Indirect Measurements of a Harmonic Oscillator
aa r X i v : . [ m a t h - ph ] D ec Indirect Measurements of a Harmonic Oscillator
Martin Fraas,
Mathematics, Virginia Tech, VA 24061, Blacksburg, U.S.A.
Gian Michele Graf, Lisa H¨anggli
Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland
January 1, 2019
Abstract
The measurement of a quantum system becomes itself a quantum-mechanicalprocess once the apparatus is internalized. That shift of perspective may result indifferent physical predictions for a variety of reasons. We present a model describ-ing both system and apparatus and consisting of a harmonic oscillator coupledto a field. The equation of motion is a quantum stochastic differential equation.By solving it we establish the conditions ensuring that the two perspectives arecompatible, in that the apparatus indeed measures the observable it is ideallysupposed to.
How long should a measurement last? Ideally, a measurement is instantaneous, sincethat is implicit in Born’s rule. Such an idealization may not be appropriate in principle,because it takes time for the pointer of the apparatus to correlate with the intendedobservable of the system. It may nonetheless be appropriate effectively if the observableis a constant of motion, making the instant of the measurement irrelevant. Actually,and quite oppositely, the longer the measurement then takes the better the correlationgets established.Quite often however observables get measured even if they are not constants ofmotion. It then appears consequential to think of the measurement as reflecting theobservable averaged over the time taken by the measurement itself. Still, one should askto which extent such a measurement is an estimator for the instantaneous measurementwhich, though impossible, remains of importance since it underlies Born’s rule. In thispaper we will discuss the available time window, which is limited from below by theneed to correlate the system to the apparatus and from above by the back-reaction ofthe latter on the former. 1hese issues are discussed in a model , to be described below, that is rich enough sothat they are not trivial, yet simple enough that its dynamics can be solved for exactly.The model features an observable and a Hamiltonian which do not commute, and theydo so to a degree tunable by a parameter α , with α = 0 corresponding to a vanishingcommutator and to a (so-called) non-demolition experiment. It moreover features anapparatus which, as appropriate for such devices, consists of a (macroscopically) largenumber of degrees of freedom. Collectively they realize a pointer which is supposed tometer the observable. In order to do its job, the (microscopic) degrees of freedom arecoupled one by one to the system proper with coupling parameter γ >
0, in guise of arepeated measurement. As a result, the pointer observable commutes at different times,in line with the classical nature of a record.Even before specifying the model in further detail, the following can be noted. Let H be the Hamiltonian of some physical system and let O be some observable one intendsto measure indirectly. By this we mean that effectively the time averaged observable O T = 1 T Z T e i Ht O e − i Ht d t , (1.1)with T large shall be measured projectively. If O is the Hamiltonian itself the time aver-age is redundant, H T = H , consistently with the fact that the projective measurementof O = H is of the non-demolition type.The situation discussed in this article is that of O = H , where H is a referenceHamiltonian (or O an observable affiliated with H ) w.r.t. which the actual Hamiltonian H is a perturbation. Such a situation can e.g. occur because full experimental controlof the Hamiltonian is lacking.The effect is twofold: First ( H ) T differs from the intended observable H , thoughthe two operators still agree in expectation for eigenstates of H ; second, the apparatussteadily demolishes the eigenstates of H , thereby heating up the system. The first erroris independent of T on eigenstates and oscillatory on (discrete) supersitions thereof andin any event as small as the control of the perturbation of H − H allows. By the seconderror however the measurement no longer reflects the properties of the initial state, atleast eventually.As we will show in the context of the model, the second error is comparatively small.More precisely the measurement time T can still be taken large enough so as to be ableto tell eigenstates of H apart, yet also small enough so as they do not become blurredby the heating.The model is as follows. The Hamiltonian of the system proper are harmonic oscil-lators H = ωa ∗ a ≡ ωN ,H = H − ω ( αa + αa ∗ )= ω (cid:0) ( a ∗ − α ) ( a − α ) − | α | (cid:1) , α -circles and 0-circles.([ a, a ∗ ] = 1, α ∈ C ), and the observable to be discussed shall be the excitation number N . The measurement apparatus will be realized later as a quantum field.The motion generated by H can be visualized classically as clockwise circular orbitsin the complex a -plane centered at α , a = α + r e − i ωt , ( α -circles, see Fig. 1). As a result the average value of N = a ∗ a is N T → | α | + r , ( T → ∞ ) , (1.2)because the mixed terms are oscillatory. Eq. (1.2) has the following semi-classical in-terpretation: The Fock state | n i , ( n ∈ N ) is associated with a 0-circle | a | = r of squareradius r = n . Its points lie on α -circles of different radii, r = | a − α | , and averagevalue h r i = | a | + | α | = n + | α | , leading by (1.2) to h n | N T | n i → h n | N ∞ | n i = n + 2 | α | , ( T → ∞ ) . (1.3)The time T needed to approach the limit is T ≫ ω − . Not surprisingly the quantummechanical calculation confirms the above. In fact, and as shown in the Appendix, N T → N ∞ = N − ( αa + αa ∗ ) + 2 | α | , (1.4)which besides (1.3) also yields h (cid:0) ∆ N ∞ (cid:1) i n = (2 n + 1) | α | . (1.5)3e conclude that the standard deviation h (cid:0) ∆ N ∞ (cid:1) i / n = √ n + 1 | α | remains smallcompared to the spacing ∆ h N ∞ i n = 1 between the expectations of N ∞ in two con-secutive states | n i and | n + 1 i ; at least for finitely many states | n i and provided α issmall.The issue to be investigated is the extent up to which that property persists when themeasurement is in itself described as a dynamical process, thus including the feedbackit exerts on the system proper. We do so by modelling the degrees of freedom of theapparatus by quantum noise. Full details will be given in the next section. For now itmay suffice that the feedback results from the perpetual observation of the system duringsome time interval [0 , T ], as it e.g. emerges from repeated measurements in the (non-trivial) limit where they become ever weaker yet ever more frequent. For an informaldiscussion it is best to postpone that limit. At the beginning of every time interval ∆ t new noise degrees of freedom are introduced in a pristine state and are then coupled tothe harmonic oscillator just for its duration. What we shall need is:i) The degrees of freedom of the noise are field quadratures P t , Q t , ( t ∈ [0 , T ]) withcommutation relations i[∆ P t , ∆ Q t ] = 2∆ t , where ∆ P t , ∆ Q t are the noises associated to [ t, t + ∆ t ].ii) The Hamiltonian of the uncoupled apparatus is trivial. In the state | Ω i of theapparatus t P t has the same distribution as a Brownian motion on the real linewith h (∆ P t ) i = ∆ t .iii) The coupling to the harmonic oscillator is H I ∆ t = γN ∆ P t (1.6)with γ >
0. Upon passing to the Heisenberg picture ( O ˆ O t = e i Ht O e − i Ht ,wherein O may carry t as a label in the Schr¨odinger picture) the change broughtabout on ∆ ˆ Q t is δ ∆ ˆ Q t = i[ ˆ H I ∆ t, ∆ ˆ Q t ] ∼ = i[ ˆ H I ∆ t, ∆ Q t ] = 2 γ ˆ N t ∆ t where ∼ = refers to the leading order in ∆ t . The changes δ ∆ ˆ Q t are additive fordifferent time intervals by (ii) and because the noise ∆ Q t of each is just transientlycoupled to the oscillator. In particular, in the limit ∆ t →
