Fluctuation-dissipation relations for continuous quantum measurements

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by  Yuriy E. Kuzovlev

Fluctuation-Dissipation Relations for Continuous Quantum Measurements
Yuriy E. Kuzovlev∗
Donetsk Physics and Technology Institute, 83114 Donetsk, Ukraine
Thegeneratingfunctionalisderivedforthefluctuation-dissipationrelationswhichresultfromthe
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unitarityandreversibilityofmicroscopic dynamicsandconnectvariousstatistical characteristics of
0
many consecutive (continuous) observations in a quantum system subjected to external perturba-
0
tions. Consequences of these relations in respect to the earlier suggested stochastic representation
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of interaction between two systems are considered.

n
a PACSnumbers: 72.70.+m,05.60.Gg
J

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2 I. INTRODUCTION sion to the thermic perturbations was developed, that is
perturbations of a probability distribution (density ma-

] trix) of the system, in addition to perturbations of its
h Both the classical and quantum mechanics is time-
c Hamiltonian which are termed dynamic ones. Exam-
reversible and unitary (conserves classical phase volume
e ples of various applications of FDR can be found e.g.
and quantum probability). In thermodynamic systems
m in [10, 12, 13, 14, 15, 16, 17, 18, 19].
(ensembles), the unitarity manifests itself in strong con-
- It is desirable to combine all the infinite variety of
t nections between noise and dissipation, while the re-
a arbitrary-order FDR into a compact visual generating
versibility in time symmetry of noise and reciprocity of
t FDR for the probabilistic functionals or corresponding
s transport processes. The famous examples are the Ein-
. characteristic functionals. In the framework of classical
t stein relation [1], the Nyquist formula [2], and the On-
a mechanics, this was realized in [11, 15] (for review, see
sager reciprocity relations [3]. All these are the relations
m [10]).
between (i) second-order (quadratic) correlators of equi-
- In the quantum theory, time-differed values of any
librium noise and (ii) linear parts of complete, possi-
d interesting variable X(t) (an operator in the Heisen-
n bly nonlinear, responses to external perturbations (the
berg representation) do not commute one with another,
o expansion of the responses in a series over powers of
c “perturbing forces” is meant). Later, the fluctuation- X(t1)X(t2) 6= X(t2)X(t1). But its measurements in
[ macroscopicdevicesaresubjectedtotheclassicaldescrip-
dissipation theorem (FDT) proved by Callen and Wel- tionlanguage,beingthoughtasacommutativestochas-
1 ton [4] and the Green-Kubo formulas [5, 6] exhausted
tic process, x(t), whose values are usual c-numbers.
v this linear theory.
Hence, neither probabilistic nor characteristicfunctional
0
The first nonlinear generalizations were obtained by
3 of x(t) hasasense,untilaconcretedefinitionofall x(t)’s
Efremov [7] who proved the quadratic FDT, which con-
6 correlators (statistical moments), in terms of the X(t),
1 nects (i) quadratic parts of the responses and (ii) third- is chosen.
0 order (cubic) equilibrium correlators ((iii) linear re- In general, the two lowest-order correlators only seem
5 sponses of quadratic correlators also take a part here,
unambiguously defined:
0 but can be excluded). The next, fourth-order, relations
t/ connect together (i) cubic components of the responses, x(t) TrX(t)ρ0 ,
ma (ii)equilibriumfourth-ordercorrelatorsand,besides,(iii) x(t1)x(th2) i≡Tr(X(t1) X(t2))ρ0 , (1)
quadratic responses of quadratic non-equilibrium corre- h i≡ ◦
d- lators ((iv) linear responses of cubic correlators are in with ρ0 being the statistical operator (density matrix),
n play too, but can be excluded). Their investigation was and designatingthesymmetricproduct(Jordanprod-
◦
o startedby Stratonovich[8]who foundthat“cubicFDT” uct), A B (AB+BA)/2. The subscript “0” of ρ0
◦ ≡
c does not exist (for details and more references see e.g. meansthat ρ0 representsaquantumstatisticalensemble
v: [9, 10, 11]). Nevertheless, the fourth-order relations are atafixedtimemoment,e.g. t=0. Theright-handsides
i much useful, for instance, when analysing low-frequency presentdefinitions ofthe anglebracketsonthe leftsides,
X
fluctuations in transport and relaxation rates, especially that is effective statistics of classical image, x(t), of the
r flicker fluctuations [12, 13, 14]. quantum variable X(t).
a
The producing formulas for the whole (infinite) chain What is for the higher-order correlators
of arbitrary-orderfluctuation-dissipation and reciprocity x(t1)x(t2)...x(tN) , unfortunately, at any N > 2,
h i
relations (FDR) were obtained in [11]. In the works there are N!/2 > 1 different symmetrized (Hermitian)
[12, 15] these results were extended to non-equilibrium expressions produced by various permutations of X(tk)
1
steady states of open systems. In [10, 12] the exten- in 2Tr(X(t1)X(t2)...X(tN) + X(tN)...X(t2)X(t1))ρ0,
plus uncountably many their weighted linear combina-
tions.
