid:48) ) ψ in ( ω (cid:48) , µ (cid:48) )d K ω (cid:48) . (25)Assuming that the coordinate systems of the optical modes are chosen such that x k = 0 ∀ k ∈ { , . . . N } , fromEqs. 22 and 25 we obtain: ψ out ( ω , µ ) = 1 N ! (cid:88) µ (cid:48) (cid:88) P K (cid:20) K (cid:89) l =1 S µ l , P K µ (cid:48) l (cid:21) ψ in ( ω , P K µ (cid:48) ) . (26)Therefore, | ψ out (cid:105) = (cid:88) µ (cid:90) R K ψ in ( ω , µ ) (cid:20) K (cid:89) l =1 ˜ a † µ l ( ω l ) (cid:21) | vac (cid:105) d K ω , (27)where ˜ a i ( ω ) = (cid:80) Ni =1 S ∗ j,i a j ( ω ). The application of the quantum scattering matrix of a linear optical device to an inputstate is equivalent to replacing the annihilation operators in the input state with S † times the annihilation operators.This is the usual procedure used to analyze the impact of a linear optical element on an input state [6, 7], and ouranalysis derives it from a Hamiltonian based description of the linear optical element and thus lends rigor to thisprocedure. C. Normal modes of the point-coupling Hamiltonian
In this subsection, we diagonalize the point-coupling Hamiltonian and explicitly calculate the normal modes of thelinear-optical device. Given that the frequency-domain annihilation operator of the n th normal mode is b n ( ω ), itshould satisfy the following commutation relations:[ b n ( ω ) , H ] = ωb n ( ω ) , (28a)[ b n ( ω ) , b † m ( ω (cid:48) )] = δ n,m δ ( ω − ω (cid:48) ) , (28b)where H is the point-coupling Hamiltonian. Eq. 28a is a consequence of the Hamiltonian H being expressible as asum of independent continua of harmonic oscillators in the normal mode basis: H = N (cid:88) n =1 (cid:90) ∞−∞ ωb † n ( ω ) b n ( ω )d ω, (29)and Eq. 28b is a result of the different normal modes being physically independent modes. As is shown in appendixB, using Eqs. 28a and 28b, the normal mode annihilation operators b n ( ω ) can be chosen to be: b n ( ω ) = (cid:90) x n −∞ e − i ω ( x − x n ) a n ( x n ) d x n √ π + N (cid:88) m =1 S ∗ m,n (cid:90) ∞ x m e − i ω ( x − x m ) a m ( x m ) d x m √ π , (30)where S n,m are the elements of the classical scattering matrix. To obtain a physical interpretation of this result,consider creating a photon in the normal mode described by b n ( ω ) and calculating its projection on the modesdescribed by a m ( x m ) — this projection is given by the expectation (cid:104) vac | a m ( x m ) b † n ( ω ) | vac (cid:105) which can be readilyevaluated using Eq. 30: (cid:104) vac | a m ( x m ) b † n ( ω ) | vac (cid:105) = (cid:40) e i ω ( x m − x m ) δ m,n if x m < x m e i ω ( x m − x m ) S m,n if x m > x m . (31)This is exactly the same field profile that would be obtained on classically exciting the linear optical device throughthe n th input port at frequency ω , and calculating the fields scattered in all the output ports — the n th normal modeis simply the continuum of quantum harmonic oscillators associated with this field profile.Finally, Eq. 30 can be inverted to relate the annihilation operators a n ( x ) to the normal mode annihilation oper-ators b n ( ω ) (refer to appendix B for details): a n ( x n ) = (cid:90) ∞−∞ d ω √ π e i ω ( x n − x n ) (cid:40) b n ( ω ) if x n > x n (cid:80) Nm =1 S n,m b m ( ω ) if x n < x n , (32)where, again, we see that the modes described by a n ( x ) are identical to the normal modes at points before thelinear-optical device (i.e. x n < x n in Eq. 32) and their linear combination at points after the linear-optical device.Eqs. 30 and 32 thus provides a connection between the point-coupling Hamiltonian for linear-optical device and adirect quantization of the normal modes of the device [16]. III. MATRIX-PRODUCT-STATE BASED SIMULATIONS OF TIME-DELAYED FEEDBACK SYSTEMS
