DIFFUSION PROCESSES AND COHERENT STATES

DFPD 94/TH/30, May 1994 DIFFUSION PROCESSES arXiv:cond-mat/9407100v1 25 Jul 1994 AND COHERENT STATES Salvatore De Martino1 , Silvio De Siena2 and Giuseppe Vitiello3 Dipartimento di Fisica, Università di Salerno, and INFN, Sezione di Napoli, Gruppo collegato di Salerno, 84081 Baronissi, Italia Fabrizio Illuminati4 Dipartimento di Fisica “G. Galilei”, Università di Padova, and INFN, Sezione di Padova, Via F. Marzolo 8, 35131 Padova, Italia Abstract It is shown that uncertainty relations, as well as coherent and squeezed states, are structural properties of stochastic processes with Fokker-Planck dynamics. The quantum mechanical coherent and squeezed states are explicitly constructed via Nelson stochastic quantization. The method is applied to derive new minimum uncertainty states in time-dependent oscillator potentials. 1 Electronic Electronic 3 Electronic 4 Electronic 2 Mail: Mail: Mail: Mail: demartino@vaxsa.dia.unisa.it desiena@vaxsa.dia.unisa.it vitiello@vaxsa.dia.unisa.it illuminati@mvxpd5.pd.infn.it 1 The theory of stochastic processes is the natural framework to discuss systems with probabilistic dynamics. As such, it is by now a most powerful tool in the study of complex structures and behaviours in physics, chemistry, and biology [1]. It is then very interesting to fully understand the relationship between probabilistic and deterministic evolution, and much work has been done in this direction [2]. In quantum mechanics, it is well known that such relationship is expressed through uncertainty relations; in particular, the states of minimum uncertainty, the coherent [3] and squeezed states [4], are viewed as the “most classical” states, closest to a deterministic time evolution. In this letter our first observation is that uncertainty relations analogous to the quantum mechanical ones are a structural property of classical stochastic processes of the diffusion type. We derive the diffusion processes of minimum uncertainty (MUDPs), and we find that a special class among them is associated to Gaussian probability distributions with time-conserved covariance and expectation value with classical time evolution: we denote them as strictly coherent MUDPs. We will also consider the other Gaussian MUDPs, those with the expectation value and the covariance in general both time-dependent, and we will denote them as broadly coherent MUDPs. By exploiting Nelson’s stochastic quantization scheme [5], we will show that MUDPs provide the stochastic image of the standard quantum mechanical coherent and squeezed states, as well as of time-dependent squeezing. Our study is motivated by the possibility that the formalism of stochastic processes offers to treat on the same footing, in a unified mathematical language, the interplay between fluctuations of different nature, for instance quantum and thermal [6]. Our scheme holds for general diffusion processes. Preliminary results are presented in ref. [7]. Without loss of generality, we consider a one-dimensional random variable q. The associated diffusion process q(t) is governed by Ito’s stochastic differential equation dq(t) = v(+) (q(t), t)dt + ν 1/2 (q(t), t)dw(t) , dt > 0 , (1) where v(+) (q(t), t), is the forward drift, ν(q(t), t) is the positive-defined diffusion coefficient, and dw(t) is a Gaussian white noise, superimposed on the otherwise deterministic evolution, with expectation E(dw(t)) = 0 and covariance E(dw 2 (t)) = 2dt. The probability density ρ(x, t) associated to the 2 process satisfies the forward and backward Fokker-Planck equations. The forward and the backward drifts v(+) (x, t) and v(−) (x, t) are defined as ! q(t + ∆t) − q(t) | q(t) = x , v(+) (x, t) = lim+ E ∆t→0 ∆t (2) v(−) (x, t) = lim+ E ∆t→0 ! q(t) − q(t − ∆t) | q(t) = x . ∆t The operational meaning of the conditional expectations E(· | ·) in eqs.