Moreover, in fact even the second correlator can be
∗Electronicaddress: [email protected] introduced in an alternative way. For example, let us
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define the characteristic functional (CF) of x(t) by the II. CLASSICAL CHARACTERISTIC
identity FUNCTIONALS
exp tv(t′)x(t′)dt′ Let a classical system has canonical variables (coor-
0 ≡ (2)
≡TrDexp hR0tv(t′)X(t′)dt′i+Eln ρ0 tdeimnatuensdaenrgdoemsoamdeynntaa)miΓc =per{tqu,rbpa}t.ioGnefnreormalliyt,stohuetssidyse-,
hR i which means that its Hamiltonian, Ht = Ht(Γ), is time
Here t > 0, and v(t) is an “arbitrary probe function”
dependent. Introduce also the corresponding Liouville
(testfunction). Doubledifferentiationof(2)by v(t1)and operator L , and the evolution operator Z :
t t
v(t2) at v(t) 0 gives
≡
t
1
x(t1)x(t2) =Tr X(t1)ρα0X(t2)ρ10−αdα , (3) Lt =(∇qHt)∇p−(∇pHt)∇q , Zt =←ex−p 0 Lτdτ
h i 0 (cid:20)Z (cid:21)
Z
Eventually,weareinterestedintheevolutionandfluctu-
and N-order moments include N different ρ0’s powers
ations of variables X which represent definite functions
whose sum equals to unit. As pointed out in [10], under
of the phase space point currently occupied by the sys-
thisspecificdefinitionoftheCFalltheFDRbetweenthe
tem: X = X(Γ) (breafly, “phase functions”).
angle brackets look absolutely similar in both classical
The essential properties of the Liouville and evolution
and quantum case.
operators are as follow:
Ofcourse,tobecomepracticallyuseful,achoiceofdef-
initionoftheCF,i.e. definitionofallstatisticalmoments
A(Γ)L B(Γ)dΓ= B(Γ)L A(Γ)dΓ ,
x(t1)...x(tN) ,shouldbebasedonanalysisofrealmea- t − t
h i Z ρ(Γ)dΓ= ρ(Γ)dΓ ,
surementprocedures. Thewellknownexamplespresents R tZ−1Γ=ΓR(Γ) , (6)
quantumtheoryofelectromagneticfieldfluctuations(see RZ−1X(tΓ)Z =RXt (Z−1Γ) ,
e.g. [20]). During last decade, in the original works by t t t
Levitov,Lesovik,Nazarovandothers[21,22,23,24](see
where A(Γ), B(Γ), X(Γ) and ρ(Γ) are any phase func-
also references therein) the continuous measurements of
tions(suchthat A(Γ)B(Γ)andρ(Γ) areintegrable),and
the charge transport in electric devices were analyzed.
Γ (Γ) stand for the current values of the canonic vari-
The results of these works allow to conclude that fre- t
ables (at time t) expressed through their initial values
quently the adequate construction rule of the CF is the
Γ (at the initial time moment t = 0). In other words,
chronologicalsymmetrized product:
Γ (Γ) is the solution of the Hamilton equations under
t
exp tv(t′)x(t′)dt′ initial condition Γ0(Γ) =Γ.
0 ≡ Importantly, the latter equality in (6) is operator
Tr ←ex−p 12 0tvD(t′)X(cid:16)(tR′)dt′ ρ0 −ex→p (cid:17)21E0tv(t′)X(t′)dt′ equality, that is both X(Γ) on the left and X(Zt−1Γ)
(cid:16) (cid:17) (cid:16) ((cid:17)4) on the right have the sense of multiplication operators.