In this section, we use the point-coupling Hamiltonian proposed in the section II to develop an MPS update for atwo-level system coupled to a waveguide with a partially transmitting mirror. Using the formulated MPS update, westudy the impact of the less than unity reflectivity of the mirror on the dynamics of the time-delayed feedback system.The system under consideration is a two-level system with a time-delayed feedback shown in Fig. 3. The Hamiltonianfor this system can be expressed as a sum of a two-level system Hamiltonian, waveguide Hamiltonian and the mirrorHamiltonian (which is modeled as a point interaction between the forward and backward propagating modes): H ( t ) = H TLS ( t ) + H wg + H mirror + H wg-TLS , (33)where H TLS ( t ) is the Hamiltonian of the two-level system including a coherent drive, H wg is the Hamiltonian de-scribing the forward and backward propagating waveguide modes, H mirror is the Hamiltonian of the mirror providingfeedback to the two-level system and H wg-TLS is the interaction Hamiltonian between the two-level system and thetwo waveguide modes: H TLS ( t ) = ω e σ † σ + Ω( t ) (cid:0) σe i ω t + σ † e − i ω t (cid:1) , (34a) H wg = (cid:88) k ∈{ + , −} (cid:90) ∞−∞ ωa † k ( ω ) a k ( ω )d ω, (34b) H mirror = 2i tan (cid:18) θ (cid:19)(cid:2) e i ϕ a + ( x + = 0) a †− ( x − = 0) − e − i ϕ a − ( x − = 0) a † + ( x + = 0) (cid:3) , (34c) H wg-TLS = (cid:2) √ γ + σ † a + ( x + = t d ) + √ γ − σ † a − ( x − = − t d ) + h.c. (cid:3) . (34d)Here σ is the de-excitation operator for a two-level system with resonant frequency ω e , a + ( ω ) and a − ( ω ) are thefrequency-domain annihilation operators for the forward and backward propagating waveguide modes respectivelyand a + ( x + ) and a − ( x − ) are the position-domain annihilation operators for the forward and backward propagatingwaveguide modes respectively. Note that the x -axis for the coordinate systems for the forward and backward propa-gating waveguide modes are chosen to be in their direction of propagation with the same origin. A mirror, modeledwith a beam splitter Hamiltonian between the forward and backward propagating waveguide modes with parameters( θ, ϕ ), is located at x ± = 0. The two-level system interacts with the waveguide (with a decay rate γ + into the forwardpropagating waveguide mode and γ − into the backward propagating waveguide mode) at a distance of t d from themirror — this corresponds to the point with coordinates x + = t d and x − = − t d in the coordinate systems of theforward and backward propagating waveguide modes. Furthermore, the two-level system is driven with a laser pulseat frequency ω and pulse-shape Ω( t ).We first go into a frame rotating as per the Hamiltonian H = ω σ † σ + H wg + H mirror . In this frame, the state of thesystem evolves as per the Hamiltonian ˜ H ( t ):˜ H ( t ) = δ e σ † σ + Ω( t ) σ x + (cid:0) √ γ + σ † a + ( x + = t d ; t ) e i ω t + √ γ − σ † a − ( x − = − t d ; t ) e i ω t + h.c. (cid:1) , (35)where δ e = ω e − ω and a ± ( x ± = ± t d ; t ) are the Heisenberg picture operators corresponding to a ± ( x ± = ± t d ) withrespect to H at time t , subject to the initial condition a ± ( x ± = ± t d ; t = 0) = a ± ( x ± = ± t d ). From Eq. 4 and for t >