0, we haveˆ Q T − Q T T = 2 γN T (1.7)whence N T = ˆ Q T / γT eventually serves as a pointer for the observable N T , asintended. However by [ H I , H ] = 0 the apparatus potentially demolishes the eigen-states of H , as announced. 4igure 2: Jump between α -circles along a 0-circle.That effect can again be discussed semi-classically, and we do so for simplicity in theweak coupling regime γ ≪ ω . (1.8)We recall that the orbits of H = ωN are 0-circles which run with frequency ω . Duringa time ∆ t the interaction (1.6) thus induces a turn by an angle ∆ ψ = γ ∆ P t along thosecircles. The resulting motion is diffusive with h (∆ ψ ) i = γ ∆ t and is in competition with the drifting dynamics of H which takes place along α -circles(see Fig. 2). However during a period 2 π/ω we have h (∆ ψ ) i = 2 πγ ω − ≪ α -circle interruptedby some rare jump along a 0-circle to the next α -circle. Let them be z ( ϕ ) = α + r e i ϕ , ˜ z ( ˜ ϕ ) = α + ˜ r e i ˜ ϕ with radii r , ˜ r , and be connected by the jump z ˜ z = z e − i ∆ ψ which occurs at uniformlydistributed ϕ . Then ˜ r = | ˜ z − α | = | z ( ϕ ) e − i ∆ ψ − α | = | α (cid:0) e − i ∆ ψ − (cid:1) + r e i ϕ e − i ∆ ψ | h ˜ r i = 2 | α | (1 − cos(∆ ψ )) + r , h ˜ r i = r + | α | h (∆ ψ ) i = r + | α | γ ∆ t upon averaging in ϕ first and then, for small ∆ t , in the jumps as well.In conclusion: At large times T Eq. (1.3) is superseded by (cf. (1.2)) h N T i ≈ | α | γ T t is T /
2. However for times T ≪ | α | − γ − the observable N T remains a good estimator of the quantum number n . The pointer observable N t associ-ated to (1.7) will then also serve its purpose, at least after some settling time that willbe seen to be T ≫ γ − .We close the introduction by making a few selected references to the literature onthe measurement process. Early on von Neumann [36] mathematically discussed theHeisenberg cut, by which the conceptual dividing line between the observer and theobserved system is meant. He showed that it can be shifted, providing examples whichnowadays would be said to correspond to non-demolition measurements. Perez andRosen [34] emphasized decoherence and the essential macroscopic character of the mea-surement apparatus, to which Hepp [26] supplied structure and examples. Merkli et al.[13, 14] rigorously exhibited decoherence for more general, not explicitly solvable exam-ples. Fr¨ohlich [15] and coworkers give a notion of events that forgoes decoherence. Froma more applied perspective, Makhlin [31] et al. discuss the limitations to an accuratereadout of superconducting qubits. They are of a similar nature than in the presentwork. The precise definition of the model will be given in terms of its propagator, ratherthan the Hamiltonian, and in fact by means of quantum stochastic differential equation(QSDE).Quantum stochastic calculus is a mathematical theory for quantum noise in openquantum systems developed by Hudson and Parthasarathy in 1984 [27]. Applicationsare wide ranging. The first ones, which provided the construction of unitary dilationsof quantum dynamical semigroups ([27] already, and [21]), were followed by other onesin quantum measurement theory [5, 6], and in quantum optics. The starting pointof the latter applications is to model a Bosonic field using creation and annihilationprocesses. The first introduction of quantum stochastic calculus in this area was in[22], but related noise models already showed up earlier [30]. One kind of application6f quantum stochastic calculus in quantum optics is in order to establish the masterequation for the system of interest (e.g. [23]), another to describe the detection of photons[6, 7, 9, 16, 17, 32], and yet another to model quantum input and output channels[1, 6, 8, 9, 20, 22, 29]. The latter is the field where we would locate our result. Otherapplications can for example be found in quantum filtering (e.g. [11, 12]).Indirect measurements of quantum systems on the other hand have been discussed,including the non-demolition case, for different setups. The focus and the methodsapplied however differ from those presented in this article. We mention the experiment[18, 25] by the Haroche group and the theoretical work of Bauer and Bernard [10], as wellas [19, 2, 3, 4, 13, 14]. In the experiment by Haroche, photons in a cavity are countedby letting them repeatedly interact with Rubidium atoms in a circular Rydberg state,which subsequently are measured. A description of such repeated indirect quantumnon-demolition measurements was given by Bauer and Bernard.The system proper we consider here is a harmonic oscillator. Its degrees of freedomare a creation and annihilation operator, a ∗ and a , with commutation relation [ a, a ∗ ] = 1irreducibly realized on some Hilbert space H . The operators H = ω ( a ∗ a − ( αa + αa ∗ )) , Γ = γa ∗ a (2.1)( ω, γ > α ∈ C ) represent the Hamiltonian and the coupling strength to the measure-ment apparatus. Because the two operators do not commute, the measurement will notbe of the non-demolition type, as pointed out earlier.The state space of the apparatus is the bosonic Fock space F = F ( L ( R + )), withvacuum state | Ω i and with creation and annihilation operators A ∗ ( f ), A ( g ),[ A ( g ) , A ∗ ( f )] = h g | f i , ( f, g ∈ L ( R + )) . We set in particular A t = A (1 [0 ,t ] ), ( t ≥ Q t = A t + A ∗ t , P t = − i( A t − A ∗ t ) , resulting in i[ P t , Q t ] = 2 t . (2.2)Finally the propagator is a solution of the QSDEd U t = − (cid:18) i H + 12 Γ (cid:19) U t d t − i Γ U t d P t , U = , (2.3)which is of so-called Hudson-Parthasarathy form. The last term reflects the interaction(1.6), except that the equation is written in the Itˆo rather than in the Stratonovichsense. Moreover, the equation is of a form that ensures the unitarity of U t , or at leastwould if the operators were bounded. This point of precision is however inconsequential.7he rescaling of time ( t εt ) on the quantum field is represented by a unitaryoperator T ε acting as the second quantization of f f ε , f ε ( t ) = √ εf ( εt ), i.e. T ε A ( f ) T ∗ ε = A ( f ε ) , T ε | Ω i = | Ω i . In particular, T ε P εt T ∗ ε = √ εP t , (2.4)and T ε d P εt T ∗ ε = √ ε d P t . This reflects the Wiener scaling, by which the probabilitydistributions of the rescaled Brownian motion W εt and √ εW t are equal. For the evolutionof the system, this implies that if U ( t ) is the solution of (2.3) for parameters ( ω, α, γ )then ˜ U ( t ) = T ε U ( εt ) T ∗ ε is the solution of the equation for ( εω, α, √ εγ ). We conclude that the model has twodimensionless scales, α and ωγ − .The propagator can be computed explicitly. To this end we introduce the Weyloperators in the form D ( z ) = e za ∗ − za , ( z ∈ C ) . (2.5)They form a unitary projective representation of C by D ( z ′ ) D ( z ) = e − i Im(¯ z ′ z ) D ( z ′ + z ) , D ( z ) − = D ( − z ) = D ( z ) ∗ . Proposition 1.
The family of operators on
H ⊗ F U t = e − i φ t a ∗ a D ( Z t ) e − i G t , ( t ≥
0) (2.6) is a solution to (2.3). Here φ t = ωt + γP t , Z t = i ωαZ t , Z t = Z t e i φ s d s , (2.7) G t = ω | α | Z t Z s Im e i ( φ s − φ s ) d s d s . Moreover, U t is manifestly unitary for all t . Remark 2. Z t and G t are well-defined operators on F , because the family ( P s ) ≤ s ≤ t iscommuting; so is D ( Z t ) because the exponent in (2.5) remains self-adjoint up to a factor i . emark 3. The field P t may be identified with a random process, namely Brownianmotion W t , see above and in more detail in Sect. 5. By (2.7) we have Z t = Z s + e i φ s Z ts e i( φ r − φ s ) d r ≡ Z s + e i φ s ˆ Z t − s , (2.8) where, for fixed s , ( ˆ Z t − s ) t ≥ s is a process independent of (e i φ t ) ≤ t ≤ s (and hence of Z s ), butequal in distribution, ( ˆ Z t − s ) t ≥ s d = ( Z τ ) τ ≥ with τ = t − s . In the limit ε → , the phasefactor e i φ s oscillates quickly and the process Z t has independent increments. Under theappropriate rescaling (2.4), Z t becomes proportional to complex Brownian motion.We also note that R t := e − i φ t Z t satisfies the renewal equation R t = e − i( φ t − φ s ) R s +ˆ R t − s . (A related, but different renewal process is treated in [24].) The next result says that Q t indeed meters the excitation number a ∗ a : Proposition 4.
We have d s ( U ∗ s Q t U s ) = ( γU ∗ s a ∗ aU s d s , (0 ≤ s < t ) , , ( s ≥ t ) . (2.9)The increments then add up as follows: Proposition 5. U ∗ t a ∗ aU t = ( a ∗ + Z ∗ t ) ( a + Z t ) , (2.10) U ∗ t Q t U t = X ,t a ∗ a + X ∗ ,t a + X ,t a ∗ + X ,t , (2.11) where X ,t = 2 γt , (2.12) X ,t = 2 i γωαY ,t , Y ,t = Z t Z s d s , (2.13) X ,t = Q t + 2 γω | α | Y ,t , Y ,t = Z t Z ∗ s Z s d s . (2.14)The next result computes the expectation and the variance of the two observables.More precisely, let us focus on the initial state | n, Ω i , where | n i ∈ H , ( n ∈ N ), is theeigenstate of the excitation number, a ∗ a | n i = n | n i , and | Ω i ∈ F is the field vacuum, A t | Ω i = 0, ( t ≥ h A i = h n, Ω | A | n, Ω i , hh A ii = h A i − h A i for any operator A on H ⊗ F . 9 roposition 6.