R R
Here exp and exp are chronological and anti- Combination of this equality with the previous one im-
←− −→
chronological exponents, respectively. According to this plies
rule (and to general properties of the trace operation
Tr), the quadratic correlator remains as in (1). For the Zt−1X(Γ)Zt =X(t,Γ)≡X(Γt(Γ)) (7)
higher-order correlators the rule (4) prescribes
Hence, the composite operator Z−1X(Γ)Z reduces to
t t
x(t1)x(t2)x(t3) =Tr(X(t1) (X(t2) X(t3)))ρ0 , operatorofmultiplicationbythetime-dependentnumber
h i ◦ ◦
x(t1)x(t2)x(t3)x(t4) = X(t,Γ), which is nothing but trajectory of the variable
h i
=Tr(X(t1) (X(t2) (X(t3) X(t4))))ρ0 , X under initial conditions Γ.
◦ ◦ ◦
(5) Next, let us be convinced that CF of any variable
and so on, where, for definiteness, the inequalities tN ≥ (phase function) X(t,Γ) ≡ X(Γt(Γ)) can be expressed
... t2 t1 are presumed. [10] by the formula
≥
Below, we will derive generating FDR for so built sta-
tistical moments. All the more this is interesting be- exp tv(τ)x(τ)dτ =
0
cause the same rule (4) naturally arose in the course of (8)
the so-called “stochastic representation of deterministic = ←ex−pD 0t[LhRτ +v(τ)X(Γ)i]dEτ ρ0(Γ)dΓ
interactions” [25, 26, 27, 28] (the general method for R (cid:16)R (cid:17)
correct construction of “Langevin equations” introduc- Here ρ0 isthestatisticaloperator,i.e. distributionfunc-
ing thermodynamic noise and dissipation into quantum tion,ofthe systemattime t=0,andagain v(t) isarbi-
or classical dynamics). trary probe function. In the angle brackets, x(t) means
The strong argument for the benefit of the rule (4) is the internal variable X(t,Γ) as perceived by an outside
thatitcanbededucedfromthecorrespondenceprinciple. observer(whichknowsnothingabout Γ)andinterpreted
To see this, firstly consider classical systems. by him as a stochastic process.
3
To justify (8), it is sufficient to make standard “dis- (but now, generally, the super-operator X(t) in none
◦
entangling” of the complex exponent in the lower row of base reduces to scalar multiplication).
(8): Thusonecomesto theconstructionofquantumCFas
follows:
t
←ex−p 0[Lτ +v(τ)X(Γ)]dτ ρ0(Γ)dΓ= t
= RZt←exh−pR 0tv(τ)Zτ−1X(Γ)Zτidτ ρ0(Γ)dΓ= (9) exp 0 v(τ)x(τ)dτ = Trρ , (11)
(cid:28) (cid:20)Z (cid:21)(cid:29)
R = ←exh−pR 0tv(τ)X(τ,Γ)dτ ρi0(Γ)dΓ
Here, the lRatter thrRansformationis miade with taking into dρ = i[ρ,Ht]+v(t)X ρ , (12)
dt ¯h ◦
account the identity (7) and the second property from
(6). Evidently,finalexpressionin(9)coincideswithwhat with the initial condition ρ(t = 0) = ρ0. One can easy
is meant in the angle brackets in (8), i.e. the CF of verify that substitution of the formal direct solution of
the path X(t,Γ) whose uncontrolled dependence on the (12) into (11) produces just the formula (4), with X(t)
initialconditions Γ turnsitintoarandomprocess, x(t). standingforHeisenbergianoperatorofthequantumvari-
Accordingto(8)-(9),evaluationoftheCFisequivalent able and x(t) its effective commutative image.
to solution of definite differential equation: The doubtless advantage of such definition, (11)-(12),
of quantum CF is its differential nature [25, 26], which
exp tv(τ)x(τ)dτ = ρdΓ , highlights its automatic agreement with the causality
0 (10)
principle too. The formulas (5) demonstrate that be-
Ddρ/dRt = [Lt+v(Et)X(RΓ)]ρ , cause of the causality all the corresponding higher-order
where the function ρ=ρ(t,Γ) satisfies the initial condi- statistical moments appear asymmetric with respect to
tion ρ(0,Γ) =ρ0(Γ). time inversion. It is natural: earlier observations (mea-
In principle, all what just was said is known. The surements) can affect later ones, but opposite influence
representation (10), or (8), used in [10, 15], is variation is impossible.
of so-called Feynman-Kac formulas [29] which connect Further, we want to extend (4) to an arbitrary set
path integrals and differential equations. In fact, this of variables under simultaneous continuous observation.
