lsu...measured quantum probability distribution functions for brownian motion g. w. ford department...

Measured quantum probability distribution functions for Brownian motion

G. W. FordDepartment of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA

R. F. O’ConnellDepartment of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803-4001, USA

�Received 1 April 2007; published 22 October 2007�

The quantum analog of the joint probability distributions describing a classical stochastic process is intro-duced. A prescription is given for constructing the quantum distribution associated with a sequence of mea-surements. For the case of quantum Brownian motion this prescription is illustrated with a number of explicitexamples. In particular, it is shown how the prescription can be extended in the form of a general formula forthe Wigner function of a Brownian particle entangled with a heat bath.

DOI: 10.1103/PhysRevA.76.042122 PACS number�s�: 03.65.Ta, 05.40.�a, 05.40.Jc

I. INTRODUCTION

The notion of a joint probability distribution is basic tothe description of classical stochastic processes. The purposehere is to describe the extension to the quantum regime, giv-ing the prescription for constructing the quantum joint prob-ability distribution associated with a sequence of measure-ments. The prescription is illustrated with a number ofexamples. These are an important part of this work, sincethey show how the prescription can be used to calculate avariety of quantities of practical interest.

The idea is simple: The system is initially in thermal equi-librium. A first measurement prepares a state that then devel-ops in time according to the underlying dynamics. Then asecond measurement prepares a new state and so on until thefinal measurement in the sequence.

In this connection it is necessary to consider the descrip-tion of quantum measurement. In its most naive form, foundin many textbooks, a quantum measurement of a dynamicalvariable is described as a projection of the system into aneigenstate of the variable, with no memory of the previousstate. In particular, for a variable with a continuous spectrum,such as the position of a Brownian particle, with no square-integrable eigenstate, this naive description is unsatisfactory.In an earlier publication of Ford and Lewis �1�, the descrip-tion of measurement as applied to quantum stochastic pro-cesses was addressed in some detail. Since that earlier pub-lication may not be accessible to all readers, in Sec. III asummary is given of the essential features of quantum mea-surement as they apply to the definition of the distributionfunctions. The reader will observe that the description giventhere involves no new theory of quantum measurement.Rather, a prescription is adopted based on that used by manyauthors, making practical calculations related to real experi-ments.

The plan of the paper is as follows. To begin, in a briefsection II the joint distribution functions of classical mechan-ics are described. The quantum joint distribution functions,which are a close analog of the classical quantities, are in-troduced in Sec. III. There the key result is the prescription�3.29� for the joint distribution function associated with nsuccessive measurements. For the case of quantum Brownianmotion this prescription can be readily evaluated to give ex-

plicit closed-form expressions. Therefore in Appendix A areview is given of those aspects of the theory of quantumBrownian motion that will be useful in the applications.There the key quantities needed for the later discussion arethe commutator and the mean-square displacement, given bythe general expressions �A7� and �A11�. Later in Appendix Athese expressions are evaluated explicitly for the Ohmic andsingle relaxation time models and the results compared withapproximate expressions obtained by master equation meth-ods. In Sec. IV the results of Appendix A are used to evalu-ate the characteristic function associated with the distributionfunction describing n successive measurements. There theimportant result is the expression �4.2� for the characteristicfunction, where it is seen explicitly how in classical mechan-ics, where the commutator vanishes, the effects of measure-ment can be separated from the underlying stochastic pro-cess. As an application of this result, an explicit expression isconstructed for the pair distribution function associated withwave packet spreading. In Sec. V we discuss the probabilitydistribution, which corresponds to what in elementary quan-tum mechanics is the “square of the wave function.” A gen-eral expression is obtained that is illustrated first with theexample of wave packet spreading and then with the ex-ample of a macroscopic quantum interference �Schrödingercat� state. In either case the discussion includes the case of afree particle as well as that of a particle in a harmonic well.Finally, in Sec. VI the Wigner function is introduced. Therethe key result is the simple formula �6.5� for the Wignercharacteristic function �the Fourier transform of the Wignerfunction�. The Wigner function corresponds to the phasespace distribution of classical mechanics, but is definitely nota probability distribution �it is seen explicitly in the examplesthat the Wigner function need not be positive� and cannot bethe result of direct quantum measurement. Nevertheless, theWigner function is useful for describing the results of themeasurement. In particular, the probability distribution in co-ordinate or momentum is obtained by integration over theconjugate variable.

II. DISTRIBUTION FUNCTIONS IN CLASSICALBROWNIAN MOTION

In the theory of classical Brownian motion a stochasticprocess is completely described by a hierarchy of probability

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distributions. Here the standard reference for the physicist isthe review article by Wang and Uhlenbeck �2�, reprinted inthe “Noisebook” �3�. For a stochastic variable y�t� one intro-duces

W�y1,t1�dy1 as probability of finding y�t1� in the interval

dy1 about y1,

W�y1,t1;y2,t2�dy1dy2 as probability of finding y�t1� in the

interval dy1 about y1,

and y�t2� in the interval dy2 about y2,

and so on. �2.1�

This hierarchy must satisfy the following more or less obvi-ous conditions:

�1� PositivityW�y1 , t1 ;y2 , t2 ; . . . ;yn , tn��0.

�2� SymmetryW�y1 , t1 ;y2 , t2 ; . . . ;yn , tn� is a symmetric function of the setof variables y1 , t1 ;y2 , t2 , . . . ,yn , tn.

�3� ConsistencyW�y1 , t1 ;y2 , t2 ; . . . ;yn , tn�

=�dyn+1W�y1 , t1 ;y2 , t2 ; . . . ;yn , tn ;yn+1 , tn+1�.Note that consistency corresponds to conservation of totalprobability,

1 =� dy1� dy2 ¯� dynW�y1,t1;y2,t2; . . . ;yn,tn� .

�2.2�

A theorem of Kolomogorov states that if the hierarchy satis-fies these conditions, there must exist an underlying classicalprocess �4�. That is, there must exist an ensemble of timetracks y�t� such that the W’s are the weighted fraction of timetracks that go through the appropriate intervals.

A natural question is, how is this description changed inthe quantum case? The answer will be seen in the followingsection, but for now one can say that the essential change isthat in the quantum case the symmetry condition no longerholds.

III. QUANTUM DISTRIBUTION FUNCTIONS

The system considered is that of a Brownian particlecoupled to a heat bath, a system with an infinite number ofdegrees of freedom. The quantum mechanical motion of thissystem is described by a microscopic Hamiltonian H andcorresponds to a unitary transformation of states in Hilbertspace,

��t� = U�t��0� , �3.1�

where ��t� is the state vector at time t and

U�t� = exp�− iHt/�� . �3.2�

However, one does not have precise knowledge of the initialstate. Instead, there is an initial density matrix. The density

matrix is defined as an operator ��t� in Hilbert space suchthat �� ,��t�� / �� ,�� is the relative probability at time tthat the system is in any given state � �5,6�. Note that con-sistent with this definition � must be a positive-definite Her-mitian operator. Its time development follows from Eq. �3.1�,

��t� = U�t��0�U†�t� . �3.3�

Before introducing the distribution functions, it is neces-sary to make some general remarks about measurement inquantum mechanics. By “measurement” here is meant “mea-surement with selection” �what Pauli in his famous Ziffer 9called “measurement of the second kind” �7�� so that mea-surement irreversibly changes the state of the system. Whendiscussed in general terms in textbooks, the accepted de-scription of this change of state is framed in terms of themeasurement of a discrete variable �Hermitian operator witha pure point spectrum�. Let B be such a variable, with b aneigenvalue and Pb its associated projection operator, so that

B = b

bPb. �3.4�

Then the effect of a measurement at time t1 whose result isthat the eigenvalue b is in the interval M is to instanta-neously transform the density matrix,

��t1� → PM��t1�PM , �3.5�

where PM is the projection operator associated with the in-terval,

PM = b�M

Pb. �3.6�

Here instantaneous means that the duration of the measure-ment is short compared with the natural periods of the sys-tem. The prescription �3.5� may be obtained from variousassumptions about the optimal character of the measurement�such that the disturbance of the state is somehow minimal�.See, e.g., Lüders �8�, Goldberger and Watson �9�, Furry �10�,or Davies and Lewis �11�.

But the prescription �3.5� is too restricted for our purpose;it represents too limited a class of measurements. We mustconsider measurements of limited precision and involvingoperators with a continuous spectrum, such as the position ofa Brownian particle. For guidance as to how to generalize theprescription, we look to such practical fields as the theory ofangular correlations �see, e.g., the article by Frauenfelder andSteffan in Ref. �12�, Sec. 3� or the theory of polarization inmultiple scattering �see, e.g., Wolfenstein �13�, Sec. 4�. Thereone associates a transition operator T �in the scattering casethis would be the Wigner T matrix� with a measurement witha given result �e.g., the observation of an emitted � ray in agiven direction� and represents the transformation of �brought about by the measurement at time t1 by

��t1� → T��t1�T†. �3.7�

Note that this is the most general transformation that pre-serves the positivity of the density matrix. In the special casewhere the transition operator is a projection operator, onerecovers the prescription �3.5�. To be consistent with the

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probabilistic interpretation of the density matrix, the transi-tion operator must satisfy the general requirement

T2 � max�

�T�,T��,��

� 1. �3.8�

The diagonal matrix elements of � formed with respect to acomplete set of states are interpreted as the probabilities offinding the system in the corresponding states. The sum ofthese probabilities over all states is the trace, and as a con-sequence of the requirement �3.8�, this is reduced by mea-surement. Thus,

Tr�T�T†� = Tr��T†T� � Tr��T2 � Tr�� . �3.9�

In fact, the ratio Tr��T†T� /Tr�� can be interpreted as theprobability that measurement will produce the given result.

This reduction of the sum over all states of the probabilityof finding the system in each state is not a unique feature ofquantum probability. In classical probability, where the prob-abilities after measurement would be interpreted as jointprobabilities of the result of the measurement and of findingthe system in the state, the sum of probabilities is also re-duced. The difference is that in the classical case none of theindividual probabilities will be increased, while in the quan-tum case some may increase.