We have the following expectations and variances in the state | n, Ω i : h U ∗ t a ∗ aU t i = n + ω | α | h Z ∗ t Z t i , (2.15) hh ( U ∗ t a ∗ aU t ) ii = ω | α | (cid:0) ω | α | hh ( Z ∗ t Z t ) ii + (2 n + 1) h Z ∗ t Z t i (cid:1) , (2.16) h U ∗ t Q t U t i = 2 γ (cid:0) tn + ω | α | h Y ,t i (cid:1) , (2.17) hh ( U ∗ t Q t U t ) ii = t + 4 γ ω | α | (cid:0) ω | α | hh Y ,t ii + (2 n + 1) h Y ∗ ,t Y ,t i (cid:1) . (2.18)Let us stress once more that (2.10) is the instantaneous excitation number, whereas(2.11), divided by 2 γt , is its time-averaged value, as sampled by the apparatus initializedin | Ω i , cf. (2.9) and h Ω | Q t | Ω i = 0. We shall thus focus on the observables N t = U ∗ t a ∗ aU t , N t = U ∗ t Q t U t γt . (2.19)We note in passing that the family ( N t ) t ≥ is commuting, as appropriate to pointerobservables. This follows because of U ∗ s Q t U s = U ∗ t Q t U t , ( s ≥ t )by (2.9) and because ( Q t ) t ≥ is commuting.The quantum number n = 0 , , , . . . labels the states | n, Ω i . Their values are spacedby one, and so are the expectations of h N t i and hN t i , cf. (2.15, 2.17). The issue howeveris as to whether the measurement of N t and, more importantly, N t can be used toreliably tell apart finitely many states n . To this end (a) their expectations ought toremain close to n , and (b) their variances ought to be small w.r.t. unity. These twoconditions will require that α is small and determine a time interval during which theyare met for N t ; as for N t , the interval should not be too short, resulting in a furthercondition on t , γ . It ensures that the initial uncertainty of the pointer of the apparatushas been effaced. A sketch of how this conclusions are reached is as follows, with detailssupplied later in Sect. 4.(a) Expectations: • Bounds 0 ≤ h N t i − n ≤ | α | (4 + γ t ) , (2.20)0 ≤ hN t i − n ≤ | α | (4 + γ t . (2.21)10 Asymptotics in the regime γ ≪ ω , cf. (1.8): h N t i − n ∼ = | α | ( ωt ) , ( t ≪ ω − ) , − cos ωt ) , ( ω − ≪ t ≪ γ − ) ,γ t , ( t ≫ γ − ) , (2.22) hN t i − n ∼ = | α | ( ωt ) , ( t ≪ ω − ) , , ( ω − ≪ t ≪ γ − ) , γ t , ( t ≫ γ − ) . (2.23)The last two cases match what was expected in (1.3, 1.9).The requirements set by condition (a), i.e. that the right hand sides of eqs. (2.20)-(2.23) are ≪
1, are therefore | α | ≪ , t ≪ | α | − γ − (2.24)for both observables.(b) Variances: • N t : Both terms on the r.h.s. of (2.16) ought to be ≪
1. For both of them,this condition does not further limit the time interval (2.24). For the secondterm this is even manifest, since it equals the one discussed in connection with(2.15). • N t : In view of the normalization of N t , the terms on the r.h.s. of (2.18) oughtto be ≪ ( γt ) . For the last two terms, this yields the upper bounds (2.24).The first term however sets the lower bound t ≫ γ − . (2.25) Summary.
The window of opportunity for the effective measurement of n through N t is set by (2.24); that for N t is further restricted by (2.25).The following result estimates the accuracy of N t as an estimator for N = a ∗ a forarbitrary initial states | ψ i of the oscillator. Proposition 7.
For any normalized | ψ i ∈ H we have h ψ, Ω | ( N t − a ∗ a ) | ψ, Ω i ≤ C | α | (cid:0) γ t (cid:1) h ψ | a ∗ a +1 | ψ i + C | α | (cid:16) (cid:0) γ t (cid:1) (cid:17) + (cid:0) γ t (cid:1) − . In particular the r.h.s. is small for (2.24, 2.25), provided the excitation number isbounded.
11e conclude this section by discussing the long time limiting regime of the dynamics.To this end we introduce a notion of convergence that allows to express the stochasticbut classical nature of the limiting dynamics. It comes in three variants (i-iii). First,(i) we say that the self-adjoint operators X ε on F tend to the R -valued random variable X on some probability space ( ˜Ω , ˜ µ ) w.r.t. the vacuum | Ω i ∈ F as ε → X ε → | Ω i X , ( ε → h Ω | f ( X ε ) | Ω i → Z ˜Ω f ( X ( ω )) d ˜ µ ( ω ) , ( ε →
0) (2.26)for every bounded continuous function f : R → C . Second, let C ( R + , R ) denote theclassical Wiener space of continuous functions R + → R . Then (ii) is an extension tocommuting families ( X ε,t ) t ≥ of self-adjoint operators on the l.h.s. of (2.26) and tostochastic processes ( X t ) t ≥ , i.e. random variables taking values in C ( R + , R ), on ther.h.s.; the functions f are now C ( R + , R ) → C . Finally, (iii), both variants can beextended by replacing F with H ⊗ F . The random variable X and the stochasticprocesses ( X t ) t ≥ , respectively, are then to be understood as multiples of the identityoperator on H ; the convergence is meant in the sense of weak convergence of boundedoperators on H . Explicitly, (2.26) is then to be read as h ψ, Ω | f ( X ε ) | ψ, Ω i → Z ˜Ω f ( X ( ω )) d ˜ µ ( ω ) h ψ | ψ i , ( ε →
0) (2.27)which expresses that the limit is (a) oblivious to the quantum state | ψ i ∈ H , ( k ψ k = 1)of the oscillator, and (b) indeed given by a classical random process.In view of the linearly growing expectation of N t at large times t , cf. (2.22), a non-trivial scaling limit ought to be given by εN t/ε , ( ε → Proposition 8 (Long time limit) . Let κ = ωαγ/c , where c = i ω − γ / . (2.28) Then ε N t | t → ε − t → | Ω i | κ | | B t | , (2.29) ε N t | t → ε − t → | Ω i | κ | t Z t | B s | d s , (2.30) as ε → , where B t = Re B t + i Im B t is complex Brownian motion. Its componentsare independent and distributed as W t / √ , where W t is real Brownian motion. (Thisnormalization ensures that | d B t | = d t , as in the real case.) The two limits are in thesense (i) and (ii), respectively, as extended in (iii). The limits hold for t > and for t ∈ I , respectively, where I ⊂ R + is any compact interval.
12n plain terms the proposition states that at large times no trace is left of the initialstate of the oscillator and that the excitation number follows the square displacementof a diffusive motion of constant | κ | / Remark 9.
1. The different kinds of limit are due to the fact that only the operatorsseen on the l.h.s. of (2.30) form a commuting family.2. The characteristic function of | W t | (Laplace transform of its measure) is (1 +2 λt ) − / , that of t − R t | W s | d s is (cosh √ λt ) − / . The first is immediate, the sec-ond was derived in [28], where that of R t | W s | d s was found to be (cosh √ λt ) − / .The characteristic function of the complex counterparts is then found by the re-placements χ ( λ ) → χ ( λ/ to be (1 + λt ) − and (cosh √ λt ) − , respectively. Proof of Prop. 1.
We first observe that, according to the claim (2.7), Z t and G t aredifferentiable, whereas φ t is a stochastic integral. We shall thus regard (2.6) as a genericansatz of that type for a solution of (2.3). Three preliminaries are in order. The firstone is dd t D ( z t ) = (cid:0) ˙ z t a ∗ − ˙ z t a + i Im( ˙ z t z t ) (cid:1) D ( z t ) , where z t takes values in C as in (2.5) or is promoted to a differentiable multiplicationoperator in the variables ( P s ) ≤ s ≤ t , as in Remark 2. That claim follows by differentiatingthe equation D ( z s ) = e i Im( z s z t ) D ( z s − z t ) D ( z t )by s at the point s = t and usingdd s D ( z s − z t ) (cid:12)(cid:12)(cid:12)(cid:12) s = t = ˙ z t a ∗ − ˙ z t a . The second preliminary ise − i φ t a ∗ a ( ˙ z t a ∗ − z t a ) e i φ t a ∗ a = ˙ z t e − i φ t a ∗ − z t e i φ t a , while the last one isd (cid:0) e − i φ t a ∗ a (cid:1) = (cid:16) − i a ∗ a d φ t −
12 ( a ∗ a ) (d φ t ) (cid:17) e − i φ t a ∗ a and follows from Itˆo’s lemma [27]. So prepared we differentiate (2.6) and obtaind U t = (cid:16) − i a ∗ a d φ t −
12 ( a ∗ a ) (d φ t ) + (cid:0) a ∗ ˙ Z t e − i φ t − a ˙ Z ∗ t e i φ t + i Im( ˙ Z ∗ t Z t ) − i ˙ G t (cid:1) d t (cid:17) U t . a ∗ (or a ), a ∗ a and ( a ∗ a ) with those in (2.1, 2.3) we obtainIm (cid:16) ˙ Z ∗ t Z t (cid:17) − ˙ G t = 0 , ˙ Z t e − i φ t = i ωα , − i d φ t = − i ω d t − i γ d P t , −
12 (d φ t ) = − γ t , as well as φ = 0, Z = 0, G = 0. The last of the four equations is a consequence ofthe third, which together with the second is solved by the expressions (2.7) for φ t and Z t ; and so is the first one by that for G t , once it is restated as˙ G t = ω | α | Z t Im e i( φ s − φ t ) d s . Proof of Prop. 4.
By the Itˆo rule d(
M M ′ ) = (d M ) M ′ + M (d M ′ ) + (d M )(d M ′ ) weobtaind s ( U ∗ s Q t U s ) = U ∗ s (cid:0)(cid:0) i[ H, Q t ] − (cid:8) Γ / , Q t (cid:9)(cid:1) d s + i (Γ(d P s ) Q t − Q t (d P s )Γ) + Γ(d P s ) Q t (d P s )Γ (cid:1) U s . Let 0 ≤ s < t . In view of [ H, Q t ] = [Γ , Q t ] = [Γ , P s ] = 0 and of [d P s , Q t ] = − s ,(d P s ) = d s , (d s )(d P s ) = 0 we get − (cid:8) Γ / , Q t (cid:9) d s + Γ(d P s ) Q t (d P s )Γ = 0and thus d s ( U ∗ s Q t U s ) = 2 U ∗ s Γ U s d s , as claimed. For s ≥ t the only change is by [d P s , Q t ] = 0. Proof of Prop. 5.
