introduction to decoherence theory

8/14/2019 Introduction to Decoherence Theory

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Introduction to decoherence theory

Klaus Hornberger

Arnold Sommerfeld Center for Theoretical PhysicsLudwig-Maximilians-Universit at M unchen,Theresienstrae 37, 80333 Munich, Germany

Dec 14, 2006

1 The concept of decoherence

This introduction to the theory of decoherence is aimed at readers with aninterest in the science of quantum information. There, one is usually contentwith simple, abstract descriptions of non-unitary quantum channels to ac-count for imperfections in quantum processing tasks. Nonetheless, in order to justify such models of non-unitary evolution and to understand their limitsof applicability it is important to know their physical basis. Accordingly, Iwill emphasize the dynamic and microscopic origins of the phenomenon of decoherence, and will relate it to concepts from quantum information whereapplicable, in particular to the theory of quantum measurement.

The study of decoherence, though based at the heart of quantum theory,is a relatively young subject. It was initiated in the 1970s and 1980s withthe work of H. D. Zeh and W. Zurek on the emergence of classicality in thequantum framework. Until that time the orthodox interpretation of quantum

mechanics dominated, with its strict distinction between the classical macro-scopic world and the microscopic quantum realm. The mainstream attitudeconcerning the boundary between the quantum and the classical was that thiswas a purely philosophical problem, intangible by any physical analysis. Thischanged with the understanding that there is no need for denying quantummechanics to hold even macroscopically, if one is only able to understandwithin the framework of quantum mechanics why the macro-world appearsto be classical. For instance, macroscopic objects are found in approximateposition eigenstates of their center-of-mass, but never in superpositions of macroscopically distinct positions. The original motivation for the study of decoherence was to explain these effective super-selection rules and the ap-parent emergence of classicality within quantum theory by appreciating thecrucial role played by the environment of a quantum system.

Hence, the relevant theoretical framework for the study of decoherenceis the theory of open quantum systems , which treats the effects of an un-
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controllable environment on the quantum evolution. Originally developed toincorporate the phenomena of friction and thermalization in the quantumformulation, it has of course a much longer history than decoherence theory.However, we will see that the intuition and approximations developed in the

traditional treatments of open quantum systems are not necessarily appropri-ate to yield a correct description of decoherence effects, which may take placeon a time scale much shorter than typical relaxation phenomena. In a sense,one may say that while the traditional treatments of open quantum systemsfocus on how an environmental bath affects the system, the emphasis indecoherence is more on the contrary question, namely how the system affectsand disturbs environmental degrees of freedom, thereby revealing informationabout its state.

The physics of decoherence became very popular in the last decade, mainlydue to advances in experimental technology. In a number of experiments thegradual emergence of classical properties in a quantum system could be ob-served, in agreement with the predictions of decoherence theory. Needles tosay, a second important reason for the popularity of decoherence is its rel-evance for quantum information processing tasks, where the coherence of arelatively large quantum system has to be maintained over a long time.

Parts of these lecture notes are based on the books on decoherence byE. Joos et al. [1] and on open quantum systems by H.-P. Breuer & F. Petruc-cione [2], and on the lecture notes of W. Strunz [ 3]. Interpretational aspects,which are not covered here, are discussed in [4, 5], and useful reviews byW. H. Zurek and J. P. Paz can be found in [ 6, 7]. This chapter deals exclusivelywith conventional, i.e., environmental decoherence, as opposed to spontaneousreduction theories [8], which aim at solving the measurement problem of quantum mechanics by modifying the Schr odinger equation. These modelsare conceptually very different from environmental decoherence, though theirpredictions of super-selection rules are often qualitatively similar.

1.1 Decoherence in a nutshell

Let us start by discussing the basic decoherence effect in a rather generalframework. As just mentioned, we need to account for the unavoidable cou-pling of the quantum system to its environment. Although these environmen-tal degrees of freedom are to be treated quantum mechanically, their statemust be taken unobservable for all practical purposes, be it due to their largenumber or uncontrollable nature. In general, the detailed temporal dynamicsinduced by the environmental interaction will be very complicated, but onecan get an idea of the basic effects by assuming that the interaction is suffi-ciently short-ranged to admit a description in terms of scattering theory. Inthis case, only the map between the asymptotically free states before and after

the interaction needs to be discussed, thus avoiding a temporal description of the collision dynamics.


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Introduction to decoherence theory 3

Let the quantum state of a the system be described by the density oper-ator on the Hilbert space H. We take the system to interact with a singleenvironmental degree of freedom at a timethink of a phonon, a polaron, or agas particle scattering off your favorite implementation of a quantum register.

Moreover, let us assume, for the time being, that this environmental particleis in a pure state E = |in in |E , with |in E HE . The scattering opera-tor Stot maps between the in- and out-asymptotes in the total Hilbert spaceHtot = HHE , and for sufficiently short-ranged interaction potentials wemay identify those with the states before and after the collision. The initiallyuncorrelated system and environment turn into a joint state,

[before collision] tot = |in in |E (1)[after collision] tot = Stot [ |in in |E ]Stot . (2)

Now let us assume, in addition, that the interaction is non-invasive withrespect to a certain system property. This means that there is a numberof distinct system states, such that the environmental scattering off these

states causes no transitions in the system. For instance, if these distinguishedstates correspond to the system being localized at well-dened sites then theenvironmental particle should induce no hopping between the locations. In thecase of elastic scattering, on the other hand, it will be the energy eigenstates.Denoting the set of these mutually orthogonal system states by {|n }H, therequirement of non-invasiveness means that Stot commutes with those states,that is, it has the form

Stot =n

|n n| Sn , (3)

where the Sn are scattering operators acting in the environmental Hilbertspace. The insertion into ( 2) yields

tot =m,n

m||n |m n|Sm |in in |E Sn (4)

m,n

mn |m n| |( m )out

(n )out |E

and disregarding the environmental state by performing a partial trace we getthe system state after the interaction.

= tr E (tot ) =m,n

|m n |mn in |Sn Sm |in E ( n )out | ( m )out E(5)

Since the Sn are unitary the diagonal elements, or populations , are indeed

unaffected,mm = mm , (6)


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4 Klaus Hornberger

while the off-diagonal elements, or coherences , get multiplied by the overlapof the environmental states scattered off the system states m and n,

mn = mn (n )out |

(m )out . (7)

This factor has a modulus of less than one so that the coherences, whichcharacterize the ability of the system state to display a superposition between

|m and |n , get suppressed 1 . It is important to note that this loss of coherenceoccurs in a special basis , which is determined only by the scattering operator,i.e., by the type of environmental interaction, and that it occurs to a degreethat is determined by both the the environmental state and the interaction.

This loss of the ability to show quantum behavior due to the interactionwith an environmental quantum degree of freedom is the basic effect of deco-herence. One may view it as due to the arising correlation between the systemwith the environment. After the interaction, the joint quantum state of systemand environment is no longer separable, and part of the coherence initially lo-cated in the system now resides in the non-local correlation between systemand the environmental particle; it is lost once the environment is disregarded.A complementary point of view argues that the interaction constitutes an in-formation transfer from the system to the environment. The more the overlapin (7) differs in magnitude from unity, the more an observer could in prin-ciple learn about the system state by measuring the environmental particle.Even though this measurement is never made, the complementarity princi-ple then explains that the wave-like interference phenomenon characterizedby the coherences vanishes as more information discriminating the distinct,particle-like system states is revealed.

To nish the introduction, here is a collection of other characteristics andpopular statements about the decoherence phenomenon. One often hears thatdecoherence ( i ) can be extremely fast as compared to all other relevant timescales, (ii ) that it can be interpreted as an indirect measurement process, amonitoring of the system by the environment, ( iii ) that it creates dynamicallya set of preferred states (robust states or pointer states) which seeminglydo not obey the superposition principle, thus providing the microscopic mech-anism for the emergence of effective super-selection rules, and ( iv ) that it helpsto understand the emergence of classicality in a quantum framework. Thesepoints will be illustrated in the following, though ( iii ) has been demonstratedonly for very simple model systems and ( iv ) depends to a fair extent on yourfavored interpretation of quantum mechanics.

1.2 General scattering interaction

In the above demonstration of the decoherence effect the choice of the interac-tion and the environmental state was rather special. Let us therefore now take

1

The value |mn | determines the maximal fringe visibility in a general interferenceexperiment involving the states |m and |n , as described by the projection on ageneral superposition |, = cos( ) |m +e i sin( ) |n .


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Stot and E to be arbitrary and carry out the same analysis. Performing thetrace (5) in the eigenbasis of the environmental state, E = p| |E ,we have

= tr E Stot [E ]S

tot = j, p j |Stot | E |Stot |j E

=k

Wk Wk , (8)

where the j |Stot | are operators in H. After subsuming the two indices j, into a single one, we get the second line with the Kraus operators Wkgiven by

Wk = pk j k |Stot |k (9)It follows from the unitarity of Stot that they satisfy

k

Wk

Wk

= . (10)

This implies that Eq. ( 8) is the operator-sum representation of a completelypositive map : (see Sect. 3.1). In other words, the scattering trans-formation has the form of the most general evolution of a quantum state thatis compatible with the rules of quantum theory. In the operational formula-tion of quantum mechanics this transformation is usually called a quantum operation [9], the quantum information community likes to call it a quantum channel . Conversely, given an arbitrary quantum channel, one can also con-struct a scattering operator Stot and an environmental state E giving rise tothe transformation, though it is usually not very helpful from a physical pointof view to picture the action of a general, dissipative quantum channel as dueto a single scattering event.