As a simple example illustrating all this, consider a spin-1/2 system initially polarized in the +z direction. This can berepresented by the 22 density matrix

�0 = �1 0

0 0 . �3.10�

Note that the diagonal elements, which are, respectively, theprobability that the spin is in the +z and −z directions, add upto 1. This is in accordance with the convention that the traceof the density matrix is normalized to 1 prior to the firstmeasurement. Suppose that a measurement is made, for ex-ample, by a Stern-Gerlach apparatus, whose result is that thespin is in the +x direction. The transition matrix correspond-ing to this result is

T =1

2�1 1

1 1 , �3.11�

in this case a projection operator. The density matrix after themeasurement is

�+x = T�0T† = � 14

14

14

14

. �3.12�

The probability that the spin is in the +z direction has de-creased from 1 to 1

4 while the probability that the spin is inthe −z direction has increased from 0 to 1

4 . Nevertheless, thesum of the probabilities is 1

2 , less than the sum before themeasurement. One might ask, what happened to the prob-ability, and how is it that the probabilities after the measure-ment do not add up to 1? The answer is that there is anotherpossible result of the measurement: the spin is in the −xdirection. Repeating the above argument for this case, wefind for the density matrix after the measurement

�−x = � 14 − 1

4

− 14

14

. �3.13�

Again the probability that the spin after the measurement isin the ±z direction is 1

4 . Thus, in either case the sum of theprobabilities is 1

2 , the probability that the measurement pro-duces the given result. These probabilities add up to 1, sooverall probability is conserved.

With these remarks as a guide, consider the measurementof a dynamical variable y. Assume that y is a variable, suchas the position or velocity of the Brownian particle, with acontinuous spectrum over all real values. In this case one canassociate with the measurement a function f�y1� such that�here y1 is a c number�

�f�y − y1��, f�y − y1��,��

dy1 �3.14�

is the conditional probability that if the system is in state �,the instrument will read in the interval dy1 about y1. Here thechoice that only the difference y−y1 appears is made forconvenience, since by choosing f�y1� to be peaked the re-quirement that the measured value be somehow close to theactual value can be satisfied in an obvious way. If this con-ditional probability is to be normalized, one must require �fneed not be real�

�−

dy1�f�y1��2 = 1. �3.15�

An example to keep in mind is that of a “Gaussian instru-ment” �14�, for which

f�y1� =1

�2��12�1/4 exp�−

y12

4�12 , �3.16�

where �1 is the experimental width. The result of a measure-ment at time t1 in which the instrument reads in the intervalI1 is therefore to instantaneously transform �,

��t1� → �I1

dy1f�y − y1��t1�f�y − y1�†. �3.17�

As remarked above, an instantaneous measurement is to beunderstood as one whose duration is short compared with thenatural periods of the motion.

The distribution functions are now constructed as follows.To begin assume that at t=0 �or in the distant past� the sys-tem is in equilibrium at temperature T,

��0� = �0 �e−H/kT

Tr�e−H/kT�, �3.18�

so � is initially normalized. If y is to be measured at a latertime t1, the system must move in time from 0 to t1,

�0 → U�t1��0U†�t1� , �3.19�

according to Eq. �3.3�. At t1 a measurement is made,

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U�t1��0U†�t1� → �I1

dy1f�y − y1�U�t1��0U†�t1�f�y − y1�†,

�3.20�

according to Eq. �3.17�. Finally, the system moves from t1 tot,

�I1

dy1f�y − y1�U�t1��0U†�t1�f�y − y1�†

→ �I1

dy1U�t − t1�f�y − y1�U�t1��0U†�t1�

f�y − y1�†U†�t − t1� . �3.21�

This last expression can be simplified somewhat by introduc-ing the time-dependent variable �Heisenberg representation�

y�t1� = U†�t1�yU�t1� . �3.22�

Noting that U�t− t1�=U�t�U�−t1�=U�t�U†�t1�, the final den-sity matrix �3.21� can be written

�I1

dy1U�t�f„y�t1� − y1…�0f„y�t1� − y1…†U†�t� . �3.23�

The probability that the measurement of y is in I1 is the traceof this final density matrix. This same probability is inter-preted as the integral over the one-point distribution W�y1 , t1�over the interval I1, so that

�I1

dy1W�y1,t1� = �I1

dy1Tr�U�t�f„y�t1� − y1…

�0f„y�t1� − y1…†U†�t�� . �3.24�

Since I1 is arbitrary, one can identify

W�y1,t1� = Tr�U�t�f„y�t1� − y1…�0f„y�t1� − y1…†U†�t�� .

�3.25�

Finally, the trace is invariant under cyclic permutation of thefactors, so one can write

W�y1,t1� = Tr�f„y�t1� − y1…�0f„y�t1� − y1…†� . �3.26�

In the same way one can show that the two-point distri-bution is

W�y1,t1;y2,t2� = Tr�f„y�t2� − y2…f„y�t1� − y1…�0f„y�t1� − y1…†

f„y�t2� − y2…†� �3.27�

and, in general, using an obvious shorthand notation

W�1, . . . ,n� = Tr�f�n� ¯ f�1��0f�1�†¯ f�n�†� .

�3.28�

Finally, note that under cyclic permutation of the factors inthe trace, one can write the expression for the n-point distri-bution in the compact form

W�1, . . . ,n� = �f�1�†¯ f�n�†f�n� ¯ f�1�� , �3.29�

where the angular brackets indicate the thermal equilibriumexpectation. That is, for a given operator O,

�O� � Tr�O�0� . �3.30�

Expression �3.29� is the key result of this section.In connection with expression �3.29� for the joint prob-

ability distribution, it should first be emphasized that it istime ordered: 0 t1 t2 ¯ tn. Also we should point outthat in our shorthand notation the label applies to all theparameters of the measurement. Thus, for example, the label“j” represents not only the value yj and the time tj but alsothe instrumental parameters such as the width � j. The sym-metry property of the classical stochastic process refers tosymmetry under permutations of the labels. In other words,the probability distributions of a classical stochastic processare symmetric under the interchange of all the parameters ofthe measurements. An inspection of expression �3.29� for thequantum probability distribution shows that it does not havethat symmetry, because the f’s at different times do not ingeneral commute. However, the quantum distributions stillhave the consistency property, providing the integral is overthe results of the last measurement. This is sometimes calledmarginal consistency.

IV. CHARACTERISTIC FUNCTIONS

In this section we consider the evaluation of the distribu-tion functions when the dynamical variable y�t� is taken to bethe position operator x�t� for quantum Brownian motion, in-troduced in Appendix A. In evaluating these distributionfunctions, it is convenient in analogy with the classical caseto introduce the corresponding characteristic functions, de-fined by

��1, . . . ,n� = �−

dx1 ¯ �−

dxnW�1, . . . ,n�exp�ij=1

n

kjxj .

�4.1�

An important result is that for quantum Brownian motion thecharacteristic function can be cast in the form

��1, . . . ,n� = K�1, . . . ,n��exp�ij=1

n

kjx�tj� � , �4.2�

where the factor K�1, . . . ,n� is given by

K�1, . . . ,n� = �j=1

n �−

dxjf*�xj −

l=j+1

n

kl�x�tj�,x�tl��

2i

f�xj + l=j+1

n

kl�x�tj�,x�tl��

2i eikjxj . �4.3�

In this expression the sums are to be taken to be zero whenj=n. The importance of this result lies first of all in the factthat the effects of measurement are completely contained inthe factor K, in which the particle dynamics enters onlythrough the commutators. In the classical limit these commu-tators vanish and K becomes a simple numerical factor. In-deed in this classical limit one generally considers measure-ments of perfect precision, for which �f�xj��2→��xj� and thefactor K is unity. Expression �4.2� then becomes the familiar


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form for classical Brownian motion �15�. On the other hand,in the quantum case, where the commutators do not vanish, itis clear that the particle dynamics is inextricably linked withmeasurement and, as a consequence, the symmetry propertyof classical stochastic processes does not hold. A second rea-son for the importance of this result is that it is convenientfor calculation, as we shall illustrate in the example below.Before that, however, we give a brief derivation.

Consider first the case n=1. Using expression �3.29� inthe definition �4.1� of the characteristic function, we canwrite

��1� =��−

dx1f*„x1 − x�t1�…f„x1 − x�t1�…eik1x1� .

�4.4�

Making the change of variable x1→x1+x�t1�, we see that

��1� = K�1��eik1x�t1�� , �4.5�

where

K�1� = �−

dx1f*�x1�f�x1�eik1x1. �4.6�

Recall that the sums in expression �4.3� are to be taken to bezero when j=n.

Next consider

��1,2� =��−

dx1�−

dx2f�1�†f�2�†f�2�f�1�ei�k1x1+k2x2��= K�2��

−

dx1f*„x1 − x�t1�…ei�k1x1+k2x�t2��

f„x1 − x�t1�…� . �4.7�

Here K�2� is exactly of the form �4.6� but with the label “2”in place of “1.” Remember also that with our shorthand no-tation the label also represents the instrumental parameters,so the measurement function f changes with the label. Next,we apply the generalized Baker-Campbell-Hausdorf formula�B4� to write

ei�k1x1+k2x�t2��f„x1 − x�t1�… = f„x1 − x�t1�

+ ik2�x�t1�,x�t2��…ei�k1x1+k2x�t2��.

�4.8�

With this and making the change of variable x1→x1+x�t1�+k2�x�t1� ,x�t2�� /2i, we obtain

��1,2� = K�1,2��eik1x�t1�eik2x�t2�ek1k2�x�t1�,x�t2�� , �4.9�

where

K�1,2� = �−

dx1f*�x1 − k2�x�t1�,x�t2��

2i

f�x1 + k2�x�t1�,x�t2��

2i eik1x1K�2� , �4.10�

which is of the form �4.3� with n=2. As a final step, we usethe Baker-Campbell-Hausdorf formula �B3� to writeeik1x�t1�eik2x�t2�ek1k2�x�t1�,x�t2��=ei�k1x�t1�+k2x�t2�� and obtain theform �4.2� with n=2.

For the general case, the argument goes in the same way.Assuming the form �4.3� for smaller n, we write the defini-tion �4.1� in the form

��1, . . . ,n� = K�2, . . . ,n��−

dx1f*„x1 − x�t1�…

exp�i�k1x1 + l=2

n

klx�tl� � f„x1 − x�t1�…� .

�4.11�

Then we bring the exponential factor to the right, using thetheorem �4.9� as in Eq. �4.8�. Then, shifting the variable ofintegration and the using the Baker-Campbell-Hausdorf for-mula in the exponential factor, we get the form �4.3�.

