Quantum trajectories under frequent measurements in non-Markovian environment
aa r X i v : . [ qu a n t - ph ] N ov Quantum trajectories under frequent measurements in non-Markovian environment
Luting Xu ∗ and Xin-Qi Li † Center for Advanced Quantum Studies and Department of Physics,Beijing Normal University, Beijing 100875, China (Dated: August 7, 2018)In this work we generalize the quantum trajectory (QT) theory from Markovian to non-Markovianenvironments. We model the non-Markovian environment by using a Lorentzian spectral densityfunction with bandwidth (Λ), and find perfect “scaling” property with the measurement frequency( τ − ) in terms of the scaling variable x = Λ τ . Our result bridges the gap between the existing QTtheory and the Zeno effect, by rendering them as two extremes corresponding to x → ∞ and x → x -dependent criterion improves the idea of using τ alone, and quantitativelyidentifies the validity condition of the conventional QT theory. PACS numbers: 03.65.Ta,03.65.Xp,73.63.-b,73.40.Gk
The quantum trajectory (QT) given by stochasticSchr¨odinger equation (SSE) for an open system asso-ciated with
Markovian dynamics can be interpreted asquantum state conditioned on continuous observation(monitoring) on the environment [1, 2]. The QT the-ory of this type has been well demonstrated and broadlyapplied [3, 4], including the recent experiments in su-perconducting solid-state circuits [5–13]. On the otherhand, associated with the non-Markovian dynamics ofopen quantum systems, similar non-Markovian stochas-tic Schr¨odinger equation (nMSSE) has been constructed[14, 15]. However, the nMSSE is largely a working toolof unraveling the non-Markovian dynamics, which can-not be interpreted as measurement-conditioned physical quantum trajectory [16–18]. After careful analysis byWiseman et al , the nMSSE might be at most interpretedas certain “hidden variable” theory, i.e., taking the com-plex Wiener variable z t involved in the nMSSE as an“objective property” which inherently exists in the envi-ronment, rather than a consequence of continuous mea-surements [18].In this work we consider the interesting problem howto construct the physical QT associated with frequentmonitoring on non-Markovian environment. To be spe-cific, we model the non-Markovian environment by usinga Lorenztian spectral density function (SDF) with finitebandwidth. We show that the result is quite differentfrom the nMSSE mentioned above. Elegantly, via slightmodification by involving a “scaling” variable, the resul-tant QT formally resembles, but essentially generalizes,the conventional QT. Our result bridges the gap betweenthe existing QT [1–4] and the quantum Zeno effect [19],by rendering them as two extremes which have quite dif-ferent predictions [20, 21].Let us consider a two-level atom (qubit) prepared ina quantum superposition of the ground state ( | g i ) andexited state ( | e i ), | Ψ(0) i = α | e i + β | g i . Now con- ∗ Electronic address: [email protected] † Electronic address: [email protected] sider its evolution under continuous (very frequent) mea-surements in the surrounding environment for the spon-taneous emission of photon. From the celebrated QTtheory [1–4], conditioned on the continuous null-result(no-register of spontaneous emission) detection, the statewould change, following the simple formula | Ψ( t ) i = (cid:16) α e − Γ t/ | e i + β | g i (cid:17) / N , (1)where Γ is the spontaneous emission rate and N denotesthe normalization factor. To interpret this result, rea-soning based on informational evolution is sometimes putforward. That is, no result is a sort of information , so thestate can change according to Bayesian inferring, similaras in classical probability theory.On the other hand, the above continuous null-resultquantum motion is prohibited by the quantum Zenoeffect [19]. We may briefly summarize the treatmentand result as follows. Starting with | Ψ(0) i , let us ex-pand the evolution operator up to the second order in τ , U ( τ ) ≃ − iHτ − H τ /
