We show that uncertainty relations, as well as minimum uncertainty coherent and squeezed states, are structural properties for diffusion processes. Through Nelson stochastic quantization we derive the stochastic image of the quantum mechanical coherent and squeezed states

Demartino, S.

Desiena, S.

Illuminati, Fabrizo

Vitiello, Giuseppe

English

NASA Technical Reports Server

N95- 13938
MINIMUM UNCERTAINTY AND SQUEEZING IN DIFFUSION
PROCESSES AND STOCHASTIC QUANTIZATION
S. De Martino, S. De Siena, F. Illuminati, and G. Vitiello
Dipartimento di Fisiea, Universitd di Salerno, and
INFN Sezione di Napoli, 84081 Baronissi (Salerno), Italia
Abstract
We show that uncertainty relations, as well as minimum uncertainty coherent and squeezed
states, are structural properties for diffusion processes. Through Nelson stochastic quanti-
zation we derive the stochastic image of the quantum mechanical coherent and squeezed
states.
1 Introduction
It is well known that the theory of stochastic processes is a powerful tool in the study of the
interplay between probabilistic and deterministic evolution [1]. In quantum mechanics, and in
particular in quantum optics, such interplay is expressed by the states of minimum uncertainty,
the coherent [2] and squeezed states [3l, which are viewed as the "most classical" states.
In this paper we report on a recent derivation [4] of uncertainty relations for classical stochas-
tic processes of the diffusion type, and we determine the diffusion processes of minimum uncer-
tainty (MUDPs). We find that a special class among them is associated to Gaussian probability
distributions with time-conserved covariance and mean value with classical time evolution: we
refer to them as strictly coherent MUDPs. We will also identify Gaussian MUDPs with time-
dependent covariance and conserved expectation value: we refer to them as broadly coherent
MUDPs. By exploiting Nelson's stochastic quantization scheme [5], we will show that the
strictly coherent MUDPs provide the stochastic image of the standard quantum mechanical
coherent and squeezed coherent states, while the broadly coherent MUDPs are associated with
the phenomenon 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].
Beyond the case of diffusion processes, it is interesting to note that coherence and squeezing
have recently emerged in other contexts wider than quantum mechanics ([7], [8]).
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https://ntrs.nasa.gov/search.jsp?R=19950007525 2020-06-16T10:40:04+00:00Z
2 Uncertainty and Coherence in Diffusion Processes
In what follows, without lack of generality, we will consider a one-dimensional random variable
q. The associated diffusion process q(t) obeys
dq(t) = v(+)(q(t),t)dt + tfl/2(q(t),t)dw(t) , dt > 0, (1)
where v(+)(q(t),t), is the forward drift, t/(q(t),t) is the 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(dw2(t)) = 2dt. The forward and the backward drifts v(+)(x,t)
and v(_)(x,t) are defined as
txt--.0+ At '
(2)
vc_)(x,t) - lira E(q(t)-q(t-At) )at-_0+ _-_ I q(t) = x .
The definitions of v(+) and v(_) are not independent, but related by [5]
v(_)(x,t) = v(+)(x,t) - 20z(t,(x,t)p(x,t)) (3)
It is now convenient to define the osmotic velocity u(z, t) and the current velocity v(x, t)
u(x,t) = v(+)(x,t) - v(_)(x,t) = 8z(t/(x,t)o(x,t))
z '
(4)
v(x,t) = v(+)(x,t) + v(_)(x,t)
2
From the former definitions it is clear that uCx, t) "measures" the non-differentiability of the
random trajectories, controlling the degree of stochasticity. In the deterministic limit u vanishes,
and v(z,t) goes to the classical velocity v(t).
Finally, we have the continuity equation
Otp(x,t) = -O,(p(x,t)v(x,t)) . (5)
It is straightforward to check that E(v(+)) = E(v(_)) = E(v), and E(u) = 0. Further,
d
E(v) = _E(q) Vt. (6)
For the product qu, we have IE(qu(q,t))] = E(_,(q,t)). By Schwartz's inequality, the r.m.s.
deviations Aq and Au satisfy
AqAu >_ E(m(q,t)). (7)
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Inequality (7) is the uncertainty relation for any diffusion process. Equality in (7) defines