We have U ∗ t a ∗ aU t = D ( Z t ) ∗ a ∗ aD ( Z t ) by (2.6) and D ( Z t ) ∗ aD ( Z t ) = a + Z t because [ a, Z t a ∗ − Z ∗ t a ] = Z t , whence Eq. (2.10) follows. Furthermore, expanding thebrackets in (2.10) and integrating (2.9) yields Eq. (2.11). Proof of Prop. 6.
Let us generically denote by V any monomial ( a ∗ ) m a n with m = n ,as well as any linear combinations thereof. Then h V i = 0. We have U ∗ t a ∗ aU t = a ∗ a + Z ∗ t Z t + V , V = a Z ∗ t + a ∗ Z t , from which (2.15) follows, as well as( U ∗ t a ∗ aU t ) = ( a ∗ a + Z ∗ t Z t ) + Z ∗ t Z t ( a ∗ a + aa ∗ ) + V .
Thus hh ( U ∗ t a ∗ aU t ) ii = hh ( a ∗ a + Z ∗ t Z t ) ii + (2 n + 1) hZ ∗ t Z t i = hh ( Z ∗ t Z t ) ii + (2 n + 1) hZ ∗ t Z t i , where in the last equality we used that a ∗ a has no variance in | n i , thus amounting to ashift of Z ∗ t Z t . So (2.16) follows by (2.7).The other observable is dealt with similarly. We have U ∗ t Q t U t = X ,t a ∗ a + X ,t + V , from which (2.17) follows by (2.12, 2.14) and h Q t i = 0, as well as( U ∗ t Q t U t ) = ( X ,t a ∗ a + X ,t ) + X ∗ ,t X ,t ( a ∗ a + aa ∗ ) + V .
Again, hh ( U ∗ t Q t U t ) ii = hh X ,t ii + (2 n + 1) h X ∗ ,t X ,t i , where we used that X ,t a ∗ a has no variance in | n, Ω i , cf. (2.12). The last term on ther.h.s. yields the corresponding one in (2.18) by (2.13). It remains to discuss the firstterm: hh X ,t ii = h P t i + 4 γ ω | α | hh Y ,t ii . (3.1)To see this, we observe that Q t + i P t = 2 A t and A t | Ω i = 0. Thus X ,t | Ω i = ˜ X ,t | Ω i ,where ˜ X ,t is obtained from X ,t in (2.14) by replacing Q t with − i P t . Thus h X ,t i = h ˜ X ∗ ,t ˜ X ,t i = h P t i + 4 γ ω | α | h Y ,t i (3.2)and (3.1) follows. Finally h P t i = t . We shall supply the details leading to the conclusions given at the end of Sect. 2. Weobserve that by the r.h.s. of (2.15-2.18) we are left with expectations and variances ofoperators A on F only, i.e. h A i = h Ω | A | Ω i . The expectations are computed as follows. Lemma 10.
Let R n ( f )( t ) (also written R n ( f ( t )) with slight abuse of notation) be theremainder of the n -th Taylor expansion in t = 0 . We set c = i ω − γ / , as in (2.28). hen h e i φ t i = e ct , (4.1) h Z t i = c − R (e ct ) , (4.2) h Z ∗ t Z t i = c − R (e ct ) + c.c. , (4.3) h Y ,t i = c − R (e ct ) , (4.4) h Y ,t i = c − R (e ct ) + c.c. , (4.5) h Y ∗ ,t Y ,t i = c − R (( ct −
1) e ct ) + c.c. . (4.6)The variances, on the other hand, are estimated as follows. We set h A ; B i = h AB i − h A ih B i , hh A ii = h A ; A i . Lemma 11. |h Z ∗ t Z t ; Z t i| ≤ Cγ ω t , (4.7) hh ( Z ∗ t Z t ) ii ≤ γ ω (2 γ t + Ct ) , (4.8) and more generally |h Z ∗ s Z s ; Z ∗ t Z t i| ≤ γ ω (2 γ s + Cs ) (4.9) for ≤ s ≤ t . In particular hh Y ,t ii ≤ γ ω (cid:0) γ t + Ct (cid:1) , (4.10) where C is a numerical constant changing from line to line. We postpone the proofs of Lemma 10 and 11 and continue towards the stated goal.The following estimates on Taylor remainders, valid for Re z ≤
0, will be used: |R (e z ) | ≤ , (cid:12)(cid:12)(cid:12)(cid:12) R (e z ) z (cid:12)(cid:12)(cid:12)(cid:12) ≤ , (cid:12)(cid:12)(cid:12)(cid:12) R (e z ) z (cid:12)(cid:12)(cid:12)(cid:12) ≤ . (4.11)The first one is elementary, the second follows by the mean value theorem, and so doesthe third, yet by Cauchy’s form. Also c − + c − = −| c | − γ , ω ≤ | c | (4.12)will be often used.We first focus on N t as defined in (2.19) and on the auxiliary quantities it calls forby (2.15, 2.16). They will be written without the prefactors | α | or | α | , which are tobe included when finalizing the estimates. 16 The first one is ω h Z ∗ t Z t i and calls in turn for (4.3). We have R (e ct ) = R (e ct ) − ct and hence h Z ∗ t Z t i = c − R (e ct ) − c − t + c.c. . (4.13)By (4.11, 4.12) we find ω h Z ∗ t Z t i ≤ γ t , confirming (2.20). In the regime γ ≪ ω the more detailed discussion goes as follows:i) For ωt ≪
1, or equivalently | c | t ≪
1, we have R (e ct ) ∼ = ( ct ) / ω h Z ∗ t Z t i ∼ = ( ωt ) c.c. . ii) For ωt ≫ ≫ γ t we use (4.13), wherein e ct ∼ = e i ωt and c − R (e ct ) ∼ = − ω (e i ωt − γ t ≫ h Z ∗ t Z t i ∼ = − c − t + c.c. = γ t . • The second auxiliary quantity is ω hh ( Z ∗ t Z t ) ii and calls for (4.8). For reasons statedbelow (2.24), it yields the condition | α | M ≪ M = γ t (2 γ t + C ). Given that | α | ≪
1, this still is (2.24).We next focus on N t and on the auxiliary quantities it calls for by (2.17, 2.18). • The first one is ω t h Y ,t i , (4.14)and calls in turn for (4.5). We have R (e ct ) = R (e ct ) − ( ct ) / h Y ,t i t = c − R (e ct ) ct − c − t c.c. . By (2.14) and (4.11, 4.12) we find0 ≤ ω t h Y ,t i ≤ γ t , (4.15)confirming (2.21). In the regime γ ≪ ω , a more detailed discussion goes as follows:i) For ωt ≪
1, or equivalently | c | t ≪
1, we have R (e ct ) ∼ = ( ct ) / ω t h Y ,t i ∼ = ( ωt ) . ωt ≫ R (e ct ) = R (e ct ) − ct − ( ct ) / |R (e ct ) | ≤ ω t (cid:12)(cid:12) c − R (e ct ) (cid:12)(cid:12) ≤ ωt ≪ . Moreover by − c − + c.c. ∼ = 2 ω − we have ω t c − ( − ct + c.c. ) ∼ = 2and ω t c − (cid:18) − ( ct ) c.c. (cid:19) ∼ = γ t . These findings are summarized in (2.23). • The second auxiliary quantity is ω t h Y ∗ ,t Y ,t i , and calls for (4.6). We have R (( ct −
1) e ct ) = R (( ct −
1) e ct ) − ( ct ) / h Y ∗ ,t Y ,t i t = c − R (( ct −
1) e ct )( ct ) − c − t c.c. . By (4.11, 4.12) we find ω t h Y ∗ ,t Y ,t i ≤ γ t . (4.16)For reasons stated before (2.21) that bound yields the condition | α | (cid:18) γ t (cid:19) ≪ , i.e. (2.24), as announced before (2.25). • The third auxiliary quantity is ω t hh Y ,t ii and calls for (4.10). By the reasons just recalled it yields the condition | α | M ≪ Proof of Prop. 7.