2, where τ is the time intervalbetween the successive null-result measurements. Eachnull-result measurement would project the wave functionon the atomic subspace. Consider n subsequent null-result measurements during time t (with n = t/τ ). Inthe limit τ → t =const, one obtains (see AppendixA for more details) | Ψ n i → α | e i + β | g i ≡ | Ψ(0) i . (2)So we find that the frequent null-result monitoring of theenvironment will prevent the change of the state, result-ing thus in the quantum Zeno effect.Actually, the QT theory leading to Eq. (1) is from un-raveling the Markovian Lindblad master equation. InMarkovian approximation, one requires a wide band-width environment (i.e., the bandwidth Λ → ∞ ). There-fore, any τ is long compared to the environment’s mem-ory time Λ − , leading thus to the exponential decay ofpopulation which destroys the possibility of Zeno effect.In the case of Λ → ∞ , the above expansion on U ( τ ) isinvalid. In order to generate Zeno effect, the physicalcondition is τ << Λ − . In the remainder of this work,we will develop a treatment to smoothly bridge these twoextremes, and construct the associated QT theory by in-troducing external drive to the atom. Spontaneous decay . — The two-level atom coupled tothe electromagnetic vacuum (environment) is describedby the Hamiltonian H = ∆ eg σ z + X r (cid:18) b † r b r + 12 (cid:19) ω r + X r (cid:2) V r b † r σ − + H . c . (cid:3) . (3)Throughout this work we set ¯ h = 1. Here we intro-duce: the two-level energy difference ∆ eg = E e − E g ,the atomic operators σ z = | e ih e | − | g ih g | , σ − = | g ih e | ,and σ + = | e ih g | . V r is the coupling amplitude of theatom with the environment. Then, consider the evolu-tion of the entire system, starting with an initial state | Ψ(0) i = ( α | e i + β | g i ) ⊗ | vac i , where | vac i stands forthe environmental vacuum with no photon. Under theinfluence of the coupling, the entire state at time t canbe written as | Ψ( t ) i = α ( t ) | e i ⊗ | vac i + X r c r ( t ) | g i ⊗ | r ; 0; · · ·i + β | g i ⊗ | vac i , (4)where | r ; 0; · · ·i describes the environment with a photonexcitation in the state “ r ” and no excitations of otherstates. The coefficients have initial conditions of α (0) = α and c r (0) = 0.Substituting Eq. (4) into the Schr¨odinger equationand performing the Laplace transform, one can obtainthe solution of α ( t ) in frequency domain (see AppendixB for more details). That is, replace P r | V r | [ · · · ] → R dω r D ( ω r )[ · · · ], where D ( ω r ) = P r ′ | V r ′ | δ ( ω r − ω r ′ ) → D Λ / [( ω r − ω ) + Λ ] is the spectral density function(SDF), approximated here by a finite-band Lorentzianspectrum with ω the spectral center and Λ the width[22]. We obtain then the time-dependent amplitude α ( t ) ≡ a ( t ) α via the inverse Laplace transform as [21] a ( t ) = 1 A + − A − ( A + e − A − t − A − e − A + t ) , (5)with A ± = [Λ − iE ± p (Λ − iE ) − /
2. Here weintroduced the energy offset E = ( E e − E g ) − ω and theusual decay rate in wide-band limit, Γ = 2 πD . Frequent null-result measurements .— The null-resultmeasurement in the environment, quantum mechanically,collapses the entire wave function onto the atomic sub-space. After n such null-result measurements with sub-sequent time interval τ = t/n , the final state of the atomis | e Ψ( t ) i = [¯ a ( t ) α | e i + β | g i ] / p N n ( t ) , (6) | () | a t x=2.0x=0.2 x=0.02 x=0.002 (units of 1/ ) t G (b) | e | g r w w L
