Quantum dissipative systems from theory of continuous measurements

Abstract

We apply the restricted-path-integral (RPI) theory of non-minimally disturbing continuous measurements for correct description of frictional Brownian motion. The resulting master equation is automatically of the Lindblad form, so that the difficulties typical of other approaches do not exist. In the special case of harmonic oscillator the known familiar master equation describing its frictionally driven Brownian motion is obtained. A thermal reservoir as a measuring environment is considered.

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arXiv:quant-ph/0212104v1 18 Dec 2002
Quantum dissipative systems from theory
of continuous measurements
Michael B. Mensky
P.N.Lebedev Physics Institute, 117924 Moscow, Russia
Stig Stenholm
Royal Institute of Technology (KTH), Stockholm
Abstract
We apply the restricted-path-integral (RPI) theory of non-minimally
disturbing continuous measurements for correct description of fric-
tional Brownian motion. The resulting master equation is automat-
ically of the Lindblad form, so that the difficulties typical of other
approaches do not exist. In the special case of harmonic oscillator
the known familiar master equation describing its frictionally driven
Brownian motion is obtained. A thermal reservoir as a measuring
environment is considered.
1Introduction
Irreversible behavior is an every day phenomenon in physics, even if the
basic theories have simple properties under time reversal. It has long been
realized, that in certain limits, an incoherent environment will provide a
reservoir for relaxation processes, which in thermal equilibrium will enforce
the proper distribution functions. More recently, it has been realized that also
observations by measurements introduce decoherence, which from the point
of view of the system evolution is equivalent to environmental dephasing.
A comprehensive summary of the recent progress in this field is found in
Ref. [1].
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In the case of a continuous but weak observation, we can design quantum
probability amplitudes conditioned on the continuously observed variable.
Such a system can be formulated in a natural way by putting restrictions
on the path integral providing the quantum propagator of the system. Such
restricted path integrals (RPI) have been found to clarify many questions
related to the continuous monitoring of a selected variable [2]. In particular,
it can incorporate the unavoidable disturbing effect of the observation back
on the system observed. This is termed a minimally disturbing measurement.
Non-minimally disturbing continuous measurements have also been suggested
[2]. In this paper we pursue the consequences of a simple model for such a
measurement.
The RPI approach is basically selective. This means that the measure-
ment readout, or history of observations, is taken into account so that the
dynamics of the measured system conditioned by this history is presented
by a state vector. It is possible to connect the path integral formulation to
the more common master equation approach [2]. This is done by going over
to the non-selective description of the continuous measurement. If we take
the time evolution conditioned by the given measurement readout (history
of observations), express it in terms of a density matrix and then perform
a functional summation over all possible observed histories, we obtain an
ensemble description comprising all possible histories of observations. The
resulting total density matrix obeys a master equation.
The approach to environmental influences making use of master equations
goes back a long time, but the recent interest has been kindled by the works
of Caldeira and Leggett, Zurek, Unruh and Zurek [3]. In many connections,
especially those related to quantum optics, these results are satisfactory, but
when applied to frictional Brownian motion they are found to be flawed [4];
see also the discussions in [5]. The reason is that they are not of the Lindblad
form, and hence they cannot preserve the positivity of the density matrix. It
is important that this is the case even if the fluctuation-dissipation theorem
is fulfilled. Some authors claim that this is of little consequence, but it does
signal a possible danger in applications. The point is that even correctly
derived master equations fulfilling the fluctuation-dissipation theorem, may
not be of the Lindblad form [6]. A discussion of the situation is found in
Ref. [7].
Here we apply the RPI theory of continuous measurements to obtain
correct description of frictional Brownian motion. We formulate the RPI de-
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scription for a non-minimally disturbing continuous observation. Obtaining
the phenomenological description of the measurement (including its back-
action on the measured system), we go over to the non-selective description
and investigate the resulting master equation. It turns out to be of Lindblad
form for any choice of the parameters of the measurement.