A. Example: Wave packet spreading

Here we consider the case of two successive measure-ments, each with a measurement function of the form �3.16�,corresponding to a Gaussian slit. First, we consider a singlemeasurement with

f�x1� =1

�2��12�1/4 exp�−

x12

4�12 . �4.12�

With this we use the standard Gaussian integral �B1�, toevaluate the integral expression �4.6�. We find

K�1� = exp�−1

2�1

2k12 . �4.13�

Using the Gaussian property �B6� to evaluate the expectationin Eq. �4.5�, we find

�eik1x�t1�� = exp�−1

2�x2�k1

2 . �4.14�

With this, we find

��1� = exp�−1

2�2k1

2 , �4.15�

where we have introduced

�2 = �x2� + �12. �4.16�

Finally, we invert the definition �4.1� of the characteristicfunction to write


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W�1� = �−

dk1

2��1�e−ik1x1. �4.17�

This again is a standard Gaussian integral, and we obtain theresult

W�1� =1

�2��2exp�−

x12

2�2 . �4.18�

Thus the probability distribution associated with a singlemeasurement is a Gaussian whose variance is the sum of thatof the instrument and that of the underlying quantum state ofthe particle. We have presented the steps leading to this resultin detail since these are the steps that will be used repeatedlyin this and our later examples.

Consider now a pair of successive measurements, the firstwith measurement function of the form �4.12� and the secondof the same form but with the index “1” replaced by “2.”Using the standard Gaussian integral to evaluate the integralin expression �4.10� for K�1,2�, we find

K�1,2� = exp�−1

2�1

2k12 −

1

2��2

2 −�x�t1�,x�t2��2

4�12 k2

2� .

�4.19�

Then, using the Gaussian property, we see that

�ei�k1x�t1�+k2x�t2�� = exp�−1

2�x2��k1

2 + k22� − c�t2 − t1�k1k2 .

�4.20�

Here we have introduced the correlation �A9�. Putting thesetogether, using the expression �4.2� for the characteristicfunction, we can write

��1,2� = exp�−1

2��2k1

2 + 2��k1k2 + �2k22� , �4.21�

where �note the misprint in the Eq. �7.18� of Ref. �1��

�2 = �x2� + �12,

�2 = �x2� + �22 −

�x�t1�,x�t2��2

4�12 ,

�� = c�t2 − t1� . �4.22�

Note that �2 is the same quantity �4.16� that appears in thesingle-measurement function. The two-measurement distri-bution function is given by

W�1,2� = �−

dk1

2��

−

dk2

2��1,2�e−i�k1x1+k2x2�. �4.23�

With the form �4.21� of ��1,2� we can perform the integra-tion using the multidimensional form �B2� of the standardGaussian integral. The result is

W�1,2� =1

2��1 − �2exp�−

�2x12 − 2��x1x2 + �2x2

2

2�2�2�1 − �2� .

�4.24�

Here we remark first of all that the lack of symmetry ofthe quantum distribution is obvious: W�1,2��W�2,1�. Theexception is when the commutator vanishes. Note that thesymmetry, or lack of it, is with respect to interchange of thelabels; that is, one must interchange not only x1�x2 andt1� t2 but also �1��2. Another aspect of this asymmetry isthat it is possible to make the last measurement one of per-fect precision—that is, put �2=0.

A second remark is that the time dependence is onlythrough the time difference t2− t1. This is a general feature,independent of the form of the measurement function. Itarises from the time-translation invariance of the equilibriumstate.

Finally, we remark that for widely separated times �t2

− t1→ � the correlation and the commutator for the oscilla-tor vanish. Then we see that W�1,2�→W�1�W�2�. This is aspecial case of the cluster property of the quantum joint dis-tribution functions, a property they share with the classicalfunctions. Whenever the time between any two successivemeasurements is large, the quantum joint distribution func-tion factors into a product of distribution functions, the onecorresponding to the earlier times, the other to the latertimes. The exception would be the case of a free particle,since in that case there is no approach to an equilibriumvalue of �x2�.

V. PROBABILITY DISTRIBUTION

In elementary quantum mechanics one interprets the ab-solute square of the wave function as the probability distri-bution of the particle position. That is, P�x , t�= ��x , t��2,where Pdx is the probability of finding the particle in theinterval dx at time t. In our discussion this probability distri-bution becomes the conditional probability, specialized suchthat the second measurement is a perfect measurement. Thatis, the probability distribution is given by

P�x2,t2 − t1� � �W�1,2�W�1�

�2=0. �5.1�

Here we exhibit only the dependence on the final particleposition and the time difference. �Due to the time-translationinvariance of the equilibrium state, only the time differenceappears.� The picture we have is that the initial state is pre-pared by the first measurement, a measurement made on theequilibrium state, and that the second �perfect� measurementsamples the state. Of course, in the special case of a particlenot interacting with the bath and at temperature zero, thisreduces to the elementary quantum mechanics prescription.

If we use expression �3.29� for the joint probability dis-tributions, we see that we can write

P�x2,t2 − t1� =�f�1�†�„x2 − x�t2�…f�1��

�f�1�†f�1��. �5.2�

While we can use the method described in Sec. IV to calcu-late W�1� and W�1,2� and then put the results in the defini-tion �5.1�, it is just as well to evaluate the expression �5.2�directly. For this purpose we introduce the integral expres-sion for the delta function,


042122-6

��x� = �−

dP

2��eixP/�, �5.3�

with which we can write

P�x2,t2 − t1� = �−

dP

2��

�f�1�†e−ix�t2�P/�f�1��f�1�†f�1��

eix2P/�.

�5.4�

Now we use the formula �B4� to write

f�1�†e−ix�t2�P/�f�1� = f*„x�t1� − x1…f„x�t1� − x1 − GP…e−ix�t2�P/�,

�5.5�

where we have used relation �A7� to write the commutator interms of the Green function, G=G�t2− t1�. Next we again usethe integral expression �5.3� for the delta function to write

f*„x�t1� − x1…f„x�t1� − x1 − GP…

= �−

dx1�f*�x1� +GP

2 f�x1� −

GP

2

�−

dP�

2��ei�x1+x1�−x�t1�+GP/2�P�/�. �5.6�

Introducing this in Eq. �5.5� and using the Baker-Campbell-Hausdorf formula �B3� we can write

�f�1�†e−ix�t2�P/�f�1�� = �−


2 f�x1� −

GP

2

�−

dP�

2��ei�x1+x1��P�/�

�e−i�x�t1�P�+x�t2�P�/�� . �5.7�

The Gaussian property �B6� allows us to write

�e�−i�x�t1�P�+x�t2��P�/�� = exp�−�x2��P�2 + P2� + 2cPP�

2�2 ,

�5.8�

where c=c�t2− t1� is the correlation �A9�. Finally, with thisresult, the integral over P� in Eq. �5.7� is a standard Gaussianintegral �B1� and we obtain

�f�1�†e−ix�t2�P/�f�1��

= exp�−��x2�2 − c2�P2

2�x2��2 �

−


2 f�x1� −

GP

2

exp�− �x1 + x1��

2/2�x2� − i�c/�x2��x1 + x1��P/��2��x2�

.

�5.9�

Here we should recall that c=c�t2− t1� is the correlation �A9�and G=G�t2− t1� is the Green function �A5�. This is the keyresult of this section, valid for any form of the measurement

function f . Dividing this result by its value for P=0 we getthe integrand in the expression �5.4� for the probability dis-tribution. We next consider some examples corresponding todifferent choices of the measurement function.

A. Example: Wave packet spreading

Again, we consider wave packet spreading with an initialmeasurement corresponding to a single Gaussian slit, themeasurement function being that given in Eq. �4.12�. Thereis no need to evaluate the general expression �5.9� since wealready have the expressions �4.18� for W�1� and �4.24� forW�1,2�. Putting these in Eq. �5.1� we can write

P�x,t� =1

�2�w2�t�exp�−

�x − x�t��2

2w2�t� . �5.10�

This is a Gaussian distribution with center x�t� and variancew2�t�, where

x�t� =��

�x1 =

c�t��x2� + �1

2x1,

w2�t� = �2�1 − �2� = �x2� −�x�0�,x�t��2

4�12 −

c2�t��x2� + �1

2 .

�5.11�

Here we should again recall that c�t� is the correlation �A9�.As a first consideration, we note that P�x ,0� is the prob-

ability distribution for the particle distribution immediatelyafter the first measurement. Since c�0�= �x2� and the commu-tator vanishes at t=0, we see that the center and variance ofthe initial distribution are

x�0� =�x2�

�x2� + �12x1,

w2�0� =�x2��1

2

�x2� + �12 . �5.12�

As one can easily verify, this initial distribution correspondsto the product of a Gaussian distribution of variance �x2�centered at the origin with one of variance �1

2 centered at x1.That is, the initial distribution corresponds to the wavepacket formed when the equilibrium state of the oscillator ispassed through a Gaussian slit of width �1 centered at x1.

1. Free particle

The free particle coupled to the bath in the absence of theoscillator potential corresponds to the limit �x2�→. Thepoint here is simple: the oscillator force can be neglectednear the center, and the motion will be that of a free particle.Noting that c�t�= �x2�−s�t� /2, where s�t� is the mean-squaredisplacement and remains finite in the limit, we find that thecenter and variance �5.11� of the probability distribution be-come

x�t� = x1,


042122-7

w2�t� = �12 + s�t� −

�x�0�,x�t��2

4�12 . �5.13�

With the commutator expressed in terms of the Green func-tion, this expression for free particle wave packet spreadingcorresponds to that obtained using path integral methods byHakim and Ambegoakar �16�. For a free particle not inter-acting with the bath and at temperature zero, in which cases�t�=0 and �x�0� ,x�t��= i�t /m, this reduces to the well-known expression for wave packet spreading found in el-ementary quantum textbooks.

2. Displaced ground-state distribution

Another limit of interest is that in which �12→, while at

the same time x1→ such that x0= �x2�x1 /�12 is fixed. In this

limit, the center and variance �5.11� become

x�t� =c�t��x2�

x0,

w2�t� = �x2� . �5.14�

The probability distribution �5.12� therefore is that of a dis-placed equilibrium state. That is,

P�x,t� = Peq„x − x�t�… , �5.15�

where

Peq�x� =1

�2��x2�exp�−

x2

2�x2� �5.16�

is the probability distribution of the equilibrium state. Thus,the wave packet moves without spreading, the variance beingthat of the equilibrium state of the oscillator. The center isinitially at x0 and asymptotically approaches the origin as theequilibrium state is reached. We shall come back to discussthis result further in Sec. V when we discuss spreading of aninitial coherent state.