20 2 20 ( ) ( ) r r DD w w wL= - + L D ( ) r D w (a) FIG. 1: (Color online) (a) Spontaneous emission of a two-level atom coupled to non-Markovian environment with finite-bandwidth Lorentzian spectrum. (b) Effective decay factorof the excited state started with a quantum superposition α | e i + β | g i , under frequent null-result measurements in theenvironment. Scaling behavior is demonstrated by the re-markable agreement between Eq. (7) (continuous lines) and a n ( τ ) (symbols) calculated using Eq. (5) with Λ = 10Γ (as anexample) and E = ω − ω = 0. Remind that t = nτ and x = Λ τ . where ¯ a ( t ) = a n ( τ ) and N n ( t ) = | ¯ a ( t ) α | + | β | .Note that, unlike the case of the wide-band-limit Marko-vian environment, | e Ψ( t ) i differs from the single-null-measurement-collapsed state at the final moment from | Ψ( t ) i . It can be proved that the normalization fac-tor N n equals also the joint probability of getting null results in all the intermediate measurements, i.e., (1 − P r | c r ( τ ) | ) n . Let us denote N n ( t ) ≡ p ( n )0 ( t ). Accord-ingly, during time (0 , t ), the probability of detecting aspontaneous photon is p ( n )1 ( t ) = 1 − p ( n )0 ( t ).Now let us consider the limit of “continuous” measure-ments, n → ∞ by taking the measurement time interval τ → t = nτ fixed. Supposing to increasethe bandwidth Λ so that the variable x = Λ τ remainsconstant, we can prove a “scaling” property that the fi-nal state becomes a function of x only. To reveal the fullscaling behavior in general case, we also assume the en-ergy offset E = c Λ (in usual treatment c = 0). One findsfrom Eq. (5) that A + = κ Λ − Γ / (2 κ ) and A − = Γ / (2 κ )(up to the order of (Γ / Λ) ), where κ = 1 − ic . Using (cid:0) − zn (cid:1) n = e − z (cid:0) z n + ··· (cid:1) and neglecting small terms ∼ Γ / Λ in exponent, we arrive to [21]¯ a ( t ) = a n ( τ ) = exp (cid:26) − (cid:20) κ − (1 − e − κx ) 1 κ x (cid:21) Γ t (cid:27) . (7)Elegantly, this result reveals an explicit scaling propertyin the x = Λ τ -variable. In Fig. 1(b), by relaxing theconditions ( n → ∞ and τ →
0) for obtaining this ana-lytic formula, we illustrate the scaling behavior in broadparameter conditions.
Some remarks about Eq. (7) .— (i) The numerical resultsin Fig. 1(b) for finite Λ and τ (e.g., the “ x = 2” curve forand τ − = 0 . n → ∞ and τ →
0. This limiting pro-cedure is only a mathematical technique leading us toobtain the analytic result Eq. (7). (ii)
The scaling behavior can be understood via thetime-energy uncertainty relation. Actually, the succes-sive measurements with time interval τ in the reser-voir will cause fluctuations of the atom’s level ( E e ) byamount ∼ τ − , since the result whether or not a sponta-neous emission detected in the reservoir allows knowingwhether or not the atom is in the excited state. Then,if we (conceptually) expand the width of the reservoir’sSDF by this same amount (i.e., by ∼ τ − ), we can expectthe same (identical) decay dynamics. This is the physicalreason of the scaling behavior shown analytically by Eq.(7) and numerically in Fig. 1(b). (iii) Note that the x -dependence of the decay dynam-ics is the same as the τ -dependence for a given band-width Λ (usually it is difficult to change Λ in real set-ups).And, this τ -dependence is the essential feature associatedwith measurements in non-Markovian reservoir, which isin sharp contrast with the conventional τ -independent Markovian case. (iv)
From Eq. (7), in the wide-band limit, x → ∞ and κ →
