Unraveling Mirror Properties in Time-Delayed Quantum Feedback Scenarios
UUnraveling Mirror Properties in Time-Delayed QuantumFeedback Scenarios
Fabian M. Faulstich , ∗ Manuel Kraft , and Alexander Carmele Institut f¨ur Theoretische Physik, Technische Universit¨at Berlin,Hardenbergstraße 36, 10623 Berlin, Germany Institut f¨ur Mathematik, Technische Universit¨at Berlin,Straße des 17. Juni 136, 10623 Berlin, Germany
Abstract
We derive in the Heisenberg picture a widely used phenomenological coupling element to treatfeedback effects in quantum optical platforms. Our derivation is based on a microscopic Hamilto-nian, which describes the mirror-emitter dynamics based on a dielectric, a mediating fully quan-tized electromagnetic field, and a single two-level system in front of the dielectric. The dielectricis modeled as a a system of identical two-state atoms. The Heisenberg equation yields a system ofdescribing differential operator equations, which we solve in the Weisskopf-Wigner limit. Due to afinite round-trip time between emitter and dielectric, we yield delay differential operator equations.Our derivation motivates and justifies the typical phenomenological assumed coupling element andallows, furthermore, a generalization to a variety of mirrors, such as dissipative mirrors or mirrorswith gain dynamics. ∗ E-mail me at: [email protected]; Visit: http://page.math.tu-berlin.de/˜faulstich/ a r X i v : . [ qu a n t - ph ] S e p . INTRODUCTION Feedback protocols are successfully applied to stabilize periodic processes in classicaland quantum mechanical systems [1, 2]. In semi-classical systems Pyragas control allowsto suppress relaxation oscillations in the switch-on dynamics of a semiconductor laser [3].In those systems, feedback control is modelled via a Maxwell-based treatment of the light-matter interaction. The paradigm for a Maxwell theory based feedback control is the Lang-Kobayashi model, where part of the laser output is fed back into the laser dynamics [4,5]. Instead of self-feedback, in quantum systems measurement-based setups of feedbackcontrol are explored. They allow to stabilize Fock states, theoretically predicted and alreadyexperimentally demonstrated, e.g., in cQED systems [6]. Quantum feedback, however, isnot restricted to a read-out and open quantum system approach, first experiments studythe many-photon quantum limit of feedback [7–9].These successful experimental implementations of coherent feedback, or non-invasive self-feedback, increase the interest for models, which allow predictions and interpretations of theobserved feedback effects. A variety of models have been proposed in the linear and nonlinearregime. For example, a cavity-QED system is driven into the strong coupling regime [10], or alaser-driven two-level system is partially interacting with its own emission statistics, showingmodified Mollow triplet signatures [11, 12]. Another promising route is feedback-inducedparametric squeezing [13, 14], and enhancing of network entanglement by phase-selectivefeedback based state addressing [15, 16]. All these models are based on a phenomenologicalcoupling of the emitters to the radiation field, namely a momentum dependent couplingstrength.In this article, we justify the assumed coupling element. In order to do this, we analyze theinteraction of an initially excited two-state atom with a quantized electromagnetic field and adielectric medium of N two-state atoms. The discussion is restricted to the one-dimensionalcase where the electromagnetic field modes wave vectors k are considered to be parallel tothe z -axis. This system is illustrated in Fig. 1. As a system, we assume a semi-infinite one-dimensional waveguide [17]. Material platform for such systems are, e.g., superconductingtransmission lines [18], diamond nanowires mediating between nitrogen-vacancy centers [19],hollow optical fibers with cold atoms [20], and photonic crystal waveguides coupling quantumdots [21], or plasmonic nanowires [22]. 