B. Example: Schrödinger cat state

Here we consider the case where the initial measurementforms two separated wave packets. The first measurementfunction then has the form

f�1� =exp�− �x1 − d/2�2/4�1

2� + exp�− �x1 + d/2�2/4�12�

�8��12�1 + e−d2/8�1

2�2�1/4

.

�5.17�

With this form of the measurement function the integrationin Eq. �5.9� involves only the standard Gaussian integral�B1�. Note that x1 is the position of the center of the instru-ment, which should be chosen to be zero if we wish the wavepacket pair to be symmetrically placed about the origin. Wethen find

� �f�1�†ei�x2−x�t2��P/�f�1��f�1�†f�1��

x1=0

=exp�− w2P2/2�2 + ix2P/��

1 + exp�−�x2�d2

8�12��x2� + �1

2��cos

Pd

2�+ exp�−

�x2�d2

8�12��x2� + �1

2� cosh

GPd

4�12 � , �5.18�

where w2=w2�t2− t1� is given in Eqs. �5.11� and d= d�t2− t1�with

d�t� =c�t�

�x2� + �12d . �5.19�

While there is no difficulty evaluating the integral expression�5.4� with this expression for the integrand, our interest willbe in the limits of a free particle or, for the oscillator, adisplaced ground-state pair, in which case it is simpler to firstevaluate the limits of the above expression and then evaluatethe integral.

1. Free particle

As in the above example of wave packet spreading, weobtain the case of a free particle coupled to the bath in theabsence of the oscillator potential by forming the limit �x2�→. Forming this limit of expression �5.18� and then puttingthe result into the expression �5.4� for the probability distri-bution and performing the integral with the standard Gauss-ian formula �B1� we find

P�x,t� =1

2�1 + e−d2/8�12�� exp�− �x − d/2�2/2w2�

�2�w2

+exp�− �x + d/2�2/2w2�

�2�w2

+ 2aexp�− �x2 + d2/4�/2w2�

�2�w2cos

�x�0�,x�t��xd

4i�12w2 ,

�5.20�

where now w2�t� is given by the free particle form �5.13� andwe have used relation �A7� to reintroduce the commutator. Inthis expression a�t� is the attenuation coefficient, given by

a�t� = exp�−s�t�d2

8�12w2�t� . �5.21�

This expression for the probability distribution is the same asthat derived in an earlier brief communication �17�.

This probability distribution is the sum of three contribu-tions, corresponding to the three terms within the braces. Thefirst two are probability distributions of the form �5.12� cor-responding to a pair of single slits positioned at ±d /2, whilethe third term �that involving the cosine� is an interference


042122-8

term. The attenuation coefficient is a measure of the size ofthe interference term and is defined as the ratio of the ampli-tude of the interference term to twice the geometric mean ofthe other two terms. The point here is perhaps best seen if welook first at the case of a particle without dissipation and atzero temperature. Then s�t�=0 and �x�0� ,x�t��= i�t /m. Theresulting probability distributions are shown in Fig. 1. ThereP�x ,0� is the initial probability distribution, which dependsonly on the initial measurement function and is therefore thesame whether or not there is dissipation. In this initial distri-bution the interference term corresponds to a miniscule peakat the origin, so small that it does not show in the plot. In thesame figure P�x , t� is the distribution at a time t such that thewidth of the individual wave packets has increased by a fac-tor of roughly 3, while P0�x , t� is the same distribution butwith the attenuation factor a�t� set equal to zero. We empha-size that at this later time the amplitude of the interferenceterm is of the order of that of the other terms, despite the factthat initially it is negligibly small. The difference betweenP�x , t�, where the attenuation factor is unity and the interfer-ence term is present, and P0�x , t�,where the interference termis absent, is what is called decoherence. Thus, the attenuationcoefficient corresponds to the traditional measure of coher-ence in terms of the relative amplitude of an observed inter-ference term �18�.

In the presence of dissipation the attenuation factor de-cays rapidly when the separation d of the wave packets islarge. Here by “large” we mean not only large comparedwith the slit width �1 but also large compared with the mean

de Broglie wavelength �=� /m��x2�. To obtain the short-time behavior in this case, we note that, as we have seen inEq. �A13�, for short times the mean-square displacements�t��x2�t2. Then, since for short times w2�t��1

2, we seethat, for short times,

a�t� � e−t2/2�d2, �5.22�

where �d, the decoherence time, is given by

�d =2�1

2

��x2�d. �5.23�

In the high-temperature case, where �x2�=kT /m, this is theresult for the decoherence time obtained previously �17�, butthe result holds equally well at zero temperature, where �x2�is given in Eq. �A29�. In either case, the decoherence time isvery short when the separation d of the pair of wave packetsis large.

2. Displaced ground-state pair

As in the above example of wave packet spreading, weobtain a relatively simple expression for the oscillator case inthe limit �1

2→ and d→ such that d0= �x2�d /�12 is fixed.

Forming this limit of the result �5.18� and then evaluating theexpression ��5.4�� for the probability distribution, we find

P�x,t� =1

2�1 + e−d02/8�x2��

�Peq�x −d

2 + Peq�x +

d

2

+ 2a�t�e−d2�t�/8�x2�Peq�x�cos�x�0�,x�t��xd0

4i�x2�2 � ,

�5.24�

where Peq is the equilibrium distribution �5.16� and now

d�t� =c�t��x2�

d0, �5.25�

while here the attenuation coefficient is given by

a�t� = exp�−d0

2

8�x2��1 −c2

�x2�2 +�x�0�,x�t��2

4�x2�2 � .

�5.26�

In connection with this result we remark first that initiallythe distribution is of the same form as in the free particlecase, with two Gaussian peaks at ±d0 /2 and a minisculecentral peak. The difference is in the motion: the two peaksdrift back and forth against each other without spreading,eventually arriving at the origin. The interference term there-fore has a different effect �sometimes called a “quantum car-pet” �19�� but we have nevertheless introduced the attenua-tion coefficient in the same way as the ratio of the coefficientof the cosine to twice the geometric mean of the first twoterms. For very short times, c�t��x2�− 1

2 �x2�t2 and�x�0� ,x�t�� i�t /m, so a�t� is of the same form �5.22� butnow with

�d =2�x2�

��x2� − �2/4m2�x2�d0

. �5.27�

Note that the uncertainty principle tells us that m2�x2��x2��2 /4, so the argument of the square root is always posi-tive.

VI. WIGNER FUNCTION

The Wigner function is the analog of the probability dis-tribution in which the second measurement is a perfect mea-

x / σ 1

- 2 0 - 1 5 - 1 0 - 5 0 5 1 0 1 5 2 0

σ 1 P

0 . 0 0

0 . 0 1

0 . 0 2

0 . 0 3

0 . 0 4

0 . 0 5

P ( x , 0 )

P ( x , t )

P 0 ( x , t )

FIG. 1. Probability distribution for a free particle without dissi-pation in the Schrödinger cat state. P�x ,0� is the initial distribution.P�x , t� is the distribution at time t, while P0�x , t� is the distributionobtained at time t by artificially setting the attenuation coefficientequal to zero.


042122-9

surement of position and momentum. Of course, a quantumparticle cannot have simultaneously a precise position andmomentum, so this last cannot be a proper quantum mea-surement of the general form �3.7�. The Wigner function istherefore not a probability distribution but rather what iscalled a “quasiprobability distribution function.” To get theWigner function corresponding to the density matrix at timet2, which we denote by W�q , p ; t2− t1�, we make the replace-ment f†�2�f�2�→“�(q−x�t2�)�(p−mx�t2�)”=F(q−x�t2� , p−mx�t2�), where

F�q,p� =1

�2��2�−

dQ�−

dPei�Pq+Qp�/�. �6.1�

That is, in place of the general formula �5.2� for the prob-ability distribution we have its generalization to quantumphase space given by the general formula

W�q,p;t2 − t1� =�f†�1�F„q − x�t2�,p − mx�t2�…f�1��

�f†�1�f�1��

= �−

dQ

2��

−

dP

2��ei�Pq+Qp�/�

�f†�1�e−i�x�t2�P+mx�t2�Q�/�f�1��

�f†�1�f�1��. �6.2�

Note first of all that as a classical function, F�q , p�=��q��p�. This expression would be unsatisfactory for ourpurposes, since in the general formula �6.2� the arguments ofthe delta functions would not commute. The integral expres-sion �6.1� corresponds to the Fourier–von Neumann repre-sentation of the classical operator. Second, we note that themomentum operator is interpreted as the mechanical mo-mentum mx. This, as we have noted above, is in accordancewith the macroscopic description of a dissipative system.The point here is that the canonical momentum is an operatorof the microscopic description, which for the same macro-scopic description may or may not be equal to the mechani-cal momentum �20�. Finally, we should emphasize that theWigner function is not a probability distribution as discussedin Sec. III. For this reason we use a calligraphic W to helpkeep this in mind. This formula for the Wigner function isunique in the sense that it satisfies certain general require-ments such as that it be a real function, that the integral overp or q must give the corresponding probability distribution inposition or momentum, etc. For a thorough discussion of theWigner function as it has appeared in the literature, see thereview article of Hillery et al. �21� �see especially their Eq.�2.45��.

It is convenient to introduce the Fourier transform of theWigner function,

W�Q,P;t� = �−

dq�−

dpe−i�Pq+Qp�/�W�q,p;t� . �6.3�

This Fourier transform is what in the literature is called the“characteristic function” �21�. We adopt this convenient ter-minology, but warn that this Wigner characteristic functionshould not be confused with the quantum analog of the char-

acteristic functions of classical probability introduced in Sec.IV. The inverse Fourier transform is

W�q,p;t� = �−

dQ

2��

−

dP

2��W�Q,P;t�ei�Pq+Qp�/�.

�6.4�

Comparing this with the general formula �6.2�, we obtain byinspection a simple formula for the Wigner characteristicfunction:

W�Q,P;t2 − t1� =�f†�1�e−i�x�t2�P+mx�t2�Q�/�f�1��

�f†�1�f�1��. �6.5�

This is the key result of this section, valid for any form of themeasurement function f . As we next shall show, it allows usto readily calculate the Wigner function for a variety of ex-amples.