By (2.11, 2.12), we have( N t − a ∗ a ) = 1(2 γt ) (cid:0) X ∗ ,t a + X ,t a ∗ + X ,t (cid:1) ≤ γt ) (cid:0) X ∗ ,t X ,t (2 a ∗ a + 1) + X ,t (cid:1) by the Cauchy-Schwarz inequality. Expectations in | Ω i are computed by (2.13) as h X ∗ ,t X ,t i = 4 γ ω | α | h Y ∗ ,t Y ,t i , and by (3.2). The proof will thus be finished by showing ω h Y ∗ ,t Y ,t i ≤ Ct (1 + γ t ) ,ω h Y ,t i ≤ Ct (1 + ( γ t ) ) . The first bound is just (4.16). The second one follows from h Y ,t i = h Y ,t i + hh Y ,t ii together with (4.15, 4.10) and γ t ≤ t (1 + ( γ t ) ).We next catch up on the proofs of Lemma 10 and 11. We begin by some preliminaries,which will be used repeatedly.a) By iterating f ( t ) = f (0) + R t f ′ ( s ) d s we get the integral form of the remainder ofthe n -th Taylor approximation ( n = 0 , , , . . . ) R n ( f )( t ) = Z t R n − ( f ′ )( s ) d s with R − ( f ) = f ; equivalently R n ( f )(0) = 0 , R n ( f ) ′ = R n − ( f ′ ) . (4.17)b) We observe that ( φ t − φ s ) ≤ s ≤ t d = ( φ t − s ) ≤ s ≤ t , where d = means equality in distribution.Thus e − i φ t Z t = Z t e − i( φ t − φ s ) d s d = Z t e − i φ s d s = Z ∗ t . (4.18)c) A continuous function on an interval, which has a continuous right derivative, isdifferentiable (see e.g. [33], Cor. 1.2). Claims on derivatives can thus be read asbeing taken from the right.d) Let t ≥ s and let F s be a functional of ( P τ ) ≤ τ ≤ s . Then by independence and (4.1), h F s e i φ t − ct i = h F s e i φ s − cs ih e i( φ t − φ s ) i e − c ( t − s ) = h F s e i φ s − cs i . In particular dd t h F s e i φ t − ct i (cid:12)(cid:12) t = s = 0 . (4.19)This last preliminary shall not be used in the proof of (4.1), since it depends on it.19 roof of Lm. 10. The basic expectation value is h e i γP t i = e − γ t/ , ( t ≥ , in view of e i γP t = e γ ( A t − A ∗ t ) = e − γA ∗ t e γA t e − γ t/ . Thus h e i φ t i = e i ωt h e i γP t i = e ct for t ≥ t h Z t i = h e i φ t i = e ct which by (4.17) implies (4.2). Nextdd t h Z ∗ t Z t i = h e − i φ t Z t i + h Z ∗ t e i φ t i = h Z ∗ t i + h Z t i by (4.18), whence (4.2) implies (4.3). Fromdd t h Y ,t i = h Z t i , dd t h Y ,t i = h Z ∗ t Z t i the equations (4.4, 4.5) follow, too. Finallydd t h Y ∗ ,t Y ,t i = h Z ∗ t Y ,t i + h Y ∗ ,t Z t i , dd t h Y ∗ ,t Z t i = h Z ∗ t Z t i + h Y ∗ ,t e i φ t i , and, by (4.19, 4.2)dd t h Y ∗ ,t e i φ t − ct i = h Z ∗ t e i φ t − ct i = e − ct h Z t i = c − (cid:0) − e − ct (cid:1) . Hence h Y ∗ ,t e i φ t i = e ct (cid:0) c − t + c − (cid:0) e − ct − (cid:1)(cid:1) = c − (cid:0) ( ct −
1) e ct +1 (cid:1) = c − R (cid:0) ( ct −
1) e ct (cid:1) since g ( x ) = ( x −
1) e x has g (0) = − g ′ (0) = 0. We concluded d t h Y ∗ ,t Y ,t i = h Z ∗ t Z t i + h Y ∗ ,t e i φ t i + c.c. = 2 c − R (cid:0) e ct (cid:1) + c − R (cid:0) ( ct −
1) e ct (cid:1) + c.c. = c − R (cid:0) ( ct + 1) e ct (cid:1) + c.c. , and hence (4.6).The proof of Lm. 11 rests on the following two lemmas.20 emma 12. dd t h Z ∗ s Z s ; Z ∗ t Z t i| t = s = h Z ∗ s Z s ; Z s + Z ∗ s i , (4.20)dd t h Z ∗ t Z t ; Z ∗ t Z t i = 2 h Z ∗ t Z t ; Z t + Z ∗ t i , (4.21)dd t h Z ∗ t Z t ; Z t i = h e − i φ t Z t ; Z t i + h Z ∗ t e i φ t ; Z t i + h Z ∗ t Z t ; e i φ t i , (4.22)dd t h Z ∗ t Z t ; e i φ t − ct i = h e − i φ t Z t ; e i φ t − ct i + h Z ∗ t e i φ t ; e i φ t − ct i ≡ g − ( t ) − g + ( t ) , (4.23)dd t h Z ∗ t e i φ t − ct ; Z t i = h Z ∗ t e i φ t − ct ; e i φ t i = − g + ( t ) , (4.24)dd t h e − i φ t − ct Z t ; Z t i = h e − i φ t − ct Z t ; e i φ t i = e ( c − c ) t g − ( t ) , (4.25) with g ± ( t ) = Z t e ± cs (cid:0) − e − γ s (cid:1) d s . The terms on the r.h.s. of (4.22) vanish for t = 0. As a result, the equations (4.23-4.25) may be rephrased as follows: h Z ∗ t Z t ; e i φ t i = e ct Z t g − ( s ) d s − e ct Z t g + ( s ) d s , h Z ∗ t e i φ t ; Z t i = − e ct Z t g + ( s ) d s , h e − i φ t Z t ; Z t i = e ct Z t e ( c − c ) s g − ( s ) d s . (4.26)The computation of the integrals yields: Lemma 13. e ct Z t g + ( s ) d s = c − e ct − ( c − γ ) − e (2 c − γ ) t + γ (2 c − γ ) c ( c − γ ) e ct + γ c ( c − γ ) t e ct , (4.27)e ct Z t g − ( s ) d s = c − − c − e − γ t − γ ω | c | e ct − γ | c | t e ct , (4.28)e ct Z t e ( c − c ) s g − ( s ) d s = | c | − (cid:0) − e − γ t (cid:1) + i γ | c | ω (cid:0) e ct − e ct (cid:1) . (4.29)Let us give the proof of this computation straight away, while that of Lm. 12 will bepostponed till after that of Lm. 11. 21 roof of Lm. 13. The computation rests on R t e αs d s = α − (e αt − Z t e αs (cid:0) − e βs (cid:1) d s = Z t (cid:0) e αs − e ( α + β ) s (cid:1) d s = α − e αt − ( α + β ) − e ( α + β ) t − βα ( α + β ) (4.30)= ( α + β ) − (cid:0) e αt − e ( α + β ) t (cid:1) + βα ( α + β ) (cid:0) e αt − (cid:1) (4.31)by α − − ( α + β ) − = βα − ( α + β ) − ; and similarly Z t Z s e αs (cid:0) − e βs (cid:1) d s d s = α − e αt − ( α + β ) − e ( α + β ) t − β (2 α + β ) α ( α + β ) − βα ( α + β ) t (4.32)by α − − ( α + β ) − = (2 αβ + β ) α − ( α + β ) − . In passing we mention a slight general-ization of (4.31), namely Z t (cid:0) c − e αs − c − β e ( α + β ) s (cid:1) d s = ( α + β ) − c − β (cid:0) e αt − e ( α + β ) t (cid:1) + ∆ (cid:0) e αt − (cid:1) (4.33)with ∆ = ( αc ) − − (( α + β ) c β ) − . Now (4.27) follows by applying (4.32) with α = c , β = − γ to its integral. Likewise does (4.28) with α = − c , β = − γ in view of c + γ = − c , − c − γ = − iω . Finally by (4.30) applied with the same α , β we havee ( c − c ) t g − ( t ) = − c − e − ct +( c + γ ) − e − ( c + γ ) t + γ c ( c + γ ) e ( c − c ) t = − c − e − ct − c − e ct − γ | c | e ( c − c ) t and Z t e ( c − c ) s g − ( s ) d s = | c | − (cid:0) e − ct − e ct (cid:1) − γ | c | ( c − c ) (cid:0) e ( c − c ) t − (cid:1) with c − c = 2 iω , proving (4.29). Incidentally the last integration can be seen as anapplication of (4.33) with c = − c , c β = c , α = − c , α + β = c , whence ∆ = 0. Proof of Lm. 11.