representation of the CF is valid not for deterministic With this purpose,let us replace v(t)X in (12) with the
evolution only, but also for Marcovian stochastic evolu- operator
tions. Inthiscase, Γdesignatesinstantstateofa(multi-
component) Marcovianrandom process, and Lt its evo- Vt = vtµνXµν , Xµν ≡|µihν| , (13)
lution (kinetic) operator [10, 12, 15]. µν
X
where µ are states, or vectors (in the Dirac’s desig-
| i
nations), which constitute a complete orthonormalbase,
III. QUANTUM CHARACTERISTIC
FUNCTIONALS and vtµν arbitrary probe functions. This is most gen-
eral form of the observation. Analogously, most general
external perturbation can be described as
Followingthe correspondenceprinciple, it seems natu-
ral to suggest formulas (8) and/or (10) be the basis for
definition of the CF in quantum case. Ht =H0−Ft , Ft = ftµνXµν , (14)
µν
Now, the Liouville operator changes to the commuta- X
tor: LtΦ = i[Φ, Ht]/¯h (with [A,B] ≡ AB −BA, and where H0 is Hamiltonian of the “free” system, and ftµν
Φ,A,B being arbitrary operators). Quantum analogue
are “perturbing forces”.
of the Z exploited in previous section is super-operator
t Then, instead of (12) and (4), we have
whose action is defined by
dρ i
Ztρ=U(t)ρU−1(t) , U(t)≡←ex−p[−h¯i 0tHτdτ] dt = ¯h[ρ,H0−Ft]+Vt◦ρ , (15)
The phase functions X(Γ) and X(Γ (Γ))Rare replaced
t
bSyhrtohdeinogpeerraatnodrsHoefiqseunabnetrugmrveparrieasbenleta(otibosne,rvraesbplee)ctinivethlye, exp 0tvτµνxµν(τ)dτ H0,Fτ = Trρ ≡Ξ(Vτ; H0,Fτ)
X and X(t), where X(t)= Z−1X = U−1(t)XU(t). D hR iE (16)
t
What is for the operation of multiplication by X(Γ) Fromhere,therepeatedindices µ or ν meansummation
in (10), it should be replaced by super-operator of the overthem; xµν(t) areeffectiveclassicalimagesofquan-
symmetric product: X(Γ)Φ(Γ) X Φ. The matter tumvariables Xµν(t)=U−1(t)XµνU(t); Ξ(Vτ;H0,Fτ)
is that it is very hard to suggest⇒some◦thing else. Under will be used as shortened designation of the CF, i.e. the
this extension, the equality (7) remains valid: angle brackets.
The second and third arguments of Ξ, as well as
Z−1(X (Z Φ))=X(t) Φ=(U−1(t)XU(t)) Φ the subscript under angle brackets, remind about eigen
t ◦ t ◦ ◦
4
Hamiltonian of the system and its perturbations. Direct the matrices (in terms of the chosen basis)
solution of (15) yields
E 2T E
νµ µν
S ,C sinh,cosh , ∆ tanh ,
Ξ(Vτ;H0, Fτ)= νµ νµ ≡ 2T µν ≡ Eµν 2T
=Tr ←ex−p 0t −h¯i(H0−Fτ)+ 12Vτ dτ ρ0× and super-operators C, S and ∆ whose action is(1d8e)-
×−ex→(cid:16)pR (cid:2)0t h¯i(H0−Fτ)+ 12Vτ(cid:3)dτ(cid:17) = (17) termined by these matrices:
=Tr ←e×(cid:16)x−Rp−exh→(cid:2)p21R021tvτ0µtνvXτµνµXν(µτν)(dττ)idτ(cid:3)ρ0×(cid:17) C =Sc=oss(cid:0)in2h¯TL2h¯T0(cid:1)L0, (,C(ΦS)Φµν)µ=ν C=µSνΦµνµΦνµ,ν , (19)
(∆Φ) =∆ Φ
h R i (cid:0) (cid:1)µν µν µν
The Ξ’s arguments V and F can be understood as
t t
eithertwosets,oftheprobefunctions vµν andtheforces Then, finally, we can formulate the generating FDR.
t
fµν, orthe whole operatorsintroducedin (13) and(14). It is expressed by the formulas as follow:
t
Of course, the pair x (t) and x (t) in (16) can be
µν νµ
interpreted as two mutually conjugated complex-valued Ξ(Vt′−τ ; H0′, Ft′−τ)=Ξ(Vτ ; H0, Fτ) , (20)
randomprocesses(atleastwhile F assumedHermitian).