0, we obtain: a + ( x + = t d ; t ) = e − i ω ( t − t d ) (cid:40) A + ( t ) if t < t d A + ( t ) cos θ + e i ϕ A − ( t − t d ) sin θ if t > t d , (36a) a − ( x − = − t d ; t ) = e − i ω ( t + t d ) A − ( t ) , (36b) FIG. 3. Schematic figure of the time-delayed feedback system analyzed in this paper. A two-level system with de-excitationoperator σ couples to a waveguide which supports both forward propagating and backward propagating waveguide modeswith frequeny-domain (position-domain) annihilation operators a + ( ω ) and a − ( ω ) ( a + ( x + ) and a − ( x − )) respectively. A mirror,modelled as a beam-splitter on the forward and backward propagating waveguide mode with parameters ( θ, ϕ ), is located as x ± = 0 and the two-level system couples to both the waveguide modes at a distance t d from the mirror. The decay rates of thetwo-level system into the forward and backward propagating waveguide modes are denoted by γ + and γ − respectively. Finally,we also consider the dynamics of this system when the two-level system is driven by an external laser pulse — Ω( t ) denotesthe complex amplitude of the laser pulse that drives the two-level system. where we have defined the operators A ± ( t ) via: A ± ( t ) = (cid:90) ∞−∞ a ± ( ω ) e − i( ω − ω )( t ∓ t d ) d ω √ π . (37)An MPS update for the state of the system in the rotating frame with respect to H can now be framed using theprocedure outlined in Ref. [22] — the first step is to discretize the waveguide Hilbert space. Using a discretizationstep ∆ t , we can define the waveguide bin operators A ± [ k ] in terms of the operators A ± ( t ) via: A ± [ k ] = (cid:90) ( k +1)∆ tk ∆ t A ± ( t ) d t √ ∆ t , (38)which satisfy the commutation relations [ A + [ i ] , A † + [ j ]] = δ i,j and [ A − [ i ] , A †− [ j ]] = δ i,j . The state of the entire system,including the waveguide and the two-level system, can be represented by a matrix-product state with the two-levelsystem corresponding to the first site in the matrix product state, and the subsequent sites corresponding to thewaveguide bins. The Hilbert space of the k th waveguide bin is spanned by the tensor product of the Hilbert spacesof the harmonic oscillators whose annihilation operators are A + [ k ] and A − [ k ]. To compute the the state at t = k ∆ t from the state at t = ( k + 1)∆ t , we act on the matrix product state with the unitary operator U [ k + 1 , k ] defined by: U [ k + 1 , k ] = e − i H [ k +1 ,k ] , (39)where H [ k + 1 , k ] = (cid:90) ( k +1)∆ tk ∆ t ˜ H ( t ) d t = δ e ∆ t σ † σ + Ω k σ x + σ † (cid:0)(cid:112) γ + ∆ t e i ω t d A + [ k ] + (cid:112) γ − ∆ t e − i ω t d A − [ k ] (cid:1) + h.c. if k ≤ n d σ † (cid:0)(cid:112) γ + ∆ t e i ω t d (cid:0) A + [ k ] cos θ + e i ϕ A − [ k − n d ] sin θ (cid:1) + (cid:112) γ − ∆ te − i ω t d A − [ k ] (cid:1) + h.c. if k > n d , (40)where n d = (cid:98) t d / ∆ t (cid:99) and Ω k = Ω( k ∆ t )∆ t . The application of U [ k + 1 , k ] on the matrix product state at time step k requires the implementation of a long-range gate, since it acts on the site corresponding to the two-level system, the k th waveguide bin and the ( k − n d ) th waveguide bin. Following the approach introduced in Ref. [22], we implementthis long range gate using a sequence of swap operations [29] followed by a short-range gate corresponding to U [ k +1 , k ][30]. The update is implemented using the tensor network state python library tncontract [31] along with qutip [32]We first validate our MPS update implementation [33] against the implementation introduced in Ref. [22] for anideal mirror (i.e. θ = π/ t ) = Ω e − αt ). We simulate both of these settings for different mirror phase ϕ —as is known in such feedback systems with ideal mirrors, a properly chosen mirror phase can result in the emitternot decaying completely into the waveguide mode, rather exciting the bound state that exists between the emitterand the waveguide mode. We observe such incomplete decay for ϕ = 0, and a complete decay of the emitter into thewaveguide mode for other mirror phases. Moreover, the MPS update implementation presented in this section agreesperfectly with the MPS update implementation introduced in Ref. [22]. This perfect agreement can be analyticallyexplained by considering the normal modes of the point-coupling Hamiltonian describing the mirror (as described insection II C). Using Eq. 30 for a perfect mirror ( θ = π/ b L,R ( ω ) can be expressed in termsof a ± ( x ) via: b L ( ω ) = (cid:90) −∞ a + ( x + ) e − i ωx + d x + √ π − e i ϕ (cid:90) ∞ a − ( x − ) e − i ωx − d x − √ π , (41a) b R ( ω ) = (cid:90) −∞ a − ( x − ) e − i ωx − d x − √ π + e − i ϕ (cid:90) ∞ a + ( x + ) e − i ωx + d x + √ π . (41b)Clearly, b † L ( ω ) only creates excitations to the left of the mirror (which do not interact with the two-level system)0 Time t . . . . . . | " ( t ) | This paper (Solid), Ref. [19] (Dashed) ' = 0.0 ⇡' = 0.25 ⇡' = 0.5 ⇡ ' = 0.75 ⇡' = 1.0 ⇡ Time t . . . . . . | " ( t ) | This paper (Solid), Ref. [19] (Dashed) ' = 0.0 ⇡' = 0.25 ⇡' = 0.5 ⇡ ' = 0.75 ⇡' = 1.0 ⇡ (a) (b)This paper (solid), Ref. [22] (dashed) This paper (solid), Ref. [22] (dashed) FIG. 4. Validation of our MPS update implementation against the implementation introduced in Ref. [22] for an ideal mirror(i.e. θ = π/
2) for two simulation settings: (a) Simulation of an undriven two-level system (Ω( t ) = 0) which is initially in itsexcited state for different mirror phases ϕ and (b) simulation of a two-level system initially in its ground state and driven by anexponentially decaying pulse (Ω( t ) = Ω e − αt for t >
0) for different mirror phases ϕ . | ε ( t ) | is the probability amplitude of thetwo-level system being in the excited state computed using | ε ( t ) | = (cid:104) σ † σ (cid:105) . It is assumed that γ + = γ − = γ/ δ e = ω e − ω = 0, ω t d = π , γt d = 2 and α = 2 γ . For the discretization into an MPS, we use γ ∆ t = 0 .
05, and truncate the dimensionality of theHilbert space of each waveguide bin to 2 for both forward and backward propagating modes. A threshold of 0 .