A. Example: Equilibrium Wigner function

As a first simple example we consider the equilibriumWigner function, which we denote by Weq�q , p� and whichwe get when we make no initial measurement. This corre-sponds to f�1�→1, and for this case the formula �6.5� for theWigner characteristic function becomes

Weq�Q,P� = �e−i�x�t2�P+mx�t2�Q�/�� . �6.6�

Using the Gaussian formula �B6� we find

Weq�Q,P� = exp�−1

2�2 ��x2�P2 + m2�x2�Q2� , �6.7�

where we have used the fact that �xx+ xx�=0. With this inEq. �6.4�, the integrals are standard Gaussian integrals andwe find, for the equilibrium Wigner function,

Weq�q,p� =1

2�m��x2��x2�exp�−

q2

2�x2�−

p2

2m2�x2� .

�6.8�

Here �x2� and �x2� are given in expressions �A30� and �A29�.The familiar weak-coupling form, well known as the equi-librium solution of the master equation, results if we recallthe relations for weak coupling given in Eq. �A39�.

B. Example: Motion of a coherent state

Coherent states are generally defined for the free oscilla-tor by operating on the oscillator ground state with the gen-eral displacement operator exp�i�mv0x−x0p� /��. The result-ing coherent state corresponds to a displaced ground state,centered at x0 and moving with velocity v0. Here we define ageneralized coherent state for an oscillator interacting with alinear passive heat bath. The corresponding density matrix isobtained by acting on the equilibrium density matrix with themeasurement function

f�1� = exp�im�v0x�t1� − x0x�t1��/�� . �6.9�

With this, the expression �6.5� for the Wigner characteristicfunction becomes


042122-10

W�Q,P;t2 − t1�

= �e−im�v0x�t1�−x0x�t1��/�e−i�x�t2�P+mx�t2�Q�/�eim�v0x�t1�−x0x�t1��/�� .

�6.10�

Use the Baker-Campbell-Hausdorf formula �B3� to write

e−im�v0x�t1�−x0x�t1��/�e−i�x�t2�P+mx�t2�Q�/�eim�v0x�t1�−x0x�t1��/�

= e−i�x�t2�P+mx�t2�Q�/�em�x�t2�P+mx�t2�Q,v0x�t1�−x0x�t1��/�2.

�6.11�

With this, we find

W�Q,P;t� = exp�− i1

��x�t�P + mv�t�Q� Weq�Q,P� ,

�6.12�

where, using Eq. �A7� to express the commutator in terms ofthe Green function,

x�t� = mG�t�x0 + mG�t�v0,

v�t� =dx�t�

dt= mG�t�x0 + mG�t�v0. �6.13�

The Wigner function is given by the inverse Fourier trans-form �6.4�. We find

W�q,p;t� = Weq„q − x�t�,p − mv�t�… . �6.14�

Thus the Wigner function corresponding to an initial coher-ent state has the form of an equilibrium Wigner functionwhose center moves according to Eqs. �6.13�.

We should point out that the motion �6.13� of this centeris not the solution of the mean of the quantum Langevinequation �A1�. Rather, it is the solution of the mean of theinitial value Langevin equation �22�

md2x

dt2 + �0

t

dt��t − t��dx�t��

dt�+ Kx = − ��t�x�0� ,

�6.15�

with initial data x�0�=x0 and v�t�=v0. The effect of the termon the right-hand side can be seen clearly in the Ohmic case,where the Green function has the form �A20�. With that formwe see that x�t+0+�=x0, while v�t+0+�=v0−�x0. Thus thecenter of initial distribution in the qp plane makes a jump inthe p direction, down �up� if x0 is positive �negative�, afterwhich the motion of the center is that of a damped harmonicoscillator. At all times the shape of the distribution is that ofa displaced thermal equilibrium state of the oscillator.

The probability distribution is obtained by integratingover p,

P�x,t� = �−

dpW�x,p;t� . �6.16�

With the expression �6.14� for the Wigner function in whichWeq is of the form �6.8� this becomes

P�x, ;t� = Peq„x − x�t�… , �6.17�

where Peq is the equilibrium distribution �5.16�. This is ex-actly the form we encountered above in the example of wavepacket spreading, where the parameters for the displacedground state are given in Eq. �5.14�. In either case the prob-ability distribution is that of a displaced ground state. Thedifference lies in the motion of the center, which is tempera-ture independent in the case of the coherent state but has atemperature-dependent form for the displaced ground state.In either case, of course, the distribution approaches that ofequilibrium for long times. The lesson we learn here is thatthe time dependence of the approach to equilibrium dependson how the initial state is formed. In Fig. 2 we plot x�t� forthe two cases, the displaced ground-state motion being cal-culated at zero temperature. The parameters chosen were� /�0=10/13 and � /�0=5, but despite this rather strongcoupling, there is not much difference between the twocurves.

C. Example: Coherent-state pair

The idea here is to form an initial state like theSchrödinger cat state discussed in Sec. V B. There the initialstate was prepared with a pair of Gaussian slits. Here weconsider instead an initial state which is a superposition oftwo separated coherent states. This is accomplished with ameasurement function of the simple form

f�1� = cosmdx�t1�

2�, �6.18�

which results in a superposed pair of generalized coherentstates, centered at x= ±d /2 and each with zero velocity. Ex-pression �6.5� for the Fourier transform of the normalizedWigner function at time t2 therefore becomes

ω 0 t

0 2 4 6 8 1 0

x_(t)/x0

- 0 . 5

0 . 0

0 . 5

1 . 0

c o h e r e n t s t a t e

d i s p l a c e d g r o u n d s t a t e

FIG. 2. The motion of the wave packet center for the displacedground state and for the coherent state, both for initial velocity zero.The displaced ground-state motion is computed at zero temperature.The parameters chosen are � /�0=10/13 and � /�0=5.


042122-11

W�Q,P;t2 − t1�

=�cos�mdx�t1�/2��e−i�x�t2�P+mx�t2�Q�/�cos�mdx�t1�/2��

�cos2�mdx�t1�/2��.

�6.19�

Now, using again the Baker-Campbell-Hausdorf formula�B3�, we see that

cosmdx�t1�

2�e−i�x�t2�P+mx�t2�Q�/�cos

mdx�t1�2�

=1

4�2e−i�x�t2�P+mx�t2�Q�/�cos

md�GP + mGQ�2�

+ e−i�x�t2�P+mx�t2�Q−mdx�t1��/� + e−i�x�t2�P+mx�t2�Q+mdx�t1��/� ,

�6.20�

where, to shorten the expression, we have used expression�A7� to express the commutator in terms of the Green func-tion. Putting this into Eq. �6.19� and using the Gaussianproperty �B6� we find

W�Q,P;t� =exp�− ��x2�P2 + m2�x2�Q2�/2�2�

1 + exp�− m2�x2�d2/2�2�

�cosmd�GP + mGQ�

2�+ exp�−

m2d2�x2�2�2

coshmd�sP + msQ�

2�2 � , �6.21�

where G=G�t� is the Green function �A5� and s=s�t� is themean-square displacement �A8�. Putting this into Eq. �6.4�,the integral is a two-dimensional standard Gaussian �B2� andwe find

W�q,p;t� =1

2�1 + exp�− m2�x2�d2/2�2��Weq�q −

mGd

2,

p −m2Gd

2 + Weq�q +

mGd

2,p +

m2Gd

2

+ 2e−A�t�Weq�q,p�cos ��q,p;t�� , �6.22�

where Weq�q , p� is the equilibrium Wigner function �6.8� andwe have introduced

A�t� =m2d2�x2�

2�2 �1 −s2�t�

4�x2��x2�−

s2�t�4�x2�2 ,

��q,p;t� = �msq

�x2�+

sp

�x2� d

2�. �6.23�

This expression for the Wigner function is identical with thatobtained by Romero and Paz using path integral methods�23�.

Viewed in the qp plane, the expression �6.22� for theWigner function shows three peaks: an outlying pair in theform of single coherent states, centered initially at q= ± d

2 ,p=0, and an interference peak, centered at the origin andmodulated by the factor cos �. Initially, since A�0�=0, theamplitude of the interference peak is twice that of either ofthe two outlying peaks. In general, the interest is in the caseof a widely separated coherent-state pair—that is, d��x2�.In that case for very short times A�t� becomes large and theinterference peak is practically zero. This disappearance ofthe interference peak is the phenomenon of decoherence asseen with the Wigner function. To be more explicit, with theexpansion �A26� in the expression �A24� for s�t�, we canreadily evaluate this expression for short times. In the limitof small bath relaxation time this takes the simple form

A�t� � −t2

2�d2 ln

t

�, �6.24�

where

�d = ��

�d2 . �6.25�

Here � is the friction constant and � is the bath relaxationtime, the parameters in the single relaxation time model. Wecan interpret �d, the time of the order of which the centralpeak in the Wigner distribution vanishes, as a decoherencetime. Note that this expression for the decoherence time isvery different from that for the displaced ground-state pairgiven in Eq. �5.27�, despite the similarity of the initial states.The qualitative nature of the phenomenon is the same: therapid disappearance of an interference term.

Since the Wigner function is not directly observable, weshould consider the probability distribution. That is, we putthe expression �6.22� for the Wigner function into the inte-gral �6.16� for the probability distribution to obtain

P�x,t� =1

2�1 + exp�− m2�x2�d2/2�2��Peq�x − mG

d

2

+ Peq�x + mGd

2

+ 2a exp�−m2G2d2

8�x2� Peq�x�cos

msdx

2��x2�� ,

�6.26�

where a=a�t� is the attenuation coefficient, now given by

a�t� = exp�− �4m2�x2��x2��2 −

m2s2�t��2 − m2G2 d2

8�x2�� .

�6.27�

As in our discussion of the Schrödinger cat state in Sec. V,the attenuation coefficient is defined as the ratio of the coef-ficient of the cosine term to the geometric mean of the firsttwo terms and corresponds to the traditional measure of co-herence. Its disappearance is the phenomenon of decoher-


042122-12

ence as seen in the probability distribution. But here the ini-tial value of the attenuation coefficient,

a�0� = exp�− �4m2�x2��x2��2 − 1 d2

8�x2�� , �6.28�

is already vanishingly small for large separations �the uncer-tainty principle tells us that the factor in the exponent isnecessarily positive�. Indeed, for the Ohmic model, themean-square velocity is logarithmically divergent, so the at-tenuation coefficient is identically zero for all times. In anyevent, we would say that as seen in the probability distribu-tion, the decoherence for a coherent state pair occurs initiallyand there is no notion of a decoherence time.