t
where V′ = vµνX′ , V = vµeνX , Fe′ = fµνX′ ,
τ τ µν τ τ µν τ τ µν
F =fµνX , and
τ τ µν
IV. THE GENERATING FDR e e
In the present paper we confine ourselves by the as- e e fvτµµνν = 2CSµν i2h¯CSµν fvτµµνν , (21)
(cid:20) τ (cid:21) (cid:20) ih¯ µν µν (cid:21)(cid:20) τ (cid:21)
sumption that the initial density matrix, ρ0, is canonic e
thermodynamically equilibrium one: ρ0 ∝ exp(−H0/T) or, in anotheer equivalent form,
(in the classical theory, not only this case already was
considered [11, 15] but also the case of non-equilibrium F C h¯S F
initial distributions [10, 12]). (cid:20) Vττ (cid:21)≡(cid:20) h¯2S −C2 (cid:21)(cid:20) Vττ (cid:21) (22)
Following the recipes of [11], let us make three trans- e
formations of the operator expression placed after the For conveniencee, the observation and perturbation vari-
trace symbol in (17). ables are unified into the column vector.
(i) Firstly, transpose this expression as a whole, with The combined transformation Φ Φ′ represents
the help of the usual rule (AB...C)′ = C′...B′A′, where the time reversal. Thus, (20) and (τ2⇔1) ort−(2τ2) state in-
the prime means the transposition: A′ = (A†)∗ = (A∗)† variance of the CF under (i) simultaneous time reversal
(symbols and willstandfortheHermitianandcom- of both the probe functions and external forces and (ii)
† ∗
plex conjugation, respectively). As is well known, this their mutual mixing as described by (21) and (22).
operation does not change the trace. Importantly, when dealing with (20)-(21) one should
(ii) Secondly, invert the direction of the time account, remember about the time translational invariance: any
by means of rewriting chronological exponents as anti- joint temporal shift of V and F does not change
t t
chronologicalones, and vice versa, as in the examples Ξ(Vτ;H0,Fτ)’s value, since ρ0 is invariant with re-
spect to free (unperturbed and unobserved) evolution
←−ex→−p 0tA(τ)dτ =−←ex→−p 0tA(t−τ)dτ ([ρ0,H0]=0).
Itshouldbeunderlinedalsothatbytheverydefinition
hR i hR i (see (13)) of the operators X we have
(iii) Thirdly, rewrite ρ0 in the form √ρ0√ρ0 and µν
“drag”oneofthesemultiplicands tothe leftandanother
X′ = ν∗ µ∗ , H′ µ∗ =E µ∗ , (23)
to the right and after that again unify them, with the µν | ih | 0| i µ| i
help of the trace property TrABC = TrCAB.
where µ∗ areeigenvectorsofthetransposedfreeHamil-
The result of these manipulations is an expression tonian|H′i,withthesameeigenvalues E asthatof µ
0 µ
which looks quite identical to the initial one, but with (one can write also µ∗ = µ ∗ = µ′). Hence, dur|inig
modified operator of the observation instead of Vt and the time reversedevo|lutiiont|heioperhat|or X′ represents
modified operator of the perturbation instead of F . To µν
t those quantum variable which is represented by opera-
write the result, of course, it is convenient to choose the
tor X inthe directprocess. Consequently,interms of
νµ
states µ as a complete set of orthonormaleigenvectors
| i their effective classical images, the left side of (20) looks
of the free Hamiltonian H0: H0|µi = Eµ|µi, with the as
eigenvalues (energies) E . Besides, introduce two oper-
µ
ators Ξ(V′ ; H′,F′ )=
t−τ 0 t−τ
= exp tvνµ x (τ)dτ (24)
L0Φ i[Φ,H0]/¯h , U0(t) exp[ iH0t/¯h] , 0 t−τ µν H′,F′
≡ ≡ − D hR iE 0 t−τ
5
Inotherwords,underthetimereversalthematrix {vτµν} d(U−1(t)H0U(t))/dt = U−1(t)PtU(t)). Hence, we can
behaves like the operator V , that is it undergoestrans- rewrite Eq.26 also as
τ
position: vµν (v′ )µν = vνµ (similarly, fµν
(ft′I−nτt)hµeνc=lasfτstνi−cµτa⇔)l.limitt−(τformally h¯t−τ 0),wehaveCτ ⇔1, Ξ −T1∆Pτ; H0, Fτ =1 , (28)
S →0, 2S/¯h→T−1L0,andformu→la(22)takesthe→form with arbitrary F(cid:16)τ and related Pet =(cid:17) h¯i[H0,Ft].