2he article is structured as follows. First, we describe briefly in the next section, Sec. II,the phenomenological model that is widely used in the literature [23, 24]. This model is ourbenchmark, and the following sections fulfill the purpose to give a microscopic justificationfor the applied quantum optical, momentum-dependent coupling. In order to do this, wepresent a microscopical Hamiltonian in Sec. III. Starting from this Hamiltonian, we employthe Heisenberg equation approach and derive operator equations of motions for the dynamicsof an atom near a plane dielectric interface, mathematically rigorously in Sec. IV. Thissection ends with an effective operator equations in analogy to the effective model of Sec.II. This is the main result of the paper. In Sec. V, we conclude the article and give a shortoutlook, about possible extensions of the model. II. PHENOMENOLOGICAL MODEL
In this section, we describe the effective and widely used phenomenological model. Com-monly, the interaction between a dielectric and an electromagnetic field is simplified byassuming a hard-wall boundary at the position of the dielectric. This assumption yields thefollowing Hamiltonian [23, 24]: H eff / (cid:126) = ω e P † P + (cid:88) k (cid:114) cK , Lπ sin( kl ) (cid:16) P † r k + r † k P (cid:17) + (cid:88) k ω k r † k r k , (1)where ω e is the transition frequency of the atom, K , is the coupling constant between atomand field, L is the length of the quantized box, ω k is the frequency of mode k and l is thedistance between the atom and the dielectric. The ground and excited state of the initiallyexcited atom at position z are described by the operators G = | (cid:105)(cid:104) | and E = | (cid:105)(cid:104) | ,respectively. Its excitation (resp. de-excitation) dynamics is denoted by the raising (resp.lowering) operator P † = | (cid:105)(cid:104) | (resp. P = | (cid:105)(cid:104) | ). The k -th mode of the electromagneticfield is described by the creation (resp. annihilation) operator r † k (resp. r k ). Note, thedynamics of the dielectric are here fully represented by the momentum-dependent couplingelement sin( kl ) and do not appear explicitly in this effective Hamiltonian.This Hamiltonian introduces an interesting quantum feedback due to the structured con-tinuum approach, namely by introducing a wavelength dependent coupling between the emit-ter and the photons of the reservoir. Known from perturbation theory, this kind of frequencydependent coupling leads to numerous non-Markovian effects and renders Weisskopf-Wigner3 IG. 1. Excited atom located in a distance l to the dielectric medium. approaches impossible to treat quantum feedback. Deriving the dynamics of the excitationoperator via the Heisenberg equation of motion − i (cid:126) ∂ t O = [ H eff , O ], the feedback mechanismbecomes apparent: ddt P † ( t ) = − K , P † ( t ) + K , e − iω e τ P † ( t − τ )Θ( t − τ )( G ( t ) − E ( t ))+ (cid:90) R dk i (cid:114) cK , Lπ sin( kl ) e i ( ω k − ω e ) t r † k (0)( G ( t ) − E ( t )) , (2)where Θ( t ) is the Heaviside step function and τ := 2 l/c is the round-trip time [23, 25, 26].To derive Eq. (2) a transformation into the rotating frame with respect to the transitionfrequency ω e was performed and without loss of generality l ≥ K , . Additionally to this decay, after a round-trip time of τ , part of the initialsignal is fed back into the dynamics of the atomic operator which is described by the secondterm in Eq. (2). Note, evaluating this equation in the many-excitation limit is a tedious taskdue to the noise contributions of the third term in Eq. (2) and increasing reservoir-systementanglement spread. Possible strategies have been proposed in the