In the expressions (4.27-4.29) we distinguish secular terms like 1, e − γ t from oscillating ones such as e αt , t e αt with α = c, c, c , including factors e − γ t . Equation224.27) only contains terms of the second type. Collecting secular terms on the r.h.s. of(4.22) we find by (4.26) S t := c − − c − e − γ t + | c | − (cid:0) − e − γ t (cid:1) = c − + | c | − − (cid:0) c − + | c | − (cid:1) e − γ t = c + c | c | (cid:0) c − c e − γ t (cid:1) and | S t | ≤ γ ω − by c + c = − γ . For later use we observe that S t + S ∗ t has a betterestimate: S t + S ∗ t = ( c + c ) | c | (cid:0) − e − γ t (cid:1) , | S t + S ∗ t | ≤ γ ω − . (4.34)As for the oscillatory terms, we distinguish between the first two terms in (4.27), O t = c − e ct − ( c − γ ) − e (2 c − γ ) t and all the rest, R t : dd t h Z ∗ t Z t ; Z t i = S t − O t + R t . We then integrate the equation and estimate terms as follows. (cid:12)(cid:12)(cid:12)(cid:12)Z t R s d s (cid:12)(cid:12)(cid:12)(cid:12) ≤ C γ ω min( ω − , t ) , because of (cid:12)(cid:12)(cid:12)(cid:12)Z t e ws d s (cid:12)(cid:12)(cid:12)(cid:12) ≤ | w | − min(1 , | w | t ) , (cid:12)(cid:12)(cid:12)(cid:12) w Z t s e ws d s (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | w | − min(1 , | w | t )for Re w ≤
0; the first minimum bounds the second one. By (4.33) we have Z t O s d s = ˜ O t + ∆ (cid:0) e ct − (cid:1) , ˜ O t = (2 c − γ ) − ( c − γ ) − (cid:0) e ct − e (2 c − γ ) t (cid:1) , where ∆ = (2 c ) − − (2 c − γ ) − ( c − γ ) − satisfies | ∆ | ≤ (3 / | c | − γ ; moreover | e ct − | ≤ | c | t , | ˜ O t | ≤ ω − (1 − e − γ t ) ≤ ω − γ t . This proves (4.7) and h Z ∗ t Z t ; Z t + Z ∗ t i = −
2( ˜ O t + ˜ O ∗ t ) + ˜ R t with | ˜ R t | ≤ γ ω t + C γ ω = γ ω (2 γ t + C )23y (4.34). A further integration will give h Z ∗ t Z t ; Z ∗ t Z t i by (4.21). That of ˜ O t is doneby (4.31) with α = 2 c , β = − γ . Both resulting terms have bounds ω − times ω − γ t .This proves (4.8). For the proof of (4.9) we make use of (2.8). We so get h Z ∗ s Z s ; Z ∗ t Z t i = h Z ∗ s Z s ; Z ∗ s Z s i + h Z ∗ s Z s ; Z ∗ s e i φ s ih ˆ Z t − s i + h Z ∗ s Z s ; e − i φ s Z s ih ˆ Z ∗ t − s i = h Z ∗ s Z s ; Z ∗ s Z s i + h Z s Z ∗ s ; Z s ih ˆ Z t − s i + h Z s Z ∗ s ; Z ∗ s ih ˆ Z ∗ t − s i . The result now follows from (4.8, 4.7, 4.2). By definition (2.14), hh Y ,t ii = Z t Z t h Z ∗ s Z s ; Z ∗ s Z s i d s d s , and thus (4.10) follows from (4.9) by integration. Proof of Lm. 12.
By straightforward differentiation and by (4.18) we getdd t h Z ∗ s Z s ; Z ∗ t Z t i| t = s = h Z ∗ s Z s ; e − i φ s Z s + Z ∗ s e i φ s i = h Z ∗ s Z s ; Z ∗ s + Z s i , which is the same as (4.20). Equation (4.21) then follows. Equation (4.22) is straight-forward. Equation (4.23) follows from (4.19); likewise for (4.24-4.25) by also notingthat h e − i φ t e i φ t − ct ; e i φ t i = e − ct h
1; e i φ t i = 0 . It remains to derive g ± ( t ). Let I = [ s , s ], ( s < s ) be an interval and set P I = P s − P s , φ I = φ s − φ s = ω | I | + γP I in line with (2.7). We observe that h e i φ I ; e − i φ I i = h e i φ I e − i φ I i − h e i φ I ih e − i φ I i = 1 − e c | I | e c | I | = 1 − e − γ | I | , h e i φ I ; e i φ I i = h e φ I i − h e i φ I i = e ω | I | e − (2 γ ) | I | / − e ω − γ / | I | = e ω | I | (cid:0) e − γ | I | − e − γ | I | (cid:1) = e c | I | (cid:0) e − γ | I | − (cid:1) . Now h Z ∗ t e i φ t ; e i φ t i = Z t h e i( φ t − φ s ) ; e i φ t i d s = Z t h e i φ s ih e i( φ t − φ s ) ; e i( φ t − φ s ) i d s = Z t e cs e c ( t − s ) (cid:0) e − γ ( t − s ) − (cid:1) d s and multiplication by e − ct followed by substitution τ := t − s gives − g + ( t ) = Z t e cτ (cid:0) e − γ τ − (cid:1) d τ . h e − i φ t Z t ; e i φ t i = Z t h e − i( φ t − φ s ) ; e i φ t i d s = Z t h e i φ s ih e − i( φ t − φ s ) ; e i( φ t − φ s ) i d s = Z t e cs (cid:0) − e − γ ( t − s ) (cid:1) d s ,g − ( t ) = Z t e − cτ (cid:0) − e − γ τ (cid:1) d τ . The proof of Prop. 8 calls for Brownian motion W = ( W t ) t ≥ in a way that is independentof the statement of the proposition itself. In fact, we shall use the Wiener-Itˆo-Segalisomorphism, denoted by ≡ , between F = F ( L ( R + )) and L (Ω , µ ), with µ the Wienermeasure on the Brownian path space Ω ∋ W . Any random variable ξ ( W ) on Ω definesa multiplication operator on that L -space, an example being ξ ( W ) = W t for some t ≥
0. The above map diagonalizes the process P = ( P t ) t ≥ , in the sense that P t ≡ W t .Moreover any such random variable, if square integrable w.r.t. µ , also naturally definesa vector ξ ∈ L (Ω , µ ). In that sense, | Ω i ≡
1, which is the constant function 1( W ) = 1on Ω. In particular h Ω | f ( P t ) | Ω i = Z f ( W t ) d µ ( W ) . Brownian motion is self-similar under diffusive scaling for any normalization of themean square displacement. The next lemma states that the diffusive scaling limit of theprocess ( Z t ) t ≥ is Brownian motion, suitably normalized. Lemma 14.
We have the convergence of processes on Ω ∋ W : ˜ Z ε,t ( W ) := √ ε T ∗ ε Z ε − t ( W ) T ε −→ − i γc B t , ( ε →
0) (5.1) in distribution, where B t is complex Brownian motion and T ε is the dilation seen in(2.4).Proof. We have Z t = 1 c (cid:0) e i φ t − (cid:1) − i γc Z t e i φ u d P u . (5.2)Indeed, by Itˆo’s lemma [27] we haved e i φ t = e i φ t (cid:18) i ω d t + i γ d P t − γ d t (cid:19) = e i φ t ( c d t + i γ d P t ) . (5.3)25ntegrating and solving for the first term on the r.h.s. yields (5.2). The first one on ther.h.s. of the latter is bounded in t (by 2 /ω ) and therefore vanishes in the diffusive scalinglimit. As for the integral it becomes under scaling √ ε T ∗ ε (cid:18)Z ε − t e i φ u d P u (cid:19) T ε d = Z t e i ψ s d W s =: X t (5.4)with ψ s = ε − ωs + ε − / γW s . In fact under the substitution u =: ε − s , P u =: ε − / W s wehave φ u = ψ s and W s is real Brownian motion. In order to prove (5.1), it thus sufficesto show X t d → B t , ( ε → . We will do so using Prohorov’s theorem (e.g. [35], Thm. 13.5), calling for the standardtwo-step procedure of proving:1. The finite-dimensional distributions converge.2. The family is tight on Wiener space.We begin with the first step. To this end we observe that d X t = e i ψ t d W t satisfiesd X t d X t = d t , (d X t ) = e ψ t d t , (5.5)which should be compared with d B t d B t = d t , (d B t ) = 0. The mixed product isd X t d W t = e i ψ t d t .Let next f = f ( z ) be any smooth function of z ∈ C . Then d f = ( ∂f ) d z + ( ∂f ) d z with ∂ = ∂/∂z , ∂ = ∂/∂z ; thusd f ( X t ) = ( ∂f ) d X t + ( ∂f ) d X t + 12 (cid:16) ∂∂f ) + (cid:0) ∂ f (cid:1) e ψ t + (cid:0) ∂ f (cid:1) e − ψ t (cid:17) d t . (5.6)We then fix times s i in 0 ≤ s ≤ · · · ≤ s n ≤ t and consider smooth functions f ( X t ) ≡ f ( X s , . . . , X s n , X t )of the values of the process X at finitely many times. By induction in n it suffices toshow: The convergence E [ f ( X t )] → E [ f ( B t )] , ( ε → , (5.7)holds true for t ≥ s n as soon as it does for t = s n . Actually, by repeating the argument,it suffices to do so for s n ≤ t ≤ s n + ∆ and small ∆ > f ). Moreoverand if later needed, it suffices to show (5.7) for f ( X t ) = e i( λX t + λX t ) , ( λ ∈ C ) , (5.8)26he expectation of which is the characteristic function of X t . By integrating (5.6) wehave E [ f ( X t )] | ts n = Z ts n E (cid:2)(cid:0) ∂∂f (cid:1) ( X s ) (cid:3) d s + 12 Z ts n E h(cid:0) ∂ f (cid:1) ( X s ) e ψ s + (cid:0) ∂ f (cid:1) ( X s ) e − ψ s i d s , (5.9)since only the terms containing d t contribute to the expectation. In passing we notethat if X s were replaced by B s the last integral would be absent, as noted below (5.5).We claim that for X s it vanishes as ε → (cid:12)(cid:12)(cid:12)(cid:12)Z tt E (cid:2) e ± ψ s g ( X s ) (cid:3) d s (cid:12)(cid:12)(cid:12)(cid:12) ≤ C √ ε (5.10)for 0 ≤ t − t ≤ C depending on the smooth function g = g ( z ). To see this weuse the integration by parts formula f g | tt = Z tt f d g + Z tt g d f + Z tt (d f )(d g ) (5.11)with (adapted) processes f = f s , g = g s and apply it to f s = e ψ s , g s = g ( X s ). Hered f is computed from (5.3) by the substitution ω → ωε − , γ → γε − / , and hence c → ˜ cε − with ˜ c = 2(i ω − γ ), i.e.d t f t = d e ψ t = e ψ t (˜ cε − d t + 2 i γε − / d W t ) . Moreover by (5.6) with g in place of f we haved t g t = ( ∂g t ) d X t + ( ∂g t ) d X t + 12 (cid:16) ∂∂g t ) + (cid:0) ∂ g t (cid:1) e ψ t + (cid:0) ∂ g t (cid:1) e − ψ t (cid:17) d t , (d t f t )(d t g t ) = 2 i γε − / (cid:0) ( ∂g t ) e ψ t +( ∂g t ) e i ψ t (cid:1) d t . We then take expectation values in (5.11). The l.h.s. is O ( ε ) and the three integrals onthe r.h.s. contribute at order ε , ε − , ε − / respectively. After multiplying by ε ˜ c − andusing | ˜ c − | ≤ (2 ω ) − we so obtain (cid:12)(cid:12)(cid:12)(cid:12)Z tt E [e ψ s g ( X s )] d s (cid:12)(cid:12)(cid:12)(cid:12) ≤ εω k g k ∞ + ε ω k D g k ∞ | t − t | + γ √ εω k Dg k ∞ | t − t | , where k D k g k ∞ := X | α | = k (cid:13)(cid:13) ( − ) α ∂ g (cid:13)(cid:13) ∞ . Eq. (5.10) follows. Applied to (5.9) we get (cid:12)(cid:12)(cid:12)(cid:12) E [ f ( X t )] − E [ f ( X s n )] − Z ts n E (cid:2)(cid:0) ∂∂f (cid:1) ( X s ) (cid:3) d s (cid:12)(cid:12)(cid:12)(cid:12) ≤ C √ ε (5.12)27or ∆ ≤