The earliest discussion of this problem of a displaced pairof coherent states at zero temperature was that of Walls andMilburn �24�, who based their discussion on the master equa-tion. As we note in the last paragraph of Appendix A, thiswould correspond to the Weisskpof-Wigner approximation.In this approximation

A�t� =m�0d2

4��1 − e−�t� �6.29�

and

a�t� = exp�− A�t�� . �6.30�

Note first of all that in this Weisskopf-Wigner approximationthe decay of coherence is identical whether viewed in theWigner function or in the probability distribution, However,the short-time behavior is very different from that of theexact result. As an illustration, in Fig. 3 we compare theshort-time behavior of the exact result with that of theWeisskopf-Wigner approximation as well as that of theweak-coupling approximation. In this range the weak-coupling expression for A�t� is just twice that from theWeisskopf-Wigner approximation, while both are muchlarger than the exact result and would give a correspondinglymuch shorter estimate of the decoherence time. The conclu-sion to be drawn is that the master equation can give mis-leading results at short times.

D. Example: Squeezed state

The squeezed state that appears in the quantum opticsliterature is obtained by operating on the ground state of thefree oscillator with the squeeze operator �25�,

S = exp� r

2�e−i�a2 − ei�a†2� , �6.31�

where r and � are real parameters and a= �m�0x+ ip� /�2m��0 is the usual annihilation operator for the freeoscillator. Here we consider the so-called ideal squeeze op-erator, corresponding to �=0, and again replace the canoni-cal momentum with the mechanical momentum. The idealsqueeze operation would therefore correspond to an initialmeasurement operator of the form

f�1� = exp�imr

2��x�t1�x�t1� + x�t1�x�t1�� . �6.32�

Since this is a unitary operator, we see that

�f†�1�e−i�x�t2�P+mx�t2�Q�/�f�1�� = �e−i�X�r,t2�P+mX�r;t2�Q�/�� ,

�6.33�

where we have introduced the operator

X�r;t2� = f†�1�x�t2�f�1� . �6.34�

To evaluate this operator, form the derivative with respectto r,

�X�r;t2��r

= f†�1�im

2��x�t2�,x�t1�x�t1� + x�t1�x�t1��f�1�

= − mG�t2 − t1�X�r;t1� + mG�t2 − t1�X�r;t1� ,

�6.35�

where we have used relation �A7� to express the commutatorin terms of the Green function. Also, we have

�X�r;t2��r

= − mG�t2 − t1�X�r;t1� + mG�t2 − t1�X�r;t1� .

�6.36�

If we set t2= t1 and use the fact that G�0�=0 and G�0�=1/m, we find from Eq. �6.35� that

X�r;t1� = x�t1�e−r, �6.37�

while from Eq. �6.36� we find that

X�r;t1� = x�t1�er. �6.38�

Here we should emphasize that we use the single relaxation

time model, for which G�0�=0. For the Ohmic model therewould be an extra term. Putting these results into Eqs. �6.35�and �6.36� and integrating, we find

X�r;t2� = x�t2� − �1 − e−r�mGx�t1� + �er − 1�mGx�t1� ,

ω 0 t

0 . 0 0 0 0 . 0 0 2 0 . 0 0 4 0 . 0 0 6 0 . 0 0 8 0 . 0 1 0

A(t)

W e i s s k o p f - W i g n e r

e x a c t

w e a k c o u p l i n g

FIG. 3. The function A�t� for a coherent state pair. The param-eters chosen are � /�0=10/13 and � /�0=5.


042122-13

X�r;t2� = x�t2� − �1 − e−r�mGx�t1� + �er − 1�mGx�t1� ,

�6.39�

where G=G�t2− t1�.Since X�r ; t2� and X�r ; t2� are linear in the operators x�t1�

and x�t1�, they have the Gaussian property and we can usethe identity �B6� to evaluate expression �6.33�. With this re-sult the Wigner characteristic function �6.5� can be written inthe form

W�Q,P;t2� = exp�−A11P2 + 2A12QP + A22Q

2

2�2 ,

�6.40�

where, for this present example,

A11 = �X2�r;t2�� = �1 − �1 − e−r�mG�2�x2� + �er − 1�2m2G2�x2�

+ �1 − e−r�mGs + �er − 1�mGs , �6.41a�

A12 =m

2�X�r;t2�X�r;t2� + X�r;t2�X�r;t2��

= − �1 − �1 − e−r�mG��1 − e−r�m2G�x2�

+ �er − 1�2m3GG�x2� +1

2�1 − e−r�m2�Gs + Gs�

+1

2�er − 1�m2�Gs + Gs� , �6.41b�

A22 = m2�X2�r;t2��

= �1 + �er − 1�2m2G2�m2�x2� + �1 − e−r�2m4G2�x2�

+ �er − 1�m3Gs + �1 − e−r�m3Gs . �6.41c�

Here we should again recall that s=s�t2− t1� is the mean-square displacement �A8� and G=G�t2− t1� is the Greenfunction �A5�.

Forming the corresponding Wigner function, we find, forthe squeezed state,

W�q,p;t� =1

2��A11A22 − A122

exp�−A11p2 − 2A12pq + A22q

2

2�A11A22 − A122 �

.

�6.42�

In Fig. 4 we plot a constant density contour for this functionin the plane of the dimensionless variables u=q / �x2� and v= p /m�x2�. The dashed circle corresponds to the equilibriumstate, the state just before the initial squeeze as well as thestate at long times. The contour marked �0� corresponds tothe initial squeezed state. In the course of time this contourrotates clockwise. The contour marked �1/4� is that corre-sponding to a quarter period, while that marked �1/2� is thatcorresponding to a half period when the squeezing is muchreduced. The relatively strong coupling chosen, � /�0=10/13, emphasizes the effect: dissipation leads to a loss ofsqueezing.

E. Example: Schrödinger cat state

Here we consider the Wigner function when the initialmeasurement corresponds to the measurement function�5.17�. The calculation goes exactly the same as that begin-ning with Eq. �5.5� in the previous section, so we shall sim-ply quote the results. For general f we find

�f†�1�e−i�x�t2�P+mx�t2�Q�/�f�1�� = exp�−�x2�P2 + m2�x2�Q2

2�2 +�cP + mcQ�2

2�x2��2 �−

dx1�f†�x1� +GP + mGQ

2 f�x1� −

GP + mGQ

2

1

�2��x2�exp�−

�x1 + x1��2

2�x2�− i�x1 + x1��

cP + mcQ

�x2�� . �6.43�

u

v

( 0 )

( 1 / 4 )

( 1 / 2 )

FIG. 4. Constant density contours of the Wigner function for asqueezed state, shown in the plane of the dimensionless variablesu=q /��x2� and v= p /m��x2�. The dashed circle corresponds to theequilibrium state, the state just before the initial squeeze as well asthe state at long times. The contour marked �0� corresponds to theinitial squeezed state. The contour marked �1/4� is that correspond-ing to a quarter period, while that marked �1/2� is that correspond-ing to a half period. The parameters chosen are � /�0=10/13,� /�0=5.


042122-14

Here c=c�t2− t1� is the correlation �A9� and G=G�t2− t1� isthe Green function �A5�. With the form �5.17� for the mea-surement function, this becomes �remember we must choosex1=0 if we wish the initial wave packet pair to be placedsymmetrically about the origin�

�f†�1�e−i�x�t2�P+mx�t2�Q�/�f�1��x1=0

=exp�− �A11P2 + 2A12PQ + A22Q

2�/2�2�

�1 + e−d2/8�12��2��x2� + �1

2�

�e−d2/8��x2�+�12� cos

�cP + mcQ�d2��x2� + �1

2��+ e−d2/8�1

2

cosh�GP + mGQ�d

4�12 , �6.44�

where in this present example

A11 = �x2� +�2G2

4�12 −

c2

�x2� + �12 ,

A12 = m��2GG

4�12 −

cc

�x2� + �12 ,

A22 = m2��x2� +�2G2

4�12 −

c2

�x2� + �12 . �6.45�

The Wigner characteristic function is therefore

W�Q,P;t� =exp�− �A11P2 + 2A12PQ + A22Q

2�/2�2�1 + exp�− �x2�d2/8�1

2��x2� + �12��

�cos�cP + mcQ�d2��x2� + �1

2��+ exp�−

�x2�d2

8�12��x2� + �1

2� cosh

�GP + mGQ�d4�1

2 � . �6.46�

With this form of the Wigner characteristic function thereis no difficulty evaluating the inverse transform �6.4� to ob-tain the corresponding Wigner function. However, the inter-est will be in the limits of a free particle or, for the oscillator,a displaced ground-state pair, in which case it is simpler tofirst evaluate the limits of the above expression and thenevaluate the inverse transform.

1. Free particle

We obtain the case of a free particle coupled to the bath inthe absence of an oscillator potential by forming the limit

�x2�→. In forming this limit we should recall that c�t�= �x2�−s�t� /2, where the mean-square displacement s�t� isfinite in the free particle limit. We find

W�Q,P;t� =exp�− �A11P2 + 2A12PQ + A22Q

2�/2�2�

1 + exp�− d2

8�12� �cos

Pd

2�

+ exp�−d2

8�12 cosh

�GP + mGQ�d4�1

2 � . �6.47�

In this free particle case, expressions �6.45� become

A11 = �12 + s +

�2G2

4�12 ,

A12 = m� s

2+

�2GG

4�12 ,

A22 = m2��x2� +�2G2

4�12 . �6.48�

Forming the inverse Fourier transform �6.4� we obtain

W�q,p;t� =1

2�1 + e−d2/8�12��W0�q −

d

2,p;t

+ W0�q +d

2,p;t

+ 2 exp�− A�t��W0�q,p;t�cos ��q,p;t�� ,

�6.49�

where W0 is the Wigner function corresponding an initialmeasurement forming a single wave packet at the origin,

W0�q,p;t2�

=exp�− �A22q

2 − 2A12qp + A11p2�/2�A11A22 − A122 ��

2��A11A22 − A122

.

�6.50�

In this example the phase � is given by

��q,p;t� =�GA22 − mGA12�q + �mGA11 − GA12�p

A11A22 − A122

�d

4�12

�6.51�

and the quantity A by

A�t� =�A11 − �2G2/4�1

2��A22 − �2m2G2/4�12� − �A12 − �2mGG/4�1

2�2

A11A22 − A122

d2

8�12 . �6.52�


042122-15

As in the case of the coherent state pair, The Wigner func-tion for the free particle Schrödinger cat state shows threepeaks, an outlying pair centered at q= ±d /2 , p=0 and aninterference peak centered at the origin. However, in this freeparticle case,

A�0� =�1

2�x2��1

2�x2� + �2/4m2

d2

8�12 =

d2

8�12 + 2�2

� 0, �6.53�

where � is the mean de Broglie wavelength in equilibrium,

� =�

m��x2�. �6.54�

Therefore, the amplitude of the interference peak will ini-tially be vanishingly small whenever the separation d of thewave packets is large compared with both the slit width �1

and the mean de Broglie wavelength �.