the MUDPs. Saturation of Schwartz's inequality yields u(x,t) = C(t)(x - E(q)), where C(t) is
an arbitrary function of time. Considering constant t, and time-dependent v, in both cases we
obtain a Gaussian minimum uncertainty density:
p(x,t) -- I exp
V/2_r(Aq) 2
(x-ECq)) 2:
2(Aq)2
, (8)
where 2(Aq) 2 -- -_(t)/C(t).
From eq. (5) we can determine the current velocity:
_E(q) + -_q -_Aq F(x,t), (o)
where
[:r, 2 - 2xE(q)
FCx, t) = x- E(q) + ECq)exp [ _-_
Eqs. (8)-(10) lead to the stochastic differential equation obeyed by any MUDP:
(10)
dq(t) = [A(t) + B(t)q(t)]dt + t/1/2(t)dw(t). (11)
It is interesting to observe that (11) defines the so-called linear processes in narrow sense.
When A(t) = 0 they are the time-dependent Ornstein-Uhlenbeck processes. These last ones
play a natural role in the theory of low noise systems [1], which are thus found to be related
with MUDPs.
The possible choices of E(q) and Aq in (8) are not independent: taking the expectation
value of in (9)-(10), and reminding (6) one has that either
Aq = const. Vt, (12)W(q) = j(t),
or
=
(13)
E(q) =0 Vt,
where j(t) and k(t) are arbitrary functions of time, and we have chosen for simplicity q(t =
0) = 0. Consider first case (12): Aq does not spread; also, it is immediate to verify that the
expectation value of the process E(q) follows a classical trajectory:
d
v(x,t) = -_ECq) = v(t), (14)
As a consequence, MUDPs of the form (8) obeying (12) and (14) are coherent in a sense
precisely analogous to that of quantum mechanical coherent states: we will refer to them as
strictly coherent MUDPs and to processes (8) obeying (13) as broadly coherent MUDPs.
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It is possible to discriminate on physical grounds the strictlycoherent MUDPs from the
broadly coherent ones by observing that (12) and (14) come as immediate consequence on
imposing the Ehrenfest condition
d
v(E(q), t)= -_E(q) , (15)
so that the strictlycoherent MUDPs can be viewed as the most deterministic semi-classical
processes.
Consider the scaletransformation x _ e-'x which authomatically impliesu _ e'u,where 8 is
the scale parameter. The Gaussian distribution(8) isform-invariant under thistransformation,
while the uncertainty product (7) isstrictlyinvariantwith Aq ---,e-'Aq and Au _ e'Au. We
willshow next that in the framework of Nelson stochastic quantization this transformation is
just the squeezing transformation of quantum mechanics. In this context, broadly coherent
MUDPs are of special interestwhen considering time-dependent squeezing.
3 Nelson Diffusions
A very important class of diffusion processes (Nelson diffusions) in physics has been introduced
by Nelson in his stochastic formulation of quantum mechanics [5].
To'each single-particle quantum state • = exp [R % _S], Nelson stochastic quantization
associates the diffusion process q(t) with
h 1 as(x,t)
= 2-_' p(_,t) = I_(_,t)i _, v(x,t) -- rn cgx ' (16)
where m is the mass of the particle. At the dynamical level, the Schroedinger equation with
potential V(x, t) is equivalent to the Hamilton-Jacobi-Madelung equation
(a.s(_,t)): as a:.p,/:(_,t) -v(_,t). (17)
o,s(.,t) + 2., - 2m p,/,(.,t) =
It is well known [9] that for Nelson diffusions the uncertainties Aq and Au are related to the
quantum mechanical uncertainties A@ and A_ of the position and momentum operators _ and
by
z_0= _q, CA_)_= -:[(_u)_+ (z_v)_],
(18)
(zX@)2(Aib): > (Aq)2(ZXrnu) 2 > --.
_ 4
Minimum uncertainty Nelson diffusions (MUNDs) are MUDPs. Correspondingly, we will
speak of strictly and broadly coherent MUNDs. By solving (17) for MUNDs we obtain V(x,t)
and the classical equations of motion for E(q). For strictly coherent MUNDs (12) we have
334
= + f(t) + Vo(t)2 tO 2 ___-
d 2
-d-_E(q) + c#2E2(q) = f(t) .
When the arbitrary constants f(t) and Vo(t) vanish, eqs.
_2
4m2(Aq) 4
(19)
(19) are those of the classical
harmonic oscillator and the associated quantum states are the standard Glauber coherent states;
when fit) =const. we have the Klauder-Sudarshan displaced oscillator coherent states; finally,