Then we derive the equations of motion for the first and second moments
which turn out to nearly coincide with the expected Ornstein-Uhlenbeck
equations. The main correction is a position diffusion coefficient which may
be derived from the quantum uncertainty relation. The conditions are found
when this term may be omitted. In case of thermal equilibrium of the mea-
sured system with thermal reservoir, the strength of the continuous mea-
surement performed by the reservoir is shown to depend on the reservoir’s
temperature. If, however, the interaction between the system and reservoir
is weak enough, the equilibrium turns out to be impossible, and the relation
between the strength of the measurement and the temperature is violated.
The paper is organized in the following way. In Sect. 2 we formulate the
RPI description of a (non-minimally disturbing) continuous measurement,
derive the corresponding master equation and show that it automatically
turns out to be of the Lindblad type. In Sect. 3 we restrict our treatment
to the familiar case of a harmonic oscillator. We find that continuously
monitoring the momentum of the oscillator (by its environment) leads to
the familiar form of the master equation [7] describing the frictionally driven
Brownian motion. This gives the Ornstein-Uhlenbeck-type equations for the
first and second moments. In Sect. 4 the equilibrium of the measured system
with the (measuring) reservoir having a definite temperature is discussed.
Finally Sect. 5 summarizes and concludes our paper.
2Non-minimally disturbing measurement
The restricted path integral (RPI) for the monitoring of a quantum observable
A(q,p) with the observed history a(t) may be formally written [2] as an
ordinary Feynman path integral (in the phase-space representation) with the
weight functional of the form
w0[a] = exp
??
dt
?
−κ(A(q,p) − a(t))2??
(1)
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included in the itegrand. In case of non-minimally disturbing monitoring,
imaginary terms are also present in the exponent [2]. In the linear approxi-
mation this gives
w[a] = exp
??
dt
?
−κ(A(q,p) − a(t))2−i
¯ h(λa(t)B(q,p) + C(q,p))
??
(2)
Here κ and λ are the real parameters which characterize correspondingly
the strength of the measurement (monitoring) and non-minimal disturbance
produced by it. The real term in the exponent leads to the restriction of the
path integral that is analogous to von Neumann’s projecting in case of instan-
taneous measurements and is necessary for observations. Imaginary terms
are responsible for adding phase in the course of the measurement which
is not necessary for the observation, hence the terminology ‘non-minimally
disturbing measurement’.
With these definitions, the time evolution conditioned on the observation
of a(t) can be written as a RPI
U[a](q,q′,t) =
?
−κ(A(q,p) − a(t))2−i
d[p]
?q′
q
d[q]exp
??t
0dt
?i
¯ h(p ˙ q − H(q,p))
¯ h(λa(t)B(q,p) + C(q,p))
??
. (3)
The corresponding evolution operator U[a](t) allows one to express the density
matrix at an arbitrary time conditioned on the observation of a(t):
ˆ ρ[a](t) = U[a](t) ˆ ρ(0)U†
[a](t).(4)
The ensemble averaged or total density matrix is obtained from this by
carrying out the functional integration over all possible observations (mea-
surement readouts) [a]:
?
As a result we obtain
ˆ ρ(t) =d[a] ˆ ρ[a](t)(5)
ˆ ρ(t) =
?
d[q′,p′]
?
d[q′′,p′′] ρ0(q′,q′′)
× exp
??t
0dt
?i
¯ h(p′˙ q′− H′− C′) −i
λ2
8κ¯ h2(B′− B′′)2+iλ
¯ h(p′′˙ q′′− H′′− C′′)
−
κ
2(A′− A′′)2−
2¯ h(A′+ A′′)(B′′− B′)
?
(6)
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where the notations A′= A(q′,p′), B′′= B(q′′,p′′) etc. are used.
From this we can calculate the master equation for the density matrix by
taking the time derivative and noting that all primed operators go to the left
of ˆ ρ and the doubly primed ones to the right.
We still need to resolve the order ofˆA andˆB when taken at the same
time. Our physical interpretation suggests a solution: asˆB is the back action
of observingˆA , the latter should act before the former. We thus obtain the
equation
∂
∂tˆ ρ = −i
¯ h
?ˆH +ˆC, ˆ ρ
−iλ
2¯ h
?
?