2. Displaced ground-state pair

Again we form the limit �12→ and d→ such that d0

= �x2�d /�12 is fixed. The coefficients �6.45� become

A11 = �x2�, A12 = 0, A22 = m2�x2� . �6.55�

The Wigner characteristic function �6.46� then becomes

W�Q,P;t� =exp�− ��x2�P2 + m2�x2�Q2�/2�2�

1 + exp�− d02/8�x2��

�cos�cP + mcQ�d0

2�x2��+ exp�− d0

2/8�x2��

cosh�GP + mGQ�d0

4�x2� . �6.56�

With this, the Wigner function is

W�q,p;t� =1

2�1 + exp�− d02/8�x2��

�Weq�q −cd0

2�x2�,

p −mcd0

2�x2� + Weq�q +

cd0

2�x2�,p +

mcd0

2�x2�

+ 2e−AWeq�q,p�cos �� , �6.57�

where Weq�q , p� is the equilibrium Wigner function �6.8� and

��q,p;t� = � �G

�x2�q +

�G

m�x2�p d0

4�x2�,

A�t� =d0

2

8�x2��1 −

�2G2

4�x2�2 −�2G2

4�x2��x2� . �6.58�

Again, as in the free particle case the initial value of A is notzero,

A�0� =d0

2

8�x2��1 −�2

4m2�x2��x2� � 0, �6.59�

and the interference term is vanishingly small.

This situation, in which the initial state is a Schrödingercat state formed by passing the particle through a pair ofGaussian slits, is to be contrasted with that described in Sec.VI C, in which the initial state is prepared by displacing theequilibrium state to form a coherent state pair. The Wignerfunctions, given by Eqs. �6.57� and �6.22�, respectively, areidentical in form, but in the Schrödinger cat case the inter-ference peak is initially vanishingly small and remains so forall time, while in that of the coherent state pair the interfer-ence peak is initially twice as high as the outlying peaks,becoming vanishingly small only after a short relaxationtime. The reverse is true for the probability distributions,given in Eqs. �5.24� and �6.26�, respectively. That is, theinterference term in the probability distribution is vanish-ingly small at all times for the coherent state pair, while forthe Schrödinger cat state the attenuation coefficient multiply-ing the interference term vanishes only after a short decoher-ence time. We must conclude that the notion of decoherencetime is arbitrary, depending on the situation and how onechooses to view it.

VII. CONCLUDING REMARKS

The quantum probability distributions are measured dis-tributions. That is, they depend explicitly on the parametersof the measurements. This is seen clearly in the general for-mula �3.29� for the n-point distribution, where f�j� is themeasurement function for the jth measurement. This is alsoseen in the in the case of quantum Brownian motion in ex-pression �4.2� for the characteristic function. There it is seenthat the characteristic function can be factored, with a factorK�1, . . . ,n� multiplying a quantum expectation independentof measurement. The factor K�1, . . . ,n� contains the mea-surement parameters and depends upon the dynamicsthrough the non-equal-time commutator. In the classicallimit, where this commutator vanishes, this factor becomes anumerical factor independent of the dynamics that can inpractice be taken to be unity. The result is the familiar ex-pression for the classical characteristic function �15�.

An important part of this work has been to demonstratethat the general expression �4.2� for the characteristic func-tion can be very useful for practical calculations. The use ofthis formula together with its specializations �5.4� to theprobability distribution and Eq. �6.5� to the Wigner charac-teristic function has been illustrated with a number of ex-amples, each of which is an important application. Amongthe results we point out expression �5.13� for wave packetspreading in the presence of dissipation, a generalization ofthe well-known expression found in elementary textbooks.Another result is illustrated in Fig. 4 where the disappear-ance of squeezing in the presence of dissipation is illustrated.Finally, a comparison of a “Schrödinger cat” state formed bypassing the particle through a pair of Gaussian slits with thenearly identical state formed with a coherent state pair showsthat a quantitative measure of decoherence. depends on howthe state is formed.

In Sec. VI we discuss the Wigner function. Some authorsprefer instead the density matrix element in the coordinaterepresentation, which is given by a kind of half-Fouriertransform


042122-16

�x��t��x�� = �−

dpW� x + x�

2,p;t ei�x−x��p/�. �7.1�

Clearly, the Wigner function and the density matrix elementcontain the same information. Indeed, it is not difficult to seethat the central interference peak in, say, the Wigner function�6.49� corresponding to a coherent-state pair becomes a pairof off-diagonal peaks in the density matrix element. We pre-fer the language of the Wigner function since it is always areal function that in the classical limit becomes a real prob-ability distribution.

The examples are all in one dimension. The reader shouldbe aware that this is not a necessary restriction, but has beenmade to keep the discussion within bounds. The generaliza-tion to higher dimensions is straightforward. All that onemust keep in mind is that the fluctuating force operator isindependent in the different directions and that the effects ofdissipation �the fluctuating force and the memory force� areindependent of the applied force �20�. Finally, we have re-stricted the discussion to the single relaxation time model ofdissipation and its limiting Ohmic case. Again this has beendone to keep the discussion within bounds. There is no prob-lem with the discussion for more general models such as thecoupling to the blackbody radiation field �20�.

APPENDIX A: QUANTUM BROWNIAN MOTION

1. Quantum Langevin equation

Quantum Brownian motion for an oscillator coupled to aheat bath at temperature T is described by the quantumLangevin equation

mx + �−

t

dt��t − t��x�t�� + Kx = F�t� . �A1�

This is a Heisenberg equation for the position operator x�t�.On the right-hand side F�t� is a Gaussian random operatorforce, with mean zero, �F�t��=0, and with autocorrelationand commutator given by

1

2�F�t�F�t�� + F�t��F�t��

=�

��

0

d� Re�� + i0+�� coth��

2kTcos ��t − t�� ,

�F�t�,F�t�� =�

i��

0

d� Re�� + i0+�� sin ��t − t�� .

�A2�

In these expressions � is the Fourier transform of thememory function,

��z� = �0

dt��t�eizt, Im�z� � 0. �A3�

It is a consequence of the second law of thermodynamicsthat ��z� must be what is called a positive real function:

analytic with real part positive everywhere in the upper halfplane.

In our present discussion we take the view that the aboveis a macroscopic description, which is complete as it stands.For a thorough discussion, including the derivation from anumber of microscopic models, we refer to a paper of Ford,Lewis, and O’Connell �20�.

The solution of the quantum Langevin equation �A1� canbe written

x�t� = �−

t

dt�G�t − t��F�t�� , �A4�

where the Green function G�t� is given by

G�t� =1

2��

−

d�� + i0+�e−i�t, �A5�

in which ��z�, the response function, is given by

��z� =1

− mz2 − iz��z� + K. �A6�

With this solution, we can obtain the following expres-sions for the commutator:

�x�t�,x�t�� =2�

i��

0

d� Im�� + i0+��sin ��t − t��

= i��G�t� − t� − G�t − t�� . �A7�

The Green function vanishes for negative times, while

G�0�=0 and G�0�=1/m. We see, therefore, that the canoni-cal commutator �x , p�= i� holds with p=mx, the mechanicalmomentum. �The canonical momentum may or may not bethe same as the mechanical momentum, depending on theform of the microscopic Hamiltonian.�

Also of interest is the mean-square displacement

s�t − t�� x�t� − x�t��2� = 2�x2� − 2c�t − t�� . �A8�

Here c�t� is the correlation

c�t� =1

2�x�t�x�0� + x�0�x�t�� . �A9�

Using the solution �A4� or, more directly without recourse tothe Langevin equation, the fluctuation-dissipation theorem ofCallen and Welton �26�, we obtain the following expressionfor the correlation:

c�t� =�

��

0

d� Im�� + i0+��coth��

2kTcos �t .

�A10�

With this, we also have

s�t� =2�

��

0

d� Im�� + i0+��coth��

2kT�1 − cos �t� .

�A11�

Here we should note first of all that for very long times thecorrelation vanishes and


042122-17

s�t� � 2�x2� . �A12�

The exception is the free particle case, where K=0 and con-sequently �x2�=. In that case s�t� grows for long timeswithout limit, with a time dependence that depends on themodel as well as the temperature �27,28�. On the other hand,for very short times we can expand the cosine to obtain thegeneral result

s�t� � �x2�t2. �A13�

2. Explicit expressions

Here we obtain explicit, closed-form expressions for theGreen function G�t� and the mean-square displacement s�t�.For this purpose, the model of choice for most applications,due to its simplicity, is the Ohmic model, for which ��z�=�, the friction constant. While it is adequate for most pur-poses, this Ohmic model is singular, particularly at shorttimes or high frequencies, so we here consider a more gen-eral model in which the higher frequencies are suppressed.The simplest of these is the single relaxation time model, forwhich

��z� =�

1 − iz�. �A14�

Here � is the relaxation time �at times called the bath corre-lation time� which we assume is small in the sense that�� /m�1. Putting this form of � in the response function�A6� and replacing the parameters �, �, and K with param-eters �, �, and �0 through the relations

� =1

� + �, � = m�

�� + �� + �02

�� + ��2 , K = m�02 �

� + �,

�A15�

we obtain the convenient form

��z� =z + i�� + ��

m�z + i��− z2 − i�z + �02�

. �A16�

Note that −i� is the pole of the response function far downon the negative imaginary axis. For small relaxation time��1/� we have the expansion

� =1

�−

�

m−

�2�

m2 + �K�

m2 − 2�3

m3 �2 + ¯ . �A17�

The Ohmic model, for which �→0, therefore corresponds to�→, while �→� /m and �0

2→K /m.

a. Green function

With the form �A16� of the response function, we evaluatethe integral in expression �A5� for the Green function bydeforming the path of integration into the lower half plane,picking up the residues at the poles of the response function.The result is

G�t� =�

m��2 − �� + �02�

�e−�t − e−�t/2cos �1t�

+�2 + �0

2 − �2/2

�2 − �� + �02 e−�t/2 sin �1t

m�1, �A18�

where

�1 = ��02 − �2/4. �A19�

In the limit, �→, this becomes the familiar Ohmic Greenfunction

GOhmic�t� = e−�t/2 sin �1t

m�1. �A20�

For reasonable choices of the parameters the OhmicGreen function is very little different from that for the singlerelaxation time model. The difference between the two mod-els is more apparent in the second derivative of the Greenfunction, which we show in Fig. 5 for the two models. The

difference is small except for short times, where G�0� van-

ishes, while GOhmic�0�=−� /m2 is finite.

b. Mean-square displacement at high temperature

First we consider the mean square displacement in thehigh-temperature limit, in which in expression �A10� we re-place the hyperbolic cotangent by the reciprocal of its argu-ment. If we compare the resulting expression with the ex-pression �A5� for the Green function, we see that in thishigh-temperature limit,

s�t� = 2kT�0

t

dt�G�t�� . �A21�

The long- and short-time behaviors of the mean-square dis-placement are of the general form given in Eqs. �A12� and�A13�. Note that in this high-temperature limit �x2� and �x2�are given by the classical equipartition formulas,

ω 0 t

0 2 4 6 8 1 0

mG''/ω0

- 1 . 0

- 0 . 8

- 0 . 6

- 0 . 4

- 0 . 2

0 . 0

0 . 2

0 . 4

O h m i c

s i n g l e r e l a x a t i o n t i m e

FIG. 5. Second derivative of the Green function for the oscilla-tor. Parameters for the single relaxation time model are � /�0

=10/13 and � /�0=5.