In the classical limit, ∆ disappears, in the sense
Fτ =Fτ , Vτ =Vτ +T−1L0Fτ (25) that ∆µν 1e. Therefore, (28) turns iento equality
→
exp( A/T) = 1 [10, 11], where A is total work pro-
At that, theeoperators Feτ and Vτ turn into phase func- h − i
ducedbyperturbationsduringtimeinterval (0,t). More
tions, L0 ⇒ (∇qH0)∇p−(∇pH0)∇q, and the prime in generally, perhaps, ∆P /T can be interpreted as opera-
(20) means replacement of q,p with q, p . t
{ } { − } tor of entropy production.
Comparisonbetween(22)and(25)revealsthattheob-
servations anyway are influenced by the perturbations,
but opposite effects exist in the quantum theory only.
V. PROBABILITY FUNCTIONALS
For illustration, let us choose F 0, that is the real
τ
≡
direct-time observations (described by V ) are made in
τ
Provided statistical moments of quantum variables
equilibriumsystem. Then, accordingto (20)-(22), detec-
tionofexactlytime-reversedresults(describedby V′ ), X(t) andtheirCFaredefined(by(15)-(17),inourcase),
t−τ
wecanintroducethe probabilityfunctional(PF)oftheir
inequilibriumsystemtoo,isincertainsensethe sameas
classical (commutative) equivalents x(t). We will des-
detection of the direct results but under specific nonzero
perturbations. This unpleasant peculiarity of the quan- ignate it by W(x(τ); H0, Fτ). As usually, is represents
functional Fourier transform of the CF:
tum theory is not surprising: since any measurement in-
fluences subsequent ones, but not vice versa, the mere
rearrangementof their results, generally speaking, could W(x(τ); H0, Fτ)= exp −i 0tvτµνxµν(τ)dτ ×
not realize under same conditions. ×Ξ(iVτR; H0h, Fτ)RdVτ , i
VariousparticularFDRcanbeobtainedfrom(20)-(22) (29)
by either some special choice of Vτ and Fτ or varia- where dVτ = τµν(dvτµν/2π).
tional differentiations with respect to vµν and fµν. Ev- Let us apply this transform to both sides of the
τ τ Q
idently, if both F and V are Hermitian, then both FDR (20), again with the indices being related to the
τ τ
transforms V and F also are Hermitian. But if V eigenstates of H0. We omit rather bulky manipulations
τ τ τ
is quite arbitrary, then one should be ready to deal with and write their result in two steps:
non-Hermitiaen perturebation F .
τ W(x′(t τ); H′, F′ )= (30)
For example, let us choose Vt 0, i.e. there is no − 0 t−τ
≡
observation at all. Accordingeto (12) or (17), of course,
in absent of observations the evolution of the statisti- t 2i E
cTahleorpefeorraetorTρrρis=unTitraρr0y u=nd1e,r wanhdateΞv(e0r;Hpe0r,tuFrτb)ation1s.. =exp(cid:20)−Z0 ¯h tanh(cid:18) 2µTν(cid:19)fτµνxµν(τ)dτ(cid:21)×
≡
Consequently, (20) and (22), together with (19), yield
the equality
W(x(τ); H0, Fτ) ,
×
1
1=Ξ ∆L0Fτ; H0, Fτ , (26) f
(cid:18)T (cid:19) W(x(τ); H0, Fτ)= (31)
or, in other equivalent designeations, e
f
t ¯h δ δ
(cid:28)exp(cid:20)−Z0t 2¯hitanh(cid:18)E2µTν(cid:19)fτµνxµν(τ)dτ(cid:21)(cid:29)H0,Fτ =1 =exp(cid:20)Z0 dτ2iSµνCµν δfτµν δxµν(τ)(cid:21)×
(27)
Theserelationsarevalidatarbeitraryperturbatioenoper- ×W(C−1x(τ); H0, C−1Fτ) ,
ator F . The subscriptunder the anglebracketreminds
τ where (C−1Φ) =Φ /C .
about the perturbation. µν µν µν
In these two formulas, both the exponents in mid-
To emake Eqs.26-27 better transparent, combine them
dle rows result from the left bottom and right top non-
with the identities
diagonal elements of matrix (21) (or (22)), respectively,
L0Ft = h¯i[Ft,H0]=−h¯i[H0−Ft,H0]=−Pt iT.eh.ufsr,otmhemeuxtpuoanlemnitxiinng(o3f1)perretfluercbtastidoinstaunrbdinogbsaecrvtiaotnioonf.