Heisenberg picture [23],in the quantum cascaded approach based on Liouvillian [12] or on the quantum stochasticSchr¨odinger equation [11, 27].Subsequently, we are interested in the development of a more detailed picture of the feedbackmechanism than described by the Hamiltonian Eq. (1). Therefore, we explicitly include theatomic dynamics of the dielectric induced by the field. The derivation is in analogy to thecalculations of P. W. Milonni and R. J. Cook [28]. However, in contrast to Milonni et al., the4quations are derived in the Heisenberg picture, to simplify the many-excitation limit. Thisis rendered possible by treating the quantum noise contributions explicitly, which is beyondthe scope of the model from Milonni et al. Our results form the backbone for further anddetailed investigations by expanding the proposed feedback mechanism to a wider familyof mirrors (metallic, dielectric, active, passive) and to acknowledge possible connections tothe regime of quantum optomechanics [29, 30]. In particular, in the limit of a continuouslydistributed dielectric we derive the proposed sin( kl ) coupling from a reflecting medium indistance l justifying the phenomenological ansatz.In the following, we will derive the polarization equation of motion in Eq. (2) with a moremicroscopic model, and discuss thereby the limits of validity for the given implementation. III. MICROSCOPIC APPROACH TO MODEL QUANTUM FEEDBACK
To derive a more general formula including the dielectric properties, we use a microscopicapproach describing quantum feedback, applying the calculation of Ref. [28] to the Heisen-berg picture. The interaction Hamiltonian H I for the system illustrated in Fig. 1 is inrotating wave and dipole approximation given by: H I = i (cid:126) (cid:88) k ∈ N C k, (cid:16) P † r k e ikz − P r † k e − ikz (cid:17) + i (cid:126) N (cid:88) j =1 (cid:88) k ∈ N C k,j (cid:16) σ ( j )2 , r k e ikz j − σ ( j )1 , r † k e − ikz j (cid:17) , (3)where C k,j = µ j (cid:126) (cid:18) π (cid:126) ω k AL (cid:19) (4)is the frequency dependent coupling element in the light-matter interaction. Here, A is aneffective area, L is the length along the z -axis of the quantized box and µ j is the magnitudeof the transition dipole moment of each two-state atom. The electronic system is describedvia σ ( j )1 , = | (cid:105) j j (cid:104) | (resp. σ ( j )2 , = | (cid:105) j j (cid:104) | ) denoting the operator of the ground state (resp.excited state) dynamics of the j -th atom in the dielectric. Its excitation (resp. de-excitation)dynamics is described by the operator σ ( j )2 , = | (cid:105) j j (cid:104) | (resp. σ ( j )1 , = | (cid:105) j j (cid:104) | ). The transitionfrequency of the atom at z is denoted ω e . We assume the dielectric to consist of identicalatoms with resp. transition frequencies ω = ω = ... = ω N . Further, we neglect contributionsorthogonal to the polarization-density, denoted P ⊥ [31]. As the dielectric consists of identical5toms the magnitude of the transition dipole moment in the dielectric can be identicallychosen to be µ which is not necessarily equal to µ the magnitude of the transition dipolemoment of the initially excited atom in z . The system Hamiltonian H is given by thenon-interacting Hamiltonian and the interacting Hamiltonian H I . Performing a unitarytransformation of H into the rotation frame with respect to ω e , the Heisenberg equationyields the following system of differential operator equations ddt P † = (cid:88) k ∈ N C k, e − ikz r † k ( G − E ) (5a) ddt σ ( j )2 , = i ( ω − ω e ) σ ( j )2 , + (cid:88) k ∈ N C k, e − ikz j r † k (cid:16) σ ( j )1 , − σ ( j )2 , (cid:17) , j ∈ { , ..., N } (5b) ddt r † k = i ( ω k − ω e ) r † k − C k, e ikz P † − N (cid:88) j =1 C k, e ikz j σ ( j )2 , , k ∈ N . (5c)We emphasize that we have already applied the rotating wave