01 is used inall the Schmidt decompositions performed while applying the swap gates and the short-range gates. Refer to appendix C forconvergence studies of the MPS simulations. while b † R ( ω ) only creates excitations to the right of the mirror (which interact with the two-level system). Indeed,using Eq. 32, it can easily be seen that the interaction Hamiltonian between the waveguide and the two-level system( H wg-TLS defined in Eq. 34d) can be entirely expressed in terms of b R ( ω ): H wg-TLS = (cid:90) ∞−∞ (cid:20)(cid:0) √ γ + σ † e i( ωt d + ϕ ) + √ γ − σ † e − i ωt d (cid:1) b R ( ω ) + h.c. (cid:21) d ω √ π , (42)which is identical to the interaction Hamiltonian used in Ref. [22].Next, we study the impact of the non-ideality (i.e. the mirror reflection being less than 1) in the mirror onthe dynamics of the feedback system. Changing the mirror reflection is equivalent to changing the parameter θ describing the mirror. Fig. 5 shows the impact of θ on the dynamics of the feedback system. Again, we simulate twodifferent settings — Fig. 5(a) in which the emitter is initially prepared in its excited state and allowed to decay intothe waveguide without any external driving, and Fig. 5(b) in which the emitter is initially in its ground state and thendriven by an exponentially decaying pulse (Ω( t ) = Ω e − αt ). We note that unlike the case of an ideal mirror, havinga less than unity reflectivity implies that there is no bound state that the emitter can decay into. Consequently,the emitter always decays to the ground state — however, as is seen in Fig. 5, the decay rate can be controlled bycontrolling the reflectivity of the mirror. While this seems reminiscent of the Purcell effect, we note that even with aless than unity reflectivity, the emitter does not decay exponentially into its ground state — this is a consequence ofthe non-Markovian nature of the feedback system.Finally, we point out that an alternative to using the point-coupling Hamiltonian for analyzing the feedback systemconsidered in this section is to directly formulate the Hamiltonian using the normal modes of the mirror-waveguidesystem. Such a formulation would also be able to capture the impact of the mirror reflectivity on the dynamics of thetwo-level system if it was accounted for in the construction of the normal modes. The point-coupling Hamiltonianprovides an alternative, and equivalent (as shown in section II C), approach for modelling linear-optical devices.Apart from the didactic importance of this result, we expect it to be of utility in understanding complicated feedbacksystems with multiple linear optical devices providing feedback.1 Time t . . . . . . | " ( t ) | cos ✓ = 0.2cos ✓ = 0.4cos ✓ = 0.6 cos ✓ = 0.8cos ✓ = 1.0 0 10 20 30 40 50 Time t . . . . . . | " ( t ) | cos ✓ = 0.2cos ✓ = 0.4cos ✓ = 0.6 cos ✓ = 0.8cos ✓ = 1.0 (a) (b) FIG. 5. Impact of mirror reflectivity on the dynamics of the time-delayed feedback system for two simulation settings: (a)Simulation of an undriven two-level system (Ω( t ) = 0) which is initially in its excited state for different mirror reflectivitiescos θ and (b) simulation of a two-level system initially in its ground state and driven by an exponentially decaying pulse(Ω( t ) = Ω e − αt for t >
0) for different mirror reflectivities cos θ . | ε ( t ) | is the probability amplitude of the two-level systembeing in the excited state computed using | ε ( t ) | = (cid:104) σ † σ (cid:105) . It is assumed that γ + = γ − = γ/ δ e = ω e − ω = 0, ω t d = π , γt d = 2, ϕ = 0 and α = 2 γ . For the discretization into an MPS, we use γ ∆ t = 0 .
05, and truncate the dimensionality of theHilbert space of each waveguide bin to 2 for both forward and backward propagating modes. A threshold of 0 .
01 is used inall the Schmidt decompositions performed while applying the swap gates and the short-range gates. Refer to appendix C forconvergence studies of the MPS simulations.
IV. CONCLULSION
This paper resolves the problem of calculating the Hamiltonian for a frequency-independent linear-optical devicefrom its classical scattering matrix. It is shown that an application of the quantum scattering matrix correspondingto the proposed Hamiltonian on an input state is equivalent to applying the inverse of the classical scattering matrixof the linear-optical device on the annihilation operators in the input state. We also diagonalize the point-couplingHamiltonian and provide a connection between the point-coupling Hamiltonian and the quantization of normal modesof the linear-optical device [16]. Finally, we demonstrate the practical utility of the proposed Hamiltonian by usingit to rigorously formulate an MPS-based update for a time-delayed feedback system wherein the linear-optical deviceproviding feedback is described by a full scattering matrix as opposed to a hard boundary condition.