where P is operator of the instant energy power being observations,whichisquiteunpleasantpeculiarityofthe
te e e
pumped by the perturbations (notice that dH0(t)/dt = quantum case. Of course, in fact this operator-valued
6
exponent acts as an integral operator. We leave it un- where Xo (t) are the operators X considered in the
µν µν
wrapped till possible separate work (though example of interaction representation, thus representing free evolu-
similar operators can be found in [28]). For the present, tion:
confine ourselves by the sad conclusion that generally
W(x(τ);H0,Fτ), and thus PF of the reversed process, Xµoν(t)=U0−1(t)XµνU0(t)=Xµνexp(iEµνt/¯h)
W(x′(t τ);H′,F′ ),relatestoPFofthedirectprocess
Wf(x(τ)−;H0,F0τ) itn−τsomenon-localway,withrespectto According to (33)-(34), if Vt ≡0 then
both fµν and x (t).
Howetver, it isµnνot hard to notice that a degree of the exp 0tfτµνyµν(τ)dτ =Ξ(0;H0,Fτ)=1 , (35)
2 o
non-locality is proportional to tanh (E /2T), hence,
µν D R E
it is negligible in respect to low-energy quantum transi- that is any statistical moment of y’s themselves is equal
tions. In the classicallimit (h¯ 0) the exponentin (31) to zero. But their correlations with x’s differ from zero:
→
turns into unit, W W , and (30)-(31) reduce to the
→
purely local relation δ
x(t )y(τ ) = x(t )
W(x′(t−τ);H0′,AfFt′=−τ)t=fµeνxpd(−xA/(τT))dWτ (,x(τ);H0,Fτ) , *Yj,m j m +o Ym δf(τm)*Yj j +FτFτ(=360)
0 τ dτ µν
(32) (indices µ,ν are omitted). Hence, cross-correlation be-
where A is again theRwork of the external forces fµν, tween N copiesof x and M copiesof y represents M-
t
and Φ′(q,p) Φ(q, p). ThisFDRisequivalenttowhat orderresponsetoperturbationsofan N-orderstatistical
was obtained≡in [11−]. Curiously, if quantum CF is de- moment of the x’s.
fined by Eq.2, instead of Eqs.15-17, then Eq.32 replaces Interestingly, the relations (36) are valid also for the
Eqs.30-31 even in general quantum case. x’s and y’scumulants(semiinvariants)whosegenerating
function is ln Ξ(Vτ;H0,Fτ). The proof is trivial:
VI. STOCHASTIC REPRESENTATION OF ln exp [vτx(τ)+fτy(τ)] dτ o =
RESPONSE TO PERTURBATIONS = exp (cid:10) dτfRτ δ/δgτ ln Ξ(Vτ; H0,(cid:11)Gτ) Gτ=0
(cid:2) (cid:0)R (cid:1) (cid:3)
The above consideration demonstrated a lot of formal (againwithoutindices). Consequently,insteadof(36)we
symmetrybetweenperturbationsandobservations. This can write
symmetry suggests that the perturbing forces fµν can
t c c
be treated as another test (probe) functions which cor- δ
x(t )y(τ ) = x(t )
respond to watching for additional ghost variables. Let * j m +  δf(τm)* j + 
the latter be namedas yµν(t). To define them, following Yj,m o Ym Yj Fτ Fτ=0
[25, 26, 27], we rewrite Eqs.15-17 in the form   (37)
with the superscript “c” marking the cumulants.
exp t[vµνx (τ)+fµνy (τ)]dτ Theunionofthetwosetsofrandomprocesses,quitere-
0 τ µν τ µν o ≡ alistic x’s and indeedrather illusive y’s, givesstochastic
=DTr ←ex−Rp 0t≡−Ξh¯i(V(Hτ;0H−0F,Fττ))+=12Vτ dτE ρ0× (33) rIfeptrheeselnatttaetrioanreofctahuesesdysbteymi’nstereraspctoinosnestwoiptherstoumrbeatoitohnesr.