−κ
2
?ˆA,
?ˆA, ˆ ρ
??
−
λ2
8κ¯ h2
?ˆB,
?ˆB, ˆ ρ
??
?
ˆB,
?ˆA, ˆ ρ
+
?
. (7)
Let us rewrite the master equation (7) by introducing the operator
ˆl =ˆA − i
λ
2κ¯ h
ˆB(8)
Solving forˆA andˆB and inserting into Eq. (7) we obtain
∂
∂tˆ ρ = −i
¯ h
?
ˆH +ˆC − iκ¯ h
4
?ˆl†2−ˆl2?
, ˆ ρ
?
−κ
2
?ˆl†ˆl ˆ ρ − 2ˆl ˆ ρˆl†+ ˆ ρˆlˆl†?
(9)
The resulting equation is of the Lindblad form. The original Hamiltonian of
the measured system is renormalized by the measurement procedure.
3 Special case of a harmonic oscillator
We shall now specialize to the case of a harmonic oscillator,
ˆH =
ˆP2
2
+1
2ω2ˆQ2. (10)
Let the momentum operator be monitored and the coordinate operator presents
non-minimal disturbance (this means that the momentum is shifted when be-
ing monitored):
ˆA =ˆP,
ˆB = ωˆQ(11)
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The weight factor (2) takes the form
w[a] = exp
??
dt
?
−κ(p − a(t))2−i
¯ h(λω a(t)q + C(q,p))
??
.(12)
Exponential of operatorˆQ is the displacement operator for observableˆP.
Therefore, the non-minimal disturbance (determined by the term propor-
tional to q in Eq. (12)) consists in this model in shifting the momentum (the
same observable which is measured).
The master equation now becomes (forˆC = 0)
∂
∂tˆ ρ = −i
¯ h
?ˆH, ˆ ρ
?
?
−κ
2
?ˆP,
?
?ˆP, ˆ ρ
.
??
−λ2ω2
8κ¯ h2
?ˆQ,
?ˆQ, ˆ ρ
??
−iλω
2¯ h
ˆQ,
?ˆP, ˆ ρ
+
?
(13)
This is the well known master equation for a Brownian motion [7]. The Brow-
nian motion of the oscillator is thus interpreted as the effect of monitoring
the momentum by a continuously acting environment (reservoir).
The equations for the first moments (mean values ofˆP andˆQ) following
from Eq. (13) are
?∂ˆP
∂t? = −ω2?ˆQ? − λω?ˆP?
?∂ˆQ
∂t? = ?ˆP? (14)
and the equations for the second moments
∂
∂t?ˆP2? = −ω2?ˆPˆQ +ˆQˆP? − 2λω?ˆP2? +λ2ω2
∂t?ˆPˆQ +ˆQˆP? = −λω?ˆPˆQ +ˆQˆP? + 2
∂
∂t?ˆQ2? = ?ˆPˆQ +ˆQˆP? + κ¯ h2.
4κ
∂
?
?ˆP2? − ω2?ˆQ2?
?
(15)
We can introduce a damping constant in Eqs.(14) and (15) by writing γ = λω.
Equations (15) are appropriate for an Ornstein-Uhlenbeck process except for
the diffusion term in position space. The same type of equations follow from
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the model introduced by Gallis [8]; see also [7]. Also Di´ osi [9] has pointed
out that such a term is necessary to obtain a Lindblad form of evolution.
In the classical limit, ¯ h → 0, Eq. (15) gives the steady-state solution
?ˆPˆQ +ˆQˆP? = 0,?ˆP2? = ω2?ˆQ2? =λω
8κ
(16)
which agrees with the virial theorem. These result in the expression for the
mean energy
?ˆH? =1
2?ˆP2+ ω2ˆQ2? =λω
8κ.
(17)
The additional diffusion term in the third of Eqs. (15) has got no simple
physical interpretation within a classical stochastic model. However, it is
easily interpreted in the scheme of measurements as a consequence of the
uncertainty relations. The fact that the term is proportional to ¯ h2signals
its quantum origin. Indeed, with time passing, momentum is measured with
better precision, and therefore the uncertainty of the coordinate becomes
larger. This may be shown to be expressed by just this term; see [10] and
[2, Sect. 4.3.4] for the dual situation when the coordinate is continuously
measured and grows more uncertain.