1, one recovers the result ¯ a ( t ) → e − Γ t/ predictedby the standard QT theory. On the other hand, in thelimit of x →
0, one finds from Eq. (7) that ¯ a ( t ) = 1, sothat the atom is frozen in its initial state, showing theZeno effect. (v) In the Zeno regime τ − >> Λ, one may encountera “negative frequency” problem if the central frequency ω is not much larger than Λ. In this case (and for thetransition energy ∆ eg > ω ) the level E e may fluctuateinto the domain of “negative frequency” of the SDF, thusviolating the condition of the symmetric Lorenztian SDFmodel and needing certain modification to Eq. (7). Inthis work, we assume a symmetric Lorentzian SDF modelunder the conditions ∆ eg > ω >> Λ, for the sake ofshowing a full transition from the Markovian behavior toZeno effect governed by the unified Eq. (7). In this case,there is no “negative frequency” difficulty to affect thevalidity of Eq. (7). (vi)
From Eq. (7), one can define an effective decay rate γ eff = Re (cid:8)(cid:2) − ( κx ) − (cid:0) − e − κx (cid:1)(cid:3) /κ (cid:9) Γ . (8)Note that for the wide-band-limit Markovian environ-ment the exponential decay process implies no-effect ofthe intermediate null-result interruptions [21]. Eq. (8), however, shows that the decay rate is influenced bythe frequent null-result measurements. This x - or τ -dependence reflects the non-Markovian effect rooted inEq. (5), despite that the frequent measurements cut offthe usual non-Markovian correlation (memory) effect be-tween different τ -period evolutions. It is right the ac-cumulation of the “small” non-Markovian contributionsover t = nτ that makes Eqs. (7) and (8) and the associ-ated QT (to be constructed) generalize the usual Marko-vian results. Quantum trajectories .— Corresponding to direct photondetection, let us first construct the Monte-Carlo wavefunction (MCWF) approach, closely along the line pro-posed in Ref. [1]. Consider the state evolution underfrequent null-result measurements between t and t + ∆ t ,with thus ∆ t = nτ . The probability with photon registerin the detector during ∆ t , is p ( n )1 (∆ t ) = | α ( t ) | γ eff ∆ t .Under the “scaling” consideration, the effective decayrate γ eff is simply given by Eq. (8), or, alternatively by γ eff = [1 − | ¯ a (∆ t ) | ] / ∆ t or γ eff = − ln[ | ¯ a (∆ t ) | ] / ∆ t . (9)For small ∆ t , which implies | ¯ a (∆ t ) | ≃
1, both definitionsare equivalent and coincide with Eq. (8).In practical simulations, generate a random number ǫ between 0 and 1. If ǫ < p ( n )1 (∆ t ), which corresponds tothe probability of having a photon register in the detector(∆ N c = 1), we update the state by a “jump” action | e Ψ( t + ∆ t ) i = σ − | e Ψ( t ) i / k • k , (10)where k • k denotes the normalization factor. On theother hand, if ǫ > p ( n )1 (∆ t ), which corresponds to theNRM with ∆ N c = 0, we update the state via the effective smooth evolution | e Ψ( t + ∆ t ) i = U (∆ t ) | e Ψ( t ) i / k • k . (11)In terms of a matrix form defined by { α ( t + ∆ t ) , β ( t +∆ t ) } T = U (∆ t ) { α ( t ) , β ( t ) } T , the effective non-unitary evolution operator reads U (∆ t ) = ¯ a (∆ t ) 00 1 ! . (12)Noting that ∆ t = nτ , as above, here we mention againthat ¯ a (∆ t ) = [ a ( τ )] n which can be Eq. (7) in the limit τ → n → ∞ , or be more generally determinedusing Eq. (5) for a ( τ ).Based on the MCWF approach proposed above, onecan simulate the (stochastic) quantum trajectories un-der frequent photon detections in the environment. En-semble average over these trajectories of quantum (pure)state corresponds to the result given by the followingmaster equation [1–4]:˙ ρ = − i [ H S , ρ ] + γ eff D [ σ − ] ρ , (13)where D [ • ] ρ ≡ • ρ • † − {• † • , ρ } . Formally, this is an x - or τ -dependent Lindblad-type master equation. How-ever, unlike its Markovian counterpart, a significant dif-ference lies in the fact that this equation does not de-scribe the reduced state ̺ ( t ) of the (open) quantum sys-tem. It is well known that ̺ ( t ) is defined by tracing theenvironment degrees of freedom from the entire (system-plus-environment) unitary wavefunction at time t . Here,“tracing” simply means performing projective measure-ments and making average only at the last moment t , onthe entire unitary wavefunction evolved from the same initial state. In contrast to ̺ ( t ), the state ρ ( t ) given byEq. (13) is the ensemble-averaged state of the system un-der successive measurement interruptions. Remarkably,the successive measurements would destroy the correla-tion effect between different τ -period evolutions, result-ing thus in the Markovian-Lindblad-type Eq. (13) with,however, an effective γ eff rather than certain “natural”decay rate.Following Refs. [1–4], we now include external drivinginto Eq. (13), via H S = ∆ eg σ z + Ω σ x . Note that the va-lidity of this procedure is rooted in the additivity of thestate changes over the very small time interval ( τ ). Asa result, there are two contributions to the state change:one is informational owing to the continuous measure-ments, and the other is physical which is caused by theexternal driving. Note also that in general the dissipativetwo-level atom under driving is not exactly solvable. Theunderlying complexity can be imagined as follows: thereare more and more photons emitted into the reservoir;and the emitted photon can re-excite the atom. How-ever, in the presence of frequent measurements , the emit-ted photon will be destroyed by detectors. During eachsuccessive measurement interval ( τ ), it is reasonable toassume that there is at most one photon in the reser-voir. Therefore, even in the presence of external driving,Eq. (13) is valid under the above considerations.Instead of the direct detection of the spontaneous emis-sion considered above, one can also adopt the so-calledhomodyne detection scheme by mixing the emitting pho-tons with a classical field with modulating phase ϕ [2, 3].The measurement result (optical current) of this type canbe expressed as [2, 3], I ϕ ( t ) = √ γ eff h σ − e − iϕ + σ + e iϕ i / ξ ( t ), where h· · ·i = Tr[( · · · ) ρ ( t )] and ξ ( t ) is the Gaussianwhite noise associated with quantum jumps. In this mea-surement scheme, the detection result is a sum of theclassical reference field and the photon emitted by theatom. The “jump” (knowledge change of atom state) as-sociated with photon register in the detector is relativelyweak, developing thus a “diffusive” regime because ofthe mix of the reference field. Through a careful analysis[2, 3], the difference of the detected result (in single re-alization) during ( t, t + dt ) from the expected one usingearlier ρ ( t ), is characterized by ξ ( t ) dt in the expression of I ϕ ( t ). Conditioned on I ϕ ( t ), the state evolution is givenby the diffusive QT equation [2, 3]:˙ ρ = − i [ H S , ρ ] + γ eff D [ σ − ] ρ + √ γ eff H [ e − iϕ σ − ] ρξ ( t ) , (14) (b) ρ ee t (units of 1/ Γ ) (a) MCWF QTE
FIG. 2: (Color online) (a) Two quantum trajectories fromthe MCWF (black) and QT equation (red) simulations. Theblue arrows indicate quantum “jumps” owing to “direct” de-tection of the spontaneous emission of the atom. (b) En-semble average of 2000 MCWF and QT equation trajecto-ries and the result (green curve) from the master equation˙ ρ = − i [ H S , ρ ] + γ eff D [ σ − ] ρ . Parameters used in the simula-tion: Ω = 0 .