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�x2� =kT

K, �x2� =

kT

m. �A22�

c. Mean-square displacement at zero temperature

We consider now the effect of zero-point oscillations onthe mean-square displacement. At temperature zero, we re-place the hyperbolic cotangent in expression �A10� by unity.Then, if we write

Im�� + i0+�� = −�

m��2 − �� + �02�

�2

��2 + �2�

+ Im� �2 − ��/2 − i�1�2

m�1��2 − �� + �02�

��/2 + i�1�2

��2 + ��/2 + i�1�2� , �A23�

we can write expression �A11� for the mean-square displace-ment in the form

s�t� =2�

m��−

�

�2 − �� + �02V��t�

+ Im� �2 − ��/2 − i�1�2

�1��2 − �� + �02�

V��t

2+ i�1t �� .

�A24�

Here we have introduced the function

V�z� � �0

dy1 − cos zy

y�1 + y2�= ln z + �E −

1

2�e−zEi�z� + ezEi�− z�� ,

�A25�

where �E=0.577 215 665 is Euler’s constant and Ei is theexponential integral �29�. Using the expansion of the expo-nential integral for small argument, we obtain the expansion

V�z� = − �ln z + �E��cosh z − 1�

−1

2�e−z

n=1

zn

n ! n+ ez

n=1

�− z�n

n ! n . �A26�

We see from this that, for small z,

V�z� � −1

2z2�ln z + �E −

3

2 −

1

24z4�ln z + �E −

25

12 + ¯ .

�A27�

Note that V�0�=0, in agreement with the definition �A25�.For large z, using the asymptotic formulas for the exponen-tial integral, we obtain the asymptotic expansion

V�z� � ln z + �E −1

z2 −3!

z4 −5!

z6 − ¯ . �A28�

With these results, we see that for very short times �t��the mean-square displacement again takes the form �A13�but now with the mean-square velocity, given by

�x2� = ��2��0

2 − �2/2� + �04�arcos��/2�0� + ��1�2 ln��/�0�

�m�1��2 − �� + �02�

��

�m�− � ln �0� +

�02 − �2/2

�1arcos

�

2�0 , �A29�

where the second form is that for small relaxation time, �→1/�. In the Ohmic limit this is logarithmically divergent,so we have here a case where the single relaxation timemakes a difference. On the other hand,

�x2� = ��2 + �0

2 − �2/2�arcos��/2�0� − ��1 ln��/�0��m�1��2 − �� + �0

2�

��

�m�1arcos

�

2�0. �A30�

Thus the Ohmic limit of �x2� is finite.

d. Free particle

The free particle corresponds to the absence of the oscil-lator potential—that is, to the limit K→0. For the Greenfunction we can obtain this limit by setting �0=0 and �1= i� /2 in expression �A18�. This gives

G�t� =�2�1 − e−�t� − �2�1 − e−�t�

��2 − �2�. �A31�

In this free particle case the parameters � and � are given bythe relations �A15� with K=0. These can then be inverted togive

� =1 + �1 − 4��/m

2�, � =

1 − �1 − 4��/m

2�. �A32�

With this expression for the Green function, the high-temperature form �A21� of the mean-square displacement be-comes

s�t� =2kT

��t −

�3�1 − e−�t� − �3�1 − e−�t��2 − �2�

�2kT

��t −

1 − e−�t

� . �A33�

Note that at long time this increases linearly with time, con-sistent with Eq. �A12� in the sense that �x2�= for the freeparticle. On the other hand, for short times, we get exactlythe short-time result �A13� with �x2� given by the equiparti-tion form �A22�. In other words, the short-time behavior ofthe oscillator is that of the free particle.

At zero temperature the result �A24� becomes, for the freeparticle,

s�t� =2�

��

�2V��t� − �2V��t��2 − �2 . �A34�

At very short times �t�� this takes the form �A13� withnow


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�x2� =��

�m�� − ��ln

�

�� −

��

�mln �� . �A35�

At very long times �t��−1�, we find

s�t� �2�

�� + �

��ln �t + �E� −

�2

�� − ��ln

�

�

�2�

��ln �t/m . �A36�

3. Weak coupling

The coupling to the heat bath is measured by the function��z�. If this is small, the response function will be sharplypeaked about �0=�K /m, the natural frequency of the oscil-lator. In the integral expression �A5� for the Green function,one is therefore led to make the replacement ��→ ��0�.The result is an expression of the Ohmic form �A20� with

� =1

mRe��0�� . �A37�

�The imaginary part gives a negligible contribution to �0.�Next, in the integral expression �A11� for the mean-squaredisplacement, we make the same approximations with, inaddition the replacement �� coth ��

2kT →��0 coth��0

2kT , to ob-tain

s�t� � 2m�x2��0

t

dt�G�t��

= 2�x2��1 − e−�t/2�cos �1t +�

2�1sin �1t � ,

�A38�

where

�x2� � �02�x2� �

��0

2mcoth

��0

2kT. �A39�

This constitutes the weak-coupling approximation �30�.We should emphasize that this weak-coupling approximationis not valid for the free particle, as should be clear from theabove argument. Note, incidentally, that for the Ohmicmodel this approximation is exact in the high-temperaturelimit. The usual statement is that it is valid in the limit��0, but consideration of the exact results given abovetells us that even in this limit the approximation is not correctfor very short times ��0t�1� or for very long times��t�1�. However, for all other times, the weak-couplingapproximation is very good for surprisingly large values ofthe coupling. As an illustration in Fig. 6 we compare themean-square displacement at zero temperature as calculatedfirst with the exact formula �A24� and then with the weak-coupling approximation. The parameters � /�0=5 and� /�0=10/13 were chosen to exaggerate the difference. Whatwe see is that the weak-coupling approximation is surpris-ingly good, even for rather strong coupling.

If one makes the further approximation of neglectingquantities of relative order � /�0, one gets what at times is

called the Weisskopf-Wigner approximation. In Eq. �A38� itwould correspond to replacing �1→�0 and dropping thesecond term after the exponential. This then would corre-spond exactly to what one obtains by solving the well-knownweak-coupling master equation �31�. Because of this, in theliterature the Weisskopf-Wigner approximation is oftencalled the weak-coupling approximation. The difference be-tween the weak-coupling approximations we have defined itand the Weisskopf-Wigner approximation is illustrated dra-matically in Fig. 3.

APPENDIX B: MATHEMATICAL FORMULAS

Here we collect some formulas used in the evaluation ofthe various examples. These formulas are all simple andmore or less well known. The first is the standard Gaussianintegral

�−

du exp�−1

2au2 + bu =�2�

aexp� b2

2a . �B1�

The generalization to d dimensions takes the form

� du exp�−1

2u · A · u + B · u

=�2��d/2

�detAexp�1

2B · A−1 · B . �B2�

Here we have used dyadic notation, with A a positive-definite symmetric matrix and B a vector �not necessarilyreal� in d dimensions. This generalization follows from thestandard integral, using the fact that a symmetric matrix canbe diagonalized by an orthogonal transformation.

The second formula is the Baker-Campbell-Hausdorf for-mula. If A and B are a pair of operators �not necessarilyHermitian� whose commutator is a c number, then

ω 0 t

0 2 4 6 8 1 0

mω0s/h

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

1 . 2

1 . 4

e x a c t w e a k c o u p l i n g

FIG. 6. Comparison of the exact and weak-coupling expressionsfor the mean-square displacement at zero temperature for the oscil-lator. The parameters chosen are � /�0=10/13 and � /�0=5.


042122-20

eAeB = eA+Be�A,B�/2 = eBeAe�A,B�. �B3�

This formula is easily checked by expanding the exponen-tials in powers of their argument. A generalization of thistheorem is the convenient formula

eAg�B� = g�B + �A,B��eA, �B4�

which holds for a general function g�B�. Again, this can beverified by expanding g in powers of its argument.

Finally, we have a couple of formulas based on the notionof a Gaussian variable. In general a set of operators �eachwith mean zero� is Gaussian if the expectation of a productof an odd number of the operators is zero while the productof an even number is equal to the sum of products of pairexpectations, the sum being over all �2n−1� ! ! pairings withthe order within the pairs preserved. A Gaussian variable—e.g., x�t�—is such a set with the members labeled with thetime. Thus, for example, with an obvious shorthand,

�1234� = �12��34� + �13��24� + �14��23� . �B5�

Note that within each pair the order is the same as the origi-nal order. A straightforward consequence of this Gaussianproperty is that for a Gaussian operator O, we have the for-mula

�eiO� = e−�O2�/2. �B6�

This is easily verifies by expanding the exponentials. An-other convenient result is

� 1�2��2

exp�−�O − a�2

2�2 �=

1

�2��2 + �O2��exp�−

a2

2��2 + �O2�� . �B7�

To obtain this result, form the Fourier transform with respectto the parameter a using the standard Gaussian integral �B1�and then form the expectation using �B6�.

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lsu...measured quantum probability distribution functions for brownian motion g. w. ford department...

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