This diffusion term causes a spreading of the position variable according
to ∆?ˆQ2? ≈ κ¯ h2t. If this is required to be much smaller than the steady state
value in (16), we find that its effect is negligible if
t ≪
λ
8κ2ω¯ h2.(18)
4 Thermal reservoir as an environment
Let us assume that the environment of our (measured) oscillator is a thermal
bath at some temperature T and the oscillator is in equilibrium with the
environment. Then the density matrix of the oscillator corresponds to the
Boltzmann distribution in energies, which may be used for calculating mean
value of the number of energy level ˆ n = a†a and therefore mean value of
energy:
¯ n =
1
exp(¯ hω/kBT) − 1,?ˆH? = ¯ hω?ˆ n +1
2? =¯ hω
2
coth
¯ hω
2kBT
(19)
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This provides the fluctuation-dissipation theorem which relates the momen-
tum diffusion coefficient and friction coefficient in the form
D =γ¯ hω
2
coth
?
¯ hω
2kBT
?
.(20)
This gives in our case
λ = 4κ¯ hcoth
?
¯ hω
2kBT
?
.(21)
Therefore, the strength of non-minimal disturbance depends on temperature
of the measuring reservoir.
From Eq. (21) the inequality
λ > 4¯ hκ(22)
follows. We see from this inequality that for λ > 4¯ hκ, the oscillator may be
in thermal equilibrium with the reservoir. It can be shown that the regime of
measurement is in this case classical, quantum effects are negligible.
The question naturally arises as to what happens if the non-minimal
disturbance of the measurement is small enough (in comparison with the
strength of the measurement) so that λ < 4¯ hκ. In this case the measurement
is performed in quantum regime, but the situation seems to contradict the
inequality (22). However, we have derived this inequality under the assump-
tion that the distribution of the oscillator over energies is of the Boltzmann
form, i.e. the oscillator is in equilibrium with the thermal reservoir. If this
inequality is violated, thermal equilibrium of the oscillator with the thermal
reservoir is impossible.
All these conclusions are made in the framework of the certain model of
the continuous measurement. In this model the momentum is continuously
monitored and non-minimally disturbed (see the remark following Eq. (2)).
Because of symmetry between momentum and coordinate, the same is valid
also if the coordinate is measured and non-minimally disturbed. The conclu-
sions are however not justified if the oscillator is continuously measured but
according to another model.
It is also interesting to evaluate the energy shift caused by the obser-
vational procedure. Reading off this shift from Eq.(9) and making use of
Eq. (11) we obtain
− iκ¯ h
4
?ˆl†2−ˆl2?
=λω
4
?ˆPˆQ +ˆQˆP
?
. (23)
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Combining this with the Hamiltonian (10) we obtain
ˆHeff
=
1
2
1
2
?ˆP2+ ω2ˆQ2?
?
+λω
4
?ˆPˆQ +ˆQˆP
?
?
?
=
ˆP +λω
2
ˆQ
?2
+1
2
ω2−λ2ω2
4
ˆQ2. (24)
As λω is the friction coefficient, the new oscillational frequency is just the
correctly shifted one for a damped oscillator:
Ω =
?
?
?
?
?
ω2−λ2ω2
4
?
.(25)
Thus we find a consistent description of the ordinary Brownian motion pro-
cess in terms of continuous measurements.
5 Conclusion
Non-minimally disturbing continuous quantum measurements were shown to
lead to dissipation of the measured system. The resulting model of dissipation
is characterized by a Lindblad master equation and thus avoids the difficulties
arising in other approaches. Although only measuring a single observable has
been considered in the present paper, generalization on the case of measuring
two or more observables is straightforward.
ACKNOWLEDGEMENT
One of the authors (M.B.M.) acknowledges support from the Royal Swedish
Academy of Sciences and the KTH where the main part of this work was
completed. The work was supported in part by the Deutsche Forschungsge-
meinschaft and Russian Foundation of Basic Research.
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