when f(t) is truly time-dependent, we obtain the Klauder-Sudarshan driven oscillator coherent
states [10].
For broadly coherent MUNDs we have instead
V(x,t) = mco2(t)z 2 + Vo(t), co'(t) = 0(t) + 292(t) 8m_(Aq)(, (20)
where a(t) = (Aq)-XdAq/dt must be such that co2(t) is positive. Eq. (20) describes the para-
metric oscillator potential, associated to the feature of time-dependent squeezing.
Furthermore, we can identify among MUNDs those corresponding either to Heisenberg or to
Schroedinger minimum uncertainty. The key relation, easy to prove, is
E(vq)- E(v)E(q) = < {0, P} >_ __ ib- < ib>_2 ; (_ = < _ >¢' /b-- m , (21)
where < {.,-} >¢ denotes the expectation of the anti-commutator in the state _, i.e. the
Schroedinger part of the quantum mechanical uncertainty.
Eqs. (18) and (21) show that the strictlycoherent MUNDs (19) exhaust the Heisenberg
minimum uncertainty states,while the broadly coherent MUNDs (20) form a subset of the
Schroedinger minimum uncertainty states.
Finally,we investigatethe possibilityof letting_,be time-dependent in the context of quan-
tum mechanics. From the firstof equations (16) thismeans lettingeither m or h be functions
of time.
This lattercase seems a bit speculative at thisstage. We thus fixour attention on the case
of time-dependent mass re(t) and constant h.
For such systems itcan be immediately verifiedthat the Nelson scheme (16)-(17)stillholds
with m(t) replacing m. Considering the most interestingcase of strictlycoherent MUNDs,
which means choosing C(t) ocv(t),and solving (17) we obtain
V(ac,t) - lr_(t)co2(t)x2 + f(t)x + Vo(t),
d2 en(t) d w2
-_E(q) + m(t) _tE(q) + (t)E(q)
_t 2
4rnz(t) (Ax) ''
_ f(t)
tact)'
(22)
335
where f(t), Vo(t) are arbitrary functions of time. Eqs. (22) supplemented with rn(t) = rn0e r(`)
define the dynamics of the damped parametric oscillator. The stochastic approach thus sheds
new light in a unified treatment on the study of quantum dissipative oscillators, for it allows to
derive for the expectation value the dynamical equation (22) that was so far unknown.
In conclusion, we have shown that the quantum mechanical concepts of uncertainty, coher-
ence, and squeezing can be imported in the probabilistic arena of diffusion processes. This
appears to be possible because of a subtle interplay between fluctuations, control, and optimiza-
tion. Conversely, we may also say that these features of quantum mechanics cart be traced back
and related to general properties of diffusion processes.
Work on this subject is in progress, and includes application of our scheme to polymer
dynamics and chemical reactions, uncertainty relations in field theory and dynamical systems
on lattices and manifolds.
References
[1] C. W. Gardiner, Handbook of Stochastic Methods, (Springer, Berlin, 1985).
[2] J. R. Klauder and B. S. Skagerstam, Coherent States, (World Scientific, Singapore, 1985).
[3] D. Stoler, Phys. Rev. D1, 3217 (1970}; H. P. Yuen, Phys. Rev. AI3, 2226 (1976).
[4] S. De Martino, S. De Siena, F. Illuminati, and G. Vitiello, Diffusion Processes and Coherent
States, Preprint, University of Salerno, June 1993.
[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. Parisi, Statistical Field Theory, (Addison-Wesley, New York,
1988).
[6] P. Ruggiero and M. Zannetti, Phys. Rev. Lett. 48, 963 (1982).
[7] E. C. G. Sudarshan, in these Proceedings.
[8] E. Celeghini, M. Rasetti, and G. Vitiello, Phys. Rev. Lett. 66, 2056 (1991); E. Celeghini,
S. De Martino, S. De Siena, M. Rasetti, and G. Vitiello, in these Proceedings.
[9] D. De Falco, S. De Martino, and S. De Siena, Phys. Rev. Lett. 49, I81 (1982); S. De
Martino and S. De Siena, Nuovo Cimento B79, 175 (1984).
[10] J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New
York, 1970).
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SECTION 6
FIELD THEORY AND GENERAL INTEREST
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Minimum uncertainty and squeezing in diffusion processes and stochastic quantization

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