1.3 Decoherence as an environmental monitoring process

We are now in a position to relate the decoherence of a quantum system tothe information it reveals to the environment. Since the formulation is basedon the notion of an indirect measurement it is necessary to rst collect someaspects of measurement theory [10, 11].

(a) Elements of general measurement theory

Projective measurements

This is the type of measurement discussed in standard textbooks of quantummechanics. A projective operator | | P = P 2 = P is attributed to each


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6 Klaus Hornberger

possible outcome of an idealized measurement apparatus. The probabilityof the outcome is obtained by the Born rule

Prob( |) = tr ( P ) = || , (11)and after the measurement of the state of the quantum system is given bythe normalized projection

M(|) =P P

tr ( P ). (12)

The basic requirement that the projectors form a resolution of the identityoperator,

P = , (13)

ensures the normalization of the corresponding probability distribution Prob ( |).If the measured system property corresponds to a self-adjoint operator Athe P are the projectors into its eigenspaces, so that the expectation value is

A = tr ( A) .

If A has a continuous spectrum the outcomes are characterized by intervalsof a real parameter, and the sum in ( 13) should be replaced by a projector-valued Stieltjes integral dP () = , or equivalently by a Lebesgue integralover a projector-valued measure (PVM) [10, 11].

It is important to note that projective measurements are not the mostgeneral type of measurement compatible with the rules of quantum mechanics.In fact, non-destructive measurements of a quantum system are usually notof the projective kind.

Generalized measurements

In the most general measurement situation, a positive (and therefore her-mitian) operator F > 0 is attributed to each outcome . Again, the collectionof operators corresponding to all possible outcomes must form a resolution of the identity operator,

F = . (14)

One speaks of a positive operator-valued measure (POVM), in particular inthe case of a continuous outcome parameter,

dF() = , and the probability

(or probability density in the continuous case) of outcome is given by

Prob( |) = tr ( F ) . (15)


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The effect on the system state of a generalized measurement is described bya nonlinear transformation

M(

|) = k

M,k M,k

tr ( F )(16)

involving a norm-decreasing completely positive map in the numerator (seeSect. 3.1) and a normalization which is subject to the consistency requirement

k

M,k M,k = F . (17)

The operators M,k appearing in Eq. ( 16) are called measurement operators ,and they serve to characterize the measurement process completely. The Fare sometimes called effects or measurement elements. Note that differentmeasurement operators M,k can lead to the same measurement element F .

A simple class of generalized measurements are unsharp measurements ,where a number of projective operators are lumped together with probabilis-

tic weights in order to account for the nite resolution of a measurement deviceor for classical noise in its signal processing. However, generalized measure-ments schemes may also perform tasks which are seemingly impossible witha projective measurement, such as the error-free discrimination of two non-orthogonal states [12, 13].

Efficient measurements

A generalized measurement is called efficient if there is only a single summandin (16) for each outcome ,

M(|) =M M

tr M M , (18)

implying that pure states are mapped to pure states. In a sense, these aremeasurements where no unnecessary, that is no classical uncertainty is in-troduced during the measurement process, see the following Sect. 1.3(b). Bymeans of a (left) polar decomposition and the consistency requirement ( 17)efficient measurement operators have the form

M = U F , (19)with an unitary operator U . This way the state after efficient measurementcan be expressed in a form which decomposes the transformation into a rawmeasurement described by the F and a measurement back-action givenby the U .

M(|) = U back-actionF

F

tr ( F )

raw measurementU

back-action(20)


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8 Klaus Hornberger

In this transformation the positive operators F squeeze the statealong the measured property and expand it along the other, complementaryones, similar to what a projector would do, while the back-action operatorsU kick the state by transforming it in a way that is reversible, in principle,

provided the outcome is known. Note that the projective measurements ( 12)are a sub-class in the set of back-action-free efficient measurements.

(b) Indirect measurements

In an indirect measurement one tries to obtain information about the sys-tem in a way that disturbs it as little as possible. This is done by lettinga well-prepared microscopic quantum probe interact with the system. Thisprobe is then measured by projection, i.e., destructively, so that one can inferproperties of the system without having it brought into contact with a macro-scopic measurement device. Let probe be the prepared state of the probe, Stotdescribe the interaction between system and probe, and P be the projectorscorresponding to the various outcomes of the probe measurement. The prob-ability of measuring is determined by the reduced state of the probe afterinteraction, i.e.,

Prob( |) = tr p P probe = tr p robe P tr sys (Stot [probe ]Stot ) . (21)By pulling out the system trace (extending the projectors to Htot = HHp )and using the cyclic permutability of operators under the trace we have

Prob( |) = tr Stot [ P ]Stot [probe ] = tr( F ) (22)with microscopically dened measurement elements

F = tr probe Stot [

P ]Stot [

probe ] > 0 (23)

satisfying F = . Since the probe measurement is projective, we canalso specify the new system state conditioned on the click at of the probedetector,

M(|) = tr probe [Mtot (tot |)]= tr probe

[ P ]Stot [probe ]Stot [ P ]

tr( P [probe ])

=k

M,k M,ktr ( F )

. (24)

In the last step a convex decomposition of the initial probe state into purestates was inserted, probe = k wk |k k | . Taking P = | | we thus geta microscopic description also of the measurement operators,


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M,k = pk |Stot |k . (25)This shows that an indirect measurement is efficient (as dened above) if theprobe is initially in a pure state, i.e., if there is no uncertainty introduced

in the measurement process, apart form the one imposed by the uncertaintyrelations on probe .If we know that an indirect measurement has taken place, but do not know

its outcome we have to resort to a probabilistic (Bayesian) description of thenew system state. It is given by the sum over all possible outcomes weightedby their respective probabilities,

=

Prob( |)M(|) =,k

M,k M,k . (26)

This form is the same as above in Eqs. ( 8) and ( 9), where the basic effect of decoherence has been described. This indicates that the decoherence processcan be legitimately viewed as a consequence of the information transfer from

the system to the environment. The complementarity principle can then beinvoked to understand which particular system properties lose their quantumbehavior, namely those complementary to the ones revealed to the environ-ment. This monitoring interpretation of the decoherence process will helpus below to derive microscopic master equations.1.4 A few words on nomenclature

Since decoherence phenomena show up in quite different sub-communities of physics, a certain confusion and lack of uniformity developed in the terminol-ogy. This is exacerbated by the fact that decoherence often reveals itself asa loss of fringe visibility in interference experimentsa phenomenon, though,which may have other causes than decoherence proper. Here is an attempt of clarication:

decoherence: In the original sense, an environmental quantum effect af-fecting macroscopically distinct states. The term is nowadays applied tomesoscopically different states, as well, and even for microscopic states, aslong as it refers to the quantum effect of environmental, i.e. in practiceunobservable, degrees of freedom.However, the term is often (ab-)used for any process other reducing thepurity of a micro-state.

dephasing: In a narrow sense, this describes the phenomenon that coher-ences, i.e., the off-diagonal elements of the density matrix, get reduced in aparticular basis, namely the energy eigenbasis of the system. It is a state-ment about the effect, and not the cause. In particular, dephasing maybe reversible if it is not due to decoherence, as revealed e.g. in spin-echoexperiments.This phrase should be treated with great care since it is used differentlyin various sub-communities. It is taken as a synonym to dispersion in


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10 Klaus Hornberger

molecular physics and in nonlinear optics, as a synonym to decoherence in condensed matter, and often as a synonym to phase averaging inmatter wave optics. It is also called a T 2-process in NMR and in condensedmatter physics (see below).

phase averaging: A classical noise phenomenon entering through the de-pendence of the unitary system evolution on external control parameterswhich uctuate (parametric average over unitary evolutions). A typicalexample are the vibrations of an interferometer grating or the uctuationsof the classical magnetic eld in an electron interferometer due to techni-cal noise. Empirically, phase averaging is often hard to distinguish fromdecoherence proper.

dispersion: Coherent broadening of wave packets during the unitary evo-lution, e.g. due to a velocity dependent group velocity or non-harmonicenergy spacings. This unitary effect may lead to a reduction of signal os-cillations, for instance, in molecular pump-probe experiments.

dissipation: Energy exchange with the environment leading to thermal-ization. Usually accompanied by decoherence, but see Sect. 3.4(b) for acounter-example.

2 Case study: Dephasing of qubits

So far, the discussion of the temporal dynamics of the decoherence process wascircumvented by using a scattering description. Before going to the generaltreatment of open quantum systems in Sect. 3 it is helpful to take a closerlook on the time evolution of a special system, where the interaction with amodel environment can be treated exactly [14, 2].

2.1 An exactly solvable model

Let us take a two-level system, or qubit, described by the Pauli spin operatorz , and model the environment as a collection of bosonic eld modes. Inpractice, such elds can yield an appropriate effective description even if theactual environment looks quite differently, in particular if the environmentalcoupling is a sum of many small contributions 2 . What is fairly non-generic inthe present model is the type of coupling between system and environment,which is taken to commute with the system Hamiltonian.