approximation. Hence, werestrict the dynamics of the electromagnetic field to be quasi-resonant with the atomictransition frequency ω e and, thus, the intensity to be sufficiently low. In consequence, onlycertain refraction and reflection coefficients are rendered possible, i.e., for other materialand included susceptibilities [32], the operator dynamics needs to be generalized to the non-rotating wave regime [31]. Our goal, however, is mainly to derive a sin( kl ) kind of couplingand for this purpose alone, we can keep this set of equations of motion as they alreadyinclude the desired feedback mechanism. IV. MICROSCOPIC THEORY OF AN ATOM NEAR A PLANE DIELECTRICINTERFACE IN THE HEISENBERG PICTURE
Having described the model in the previous section, we aim to deduce a valid feedbackequation. We start by formally eliminating r † k ( t ) in Eqs. (5a) and (5b). This is achieved byapplying Duhamel’s formula [33] to Eq. (5c) and substitute the obtained solution in Eqs. (5a)and (5b). For r † k ( t ) we obtain: r † k ( t ) = e i ( ω k − ω e ) t r † k (0) − (cid:90) t dt (cid:48) e − i ( ω k − ω e )( t (cid:48) − t ) C k, e ikz P † ( t (cid:48) ) − N (cid:88) j =1 (cid:90) t dt (cid:48) e − i ( ω k − ω e )( t (cid:48) − t ) C k, e ikz j σ ( j )2 , ( t (cid:48) ) , (6)6hich implies the differential operator equations ddt P † ( t ) = − (cid:90) t dt (cid:48) (cid:88) k ∈ N C k, e − i ( ω k − ω e )( t (cid:48) − t ) P † ( t (cid:48) ) ( G − E ) − N (cid:88) j =1 (cid:90) t dt (cid:48) (cid:88) k ∈ N C k, C k, e − ik ( z − z j ) e − i ( ω k − ω e )( t (cid:48) − t ) σ ( j )2 , ( t (cid:48) ) ( G − E )+ (cid:88) k ∈ N C k, e − ikz e i ( ω k − ω e ) t r † k (0) ( G − E ) (7)and ddt σ ( j )2 , ( t ) = i ( ω − ω e ) σ ( j )2 , ( t ) + (cid:88) k ∈ N C k, e − ikz j e i ( ω k − ω e ) t r † k (0) (cid:16) σ ( j )1 , ( t ) − σ ( j )2 , ( t ) (cid:17) − (cid:90) t dt (cid:48) (cid:88) k ∈ N C k, C k, e ikz − ikz j e − i ( ω k − ω e )( t (cid:48) − t ) P † ( t (cid:48) ) (cid:16) σ ( j )1 , ( t ) − σ ( j )2 , ( t ) (cid:17) − N (cid:88) J =1 (cid:90) t dt (cid:48) (cid:88) k ∈ N C k, e ik ( z J − z j ) e − i ( ω k − ω e )( t (cid:48) − t ) σ ( J )2 , ( t (cid:48) ) (cid:16) σ ( j )1 , ( t ) − σ ( j )2 , ( t ) (cid:17) (8)for j ∈ { , ..., N } . We emphasize that up to this point, no further approximation havebeen made. To solve this system, we use two approximations. We start with the narrow-band approximation, which states that the emission spectrum is centered around the atomictransition frequency ω e . Consequently, we can restrict the following analysis on a frequencyinterval [ ω e − υ, ω e + υ ] on which the variation of the coupling constants is chosen to besmall. Therefore, the dependency of C k,j on the frequency ω k is negligible. We yield thevacuum field amplitude with: C k,j ≈ µ j (cid:126) (cid:18) π (cid:126) ω e AL (cid:19) =: C ,j , (9)for any j ∈ { , } . This approximation is within the range of the previously used rotatingwave approximation and therefore does not contradict previous assumptions to the system.Further, we pass to the Weisskopf-Wigner approximation. Here, two assumptions are made.First, the modes of the field are closely spaced in frequency. Hence, we will integrate overthe frequencies instead of summing. Second, the expectation value of the integrand oscillatesrapidly for very small times t (cid:48) (cid:28) t . Therefore, there is no significant contribution to thevalue of the integral. We derive: (cid:90) t dt (cid:48) (cid:88) k ∈ N C , e − i ( ω k − ω e )( t (cid:48) − t ) f ( t (cid:48) ) −→ πµ ω e A (cid:126) c (cid:104) δ ◦ g, f (cid:105) [0 ,t ] (10)7nd (cid:90) t dt (cid:48) (cid:88) k ∈ N C ,j C ,l e − ikZ e − i ( ω k − ω e )( t (cid:48) − t ) f ( t (cid:48) ) −→ πµ j µ l ω e A (cid:126) c (cid:0) (cid:104) exp( iω e g ) δ ◦ h , f (cid:105) [0 ,t ] + (cid:104) exp( iω