(2) is the following: v(+) is the mean slope of sample paths leaving point x at time t; v(−) is the mean slope of sample paths entering point x at time t. Therefore, if q(t) is a configurational process, for instance a particle of mass m performing a random motion on the real line according to eq.(1), the forward (backward) drift is the mean forward (backward) velocity field. The relation between v(+) and v(−) is the following (see also the paper by F. Guerra quoted in ref. [5]): v(−) (x, t) = v(+) (x, t) − 2∂x (ν(x, t)ρ(x, t)) . ρ(x, t) (3) It is convenient to introduce the osmotic velocity u(x, t) and the current velocity v(x, t) u(x, t) = v(+) (x, t) − v(−) (x, t) ∂x (ν(x, t)ρ(x, t)) = , 2 ρ(x, t) (4) v(+) (x, t) + v(−) (x, t) . 2 From the former definitions it is clear that u(x, t) “measures” the nondifferentiability of the random trajectories, thus controlling the degree of stochasticity. In the deterministic limit u vanishes and v(x, t) goes to the classical velocity v(t). Finally, we have the continuity equation v(x, t) = ∂t ρ(x, t) = −∂x (ρ(x, t)v(x, t)) . 3 (5) Eqs.(3)-(5) are all direct consequences of Fokker-Planck equation. It is straightforward to check that E(v(+) ) = E(v(−) ) = E(v), E(u) = 0, and that d ∀t . (6) E(v) = E(q) dt The absolute value of the expectation of the process q times the osmotic velocity u(q, t) reads |E(qu)| = E(ν(q, t)). Reminding that E(u) = 0 and by use of q Schwartz’s inequality it follows q that the root mean square deviations 2 2 ∆q = E(q − (E(q)) ) and ∆u = E(u2 − (E(q))2 ) satisfy the relation ∆q∆u ≥ E(ν). (7) Eq.(7) is the uncertainty relation for any stochastic process of the diffusion type defined by eq.(1). Equality in (7) defines the MUDPs. Saturation of Schwartz’s inequality yields u(x, t) = C(t)(x − E(q)), (8) with C(t) an arbitrary process-independent function. Eqs.(8) and (4) give " ρ(x, t) = N (t) exp C(t) Z x 0 # − E(q) − ln ν(x, t) , dx ν(x′ , t) ′ ′x (9) where N (t) denotes the normalization function. The density ρ given by eq.(9) is associated to a large variety of different processes. In this letter we consider two cases: constant ν and timedependent ν. In both cases the minimum uncertainty density is the Gaussian (x − E(q))2 ρ(x, t) = q , exp − 2(∆q)2 2π(∆q)2 " 1 # (10) with (∆q)2 = −ν(t)/C(t). The continuity equation, eq.(5), forces the current velocity v(x, t) to be of the form d d (11) v(x, t) = E(q) + (x − E(q)) ln ∆q . dt dt 4 The stochastic differential equation obeyed by the MUDPs is now completely determined by eqs.(10)-(11) and it reads dq(t) = [A(t) + B(t)q(t)] dt + ν 1/2 (t)dw(t) . (12) It is interesting to observe that eq.(12) has a drift part which is linear in the process. The above equation in fact defines the so-called linear processes in narrow sense [1]; when A(t) = 0 they are the time-dependent OrnsteinUhlenbeck processes. We can now divide the MUDPS in two general classes, the first one with ( ∆q = const. E(q) = a(t) , ∀t , (13a) and the second one with ( ∆q = F (t) , E(q) = b(t) , (13b) where a(t), b(t) and F (t) are arbitrary functions of time. In the case (13a), ∆q does not spread and E(q) follows a classical trajectory: d v(x, t) = E(q) = v(t) . (14) dt As a consequence, MUDPs of the form (10) obeying eqs.(13a)-(14) behave exactly as the quantum mechanical coherent states: we will denote them as strictly coherent MUDPs, as opposed to the ones obeying eq.(13b) which we call broadly coherent MUDPs. It is possible to discriminate on physical grounds the strictly coherent MUDPs from the other MUDPS by observing that eqs.(13a)-(14) are immediate consequences of the Ehrenfest condition v(E(q), t) = d E(q) , dt (15) so that the strictly coherent MUDPs can be viewed as the most deterministic semi-classical processes. We shall now consider a very important class of conservative diffusion processes (Nelson diffusions) which has been introduced by Nelson in the stochastic formulation of quantum mechanics [5]. 