The total Hamiltonian thus reads

Htot =2

z +k

k bk bk

H 0

+ zk

gk bk + gk bk

H int

(27)

2 A counter-example would be the presence of a degenerate environmental degreeof freedom, such as a bistable uctuator.


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with the usual commutation relation for the mode operators of the bosoniceld modes, [bi , bk ] = ik , and coupling constants gk . The fact that the sys-tem Hamiltonian commutes with the interaction, guarantees that there is noenergy exchange between system and environment so that we expect pure

dephasing.By going into the interaction picture one transfers the trivial time evo-lution generated by H0 to the operators (and indicates this with a tilde). Inparticular,

Hint (t) = e iH 0 t/ Hint e iH 0 t/ = zk

(gk eik t bk + g

k e ik t bk ) , (28)

where the second equality is granted by the commutation [ z , Hint ] = 0. Thetime evolution due to this Hamiltonian can be formally expressed as a Dysonseries,

U(t) = T exp i

t

0dt Hint (t )

=

n =0

1n!

1i

n

t

0dt1 dtn T Hint (t1) Hint (tn ) , (29)

where T is the time ordering operator (putting the operators with largertime arguments to the left). Due to this time ordering requirement the seriesusually cannot be evaluated exactly (if it converges at all). However, in thepresent case the commutator of Hint at different times is not an operator, but just a c-number,

[Hint (t) , Hint (t )] = 2ik

|gk |2 sin( k (t t)) . (30)

As a consequence, the time evolution differs only by a time-dependent phase

from the one obtained by casting the operators in their natural order,

U(t) = e i( t ) exp i t0 dt Hint (t ) . (31)

This is discussed in the note 3 , which also shows that the phase is given by

(t) =i

2 2 t0 dt1 t0 dt2 (t1 t2) Hint (t1) , Hint (t2) . (32)3 To obtain the time evolution eU (t) for the case [eH (t) , eH (t )] = c dene theoperators

(t) = 1 Z t0

dt

eH

`t

and U (t) = exp[i (t)]eU (t). This way U (t) describes the additional motion dueto the time ordering requirement. It satises


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12 Klaus Hornberger

One can now perform the integral over the interaction Hamiltonian to get

U(t) = e i( t ) exp12

zk

k (t)bk k (t)bk (33)

with complex, time-dependent functions

k (t) := 2 gk1 eik t

k. (34)

The operator U(t) is diagonal in the eigenbasis of the system, and it describeshow the environmental dynamics depends on the state of the system. In par-ticular, if the system is initially in the upper level, | = | , one has

U(t)| |0 E = e i( t ) | k

Dk k (t)

2 |0 =: ei( t ) | | (t) E , (35)

and for the lower state

U(t)| |0 E = e i( t )| k

Dk k (t)

2 |0 =: ei( t )| | (t) E . (36)

Here we introduced the unitary displacement operator s for the k-th eld mode,

Dk () = exp( bk bk ), (37)which effect a translation of the eld state in its attributed phase space. Inparticular, the coherent state | k of the eld mode k is obtained from itsground state |0 k by | k := Dk ()|0 k [15]The equations ( 35) and ( 36) show that the collective state of the eldmodes gets displaced by the interaction with the system and that the sense

of the displacement is determined by the system state. t U (t) = ddt ei( t )e i( t ) + 1i ei ( t )eH (t) e

i ( t )U (t) .The derivative in square brackets has to be evaluated with care since the eH (t) donot commute at different times. By rst showing that [ A , t A ] = c implies t A n = nA n 1 t A 12 n (n 1) cA n 2 one nds

ddt

ei( t ) = 1i

ei ( t )eH (t) +1

2 2ei( t ) Z t

0dt eH `t

, eH (t).Therefore, we have t U (t) = `2 2 1 R t0 dt [eH (t ) , eH (t)] U (t), which can be inte-grated to yield nally

eU (t) = exp 12 2 Z t

0dt1 Z

t 1

0dt2 heH (t1 ) , eH (t2 )ie

i ( t ) .


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Assuming that the states of system and environment are initially uncor-related, tot (0) = E , the time-evolved system state reads

4

(t) = tr E U(t)[E ]U(t) . (38)

It follows from (35) and ( 36) that the populations are unaffected,

|(t)| = |(0)| , |(t)| = |(0)| ,

while the coherences are suppressed by a factor which given by the trace overthe displaced initial eld state,

|(t)| = |(0)| tr Ek

Dk ( k (t))E

( t )

. (39)

Incidentally, the complex suppression factor (t) is equal to the Wigner char-acteristic function of the original environmental state at the points k (t), i.e.,it is given the Fourier transform of its Wigner function [16].

(a) Initial vacuum state

If the environment is initially in its vacuum state, E = k |0 0|k , the| (t) E and | E dened in ( 35), (36) turn into multi-mode coherent states,and the suppression factor can be calculated immediately to yield:

vac (t) =k

0|Dk ( k (t))|0 k =k

exp | k (t)|

2

2

= exp k

4 |gk |2 1 cos(k t)22k (40)

For times that are short compared to the eld dynamics, t 1k , one ob-

serves a Gaussian decay of the coherences. Modications to this become rele-vant at k t= 1, provided (t) is then still appreciable, i.e., for 4 |gk |

2 / 22k1. Being a sum over periodic functions, vac (t) is quasi-periodic, that is, it willcome back arbitrarily close to unity after a large period (which increases ex-ponentially with the number of modes). These somewhat articial Poincarerecurrences vanish if we replace the sum over the discrete modes by an integralover a continuum with mode density ,

4

In fact, the assumption tot (0) = E is quite unrealistic if the coupling isstrong, as discussed below. Nonetheless, and it certainly represents a valid initial

state.


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14 Klaus Hornberger

k

f (k ) 0 d()f (), (41)for any function f . This way the coupling constants gk get replaced by thespectral density

,J () = 4 () |g()|

2 . (42)

This function characterizes the environment by telling how effective the cou-pling is at a certain frequency.(b) Thermal state

If the environment is in a thermal state with temperature T ,

E = th =e H E /k B T

tr e H E /k B T =

k

1 e k /kT

n =0e k n/k B T |n n|k

= ( k )th

,

(43)the suppression factor reads 5

(t) =k

tr Dk ( k (t))(k )th =

k

exp | k (t)|

2

2coth

k2kB T

. (44)

This factor can be separated into its vacuum component ( 40) and a ther-mal component, (t) = e F vac ( t ) e F th ( t ) , with the following denitions of thevacuum and the thermal decay functions:

F vac (t) :=k

4 |gk |21 cos(k t)22k (45)

F th (t) :=k

4

|gk

|2 1 cos(k t)2

2k

cothk

2kB T 1 (46)

2.2 The continuum limit

Assuming that the eld modes are sufficiently dense we replace their sum byan integration. Noting ( 41), (42) we have

F vac (t) 0 dJ () 1 cos(t)22 (47)F th (t) 0 dJ () 1 cos(t)22 coth 2kB T 1 . (48)

5 This can be found in a small exercise by using the Baker-Hausdorff relation withexp

` b b

= exp( | |2 /2)exp

` b

exp( b ), and the fact that coher-

ent states satisfy the eigenvalue equation b | = | , have the number rep-resentation n | = exp( | |2 / 2) n / n!, and form an over-complete set with = 1 R d2 | |.


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So far, the treatment was exact. To continue we have to specify the spectraldensity in the continuum limit. A typical model takes g

, so that thespectral density of a d-dimensional eld can be written as [17]

J () = a c

d 1

e / c (49)

with damping strength a > 0. Here, c is a characteristic frequency cutoffwhere the coupling decreases rapidly, such as the Debye frequency in the caseof phonons.

(a) Ohmic coupling

For d = 1 the spectral density ( 49) increases linearly at small (Ohmiccoupling ). One nds

F vac (t) =a

2 2log(1 + 2c t

2), (50)

which bears a strong c dependence. Evaluating the second integral requires toassume that the cutoff c is large compared to the thermal energy, kT c :

F th (t)a2 log

sinh( t/t T )t/t T

(51)

Here tT = / (k B T ) is a thermal quantum time scale. The correspondingfrequency 1 = 2 /t T is called the (rst) Matsubara frequency , which alsoshows up if imaginary time path integral techniques are used to treat theinuence of bosonic eld couplings [17].

For large times the decay function F th (t) shows the asymptotic behavior

F th (t)

a2

t

tT [as t

]. (52)

It follows that the decay of coherence is characterized by rather differentregimes. In the short time regime ( t < 1c ) we have the perturbative behavior

F (t)a

2 22c t

2 [for t 1c ], (53)

which can also be obtained from the short-time expansion of the time evolutionoperator. Note that the decay is here determined by the overall width c of the spectral density. The intermediate region, 1c < t <

11 , is dominated

by F vac (t) and called the vacuum regime ,

F (t)a2 log(c t) [for

1c t

11 ].

Beyond that, for large times the decay is dominated by the thermal suppres-sion factor,


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16 Klaus Hornberger

F (t)a2

ttT

[for 11 t]. (54)

In this thermal regime the decay shows the exponential behavior typical forthe Markovian master equations discussed below. Note that the decay ratefor this long time behavior is determined by the low frequency behavior of the spectral density, characterized by the damping strength a in (49), and isproportional to the temperature T .