1. By the remark made there the quantity inside the modulus would vanishfor B s instead of X s . In order to prove (5.7) we may limit ourselves to (5.8), whence ∂∂f = −| λ | f . This prompts us to define the map M : C → C , h M h for C = C ([ s n , s n + ∆]) given by ( M h )( t ) = h ( s n ) − | λ | Z ts n h ( s ) d s , where h ( t ) = E [ f ( B t )]. Clearly M is a contraction k M h − M h ′ k ∞ ≤ | λ | ∆ k h − h ′ k ∞ , provided | λ | ∆ <
1. Its (unique) fixed point is h = M h , as remarked. Now (5.12)states for h ( t ) := E [ f ( X t )] that h ( t ) = ( M h )( t ) + ( h ( s n ) − h ( s n )) + O ( √ ε ) , where the middle term is o (1) by the hypothesis of (5.7). It thus implies k h − h k ∞ ≤ k M h − M h k ∞ + o (1) ≤ | λ | ∆ k h − h k ∞ + o (1)and so k h − h k ∞ → E (cid:2) | X t − X s | (cid:3) ≤ C | t − s | , (0 ≤ s ≤ t ) . We consider X t − X s as a process in t , to which we apply (5.9) for f ( z ) = | z | m ,( m = 1 , , . . . ). Observing that the second derivatives appearing there are all boundedby a constant times | z | m − , we obtaindd t E (cid:2) | X t − X s | m (cid:3) ≤ C m E (cid:2) | X t − X s | m − (cid:3) , and then, recursively, E (cid:2) | X t − X s | m (cid:3) ≤ C ′ m | t − s | m . (5.13)In preparation of the proof of Prop. 8 we rewrite (2.14) as X ,t = Q t + ˜ X ,t ( P ) with P = ( P s ) ≤ s ≤ t and hence (2.11) as U ∗ t Q t U t = F t ( P ) + Q t ,F t ( P ) = 2 γta ∗ a + X ∗ ,t ( P ) a + X ,t ( P ) a ∗ + ˜ X ,t ( P ) = F t ( P ) ∗ , (5.14)28here the dependence on P occurs through ( Z s ) ≤ s ≤ t , Z s = Z s ( P ). We moreover findit convenient to restate (2.30) for M t := U ∗ t Q t U t = 2 γt N t instead of N t . In view of M t → ε − t = 2 γε − t N t → ε − t that restatement is ε M t → ε − t → | Ω i γ | κ | Z t | B s | d s . (5.15)The proof itself will be carried out by means of three lemmas. The first one addresses theprecise meaning of the l.h.s. of (2.27) in the context of (5.15), yet foregoing scaling forthe time being. Specifically, we are going to construct a functional calculus associatedto the family of commuting operators ( M t ) ≤ t< ∞ .Let C I = C ( I, R ) be the Wiener space on I ⊂ R + and C ( C I ) the continuous functionson C I . Lemma 15.
For any | ψ i ∈ H there is a probability measure µ ψ on C R + and thus anoperator f ( M ) defined by h ψ, Ω | f ( M ) | ψ, Ω i := Z C I d µ ψ ( ω ) f ( ω ) ≡ E [ f ] , ( f ∈ C ( C R + )) . (5.16) The map f f ( M ) extends the case where f are functions of the (commuting) pro-cess at finitely many times, f ( M ) = f ( M t , . . . , M t n ) . (The latter are defined by thefunctional calculus.)Proof. By the Riesz-Markov theorem, the functional calculus for f ( M t , . . . , M t n ) de-fines a measure on R n ∼ = R I , I = { t , . . . , t n } , and in fact a consistent set of such indexedby finite subsets I ⊂ R + . By the Kolmogorov extension theorem this defines a measure µ ψ on some probability space that remains to be identified with C R + . We will (a) doso for | ψ i ∈ D with D ⊂ H a dense subspace; then (b) (5.16) defines the l.h.s. as abounded quadratic form in | ψ i ∈ D and hence f ( M ) as an operator on H . Finally (c),those operators f ( M ) define a (spectral) measure µ ψ on C R + for any | ψ i ∈ H , again bythe Riesz-Markov theorem.We are thus left with (a). That identification will be done for D = {| ψ i ∈ H| ( ψ, ( a ∗ a ) ψ ) < ∞} and by means of the Kolmogorov continuity theorem (e.g. [35], Thm. 5.1), by which itis enough to show E (cid:2) |M t − M s | (cid:3) ≤ C | t − s | , (0 ≤ s ≤ t ≤ t ) , for any t , uniformly in s, t . The constant C may depend on t . Using(( A + B ) ∗ ( A + B )) ≤ (cid:0) ( A ∗ A ) + B ∗ AA ∗ B + A ∗ BB ∗ A + ( B ∗ B ) (cid:1) (5.17)29n relation with (5.14) it becomes enough to establish the required bound for E [ T ] := h ψ, Ω | T | ψ, Ω i with T any of the following operators: (i) (∆ Q ) , (ii) (∆ F )(∆ Q ) (∆ F ),(iii) (∆ Q )(∆ F ) (∆ Q ), (iv) (∆ F ) , where∆ Q = Q t − Q s , ∆ F = F t ( P ) − F s ( P ) . Let K = i[∆ Q, ∆ F ] = i[ Q t − Q s , F t ( P )] . (5.18)Then (ii) may be replaced by K ∗ K , besides of (iii). As for the latter, it may be replacedby (∆ P )(∆ F ) (∆ P ), because of A [ s,t ] | Ω i = 0 with 2 A [ s,t ] = ∆ Q + i ∆ P . Moreover,because of commuting factors, (∆ P )(∆ F ) (∆ P ) ≤ ((∆ P ) + (∆ F ) ) /
2, and (∆ P ) hasthe same expectation as (i). We may thus update the above list of operators to: (∆ Q ) ,(∆ F ) , K ∗ K . The first one is computed easily by Wick’s lemma: E [(∆ Q ) ] = h Ω | (∆ Q ) | Ω i = 3 h Ω | (∆ Q ) | Ω i = 3(∆ t ) . Applying (5.17) to (5.14) yields(∆ F ) ≤ C (cid:0) (∆ t ) (( a ∗ a ) + 1) + | ∆ X | (( a ∗ a ) + 1) + (∆ ˜ X ) (cid:1) with | ∆ X | = | C | (cid:12)(cid:12)(cid:12)(cid:12)Z ts Z τ d τ (cid:12)(cid:12)(cid:12)(cid:12) ≤ | C | Z ts ( Z ∗ τ Z τ ) d τ · (∆ t ) , (∆ ˜ X ) = C (cid:18)Z ts Z ∗ τ Z τ d τ (cid:19) ≤ C Z ts ( Z ∗ τ Z τ ) d τ · (∆ t ) , ( C = 2 i γωα , C = 2 γω | α | , cf. (2.13, 2.14)). We then observe that E (cid:2) ( Z ∗ t Z t ) (cid:3) ≤ C ( t + 1) , E (cid:2) ( Z ∗ t Z t ) (cid:3) ≤ C ( t + 1) , (5.19)which can be seen as follows: In (5.2), i.e. Z t ≡ Z t + X t , we have | Z t | ≤ C and themoments of X t are estimated in (5.13) with s = 0. We conclude E [(∆ F ) ] ≤ C ( t + 1)(∆ t ) . Finally, we come to K ∗ K . The drifted fieldˆ P λ,s := P λ − λs appears in the identity P s e − i λQ t = e − i λQ t ˆ P λ,s , (0 ≤ s ≤ t ) , (5.20)30hich is easily verified on the basis of (2.2) and in turn implies f ( P ) e − i λQ t = e − i λQ t f ( ˆ P λ )for functions f of ( P s ) s ≤ t . In particulari[ Q t , f ( P )] = dd λ f ( ˆ P λ ) (cid:12)(cid:12)(cid:12) λ =0 . We observe from (2.7) that φ t ( ˆ P λ ) = ( ω − γλ ) t + γP t , whencei[ Q t , Z t ( P )] = − γ Z t s e i φ s d s . The commutator (5.18) is computed likewise, except that the field P τ drifts only for s ≤ τ ≤ t , i.e. i[ Q t − Q s , f ( P )] = δf ( P )with e.g. δZ τ = − γ Z τs ( ν − s ) e i φ ν d ν . Clearly, | δZ τ | ≤ C (∆ t ) . We then get K = δF = δX ∗ ( P ) a + δX ( P ) a ∗ + δ ˜ X ( P )with δX = C Z ts δZ τ d τ , δ ˜ X = C Z ts ( Z ∗ τ δZ τ + ( δZ τ ) ∗ Z τ ) d τ , being estimated as | δX | ≤ C (∆ t ) , | δ ˜ X | ≤ C Z ts | Z τ | d τ · (∆ t ) , (cid:0) δ ˜ X (cid:1) ≤ C Z ts Z ∗ τ Z τ d τ · (∆ t ) . We conclude by (5.19) that K ∗ K ≤ C | δX | ( a ∗ a + 1) + (cid:0) δ ˜ X (cid:1) , E [ K ∗ K ] ≤ C ( t + 1)(∆ t ) . The next lemma essentially computes functions of the observable U ∗ t Q t U t despitethat its two terms on the r.h.s. of (5.14) do not commute.31 emma 16. Let ˆ P λ be the drifted field ˆ P λ,s = P s − λs . Then e i λU ∗ t Q t U t = G λ,t ( P ) e i λF t ( P ) e i λQ t (5.21) where G ,t ( P ) = 1 , − i dd λ G λ,t ( P ) = ( F t ( ˆ P λ ) − F t ( P )) G λ,t ( P ) . The lemma will be used in the scaling regime where the two terms just mentionedcommute to leading order. The drift will be small, making ˆ P λ close to P , and G λ,t ( P )to 1. Proof.