1, Γ = 1 . x = 0 . E = 0. where H [ • ] ρ ≡ • ρ + ρ • † −h• + • † i ρ . Essentially, Eq. (14)generalizes the existing QT equation by accounting forthe measurement frequency ( ν = 1 /τ ) in the effective“spontaneous” emission rate γ eff .In Fig. 2(a) we display two representative quantum tra-jectories from the MCWF and the diffusive QT equation(14). We see that the former type of quantum trajec-tory reveals drastic “quantum jump” owing to the direct detection for the spontaneous emission, while the lat-ter type has no such “jump” onto the ground state | g i .However, as expected, ensemble average of each type ofquantum trajectories (over 2000) gives the same result ofEq. (13), as demonstrated in Fig. 2(b). Summary and discussions .— We have constructed ascheme to generalize the QT theory from Markovian tonon-Markovian environments. Taking the specific modelof Lorentzian SDF, we revealed a perfect scaling prop-erty between the spectral bandwidth and the measure-ment frequency. Our result bridges the gap between theexisting QT and the quantum Zeno effect by renderingthem as two extremes.While leaving the possible existence of scaling behav-ior an open question for some non-Lorentzian SDFs, themain conclusion above is valid in general. Following theprocedures in this work, one can develop similar gener-alized QT theory by numerically obtaining the ¯ a (∆ t ) inEq. (9), rather than using the analytic Eqs. (7) and (8).In Appendix C, we outline the solution scheme for arbi-trary SDF.Unlike the Markovian counterpart, ensemble averageof the proposed QTs does not describe the reduced stategiven by tracing the environment degrees of freedom fromthe entire (system-plus-environment) unitary wavefunc-tion. Since the successive measurements in the QT de-stroy the correlation (memory) effect between different free evolutions, the ensemble-average state also differsfrom the one resulted from averaging the nMSSE dis-cussed in literature [14–18]. For non-Markovian envi-ronment, as pointed out by Wiseman et al [16, 18], thenMSSE is not consistent with any physical quantum tra-jectories (i.e., having no physical interpretations).For the relevance of the present work to possible exper-iment, we may refer to the partial collapse quantum mea-surement of the solid-state superconducting qubit [23–25]. The changed state reported there is conditioned on aprojective null-result at the final time t , but not on “con-tinuous” or “frequent” null-result over the interval (0 , t ).For Markovian environment, both results are identical;however, for non-Markovian case, this is not true. Possi-ble experiment may be guided by the formula Eq. (7) or(8), via the scaling variable x . As alternative demonstra-tion, one may perform a large-derivation analysis on theemitted photons from driven atoms [26]. From presentwork, we expect that if altering the detection interval τ for the spontaneous emissions, the statistics of the emit-ted photons will be drastically different. We would liketo leave this interesting problem for future investigation. Acknowledgements. — This work was supported by theBeijing Natural Science Foundation under grant No.1164014, the Fundamental Research Funds for the Cen-tral Universities, the NNSF of China under grants No.91321106 & 210100152, and the State “973” Project un-der grant No. 2012CB932704.
Appendix A: Zeno effect for superposition state