(b) Super-Ohmic coupling

For d = 3, the case of a super-Ohmic bath, the integrals ( 47), (48) canbe calculated without approximation. We note only the long-time behavior of the decay.

limt

F (t) = 2 akB T

c

2

1 +kB T

c< (55)

Here (z) stands for the Digamma function, the logarithmic derivative of thegamma function. Somewhat surprisingly, the coherences do not get completelyreduced as t , even at a nite temperature. This is due to the suppressedinuence of the important low-frequency contributions to the spectral densityin three dimensions (as compared to lower dimensions). While such a suppres-sion of decoherence is plausible for intermediate times, the limiting behavior(55) is clearly a result of our simplied model assumptions. It will be absent if there is a small anharmonic coupling between the bath modes [ 18] or if thereis a small admixture of different couplings to Hint .

2.3 Dephasing of N qubits

Let us now discuss the generalization to the case of N qubits, which do notinteract directly among each other. Each qubit may have a different couplingto the bath modes.

Htot =N 1

j =0

j2

( j )z +k

k bk bk

H 0+

N 1

j =0

( j )zk

g( j )k bk + [g

( j )k ]

bk (56)

Similar to above the time evolution in the interaction picture reads

U(t) = e i( t ) exp12

N 1

j =0

( j )zk

( j )k bk [

( j )k ]

bk ,

where the displacement of the eld modes now depends on the N -qubit state.As an example, take N = 2 qubits and only a single vacuum mode. Forthe states


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| = c11 |+ c00 | (57)and

| = c10

|+ c01

|(58)

we obtain

U| |0 E = c11 | | (1) (t) + (2) (t)

2 E+ c00 | |

(1) (t) (2) (t)2 E

and similarly

U| |0 E = c10 | | (1) (t) (2) (t)

2 E+ c01 | |

(1) (t) + (2) (t)2 E

,

where the (1) (t) and (2) (t) are the eld displacements ( 34) due to the rstand the second qubit.

If the couplings to the environment are equal for both qubits, say, because

they are all sitting in the same place and seeing the same eld, we have (1) (t) = (2) (t) (t). In this case states of the form | are decoheredonce the factor (t) | (t) E = exp( 2 | (t)|

2 ) 0. States of the form

| , on the other hand, are not affected at all, and one says that the {| }span a (two-dimensional) decoherence-free subspace . It shows up because theenvironment cannot tell the difference between the states |and |if itcouples only to the sum of the excitations.

In general, using the binary notation, e.g., | |1012 = |5 , one hasm|(t)|n = m|(0)|n (59)

tr expN 1

j =0

(m j n j )k

( j )k (t)bk [

( j )k (t)]

bk th

where m j {0, 1}indicates the j -th digit in the binary representation of thenumber m.We can distinguish different limiting cases:

(a) Qubits feel the same reservoir

If the separation of the qubits is small compared to the wave lengths of theeld modes they are effectively interacting with the same reservoir, ( j )k = k .One can push the j -summation to the s in this case, so that, compared tothe single qubit, one merely has to replace k by (m j n j ) k . We nd

mn (t) = exp

N 1

j =0(m j

n j )

2(F vac (t) + F th (t)) (60)

with F vac (t) and F th (t) given by Eqs. ( 47) and (48).


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18 Klaus Hornberger

Hence, in the worst case, one observes an increase of the decay rate byN 2 compared to the single qubit rate. This is the case for the coherencebetween the states |0 and |2N 1 , which have the maximum difference inthe number of excitations. On the other hand, the states with an equal numberof excitations form a decoherence-free subspace in the present model, with amaximal dimension of N N/ 2 .

(b) Qubits see different reservoirs

In the other extreme, the qubits are so far apart from each other that eacheld mode couples only to a single qubit. This suggests a re-numbering of theeld modes,

( j )k k j ,and leads, after transforming the j -summation into a tensor-product, to

mn (t) =N 1

j =0tr

k j

Dk j (m j

n j ) k j (t)

(k j )

th

=N 1

j =0

exp |m j n j |2

= | m j n j |(F vac (t) + F th (t))

= exp N 1

j =0|m j n j |

Hamming distance(F vac (t) + F th (t)) . (61)

Hence, the decay of coherence is the same for all pairs of states with thesame Hamming distance. In the worst case, we have an increase by a factorof N compared to the single qubit case, and there are no decoherence-freesubspaces.

An intermediate case is obtained if the coupling depends on the position r jof the qubits. A reasonable model, corresponding to point scatterings of eldswith wave vector k , is given by g(j )k = gk exp(i k r j ), and its implications arestudied in [ 19].

The model for decoherence discussed in this section is rather exceptionalin that the dynamics of the system can be calculated exactly for some choicesof the environmental spectral density. In general, one has to resort to approx-imate descriptions for the dynamical effect of the environment, and we turnto this problem in the following section.


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3 Markovian dynamics of open quantum systems

Isolated systems evolve, in the Schr odinger picture and for the general case of mixed states, according to the von Neumann equation,

t =1i

[H, ]. (62)

One would like to have a similar differential equation for the reduced dynamicsof an open quantum system, which is in contact with its environment. If weextend the description to include the entire environment HE and its couplingto the system, then the total state in Htot = HHE evolves unitarily. Thepartial trace over HE gives the evolved system state, and its time derivativereads

t =ddt

tr E Utot (t)tot (0)Utot (t) =1i

tr E ([Htot , tot ]) . (63)

This exact equation is not closed and therefore not particularly helpful as itstands. However, it can be used as the starting point to derive approximatetime evolution equations for , in particular, if it is permissible to take theinitial system state to be uncorrelated with the environment.

These equations are often non-local in time, though, in agreement withcausality, the change of the state at each point in time depends only on thestate evolution in the past. In this case, the evolution equation is called ageneralized master equation . It can be specied in terms of superoperator-functionals, i.e., linear operators which take the density operator with itspast time-evolution until time t and map it to the differential change of theoperator at that time,

t = K[{ : < t }] . (64)An interpretation of this dependence on the systems past is that the environ-ment has a memory , since it affects the system in a way which depends onthe history of the system-environment interaction. One may hope that on acoarse-grained time-scale, which is large compared to the inter-environmentalcorrelation times, these memory effects might become irrelevant. In this case, aproper master equation might be appropriate, where the innitesimal changeof depends only on the instantaneous system state, through a Liouvillesuper-operator L, t = L. (65)Master equations of this type are also called Markovian , because of their re-semblance to the differential Chapman-Kolmogorov equation for a classicalMarkov process. However, since a number of approximations are involved intheir derivation, it is not clear whether the corresponding time evolution re-

spects important properties of a quantum state, such as its positivity. We willsee that these constraints restrict the possible form of Lrather strongly.


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20 Klaus Hornberger

3.1 Quantum dynamical semigroups

The notion a of quantum dynamical semigroup permits a rigorous formulationof the Markov assumption in quantum theory. To introduce it we rst need anumber of concepts from the theory of open quantum systems [ 11, 20, 21, 22].

(a) Dynamical maps

A dynamical map is a one-parameter family of trace-preserving, convex linear,and completely positive maps (CPM)

W t : 0 t , for t +0 (66)satisfying W 0 = id. As such it yields the most general description of a timeevolution which maps an arbitrary initial state 0 to valid states at later times.

Specically, the condition of trace preservation guarantees the normaliza-tion of the state,

tr ( t ) = 1 ,and the convex linearity, i.e.,

W t ( 0 + (1 ) 0) = W t (0) + (1 ) W t (0) for all 0 1ensures that the transformation of mixed states is consistent with the classicalnotion of ignorance. The nal requirement of complete positivity is strongerthan mere positivity of W t (0). It means that in addition all the tensor prod-uct extensions of W t to spaces of higher dimension, dened with the identitymap id ext , are positive,

W tidext > 0,that is, the image of any positive operator in the higher dimensional spaceis again a positive operator. This guarantees that the system state remainspositive even if it is the reduced part of a non-separable state evolving in ahigher dimensional space.

(b) Kraus representation

Any dynamical map admits a operator-sum representation of the form ( 8)[22],

W t () =N

k=1

Wk (t)Wk (t) , (67)

with the completeness relation 6

6

In case of a trace-decreasing , convex linear, completely positive map the condition(68) is replaced by Pk W k (t)W k (t) < , i.e., the operator Pk W k (t)W k (t)must be positive.


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N

k=1

Wk (t)Wk (t) = . (68)

The number of the required Kraus operators Wk (t) is limited by the dimensionof the system Hilbert space, N dim(H)2 (and conned to a countable setin case of an innite-dimensional, separable Hilbert space), but their choice isnot unique.

(c) Semigroup assumption

We can now formulate the assumption that the {W t : t +0 }form acontinuous dynamical semigroup 7 [20, 22]:

W t 2 (W t 1 ())!= W t 1 + t 2 for all t1 , t 2 > 0 (69)

This statement is rather strong, and it is certainly violated for truly micro-scopic times. But it seems not unreasonable on the level of a coarse-grainedtime scale, which is long compared to the time it takes for the environmentto forget the past interactions with the system due to the dispersion of correlations into the many environmental degrees of freedom.