e g ) δ ◦ h , f (cid:105) [0 ,t ] (cid:1) . (11)In the above limits g ( t (cid:48) ) := t − t (cid:48) , h ( t (cid:48) ) := t + Z/c − t (cid:48) , h ( t (cid:48) ) := t − Z/c − t (cid:48) and f , arearbitrary but sufficiently smooth functions. Here, we have introduced the constant Z ∈ R which later will be replaced by differences of the particle positions along the Z -axis, i.e., z i − z j for i.j ∈ { , ..., N } . We further used the standard notation where (cid:104)· , ·(cid:105) [0 ,t ] denotes the L -scalar product restricted on the domain [0 , t ]. For the sake of simplicity, we used calculusnotation in the dual pairing, e.g., exp( iω e g ) δ ◦ h describes the composition of δ with h multiplied by exp( iω e g ).The above notation is used to emphasize that the delta-distribution is a functional gen-erated by the Dirac measure. Hence, the evaluation in the point t (cid:48) = 0 is only possiblefor a domain in which zero is an inner point. As this is not the case for the given domainthe dual pairing of the delta distribution and the respective functions are not well-defined.Expanding the respective functions by the Heaviside step function yields that the result ofthe dual pairing can be multiplied by any constant α ∈ [0 ,
1] and is therefore not unique.However, the only choice ensuring the commutator relation to hold for any t is α = 1 / f ( t (cid:48) ) = P † ( t (cid:48) )( G ( t ) − E ( t )) and f ( t (cid:48) ) = σ ( j )2 , ( t (cid:48) )( G ( t ) − E ( t )) (respectively f ( t (cid:48) ) = σ ( j )2 , ( t (cid:48) )( σ ( j )1 , ( t ) − σ ( j )2 , ( t )) and f ( t (cid:48) ) = P † ( t (cid:48) )( σ ( j )1 , ( t ) − σ ( j )2 , ( t ))) for j ∈ { , ..., N } , wededuce the following system of delay differential operator equations: ddt P † ( t ) = − K , P † ( t ) − N (cid:88) j =1 K , e − ik l j σ ( j )2 , ( t − l j /c ) ( G ( t ) − E ( t )) Θ( t − l j /c )+ K , ∆ B † (0 , , t ) ( G ( t ) − E ( t )) (12a) ddt σ ( j )2 , ( t ) = − ( i ( ω e − ω ) + K , ) σ ( j )2 , ( t ) − K , e − ik l j P † ( t − l j /c ) (cid:16) σ ( j )1 , ( t ) − σ ( j )2 , ( t ) (cid:17) Θ( t − l j /c ) − (cid:88) J ∈{ ,...,N }\{ j } K , e − ik l j,J σ ( J )2 , ( t − l j,J /c ) (cid:16) σ ( j )1 , ( t ) − σ ( j )2 , ( t ) (cid:17) Θ( t − l j,J /c )+ K , ∆ B † (1 , j, t ) (cid:16) σ ( j )1 , ( t ) − σ ( j )2 , ( t ) (cid:17) , (12b)8here k = ω e /c , l j,J := | z j − z J | , with the special case l ,j =: l j , K i,j := πµ i µ j ω e / ( (cid:126) Ac ) and∆ B † ( l, j, t ) := (cid:18) LA (cid:126) π ω e µ l (cid:19) (cid:90) ∞−∞ dω (cid:48) e i ( ω (cid:48) z j /c − ( ω e − ω (cid:48) ) t ) r † k (cid:48) (0) . (13)Eqs. (12) explicitly expose the delay effect in the atom-atom coupling via a mediatingelectromagnetic field. Subsequently, we use the standard notation τ j := 2 l j /c and τ := 2 l/c .Further, we consider a dielectric in which the scattered field is small compared to theincident field on the scatterer. This level of treatment is consistent with the Born approxima-tion, which neglects the interaction of the atoms within the dielectric via photon exchange.Assuming that the dielectric and the initially excited atom are off-resonant yields that allatoms in the dielectric remain in their ground state. This and K , (cid:28) | ω − ω e | implies ddt σ ( j )2 , ( t ) = − i ( ω e − ω ) σ ( j )2 , ( t ) − K , e − iω e τ j / P † ( t − τ j / t − τ j /
2) + K , ∆ B † ( j, t ) (14)where Duhamel’s formula yields the solution σ ( j )2 , ( t ) = e − i ( ω e − ω ) t σ ( j )2 , (0) − K , e − iω e τ j / (cid:90) t dt (cid:48) e i ( ω e − ω )( t (cid:48) − t ) P † ( t (cid:48) − τ j /