5 Nelson stochastic quantization associates to each single-particle quantum state Ψ = exp [R + h̄i S], the diffusion process q(t) with 1 ∂S(x, t) h̄ , ρ(x, t) = |Ψ(x, t)|2 , v(x, t) = , (16) 2m m ∂x where m is the mass of the particle, and R is related to ρ by the obvious relation ρ = exp[2R]. The Schrödinger equation with potential V (x, t) leads to the HamiltonJacobi-Madelung equation ν= ∂t S(x, t) + (∂x S(x, t))2 h̄2 ∂x2 ρ1/2 (x, t) − = −V (x, t) . 2m 2m ρ1/2 (x, t) (17) It is well known [8] that for Nelson diffusions the correspondences between the expectations of stochastic and operatorial observables are hq̂i = E(q) , hp̂i = mE(v) , ∆q̂ = ∆q , (18) (∆p̂)2 = m2 [(∆u)2 + (∆v)2 ] , h̄2 , (∆q̂) (∆p̂) ≥ m (∆q) (∆u) ≥ 4 2 2 2 2 2 where the hat denotes the operatorial observables, h·i denotes the expectation value in a given state Ψ and ∆(·) denotes the root mean square deviation. Minimum uncertainty Nelson diffusions (MUNDs) are MUDPs, and we correspondingly extend to them the denominations of strictly and broadly coherent MUNDs. Relation (17) can be regarded as an equation for the potential V (x, t). In the case of strictly coherent MUNDs (13a), we obtain V (x, t) = m 2 2 ω x + f (t)x + V0 (t) , 2 6 (19a) d2 f (t) E(q) = −ω 2 E(q) + , (19b) 2 dt m with f (t) and V0 (t) arbitrary time-dependent functions and the constant frequency h̄2 ω2 = . (20) 4m2 (∆q)4 For broadly coherent MUNDs (12b) we again obtain eqs.(19a)-(19b) but now with a frequency ω(t) depending on time through the spreading ∆q: ω 2 (t) = 1 d2 h̄2 − ∆q . 4m2 (∆q)4 ∆q dt2 (21) We now address the case of time-dependent ν. From the first of eqs. (16) this means letting either m or h̄, or both, be functions of time. We focus our attention on the case of time-dependent mass m(t) and constant h̄, leaving apart other more speculative situations. The Nelson scheme (16)-(17) still holds with m(t) replacing m. Considering the general case of broadly coherent MUNDs, and solving eq.(17) for V (x, t) we obtain 1 V (x, t) = m(t)ω 2 (t)x2 + f (t)x + V0 (t) , 2 ω 2 (t) = h̄2 1 d2 ṁ(t) d − ∆q − ∆q , 2 4 2 4m (t)(∆q) ∆q dt m(t)∆q dt (22) ṁ(t) d f (t) d2 E(q) = − E(q) − ω 2(t)E(q) + , 2 dt m(t) dt m(t) where f (t), V0 (t) are arbitrary functions of time and ṁ denotes the time derivative of m. The subcase of strictly coherent MUNDs for systems with a time-dependent mass is obviously recovered by putting ∆q = const. in eqs.(22). To illustrate the advantages of the formalism presented in this paper we observe that it is computationally convenient in many cases of practical interest. For instance, when the arbitrary functions f (t) and V0 (t) vanish, eqs.(19a)-(19b) are those of the classical harmonic oscillator, and the associated quantum states are the standard Glauber coherent states; when 7 f (t) = const. we have the Klauder-Sudarshan displaced oscillator coherent states; finally, when f (t) is time-dependent, we obtain the KlauderSudarshan driven oscillator coherent states [9]. We thus see that this formalism at once provides the full set of coherent states so widely exploited in physical applications. Moreover, we observe that eqs.(19a)-(19b) supplemented by eq.(21) describe the dynamics of the parametric oscillator with the associated feature of time-dependent squeezing. Finally, eqs.(22), supplemented with m(t) = m0 eΓ(t) , define the dynamics of the damped parametric oscillator, a result which sheds new light on the study of dissipative quantum mechanical systems ([10], [11]) also in view of the relation among squeezing and dissipation [12]. Besides the computational simplicity, we would like to stress that the stochastic formalism also provides new insights beyond the conventional operatorial framework: it is in fact most remarkable that coherent and squeezed states of different types, time-dependent oscillators and dissipative systems may all be described in terms of, and associated to, diffusion processes via Nelson stochastic quantization. We note that eqs.