For a given dynamical semigroup there exists, under rather weak condi-tions, a generator, i.e., a superoperator Lsatisfying

W t = e L t for t > 0. (70)In this case W t () is the formal solution of the Markovian master equation(65).(d) Dual maps

So far we used the Schr odinger picture, i.e., the notion that the state of anopen quantum system evolves in time, t = W t (0). Like in the description of closed quantum systems, one can also take the Heisenberg point of view, wherethe state does not evolve, while the operators A describing observables acquirea time dependence. The corresponding map W t : A0 At is called the dual map , and it is related to W t by the requirement tr( AW t ()) = tr( W t (A)).In case of a dynamical semigroup, W t = exp Lt , the equation of motiontakes the form t A = LA, with the dual Liouville operator determined bytr( AL()) = tr( L (A)). From a mathematical point of view, the Heisenbergpicture is much more convenient since the observables form an algebra, andit is therefore preferred in the mathematical literature.

7 The inverse element required for a group structure is missing for general, irre-versible CPMs.


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22 Klaus Hornberger

3.2 The Lindblad form

We can now derive the general form of the generator of a dynamical semigroup,taking dim( H) = d < for simplicity [2, 22]. The bounded operators on Hthen form a d

2

-dimensional vector space which turns into a Hilbert space, if equipped with the Hilbert-Schmidt scalar product ( A, B) := tr( AB).Given an orthonormal basis of operators {Ej : 1 j d2}L (H),

(Ei , Ej ) := tr( Ei Ej ) = ij , (71)

any Hilbert-Schmidt operator Wk can be expanded as

Wk =d2

j =1

(Ej , Wk )Ej . (72)

We can choose one of the basis operators, say the d2-th, to be proportionalto the identity operator,

Ed2 =1d (73)

so that all other basis elements are traceless,

tr( Ej ) =0 for j = 1 , . . . , d 2 1d for j = d2 . (74)

Representing the superoperator of the dynamical map ( 67) in the {Ej }basiswe haveW t () =

d2

i,j =1

cij (t)Ei Ej (75)

with a time dependent, hermitian and positive coefficient matrix,

cij (t) =N

k=1

(Ei , Wk (t))( Ej , Wk (t)) . (76)

(Positivity can be checked in a small calculation.) We can now calculate thesemigroup generator in terms of the differential quotient by writing the termsincluding the element Ed2 separately.


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L = lim 0 W ()

= lim 0

1d cd2 d2 ( ) 1

c0 + lim

0

d2 1

j =1

cjd 2 ( )d Ej

BL (H ) + lim

0

d2 1

j =1

cd2 j ( )d Ej

B L (H )+

d2 1

i,j =1

lim 0

cij ( )

ij Ei Ej

= c0 + B + B +d2 1

i,j =1

ij Ei Ej

=1i

[H, ] +1

(G + G) +d2 1

i,j =1

ij Ei Ej (77)

In the last equality the following hermitian operators with the dimension of an energy were introduced:

G =2

(B + B + c0)

H =2i

(B B)By observing that the conservation of the trace implies tr( L) = 0 one canrelate the operator G to the matrix = ( ij ), since

0 = tr( L) = 0 + tr2G

+d2 1

i,j =1

ij Ej Ei

must hold for all . It follows that

G = 2d2 1

i,j =1 ij Ej Ei .

This leads to the rst standard form for the generator of a dynamical semi-group:

L =1i

[H, ]

unitary part

+d2 1

i,j =1

ij Ei Ej 12

Ej Ei 12

Ej Ei

incoherentpart

(78)

The complex coefficients ij have dimensions of frequency and constitute apositive matrix .


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24 Klaus Hornberger

The second standard form or Lindblad form is obtained by diagonaliz-ing the coefficient matrix . The corresponding unitary matrix U satisfy-ing UU = diag( 1 , . . . , d2 1) allows to dene the dimensionless operatorsLk :=

d2 1j =1 Ej U

jk so that Ej =

d2 1k=1 Lk U kj and therefore

8

L =1i

[H, ] +N

k=1

k Lk Lk 12

Lk Lk 12

Lk Lk (79)

with N d2 1. This shows that the general form of a generator of a dy-namical semigroup is specied by a single hermitian operator H, which is notnecessarily equal to the Hamiltonian of the isolated system, see below, and atmost d2 1 arbitrary operators Lk with attributed positive rates k . Theseare called Lindblad operators or jump operators , a name motivated in thefollowing section.

Lindblad showed in 1976 that this is also valid for innite-dimensionalsystems as long as the generator Lis bounded (which is usually not the case).

The Lindblad operators for a given generator are not unique. In fact theequation is invariant under the transformation

Lk Lk + ck (80)H H + 2i

j

cj Lj cj Lj + |c0|, (81)

so that the Lk can be chosen traceless. In this case the only remaining freedomis unitary rotation,

i Li j

U ij j Lj . (82)

If Lshows an additional invariance (e.g. with respect to rotations or transla-tions) the form of the Lindblad operators will be further restricted [23].

3.3 Unravelling of Markovian master equations

Generally, if we write the Liouville superoperator Las the sum of two parts,L0 and S , then the formal solution ( 70) of the master equation (65) can beexpressed as

8 It is easy to see that the dual Liouville operator discussed in Sect. 3.1(d) reads,in Lindblad form,

L (A ) = 1

i[A , H ] +

N

Xk =1

k

L

kAL k

1

2

L k

L k A

1

2

AL k

L k

.

Note that this implies L ( ) = 0, while L( ) = Pk k [L k , L k ] and tr( LX ) = 0.


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W t = e (L 0 + S ) t =

n =0

tn

n!(L0 + S )

n

=

n =0

k0 ,...,k n =0

tn + P j k j

n + j kj !Lkn0

SLkn 10

S SLk10

SLk00

n times= n =0 t0 dtn t n0 dtn 1 t 20 dt1

k0 ,...,k n =0

(t tn )kn

kn !(tn tn 1)

kn 1

kn 1! (t1 0)

k0

k0 !

Lkn0 SLkn 10 S SLk10 SLk00

= e L 0 t +

n =1 t0 dtn t n0 dtn 1 t 20 dt1eL 0 ( t t n )S eL 0 ( t n t n 1 )S eL 0 ( t 2 t 1 )S eL 0 t 1 . (83)

The step from the second to the third line, where tn + P j k j / (n + j kj )! isreplaced by n time integrals can be checked by induction.

The form (83) is a generalized Dyson expansion, and the comparison withthe Dyson series for unitary evolutions suggests to view exp ( L0 ) as an un-perturbed evolution and S as a perturbation, such that the exact timeevolution W t is obtained by an integration over all iterations of the perturba-tion, separated by the unperturbed evolutions.

The particular Lindblad form ( 79) of the generator suggests to introducethe completely positive jump superoperators

Lk = k Lk Lk , (84)

along with thenon-hermitian

operator

H

= H i2

N

k=1

k Lk Lk . (85)

The latter has a negative imaginary part, Im( H

) < 0, and can be used toconstruct

L0 =1i

H

H . (86)It follows that the sum of these superoperators yields the Liouville operator(79)

L= L0 +N

k=1Lk . (87)


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26 Klaus Hornberger

Of course, neither L0 nor S = N k=1 Lk generate a dynamical semigroup.Nonetheless, they are useful since the interpretation of Eq. ( 83) can now betaken one step further. We can take the point of view that the Lk with k 1describe elementary transformation events due to the environment (jumps),which occur at random times with a rate k . A particular realization of n suchevents is specied by a sequence of the form

R tn = ( t1 , k1 ; t2 , k2 ; . . . . ; tn , kn ). (88)

The attributed times satisfy 0 < t 1 < . . . . < t n < t , and the kj {1, . . . , N }indicate which kind of event took place. We call R tn a record of length n.The general time evolution W t can thus be written as an integration overall possible realizations of the jumps, with the free evolution exp( L0 ) inbetween.

W t = e L 0 t +

n =1 t0 dtn t n0 dtn 1 t 20 dt1 (89){R n } e

L 0 ( t t n )

Lkn eL 0 ( t n t n 1 )

Lkn 1 eL 0 ( t 2 t 1 )

Lk1 eL 0 t 1

K R tnAs a result of the negative imaginary part in ( 85) the exp( L0 ) are tracedecreasing 9 completely positive maps,

eL 0 = exp i

H

expi

H

> 0 (90)

dd

tr eL 0 = tr L0 eL 0 = N

k=1

tr( Lk eL 0 > 0) < 0 (91)

It is now natural to interpret tr eL 0 t as the probability that no jump occurs

during the time interval t ,Prob R t0 | := tr eL 0 t . (92)

To see that this makes sense, we attribute to each record of length n a n-timeprobability density. For a given record R tn we dene

prob R tn | := tr KR tn , (93)in terms of the superoperators from the second line in ( 89),

KR tn := e L 0 ( t t n )Lkn eL 0 ( t n t n 1 )Lkn 1 eL 0 ( t 2 t 1 )Lk1 eL 0 t 1 . (94)This is reasonable since the KR tn are completely positive maps that do notpreserve the trace. Indeed, the probability density for a record is thus de-termined both by the corresponding jump operators, which involve the rates

9 See the note in Sect. 3.1(b).


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k , and by the e L 0 , which account for the fact that the likelihood for theabsence of a jump decreases with the length of the time interval.