2) Θ( t (cid:48) − τ j / (cid:90) t dt (cid:48) e i ( ω e − ω )( t (cid:48) − t ) K , ∆ B † ( j, t (cid:48) ) . (15)The following adiabatic approximation is obtained by partial integration and using that theexpectation value of the integrand oscillates rapidly, as assumed in the Weisskopf-Wignerapproximation. We find: σ ( j )2 , ( t ) = e − i ( ω e − ω ) t σ ( j )2 , (0) + i K , ( ω e − ω ) e − iω e τ j / P † ( t − τ j / t − τ j / − i K , ( ω e − ω ) e − iω e τ j / e − i ( ω e − ω )( t − τ j / P † (0)Θ( t − τ j / (cid:90) t dt (cid:48) e − i ( ω e − ω )( t (cid:48) − t ) K , ∆ B † ( j, t (cid:48) ) . (16)Eliminating the dependency of σ ( j )2 , in Eq. (12a) by substituting the above solution we obtain:9 dt P † ( t ) = − K , P † ( t ) − i K , ( ω e − ω ) N (cid:88) j =1 e − iω e τ j P † ( t − τ j ) ( G ( t ) − E ( t )) Θ( t − τ j ) (17a) − K , N (cid:88) j =1 e − iω e τ j / e − i ( ω e − ω )( t − τ j / σ ( j )2 , (0) ( G ( t ) − E ( t )) Θ( t − τ j /
2) (17b)+ i K , ( ω e − ω ) N (cid:88) j =1 e − iω e τ j e − i ( ω e − ω )( t − τ j ) P † (0) ( G ( t ) − E ( t )) Θ( t − τ j ) (17c) − K , K , N (cid:88) j =1 e − iω e τ j / × (cid:90) t − τ j / dt (cid:48) e − i ( ω e − ω )( t (cid:48) − t + τ j / ∆ B † ( j, t (cid:48) ) ( G ( t ) − E ( t )) Θ( t − τ j /
2) (17d)+ K , ∆ B † (1 , , t ) ( G ( t ) − E ( t )) . (17e)This delay differential operator equation describes the effect of the dielectric on the po-larization of the atom outside the dielectric. It is the central equation used to describeoccupation expectation values of the initially excited atom, i.e., the system of interest. Weemphasize that the influence of quantum noise terms like Eq. (17d) makes this equationdifficult to handle.We now deduce a formula that describes an idealized mirror system similar to Eq. (2). Analo-gously to the calculations in [28], we pass to the limit of a continuously distributed dielectric.This will simplify the system by eliminating the sums as we are using the Weisskopf-Wignerapproximation. Assuming that the dielectric contains N A dz identical atoms in the slice[ z, z + dz ] yields N (cid:88) j =1 e − iω e τ j P † ( t − τ j )Θ( t − τ j ) → − i N A k P † ( t − τ )Θ( t − τ ) e − iω e τ , (18)where a coordinate system was chosen such that z = 0. We now restrict the system Eq. (17)by only taking terms that scale linearly in K into account. Hence, we neglect (17b), (17c)and (17d). The remaining differential equation is given by ddt P † ( t ) = − K , P † ( t ) + K , ∆ B † (1 , , t ) ( G ( t ) − E ( t ))+ K , ( ω − ω e ) N A k e − iω e τ P † ( t − τ )Θ( t − τ ) ( G ( t ) − E ( t )) . (19)10his delay differential operator equation is indeed similar to Eq. (2). To give a more explicitsimilarity, we identify the Fresnel reflection coefficient in the rotating wave approximation(see Appendix A) in Eq. (19). Since K , = K , K , , we observe K , ( ω − ω e ) N A k = K , N πµ (cid:126) ( ω − ω e ) . (20)The refraction index of a dielectric of N two-state atoms per unit volume, each of transitionfrequency ω and transition dipole moment µ without local field effects, in the rotating waveapproximation can be characterized by n ( ω e ) − ≈ Re( χ ( ω e )) ≈ N πµ (cid:126) ( ω − ω e ) . (21)For a more detailed derivation see Appendix A. We emphasize that for the deduction ofprimary Eq. (2) a negligible absorption of the dielectric is assumed. Therefore χ ( ω e ) = n ( ω e ) − R = − ( n − / ( n + 1). In the case ω ≈ ω e and n ( ω e ) ≈ R = − N πµ (cid:126) ( ω − ω e ) , (22)which yields K , ( ω − ω e ) N A k = − K , R . (23)The quantum noise term can be written as K , ∆ B † (1 , , t ) = cK , (cid:18) LA (cid:126) π ω e µ (cid:19) (cid:90) ∞−∞ dk i sin( kl ) e i ( ω k − ω e ) t r † k (0) . (24)Using the relation cK , (cid:18) LA (cid:126) π ω e µ (cid:19) = (cid:114) cK , π (cid:18) L π (cid:19) µ µ , (25)we obtain the delay differential operator equation for the polarization ddt P † ( t ) = − K , P † ( t ) − K , Re − iω e τ P † ( t − τ )Θ( t − τ ) ( G ( t ) − E ( t ))+ R N (cid:90) ∞−∞ dk i (cid:114) cK , Lπ sin( kl ) e i ( ω k − ω e ) t r † k (0) ( G ( t ) − E ( t )) , (26)where R N := µ / ( µ √ π ).This equation is similar to the Eq. (2) but differs in the appearance of the reflection coeffi-cients R and R N . As for perfect mirrors R = − | R | (cid:28)