(16) and (17) not only yield the potential of the quantum state associated to the MUND, but also allow to compute explicitly the wave function. In the following we exhibit two cases; the first is the case of the familiar Glauber state, the second one is the case of the coherent state associated to a dissipative dynamics of the Caldirola-Kanai type. The wave function for the Glauber state which we indeed obtain from eqs.(10),(14) and (17), together with the maps (16) and (18), is (x − hq̂i)2 i ΨG (x, t) = + xhp̂i , 1 exp − 2 2 4(∆q̂) h̄ (2π(∆q̂) ) 4 " 1 # (23) where hq̂i(t) and hp̂i(t) = mv(t) = m(dhq̂i/dt) are the solutions of the classical equations of motion of the harmonic oscillator (see eqs.(19a)-(19b) with the choice f (t) = V0 (t) = 0); the wave funtion (23) describes, as it is well known, both the coherent and the squeezed states, since it is form invariant under the scale transformation q̂ → es q̂ and p̂ → e−s p̂. For the more intriguing case of the damped parametric oscillator, we 8 obtain, through the same procedure, the wave function (x − hq̂i)2 hq̂ p̂i − hq̂ihp̂i i exp − ΨD (x, t) = xhp̂i + + (x − hq̂i)2 1 2 4(∆q̂) h̄ 2(∆q̂)2 (2π(∆q̂)2 ) 4 1 ( " (24) where now hq̂i(t), and hp̂i(t) = m(t)v(t) = m(t)(dhq̂i/dt) are the solutions of the classical equation of motion of the damped parametric oscillator (see eqs.(22) where we have put for simplicity f (t) = V0 (t) = 0), and we have exploited the property, easy to verify, that m(∆q)2 d ln ∆q = m [E(qv) − E(q)E(v)] = hq̂ p̂i − hq̂ihp̂i . dt (25) Equation (25) yields the correspondence between the stochastic and the operatorial correlations among the quantum observables; by exploiting it, we have that h{Q̂, P̂ }i m [E(qv) − E(q)E(v)] = , (26a) 2 where Q̂ = q̂ − hq̂i , P̂ = p̂ − hp̂i . (26b) By exploiting the above relations (26a)-(26b) we can immediately verify that the states corresponding to eq.(20) are Heisenberg minimum uncertainty (m.u.) states, and those corresponding to eq.(21) are Schrödinger ones. Indeed, one can prove that the strictly coherent MUNDs are in one to one correspondence with the Heisenberg m.u. states, while broadly coherent MUNDs are all and only Schrödinger m.u. states. The m.u. states already known for the damped parametric oscillator are Schrödinger ones and there was in the literature a widespread belief that this physical system cannot have Heisenberg m.u. states [10]; by eqs.(24)(26) we see instead that it can also exhibit Heisenberg m.u. states, strictly coherent and harmonic oscillator-like. Thus, for the dissipative oscillator the stochastic approach allows to determine not only all the known Schrödinger m.u. states, but also a whole new set of Heisenberg m.u. states. In conclusion it seems to us interesting and stimulating the possibility to relate coherent states with diffusion processes and probabilistic methods, which provide the proper definition of functional integration techniques, and the appropriate framework for the study of dissipative systems. 9 #) , References [1] See e.g.: C. W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, 1985). [2] Stochastic Processes in Classical and Quantum Systems, edited by S. Albeverio, G. Casati, and D. Merlini (Lecture Notes in Physics no.262, Springer, Berlin, 1986). [3] See e.g.: J. R. Klauder and B. S. Skagerstam, Coherent States (World Scientific, Singapore, 1985). [4] D. Stoler, Phys. Rev. D1, 3217 (1970); H. P. Yuen, Phys. Rev. A13, 2226 (1976). [5] E. Nelson Dynamical Theories of Brownian Motion (Princeton University Press, Princeton, 1967); Quantum Fluctuations (Princeton University Press, Princeton, 1985); F. Guerra, Phys. Rep. 77, 263 (1981); G. 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