This notion of probabilities is consistent, as can be seen by adding theprobability ( 92) for no jump to occur during the interval (0; t) to the integral

over the probability densities ( 93) of all possible jump sequences. As required,the result is unity,

Prob R t0 | +

n =1 t0 dtn t n0 dtn 1 t 20 dt1 {R tn } prob R tn | = 1for all and t 0. This follows immediately from the trace preservation of the map (89).

It is now natural to normalize the transformation dened by the KR tn .Formally, this yields the state transformation conditioned to a certain recordR tn . It is called a quantum trajectory ,

T |Rtn := K

R tn

tr KR tn . (95)Note that this denition comprises the trajectory corresponding to a null-record R t0 , where KR t0 = exp ( L0 t). These completely positive, trace-preserving,non linear maps T (|R tn ) are dened for all states that yield a niteprobability (density) for the given record R tn , i.e., if the denominator in ( 95)does not vanish.

Using these notions the exact solution of a general Lindblad master equa-tion ( 83) may thus be rewritten in the form

t = Prob R t0 T |R t0+

n =1

t

0

dtn

t n

0

dtn 1

t 2

0

dt1{R n }

prob R tn

|

T

|R tn .(96)

It shows that the general Markovian quantum dynamics can be understoodas a summation over all quantum trajectories T (|R tn ) weighted by theirprobability (density). This is called a stochastic unravelling of the masterequation. The set of trajectories and their weights are labeled by the possiblerecords ( 88), and determined by the Lindblad operators of the master equation(79).

The semigroup property described by the master equation shows up if arecord is formed by joining the records of adjoining time intervals, (0; t ) and(t ; t),

R (0; t )n + m := ( R(0; t )n ; R

(t ;t )m ). (97)

As one expects, the probabilities and trajectories satisfy


28/57

28 Klaus Hornberger

prob( R (0; t )n + m |) = prob( R(0; t )n |) prob( R

(t ;t )m |T (|R

(0; t )n )) (98)

and

T (|R(0; t )n + m ) = T (T (|R

(0; t )n )|R

(t ;t )m ). (99)

Note nally, that the concept of quantum trajectories ts seamlessly intothe framework of generalized measurements discussed in Sect. 1.3(a). In par-ticular, the conditioned state transformation T (|R tn ) has the form ( 18) of anefficient measurement transformation,

T |R tn =MR tn M

R tn

tr MR tn MR tn (100)

with compound measurement operators

MR tn

:= e iH ( t t n ) / Lkn

Lk2 e iH

( t 2 t 1 ) / Lk1 e iH

t 1 / . (101)

This shows that we can legitimately view the open quantum dynamics gen-erated by Las due to the continuous monitoring of the system by the en-vironment. We just have to identify the (aptly named) record R tn with thetotal outcome of a hypothetical, continuous measurement during the interval(0; t). The jump operators Lk then describe the effects of the correspondingelementary measurement events 10 (clicks of counter k). Since the absenceof any click during the waiting time may also confer information aboutthe system, this lack of an event constitutes a measurement, as well, whichis described by the non-unitary operators exp ( iH / ). A hypothetical de-mon, who has the full record R tn available, would then be able to predict thenal state T (|R tn ). In the absence of this information we have to resort tothe probabilistic description ( 96) weighting each quantum trajectory with its(Bayesian) probability.

We can thus conclude that the dynamics of open quantum dynamics, andtherefore decoherence, can in principle be understood in terms of an infor-mation transfer to the environment. Apart from this conceptual insight, theunravelling of a maser equation provides also an efficient stochastic simula-tion method for its numerical integration. In these quantum jump approaches[24, 25, 26], which are based on the observation that the quantum trajectory(95) of a pure state remains pure, one generates a nite ensemble of trajec-tories such that the ensemble mean approximates the solution of the masterequation.

10 Keep in mind that the Lk are not uniquely specied by a given generator L, seeEqs. ( 80)-( 82). Different choices of the Lindblad operators lead to different un-ravellings of the master equation, so that these hypothetical measurement eventsmust not be viewed as real.


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3.4 Exemplary master equations

Let us take a look at a number of very simple Markovian master equations 11 ,which are characterized by a single Lindblad operator L (together with a her-

mitian operator H). The rst example gives a general description of dephasing,while the others are empirically known to describe dissipative phenomena re-alistically. We may then ask what they predict about decoherence.

(a) Dephasing

The simplest choice is to take the Lindblad operator to be proportiona l12 tothe Hamiltonian of a discrete quantum system, i.e., to the generator of theunitary dynamics, L = H. The Lindblad equation

t t =1i

[H, t ] + Ht H 12

H2t 12

t H2 (102)

is immediately solved in the energy eigenbasis, H =m

E m

|m m

|:

mn (t) m|t |n = mn (0)exp iE m E n t

2

(E m E n )2 t (103)As we expect from the discussion of qubit dephasing in Sect. 2, the energyeigenstates are unaffected by the non-unitary dynamics if the environmentaleffect commutes with H. The coherences show the exponential decay thatwe found in the thermal regime (of times t which are long compared tothe inverse Matsubara frequency). The comparison with ( 54) indicates that should be proportional to the temperature of the environment.

(b) Amplitude damping of the harmonic oscillator

Next we choose H to be the Hamiltonian of a harmonic oscillator, H = aaand take as Lindblad operator the ladder operator, L = a. The resultingLindblad equation is known empirically to describe the quantum dynamics of a damped harmonic oscillator.

Choosing as initial state a coherent state, see Sect. 2.1 and Eq. ( 114),

0 = | 0 0| D( 0)|0 0|D(0 )= e | 0 |

2exp( 0a)|0 0|exp( 0a) (104)

we are faced with the exceptional fact that the state remains pure during theLindblad time evolution. Indeed, the solution of the Lindblad equation reads11 See also Sect. ?.1.3 in Cord Mullers contribution for a discussion of the master

equation describing of spin relaxation.12 As an exception, does not have the dimensions of a rate here (to avoid clumsynotation).


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30 Klaus Hornberger

t = | t t | (105)with

t = 0 exp it 2

t (106)

as can be veried easily using ( 104). It describes how the coherent state spiralsin phase space towards the origin, approaching the ground state as t .The rate is the dissipation rate since it quanties the energy loss, as shownby the time dependence of the energy expectation value,

t |H| t = e t 0|H| 0 . (107)Superposition of coherent states

What happens if we start out with a superposition of coherent states,

|0 =1

N (| 0 + | 0 ) (108)

with N = 2 + 2 Re 0| 0 , in particular, if the separation in phase space islarge compared to the quantum uncertainties, | 0 0| 1 ? The initialdensity operator corresponding to ( 108) reads0 =

1

N (| 0 0 |+ | 0 0 |+ c0 | 0 0|+ c0 | 0 0 |) (109)

with c0 = 1. One nds that the ansatz

t =1

N (| t t |+ | t t |+ ct | t t |+ ct | t t |) (110)

solves the Lindblad equation with ( 106) provided

ct = c0 exp

1

2 | 0

0

|2 + i Im( 0

0) 1

e t . (111)

That is, while the coherent basis states have the same time evolution as in(105), the initial coherence c0 gets additionally suppressed. For times that areshort compared to the dissipative time scale, t

1 , we have an exponentialdecay

|ct | = |c0|exp 2 | 0 0|

2

decot (112)

with a rate deco . For macroscopically distinct superpositions, where the phasespace distance of the quantum states is much larger than their uncertainties,

| 0

0

|1, deco can be much greater than the dissipation rate,

deco

=12 | 0 0|

21. (113)


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This quadratic increase of the decoherence rate with the separation betweenthe coherent states has been conrmed experimentally in a series of beautifulcavity QED experiments in Paris, using eld states with an average of 5 9photons [ 27, 28].

Given this empiric support we can ask about the prediction for a material,macroscopic oscillator. As an example, we take a pendulum with a mass of m =100 g and a period of 2 / = 1s, and assume that we can prepare it ina superposition of coherent states with a separation of x =1 cm. The modevariable is related to position and momentum by

= m2 x + i pm (114)so that we get the prediction

deco 1030 .

This purports that even with an oscillator of enormously low friction corre-sponding to a dissipation rate of =1/year the coherence is lost on a timescaleof 10 22 secin which light travels the distance of about a nuclear diameter.

This observation is often evoked to explain the absence of macroscopic su-perpositions. However, it seems unreasonable to assume that anything physi-cally relevant takes place on a timescale at which a signal travels at most bythe diameter of an atomic nucleus. Rather, one expects that the decoherencerate should saturate at a nite value if one increases the phase space distancebetween the superposed states.