By the identity (5.20) we have − i dd λ (cid:0) e i λU ∗ t Q t U t e − i λQ t e − i λF t ( P ) G λ,t ( P ) ∗ (cid:1) =e i λU ∗ t Q t U t (cid:0)(cid:0) F t ( P ) e − i λQ t − e − i λQ t F t ( P ) (cid:1) e − i λF t ( P ) − e − i λQ t e − i λF t ( P ) ( F t ( ˆ P λ ) − F t ( P )) (cid:1) G λ,t ( P ) ∗ = 0 . Lemma 17. i) Let N ε,t = ε T ∗ ε N t T ε | t → ε − t , ˜ N ε,t = ω | α | | ˜ Z ε,t | , (5.22) where ˜ Z ε,t is defined in (5.1). Then, for any t ≥ , f ( N ε,t ) − f ( ˜ N ε,t ) → , ( ε → )in the sense of strong convergence of operators on H ⊗ F with
F ≡ L (Ω , µ ) , for anycontinuous bounded function f on R .ii) Let M ε,t = ε T ∗ ε M t T ε | t → ε − t , ˜ M ε,t = 2 γω | α | Z t | ˜ Z ε,s | d s . (5.23) Then f ( M ε,t ) − f ( ˜ M ε,t ) → in the same sense as in (i) and likewise for functions f of the processes at finitely many times.Proof. i) In analogy with (5.14) we rewrite (2.19, 2.10) as N t = ( a + i ωαZ t ( P )) ∗ ( a +i ωαZ t ( P )); we also recall (2.4), by which we have T ∗ ε Q ε − t T ε = ε − / Q t , T ∗ ε P ε − t T ε = ε − / P t , T ∗ ε Z ε − t T ε = ε − / ˜ Z ε,t , (5.24)and thus N ε,t = ε T ∗ ε N t T ε | t → ε − t = εa ∗ a + i ωαε / ˜ Z ε,t a − i ωαε / ˜ Z ε,s a ∗ + ω | α | | ˜ Z ε,t | .
32t suffices to prove the convergence for exponentials f ( x ) = e i λx , ( λ ∈ R ). The expo-nential of the first three terms tends to 1 strongly because of Lm. 14; that of the fourthterm is seen in (5.22).ii) Besides of (5.24) we recall (2.13, 2.14), by which we have T ∗ ε Y ,t T ε | t → ε − t = ε − / Z t ˜ Z ε,s d s , T ∗ ε Y ,t T ε | t → ε − t = ε − Z t | ˜ Z ε,s | d s . (5.25)It suffices to prove the convergence for exponentials f ( x ) = e i λx , ( λ ∈ R ) with x = M ε,t or, in case of many times, with P ni =1 λ i x i in place of λx . But actually it suffices todo so for the process at a single time, because the exponential is multiplicative andstrong convergence is inherited under multiplication. By (5.21, 2.19) with replacement λ → ˜ λ = λε we obtaine i λε T ∗ ε M t T ε = T ∗ ε G ˜ λ,t ( P ) T ε e i λT ∗ ε ( ε F t ( P )) T ε e i λT ∗ ε ( ε Q t ) T ε . Upon making the substitution t → ε − t dictated by (5.23) the third factor tends stronglyto 1, since ε T ∗ ε Q t T ε | t → ε − t = ε / Q t . (5.26)As for the middle factor,e i λT ∗ ε ( ε F t ( P )) T ε (cid:12)(cid:12)(cid:12) t → ε − t − e i λ · γω | α | R t | ˜ Z ε,s | d s s → . (5.27)In fact, by (5.14) ε F t ( P )2 γ = ε a ∗ at + i ωαε Y ,t ( P ) a − i ωαε Y ,t ( P ) a ∗ + ω | α | ε Y ,t ( P ) (5.28)and hence by (5.25) T ∗ ε ε F t ( P )2 γ T ε (cid:12)(cid:12)(cid:12)(cid:12) t → ε − t = εa ∗ at + i ωαε / Z t ˜ Z ε,s d sa − i ωαε / Z t ˜ Z ε,s d sa ∗ + ω | α | Z t | ˜ Z ε,s | d s . The exponential of the first three terms tends to 1 strongly because of Lm. 14; thatof the fourth term is seen in (5.27). It remains to show T ∗ ε G ˜ λ,t ( P ) T ε (cid:12)(cid:12) t → ε − t s →
1. Thegenerator of G ˜ λ,t ( P ) w.r.t. λ is ε ( F t ( ˆ P ˜ λ ) − F t ( P )) . By comparison with (5.28) we are led to discuss Z t ( ˆ P ˜ λ ) − Z t ( P ) . (5.29)33ince the drift is − λs = − λε s , the first term amounts to the second up to a shiftof γP s by − λγε s in (2.7), or equivalently of ω by − λγε . Finally the above rescalingof G ˜ λ,t ( P ) calls for the difference (5.29) to vanish when its two terms are rescaled asseen in (5.1). It does in probability for ε →
0, as can be seen by the integration byparts formula (5.2) and by applying stochastic dominated convergence to (5.4). In fact − λγε · ε − → Proof of Prop. 8.
The convergence (2.26) claimed in (2.29) is an immediate consequenceof Lm. 14 and 17 together with h Ω | f ( T ∗ ε XT ε ) | Ω i = h Ω | f ( X ) | Ω i by T ε | Ω i = | Ω i . Theconvergence (2.27) for functions f ∈ C ( C I ) claimed in (2.30) and again in (5.15) followson the same grounds for functions f depending on the processes at finitely many times.For the general case, tightness of M ε,t , cf. (5.23), has to be shown: We observe that˜ Z ε,t d = εZ ε − t , as defined in (5.1), obeys the same moments and tightness bounds asits unscaled counterpart Z t . This is so because in (5.2) the bound on the first termimproves by a factor ε and the scaling of the second term, X t , was already incorporatedin bounds like (5.13). Moreover, all terms in (5.26, 5.28) are as in (5.14), except for Z t replaced by ˜ Z ε,t and for additional prefactors ε n ( n = 3 / , , / , A Appendix
We shall derive (1.4, 1.5). The commutation relation [ a, a ∗ ] = 1 implies i[ H , a ] = − i ωa and thus e i H t a e − i H t = a e − i ωt . Since the relation remains true upon replacing a by a − α , and a ∗ accordingly, we also havee i Ht a e − i Ht = α + ( a − α ) e − i ωt . By (1.1) that expression has to be multiplied from the left by its adjoint and then timeaveraged in order to obtain M T . As a result M ∞ = | α | + ( a ∗ − α )( a − α ) , because terms ∼ e ± i ωt do not contribute to the limit. This proves (1.4), which in turnimplies h n | M ∞ | n i = h n | ( N + 2 | α | ) | n i + | α | h n | a ∗ a + aa ∗ | n i , because monomials ( a ∗ ) l a m with l = m have vanishing expectation. The first term onthe r.h.s. equals h n | M ∞ | n i and (1.5) follows. Acknowledgment.
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