V. ACKNOWLEDGEMENTS
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The Heisenberg equation of motion (Eq. 3) can be easily integrated from t to t + τ (where τ >
0) to obtain: a n ( ω ; t + τ ) = a n ( ω ; t ) exp( − i ωτ ) − i N (cid:88) m =1 V n,m e − i ωx n (cid:90) t + τt a m ( x m ; t (cid:48) ) e − i ω ( t + τ − t (cid:48) ) d t (cid:48) √ π (A1)Therefore, it follows from Eq. 2 that a n ( x n ; t + τ ) is given by: a n ( x n ; t + τ ) = a n ( x n − τ ; t ) − i N (cid:88) m =1 V n,m (cid:90) t + τt a m ( x m ; t (cid:48) ) δ ( x n − x n − t − τ + t (cid:48) )d t (cid:48) = a n ( x n − τ ; t ) − i (cid:20) N (cid:88) m =1 V n,m a m ( x m ; t + τ − ( x n − x n )) (cid:21) Θ( x n ≤ x n ≤ x n + τ ) (A2)where the function Θ( x ≤ x ≤ x ) is defined in Eq. 5. Imposing Eq. A2 at x n = x n ∀ n ∈ { , . . . N } , we obtain: a n ( x n ; t + τ ) = a n ( x n − τ ; t ) − i2 N (cid:88) m =1 V n,m a m ( x m ; t + τ ) ∀ n ∈ { , . . . N } (A3)from which we can obtain a n ( x n ; t ) in terms of operators at t = t : a ( x ; t + τ ) a ( x ; t + τ )... a N ( x N ; t + τ ) = (cid:18) I + i V (cid:19) − a ( x − τ ; t ) a ( x − τ ; t )... a N ( x N − τ ; t ) (A4)where V is a N × N Hermitian matrix formed by V m,n as its elements and I is the identity matrix of size N .Substituting Eq. A4 into Eq. A2 together with the substitution x n = x n + y , we obtain Eq. 4 Appendix B: Calculating normal modes of the point coupling Hamiltonian
Since the normal mode annihilation operator b n ( ω ) is expressible as a linear combination of the annihilation operators a n ( x n ), we assume the following ansatz for b n ( ω ): b n ( ω ) = N (cid:88) m =1 (cid:90) ∞−∞ F n,m ( ω, x ) a m ( x + x m ) d x √ π , (B1)where F n,m ( ω, x ) is a function that is to be determined. Using Eq. 28a, we obtain the following differential equationfor F n,m ( ω, x ): i ∂ F ( ω, x ) ∂x + δ ( x ) F ( ω, V = ω F ( ω, x ) . (B2)Here F ( ω, x ) is a N × N complex matrix with the functions F n,m ( ω, x ) as its elements. The solution for Eq. B2 is ofthe form: F ( ω, x ) = (cid:40) F + ( ω ) e − i ωx x > F − ( ω ) e − i ωx x < . (B3)To evaluate the matrices F + ( ω ) and F − ( ω ), we use the boundary condition obtained on integrating Eq. B2 across asmall interval around x = 0: − i (cid:2) F ( ω, x = 0 + ) − F ( ω, x = 0 − ) (cid:3) + 12 (cid:2) F ( ω, x = 0 + ) + F ( ω, x = 0 − ) (cid:3) V = 0 , (B4)4from which it immediately follows that F + ( ω ) = F − ( ω ) S † , where S is the classical scattering matrix of the linear-optical device. Moreover, using Eq. B2 along with Eq. 28b, we obtain F †− ( ω ) F − ( ω ) = I – any choice for F ( ω ) isvalid as long as it satisfies this constraint. Choosing F − ( ω ) = I and F + ( ω ) = S † , we obtain Eq. 30.To derive Eq. 32, we assume the following anstaz for a n ( x n ) in terms of b n ( ω ): a n ( x + x n ) = (cid:90) ∞−∞ (cid:88) n,m G n,m ( x, ω ) b m ( ω ) d ω √ π , (B5)where G n,m ( x, ω ) is a function that is to be determined. Note that G n,m ( x, ω ) = √ π [ a n ( x + x n ) , b † m ( ω )] and thereforefrom Eq. B1 it follows that G n,m ( x, ω ) = F ∗ m,n ( ω, x ). We thus immediately obtain: G n,m ( x, ω ) = (cid:40) δ n,m e i ω ( x − x n ) if x < x n S n,m e i ω ( x − x n ) if x > x n . (B6)Substituting this into Eq. B5, we immediately obtain Eq. 32. Appendix C: Convergence of Matrix-product-state simulations