(c) Quantum Brownian motion

Next, let us consider a particle in one dimension. A possible choice forthe Lindblad operator is a linear combination of its position and momentumoperators,

L =pth x +

i pth

p. (115)

Here pth is just a momentum scale, which will be related to the temperatureof the environment below. The hermitian operator is taken to be the Hamil-tonian of a particle in a potential V (x), plus a term due to the environmentalcoupling,

H =p2

2m+ V (x) +

2

(xp + px). (116)

This additional term will be justied by the fact that the resulting Lindbladequation is almost equal to the Caldeira-Leggett master equation . The latteris the high-temperature limit of the exact evolution equation following from aharmonic bath model of the environment [29, 30], see Sect. 4.1. It is empiricallyknown to describe the frictional quantum dynamics of a Brownian particle,


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32 Klaus Hornberger

and, in particular, it leads to the correct thermal state in case of quadraticpotentials.

The choices ( 115) and (116) yield the following Lindblad equation:

t t =

Caldeira Leggettmaster equation

1i [ p22m + V (x), t ] + i [x, pt + t p] dissipation

2

p2th2 [x, [x, t ]]

position localization

2

1 p2th

[p, [p, t ]] (117)

The three terms in the upper line (with pth from Eq. ( 124)) constitute theCaldeira-Leggett master equation. It is a Markovian, but not a completelypositive master equation. In a sense, the last term in ( 117) adds the mini-mal modication required to bring the Caldeira-Leggett master equation intoLindblad form [ 2].

To see the most important properties of (117) let us take a look at the timeevolution of the relevant observables in the Heisenberg picture. As discussedin Sect. 3.1(d), the Heisenberg equations of motion are determined by thedual Liouville operator L. In the present case, it takes the form

L (A) =1i

[A,p2

2m+ V (x)]

i

(p [x, A] + [x, A]p) 2

p2th2 [x, [x, A]]

2

1 p2th

[p, [p, A]]. (118)

It is now easy to see that

L (x) =pm

L

(p) = V

(x) 2 p. (119)Hence, the force arising from the potential is complemented by a frictionalforce which will drive the particle into thermal equilibrium. The fact that thisfrictional component stems from the second term in ( 117) indicates that thelatter describes the dissipative effect of the environment.

In the absence of an external potential, V = 0, the time evolution deter-mined by ( 119) is easily obtained, since L

n (p) = ( 2 )n p for n

.

pt = e L t p =

n =0

(2t )n

n!p = e 2t p [for V = 0] (120)

xt = e L t x = x +

1m

n =1

tn

n! Ln 1

(p)

= x +p pt2m

[for V = 0] (121)


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Note that, unlike in closed systems, the Heisenberg operators do not retaintheir commutator, [ xt , pt ] = i for t > 0 (since the map W t = exp Lt isnon-unitary). Similarly, ( p2)t = ( pt )

2 for t > 0, so that the kinetic energyoperator T = p2/ 2m has to be calculated separately. Noting

L (T ) = p2th2m 4 T [for V = 0] (122)

we nd similarly

T t =p2th8m

+ T p2th8m

e 4t [for V = 0] . (123)

This shows how the kinetic energy approaches a constant determined by themomentum scale pth . We can now relate pth to a temperature by equating thestationary expectation value tr ( T ) = p2th / 8m with the the average kineticenergy in a one-dimensional thermal distribution. This leads to

pth = 2 mkB T . (124)

Not always is one able to state the operator evolution in closed form. Inthose cases it may be helpful to take a look at the Ehrenfest equations fortheir expectation values. For example, given p2 t = 2 m T t , the other secondmoments, x2 t and px + xp t form a closed set of differential equations.Their solutions, given in [2], yield the time evolution of the position variance2x (t) = x2 t x 2t . It has the asymptotic form

2x (t)kB T m

t as t , (125)which shows the diffusive behavior expected of a (classical) Brownian parti-cle13 .

Let us nally take a closer look at the physical meaning of the third term in(117), which is dominant if the state is in a superposition of spatially separatedstates. Back in the Schr odinger picture we have in position representation,t (x, x ) = x|t |x ,

t t (x, x ) = 2

p2th2 (x x )2

decot (x, x ) + [the other terms]. (126)

The diagonal elements (x, x ) are unaffected by this term, so that it leavesthe particle density invariant. The coherences in position representation, onthe other hand, get exponentially suppressed,

t (x, x ) = exp ( deco t) 0 (x, x ). (127)13 Note that the denition of differs by a factor of 2 in part of the literature.


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34 Klaus Hornberger

Again the decoherence rate is determined by the square of the relevant dis-tance |x x |, deco

= 4

(x x )22th

. (128)

Like in Sect. 3.4(b), the rate deco will be much larger than the dissipativerate, provided the distance is large on the quantum scale, here given by thethermal de Broglie wavelength ,

2th =2 2

mk B T . (129)

In particular, one nds deco if the separation is truly macroscopic.Again, it seems unphysical that the decoherence rate does not saturate as

|x x | , but grows above all bounds. One might conclude from thisthat non-Markovian master equation are more appropriate on these shorttimescales. However, I will argue that (unless the environment has very specialproperties) Markovian master equations are well suited to study decoherenceprocesses, provided they involve an appropriate description of the microscopicdynamics.

4 Microscopic derivations

In this section we discuss two important and rather different strategies toobtain Markovian master equations based on microscopic considerations.

4.1 The weak coupling formulation

The most widely used form of incorporating the environment is the weakcoupling approach. Here one assumes that the total Hamiltonian is known

microscopically, usually in terms of a simplied model,Htot = H + HE + Hint

and takes the interaction part Hint to be weak so that a perturbative treat-ment of the interaction is permissible.

The main assumption, called the Born approximation , states that Hintis sufficiently small so that we can take the total state as factorized, bothinitially, tot (0) = (0)E , and also at t > 0 in those terms which involveHint to second order.

Assumption 1 : tot (t)(t)E [to 2nd order in Hint ] (130)

Here E is the stationary state of the environment, [ HE , E ] = 0. Like above,the use of the interaction picture is indicated with a tilde, cf. Eq. ( 28), so thatthe von Neumann equation for the total system reads


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t tot =1i

[Hint (t), tot (t)]

=1i

[Hint (t), tot (0)] +1

(i )2 t0 ds [Hint (t), [Hint (s), tot (s)]].(131)In the second equation, which is still exact, the von Neumann equation in itsintegral equation version was inserted into the differential equation version.Using a basis of Hilbert-Schmidt operators of the product Hilbert space, seeSect. 3.2, one can decompose the general Hint into the form

Hint (t) =k

Ak (t)Bk (t) (132)

with Ak = Ak , Bk = Bk . The rst approximation is now to replace tot (s) by

(s)E in the double commutator of (131), where Hint appears to secondorder. Performing the trace over the environment one gets

t (t) = tr E ( t tot )

=1i

k

Bk (t) E [Ak (t), (0)]

+1

(i )2k t0 ds Bk (t)B(s) E C k ( t s )

{Ak (t)A(s)(s) A(s)(s)Ak (t)}

+ h.c. (133)

All the relevant properties of the environment are now expressed in terms of the (complex) bath correlation functions C k (t s). They depend only on thetime difference t

s, since [HE , E ] = 0,

C k ( ) = tr eiH E Bk e iH E BE eiHE Bk e iH E B E . (134)This function is determined by the environmental state alone, and it is typi-cally appreciable only for a small range of around = 0.

Equation ( 133) has the closed form of a generalized master equation, butit is non-local in time, i.e., non-Markovian. Viewing the second term as asuperoperator K, which depends essentially on t s we have

t (t) =1i

[ HI (t) E , (0)]

disregarded+ t0 dsK(t s)(s) , (135)

where Kis a superoperator memory kernel of the form ( 64). We may disregardthe rst term since the model Hamiltonian HE can always be reformulatedsuch that Bk (t) E = 0.


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with k () =

12 0 dseis Bk (s)B(0) E . (141)

For times t which are large compared to the time scale given by the smallest

system energy spacings it is reasonable to expect that only equal pairs of frequencies , contribute appreciably to the sum in ( 140), since all othercontributions are averaged out by the wildly oscillating phase factor. Thisconstitutes the rotating wave approximation , our third assumption

Assumption 3 :

ei( ) t f (, )

f (, ). (142)

It is now useful to rewrite

k () =12

k () + i S k () (143)

with k () given by the full Fourier transform of the bath correlation func-tion,

k () = k () + k () = 12

dteit Bk (t)B(0)

E, (144)

and the hermitian matrix S k () dened by

S k () =12i

( k () k ()). (145)The matrix k () is positive 14 so that we end up with a master equation of the rst Lindblad form ( 78),

t (t) =1i

[HLamb , (t)] +k

k () A()(t)Ak ()

1

2A

k()A()(t)

1

2(t)A

k()A() .(147)

14 To see that the matrix () ( k ()) k, is positive we write(v , v ) = Xk vk k ()v = 12 Z dteit Xk DeiHE t/ B k (0) vk e iHE t/ B (0) vE E

= Z dteit DeiHE t/ C e iHE t/ CE E (146)with C := 1 P vB (0). One can now check that due to its particular form thecorrelation function

f (t) = DeiHE t/ C e iHE t/ CE Eappearing in ( 146) is of positive type , meaning that the n n-matrices(f (t i t j )) ij dened by an arbitrary choice of t1 , . . . . , t n and n

are pos-itive. According to Bochners theorem [31] the Fourier transform of a functionwhich is of positive type is positive, which proves the positivity of Eq. ( 146).