1. How-ever, the results calculated using the model described in Sec. II can be generalized to higherreflectivities, including counter-rotating contributions in the Hamiltonian. The followinggraph Fig. 2 depicts qualitatively the behavior of the expectation value (cid:104) E (cid:105) for different R on [0 , τ ). Our derivation allows to choose a reflectivity, and we see the influence of the mir-ror properties in the degree of feedback that is measurable on the emitter’s dynamics. For R = −
1, the dielectric is a perfect mirror, and the usual results from [23, 24] are rederived,however, now within a microscopic model. This is the main result of the paper. Since nowthe mirror dynamics are explicitly included, we can also start to investigate regimes, wherethe mirror is not a passive optical element anymore but is for example driven. This allowsto include gain into the system. Just by hand, we choose a
R < − FIG. 2. Expectation value (cid:104) E (cid:105) = (cid:104) P † P (cid:105) of the atom outside the dielectric at z for different R .The values of R increase in the direction indicated by the arrow. K , = 1 . τ = 3 and ω e = π/τ . V. CONCLUSION
The presented microscopic approach to quantum feedback yields a delay differential op-erator equation for the polarization P † Eq. (17) including dielectric characteristics. Thisexpands the analysis of quantum coherent time-delayed feedback to a wider class of mirrors,e.g., metallic, dielectric, active, passive etc. Using an equation of motion approach we pro-vide the possibility to describe an excited reflecting dielectric under stimulated emission, i.e.,12 description of quantum optical gain in mirror-emitter setups. In addition, the presentedanalysis gives access to a wider class of active quantum coherent feedback control where theintrinsic mirror properties are externally steered. This could have significance both in tai-lored control of external quantum emitters and multi-photon selective reflection properties.Furthermore, we have shown that the microscopic approach justifies the sin( kl ) like couplingused in the effective Hamiltonian Eq. (1). In conclusion, the microscopic approach is verypromising for further investigations as it takes properties of the dielectric into account anddoes not contradict the qualitatively motivated model. [1] Wiseman, H.M.; Milburn, G.J. Quantum measurement and control ; Cambridge universitypress, 2009.[2] Sch¨oll, E.; Schuster, H.G.
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In this section we show how to compute the dual-pairing of the delta-distribution withfunctions over domains not containing zero in physical systems. The idea is to expand thefunction in the dual pairing with a Heaviside step function. This yields the problem that thestep function can be defined in zero with any number α ∈ [0 ,
1] multiplied by the function inthe dual pairing evaluated in zero. We show that α has to be α = 1 / t ∈ [0 , τ ) theoperator P † ( t ) = e iω e t e − κt P † (0) + ig (cid:90) t dt (cid:48) e κ ( t (cid:48) − t ) (cid:90) R dk e − i (( ω e − ω k ) t (cid:48) − ω e t ) r † k (0)∆( t (cid:48) ) , (B1)where κ = g πα/c and ∆( t (cid:48) ) = G ( t (cid:48) ) − E ( t (cid:48) ). We find a similar solution for P . Assuming asystem with weak decay yields ∆( t (cid:48) ) ≈ −
1. We then find[ P ( t ) , P † ( t )] + = e − κt + g π cκ (cid:0) − e − κt (cid:1) . (B2)As [ P ( t ) , P † ( t )] + = G + E = 1 we know that g π cκ ! = 1 . (B3)Inserting the definition of κ yields g π cκ = cg π cg πα = 12 α . (B4)Hence, the only value of α ∈ [0 ,
1] for which [ P ( t ) , P † ( t )] + = G + E = 1 is α = 1 //