Simulating time-delayed feedback with MPS, as described in section III, requires discretizing the waveguide modeswhich are described as continuas of harmonic oscillators into waveguide bins (Eq. 38). This introduces a simulationparameter δt which controls the coarseness of this discretization. Moreover, application of a unitary impactingmultiple bins in the MPS requires us to perform Schmidt decompositions on the state, and neglect Schmidt vectorswith coefficients smaller than a specified tolerance. For the MPS-based simulation to be accurate, it is necessary toensure that it has converged with respect to these two parameters. In this appendix, we present convergence studieswith respect to these parameters to show that the choice of ∆ t (= 0 . /γ ) and tolerance (0.01) assumed in sectionIII indeed results in accurate simulations.For the convergence study, we study the problem of an emitter decaying into the waveguide with a time-delayedfeedback provided by the mirror and calculate the probability amplitude | ε ( t ) | of the two-level system being in itsexcited state at time t calculated using | ε ( t ) | = (cid:104) σ † σ (cid:105) . We also assume that the mirror is perfect with phase ϕ = 0(which, as can be seen from Fig. 4(a) results in the two-level system not completely decaying into the ground state).In this case, the time-dependence of the complex amplitude of the excited state ε ( t ) is given by the following ordinarydifferential equation (ODE) [34]: d ε ( t )d t = − γ ε ( t ) − γ e i(2 ω t d + ϕ ) ε ( t − t d ) , (C1)where γ = γ + + γ − is the total decay rate of the two-level system into the forward and backward propagating waveg-uide modes and this equation is solved subject to the initial condition ε (0) = 1. ε ( t ) obtained on solving this ODEthus provides a benchmark simulation that the MPS simulation can be compared against to gauge its convergence.Fig. 6 shows the results of the convergence studies. The dependence of | ε ( t ) | obtained from the MPS update onthe discretization time-step ∆ t is shown in Fig. 6(a) — it can be seen that choosing ∆ t < . /γ is sufficient to ensurethat the MPS simulation has converged and agrees with the ODE simulation. A similar study performed with respectto the tolerance used in the Schmidt decomposition is shown in Fig. 6(b) from which it can be seen that a tolerancesmaller than 0.1 is sufficient to ensure that the MPS simulation has converged and agrees with the ODE simulation.5 Time t . . . . . | " ( t ) | t =0.05 t =0.1 t =0.15 t =0.5 t =1.0ODE 0 10 20 30 40 50 Time t . . . . . . | " ( t ) | tol = 0.01tol = 0.05tol = 0.1 tol = 0.25tol = 0.5ODE (a) (b) FIG. 6. Convergence studies on the MPS simulations for the time-delayed feedback system introduced in section III. Theundriven two-level system (Ω( t ) = 0) is initially in its excited state and is allowed to decay into the forward and backwardpropagating waveguide modes with an ideal mirror ( θ = π/ , ϕ = 0) providing feedback. | ε ( t ) simulated using MPS update for(a) different discretization time-steps ∆ t and (b) different tolerances that govern the number of Schmidt vectors retained afterevery Schmidt decomposition on the MPS. The dotted black line shows | ε ( t ) | obtained on solving the ODE (Eq. C1) with a verysmall time-step ( γ ∆ t = 0 . | ε ( t ) | is the probability amplitude of the two-level system being in the excited state computedusing | ε ( t ) | = (cid:104) σ † σ (cid:105) . In the simulations shown in (a) the Schmidt tolerance is assumed to be 0.01 and in the simulations shownin (b) γ ∆ t = 0 .
05. It is also assumed that γ + = γ − = γ/ δ e = ω e − ω = 0, ω t d = π and γt dd