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38 Klaus Hornberger

The hermitian operator

HLamb =k

S k ()Ak ()A() (148)

describes a renormalization of the system energies due to the coupling withthe environment, the Lamb shift . Indeed, one nds [ H, HLamb ] = 0.

Reviewing the three approximations ( 130),(136),(142) in view of the de-coherence problem one comes to the conclusion that they all seem to be well justied if the environment is generic and the coupling is sufficiently weak.Hence, the master equation should be alright for times beyond the short-timetransient which is introduced due to the choice of a product state as initialstate. Evidently, the problem of non-saturating decoherence rates encounteredin Sects. 3.4(b) and 3.4(c) is rather due to the linear coupling assumption,corresponding to a dipole approximation, which is clearly invalid once thesystem states are separated by a larger distance than the wavelength of theenvironmental eld modes.

This shows the need to incorporate realistic, nonlinear environmental cou-plings with a nite range. A convenient way of deriving such master equationsis discussed in the next section.

4.2 The monitoring approach

The following method to derive microscopic master equations differs consid-erably from the weak coupling treatment discussed above. It is not based onpostulating an approximate total Hamiltonian of system plus environment,but on two operators, which can be characterized individually in an opera-tional sense. This permits to describe the environmental coupling in a non-perturbative fashion, and to incorporate the Markov assumption right fromthe beginning, rather than introducing it in the course of the calculation.

The approach is motivated by the observation made in Sect. 1.3 andSect. 3.3 that environmental decoherence can be understood as due to theinformation transfer from the system to the environment occurring in a se-quence of indirect measurements. In accordance with this, we will picture theenvironment as monitoring the system continuously by sending probe particleswhich scatter off the system at random times. This notion will be applicablewhenever the interaction with the environment can reasonably be described interms of individual interaction events or collisions, and it suggests a formu-lation in terms of scattering theory, like in Sect. 1.2. The Markov assumptionis then easily incorporated by disregarding the change of the environmentalstate after each collision.

When setting up a differential equation, one would like to write the tem-poral change of the system as the rate of collisions multiplied by the statetransformation due to an individual scattering. However, in general not onlythe transformed state will depend on the original system state, but also the


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Fig. 1. (a) In the monitoring approach the system is taken to interact at most withone environmental (quasi-)particle at a time, so that three-body collisions are ex-cluded. Moreover, in agreement with the Markov assumption, it is assumed that theenvironmental particles disperse their correlation with the system before scattering

again. (b) In order to consistently incorporate the state-dependence of the collisionrate into the dynamic description of the scattering process, we imagine that thesystem is monitored continuously by a transit detector, which tells at a temporalresolution t whether a particle is going to scatter off the system, or not.

collision rate, so that such a naive ansatz would yield a nonlinear equation.To account for this state dependence of the collision rate in a proper waywe will apply the concept of generalized measurements discussed in Sect. 1.3.Specically, we shall assume that the system is surrounded by a hypotheti-cal, minimally invasive detector, which tells at any instant whether a probeparticle has passed by and is going to scatter off the system, see Fig. 1.

The rate of collisions is then described by a positive operator acting inthe the system-probe Hilbert space. Given the uncorrelated state tot =

Eit determines the probability of a collision to occur in a small time intervalt ,

Prob(C t |E ) = t tr ( [E ]) . (149)Here, E is the stationary reduced single particle state of the environment.The microscopic denition of will in general involve the current densityoperator of the relative motion and a total scattering cross section, see below.

The important point to note is that the information that a collision is goingto take place changes our knowledge about the state, that is, the descriptionof the state according to the generalized measurement transformation ( 16).At the same time, we have to keep in mind that the detector is not real, butis introduced here only for enabling us to account for the state dependence

of the collision probability. It is therefore reasonable to take the detectionprocess as efficient , see Sect. 1.3(a), and minimally-invasive , i.e., U = inEq. (20), so that neither unnecessary uncertainty nor a reversible back-action


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40 Klaus Hornberger

is introduced. This implies that after a (hypothetical) detector click, but priorto scattering, the system-probe state will have the form

M( tot

|Ct ) =

1/ 2 tot 1/ 2

tr ( tot ). (150)

This measurement transformation reects our improved knowledge about theincoming two-particle wave packet, and it may be viewed as enhancing thoseparts which are heading towards a collision. Also the absence of a detec-tion event during t changes the state. This occurs with the probabilityProb Ct = 1 Prob (C t ), and the state conditioned on a null-event isgiven by 15

M tot |Ct = tot t 1/ 2 tot 1/ 2

1 t tr ( tot ). (151)

We can now form the unconditioned system-probe state after time t by tak-ing into account that the detection outcomes are not really available. Theinnitesimally evolved state is then given by the mixture of the collidingstate transformed by the S-matrix and the untransformed non-colliding one,weighted with their respective probabilities,

tot (t ) = Prob (C t | tot ) SM( tot |Ct ) S + Prob Ct | tot M tot |Ct

= S1/ 2 tot 1/ 2St + tot 1/ 2 tot 1/ 2t. (152)Splitting off the nontrivial part of the two-particle S-matrix S = + i T onends that the unitarity of S implies that

i T T = T T . (153)Using this relation we can write the differential quotient as

tot (t ) tot

t= T 1/ 2 tot 1/ 2T

12

T T 1/ 2 tot 1/ 2

12

1/ 2 tot 1/ 2 T T +i2

T + T , 1/ 2 tot 1/ 2 . (154)

It is now easy to arrive at a closed differential equation. We trace out theenvironment, assuming, in accordance with the Markov approximation, thatthe factorization tot = E is valid prior to each monitoring interval t .Taking the limit of continuous monitoring t 0 and adding the generatorH of the free system evolution we arrive at [32]

15 Although more general transformations are conceivable, this one is distinguishedby the fact that it introduces no further (time-dependent) operators.


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ddt

=1i

[H, ] +i2

Tr E T + T , 1/ 2 [E ]1/ 2

+ Tr E T 1/ 2 [E ]1/ 2T

12 Tr E 1/ 2T T 1/ 2 [

E ]

12

Tr E [E ]1/ 2T T 1/ 2 . (155)

This general evolution equation generates a dynamical semigroup since thetransformation ( 152) is completely positive. It incorporates the state depen-dence of the collision rate in terms of the operators 1/ 2 , while the operatorsT describe the individual microscopic interaction process without approxima-tion. We will see that the second term in ( 155), which involves a commutator,accounts for the renormalization of the system energies due to the couplingto the environment, just like ( 148).

So far, the discussion was very general. To obtain concrete master equa-tions one has to specify system and environment, along with the operators and S describing their interaction. In the following applications, we willassume the environment to be an ideal Maxwell gas, whose single particlestate

gas =3th

exp p2

2m(156)

is characterized by the inverse temperature , the normalization volume ,and the thermal de Broglie wave length th dened in ( 129).

4.3 Collisional decoherence of a Brownian particle

As a rst application of the monitoring approach, let us consider the local-ization of a mesoscopic particle by a gaseous environment. Specically, wewill assume that the mass M of the Brownian particle is much greater thanthe mass m of the gas particles. In this limit the energy exchange during anelastic collision vanishes, so that the mesoscopic particle will not thermalize,but we expect that the off-diagonal of the position state will get reduced bydecoherence, as discussed in Sect. 3.4(c).

This can be seen by taking the S-matrix in the limit m/M 0. In general,a collision keeps the center-of-mass invariant, and only the relative coordinatesare affected. Writing S0 for the S-matrix in the center of mass frame anddenoting the momentum eigenstates of the Brownian and the gas particle by

|P and | p , respectively, we have [ 33]S|P | p =

d3 Q |P Q | p + Q

mm

p mM

P + Q |S0|mm

p mM

P ,

where m= Mm/ (M + m) is the reduced mass. In the limit of a large Brow-nian mass we have m/m 1 and m/M 0, so that


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42 Klaus Hornberger

S|P | p d3Q |P Q | p + Q p + Q |S0| p [for M m]. (157)It follows that a position eigenstate |X of the Brownian particle remainsunaffected by a collision,

S|X |in E = |X e ip X / S0eip

X / |in E

| ( X )out E, (158)

as can be seen by inserting identities in terms of the momentum eigenstates.Here, |in E denotes an arbitrary single-particle wave packet state of a gasatom. The exponentials in ( 158) effect a translation of S0 from the origin tothe position X , so that the scattered state of the gas particle |

( X )out E depends

on the location of the Brownian particle.Just like in Sect. 1.1, a single collision will thus reduce the spatial coher-

ences X , X = X ||X by the overlap of the gas states scattered atpositions X and X , X , X = X , X (

X )out |

( X )out E . (159)

The reduction factor will be the smaller in magnitude the better the scatteredstate of the gas particle can resolve between the positions X and X .

In order to obtain the dynamic equation we need to specify the rate oper-ator. Classically, the collision rate is determined by the product of the currentdensity j = ngas vrel and the total cross section ( prel ), and therefore shouldbe expressed in terms of the corresponding operators. This is particularly sim-ple in the large mass limit M , where vrel = | p /m P /M | | p| /m , sothat the current density and the cross section depend only on the momentumof the gas particle, leading to

= ngas |p|m (p) . (160)If the gas particle moves in a normalized wave packet heading towards theorigin then the expectation value of this operator will indeed determine

introduction to decoherence theory

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