Consider the superposition state | Ψ(0) i = ( α | e i + β | g i ) ⊗ | vac i . After small time τ the wave function be-comes | Ψ( τ ) i = [ α (1 − iHτ − H τ / · · · ) | e i + β | g i ] ⊗ | vac i . (A1)The null-result measurement in the environment impliesthat the wave function is projected on the atomic sub-space, | Ψ( τ ) i → ˆ Q | Ψ( τ ) i , where ˆ Q = (cid:0) | e ih e | + | g ih g | (cid:1) / N and N is a normalization factor. Therefore | Ψ i = ˆ Q | Ψ( τ ) i = h α (cid:0) − Kτ (cid:1) | e i + β | g i i / N , (A2)where K = P r V r and N = 1 − α Kτ , with V r theatom-environment (the r -th mode) coupling amplitude.After n subsequent null-result measurements during time t , with n = t/τ , we find | Ψ n i = h α (cid:0) − Kτ (cid:1) n | e i + β | g i i / N n , (A3) where N n = p − n α K τ . Thus in the limit τ → t =const, we obtain the result Eq. (2) in the maintext , | Ψ n i → | Ψ(0) i . Appendix B: Solution for spontaneous emission
Substituting Eq. (4) in the main text into the Schr¨odingerequation, i∂ t | Ψ( t ) i = H | Ψ( t ) i and performing theLaplace transformation, ˜ f ( ω ) = R ∞ f ( t ) exp( iωt ) dt , weobtain the following system of algebraic equations:( ω − E e )˜ α ( ω ) − X r V r ˜ c r ( ω ) = iα , (B1a)[ ω − ( E g + ω r )]˜ c r ( ω ) − V ∗ r ˜ α ( ω ) = 0 . (B1b)The r.h.s. of the first equation reflects the initial condi-tion. Substituting ˜ c r ( ω ) from Eq. (B1b) into Eq. (B1a),we obtain ( ω − E e )˜ α ( ω ) − F ( ω )˜ α ( ω ) = iα , (B2)where F ( ω ) = Z D ( ω r ) ω − ( E g + ω r ) dω r . (B3)Rather than the wide-band limit for the “Markovian”reservoir, in this work we consider a finite-band spectrumby taking the spectral density function (SDF) D ( ω r ) inthe Lorentzian form, D ( ω r ) ≡ X r ′ | V r ′ | δ ( ω r − ω r ′ ) → D Λ / [( ω r − ω ) + Λ ] , (B4)with ω the spectral center, D the spectral height, andΛ the spectral width. We obtain then F ( ω ) = ΛΓ / ω − ω − E g ) + i Λ , where Γ = 2 πD . (B5)Substituting this result into Eq. (B2), we find theamplitude ˜ α ( ω ). The time-dependent amplitude isobtained via the inverse Laplace transform, α ( t ) = R ∞−∞ ˜ α ( ω ) e − iωt dω/ (2 π ). Then, we obtain α ( t ) = a ( t ) α ,with an explicit expression of a ( t ) given by Eq. (5) in themain text. Appendix C: Solution scheme for non-LorentzianSDF
For Lorentzian SDF, as shown above, we can first solveEq. (B2) in frequency domain, then obtain the analyticsolution of α ( t ) by means of inverse-Laplace transfor-mation. However, for arbitrary SDF D ( ω r ), this strat-egy does not work. Instead, we can solve Eq. (B2) for α ( t ) numerically (and directly) in time domain. For thispurpose, an inverse-Laplace transformation to Eq. (B2)yields i ˙ α ( t ) = E e α ( t ) + Z t dt ′ F ( t − t ′ ) α ( t ′ ) , (C1)where F ( t − t ′ ) = Z ∞−∞ dω π e − iω ( t − t ′ ) F ( ω )= − i Z dω r D ( ω r ) e − i ( ω r + E g )( t − t ′ ) . (C2)Here we have employed the well known convolution for-mula in Laplace transformation, and the following resultrelated to inverse-Laplace transformation Z ∞−∞ dω π e − iωt [ ω − ( ω r + E g )] − = − i e − i ( ω r + E g ) t . (C3) In practice, for a given SDF D ( ω r ), one can first carryout F ( t − t ′ ) in advance, via Eq. (C2); then, numericallyintegrate Eq. (C1) to obtain a ( t ). With this result athand, it is straightforward to develop the generalized QTtheory, by numerically generating the ¯ a (∆ t ) in Eq. (9),rather than using the analytic Eqs. (7) and (8). We haveexamined this numerical scheme on the Lorentzian SDFand found excellent agreement with the analytic solution.The same success can be anticipated when applying toarbitrary non-Lorentzian SDFs. [1] J. Dalibard, Y. Castin, and K. Molmer, Phys. Rev. Lett. , 580 (1992).[2] H. M. Wiseman and G. J. Milburn, Phys. Rev. A , 642(1993).[3] H. M. Wiseman and G. J. Milburn, Quantum Measure-ment and Control (Cambridge Univ. Press, Cambridge,2009).[4] K. Jacobs,
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