×−e(cid:16)x→Rp (cid:2)0t h¯i(H0−Fτ)+ 21V(cid:3)τ d(cid:17)τ dthyenasmtoicchaalsstyicstreempr“eDse”n,tathtieonnwoef dmeatekremfiirnsitstsitcep(qtuoawnaturdms
(cid:16)R (cid:2) (cid:3) (cid:17) orclassical)interactions,whichwassuggestedin[25]and
Essentially, it is assumed here that an imaginative prob-
developed in [26, 27, 28]. For instance, our system can
abilitymeasurehiddenbehindthe anglebracketsisitself
serveasthermostatfor“D”.Generalstochasticequations
independentoftheforces. Inotherwords,alltherandom
whichdescribe“D”underinfluencebythethermostatin-
processes x (t) and y (t) aremeantbecharacteristics
µν µν evitably include the y’s whose main effect is dissipation.
of the unperturbed dynamics governed by the Hamilto-
nian H0. The subscript “o” serves to remind of this
circumstance.
VII. TIME REVERSAL AND GENERATING
Therefore, it may be convenient to rewrite (33) once
FDR IN THE STOCHASTIC REPRESENTATION
more, in the form
Combining Eqs.20-22 with Eq.33, one can simply re-
exp t[vµνx (τ)+fµνy (τ)]dτ =
0 τ µν τ µν formulate the generating FDR in terms of x’s and y’s :
o
D R =Ξ(Vτ; H0,Fτ)= E
=Tr ←ex−p 0t h¯ifτµν + 21vτµν Xµoν(τ)dτ ρ0× (34) exp 0t vτµνx′µν(t−τ)+fτµνyµ′ν(t−τ) dτ o = (38)
×−ex→p(cid:16)R0t(cid:2)−h¯ifτµν + 21vτµ(cid:3)ν Xµoν(τ)d(cid:17)τ , D = Rex(cid:2)p 0t[vτµνxµν(τ)+fτµνyµν(τ)]d(cid:3)τ oE,
(cid:16)R (cid:2) (cid:3) (cid:17) D R E
e e
7
where,ofcourse, x′µν =xνµ, yµ′ν =yνµ,and x, y relate ηνµ(t0−t) ≍ exp(−Eµν/2T)ηµν(t) (43)
to x, y absolutely similar to (21):
To some extent, the ξ’s and η’s can be thought like
e e
x (τ) C ih¯S x (τ) amplitudes of quantum jumps, hence, their squares, ξ 2
(cid:20) yµµνν(τ) (cid:21)=(cid:20) i2h¯Sµµνν 2Cµµνν (cid:21)(cid:20) yµµνν(τ) (cid:21) (39) and |η|2, like corresponding probabilities. Then, form| u|-
las (42) and (43) reduce to familiar relations between
e
Evidently,itthe matrices x= x and y = y are probabilities of mutually reversedjumps.
Hermitiean, x† =x and y† =y{, tµhνe}n the trans{forµmν}ation It is necessary to remember, of course, that generally
(39) does not damage this their property. (when H0′ = H0) the left and right-hand processes in
6
Taking into account ρ0’s invariance with respect to (40), (42) and (43) relate to somehow different systems.
freeevolution,theFDR(38)canbeexpressedalsointhe Combining these statistical equalities with (36) (or
form (37)) and (33) (or (34)), and besides with the causal-
ity principle, as well as with independence of statistical
xνµ(t0−t)≍xµν(t) , yνµ(t0−t)≍yµν(t) , (40) momentson t0,onecanconstructrelativelysimplealgo-
rithms for derivation of many particular FDR. However,
where t0 is arbitreary time shift, and symebol ≍ means that are tasks for separate work.
statistical equivalence of left- and right-handed random
processes (i.e. equivalence in the sense of statistical mo-
ments, x′(t0 t1)...x′(t0 tN) = x(t1)...x(tN) , and VIII. CONCLUSION
h − − i h i
so on).
Following [25], it may be convenient to introduce
e e To resume, we obtained generating fluctuation-
another random processes, whose matrices are non-
dissipation relations (FDR) for a dynamically perturbed
Hermitian:
quantum system, assuming special but theoretically and
ξ(t) x(t)+ ih¯y(t) , practically important definition of its quantum statisti-
2
≡ (41) calmoments (or correspondingcharacteristicfunctional)
η(t) ξ†(t)=x(t) ih¯y(t) which describe consecutive or continuous measurements
2
≡ − of the system.
In their terms the generating FDR look most simple: In addition, short and expressive formulation of the
FDRintermsofthestochasticrepresentationofquantum
ξνµ(t0 t) exp(Eµν/2T)ξµν(t) , (42) interactions was done.
− ≍
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