Physics Letters A 308 (2003) 135–139

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General theorems on decoherence in the thermodynamic limit

Marco Frasca

Via Erasmo Gattamelata, 3, 00176 Roma, Italy

Received 3 November 2002; received in revised form 23 December 2002; accepted 27 December 2002

Communicated by P.R. Holland

Abstract

We extend the results on decoherence in the thermodynamic limit [Phys. Lett. A 283 (2001) 271] to general Hamiltonians.It is shown thatN independent particles, initially properly prepared, have a set of observables behaving classically in thethermodynamic limit. This particular set of observables is then coupled to a quantum system that in this way decoheres so tohave the density matrix in a mixed form. This gives a proof of the generality of this effect. 2003 Elsevier Science B.V. All rights reserved.

PACS: 03.65.Yz; 05.70.Ln

Keywords: Decoherence; Thermodynamic limit; Measurement theory

Decoherence [1] appears as a rather ubiquitouseffect in quantum systems. Due to the interactionwith a large environment, a quantum system tends tomodify its quantum evolution from unitary, that meanscoherent, to a decaying form. In this terms, we cansay that decoherence, as commonly understood, is adissipative effect. Then, the density matrix containinginterference terms, after a trace procedure on theenvironment degrees of freedom gets a mixed formso to have probabilities attributed to the outcome ofpossible measurements on the system. As a matter offact, this does not solve the measurement problem inquantum mechanics [2]. The main question is that,after a measurement, one gets a pure state for thequantum system and not just the mixed form of thedensity matrix that is obtained by decoherence.

E-mail address: marcofrasca@mclink.it (M. Frasca).

Anyhow, the appearance of decaying effects onunitary evolution of a quantum system is generallyseen in experiments and one can safely affirm thatdecoherence is a rather well verified phenomenon [3].In this Letter we want to go one step beyond onthe basis of a recent proposal for nondissipativedecoherence in the thermodynamic limit [4]. We willprove that a pure state is indeed obtained when a largenumber of quantum systems interacts with anotherone, washing out superposition states and approachingin this way a possible solution to the measurementproblem, for a single event, self consistently insidequantum mechanics. The essential characteristic ofour approach is that unitary evolution is preserved anddecoherence is dynamically produced.

First of all, we will prove a general theorem onthe appearance of classical states in the dynamicalevolution of an ensemble of noninteracting quantumsystems. We show that:

0375-9601/03/$ – see front matter 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0375-9601(03)00028-8

136 M. Frasca / Physics Letters A 308 (2003) 135–139

Theorem 1 (Classicality).An ensemble of N nonin-teracting quantum systems, for properly chosen ini-tial states, has a set of operators {Ai,Bi, . . . : i =1, . . . ,N} from which one can derive a set of observ-ables behaving classically.

Proof. To prove this theorem we consider a Hamil-tonian of the form

(1)H =N∑

i=1

Hi

and a set of observables{Ai,Bi, . . . : i = 1, . . . ,N}.For each system we take a set of states of singleparticle|ψi〉 in a such a way to have the initial productstate

(2)|ψ(0)〉 =N∏

i=1

|ψi〉

not being an eigenstate of the Hamiltonian (1). Firstly,we prove that the HamiltoniansHi belong to the setof operators{Ai,Bi, . . . : i = 1, . . . ,N}, being thecorresponding observable the HamiltonianH . Thismeans that the mean value ofH on the state (2)is overwhelming large with respect to its quantumfluctuations in the thermodynamic limit. This provesthat the set{Ai,Bi, . . . : i = 1, . . . ,N} is not empty.Indeed, by straightforward algebra one has

(3)〈H 〉 =N �Hhaving set

(4)�H = 1

N

N∑

i=1

〈ψi |Hi |ψi〉

that is an average of the mean values of eachHi . Thisproves that the mean value ofH is proportional toN .Without much more difficulty we get

(5)⟨H 2⟩ =NH 2 +

∑

i =j〈ψ(0)|HiHj |ψ(0)〉

being

(6)H 2 = 1

N

N∑

i=1

〈ψi |H 2i |ψi〉

again an average of the mean values of the square ofeachHi . We note at this stage that the last term in

Eq. (5) gives a contribution also proportional toN2.This gives at last the fluctuation

(7)(�H)2 = ⟨H 2⟩ − 〈H 〉2 =N(�H)2,

the mean value ofH now removes theN2 term, with

(8)(�H)2 = 1

N

N∑

i=1

[〈ψi |H 2i |ψi〉 − 〈ψi |Hi|ψi〉2]

showing that the square of the fluctuation ofH is pro-portional to the average of the fluctuations of eachHi .So, one has that the mean value ofH is proportionalto N while the fluctuation is proportional to

√N and

then the former is overwhelming large with respectto the latter, if the initial Hamiltonian (1) has a largeensemble of systems composing it (thermodynamiclimit). Being the quantum fluctuations negligible, onecan say that the quantum system we are considering,in the thermodynamic limit, behaves classically withrespect to the observableH , if properly prepared inthe state|ψ(0)〉. So, the ensemble{Ai,Bi, . . . : i =1, . . . ,N} is not empty.

Now, we extend the proof to any other operatorthat can belong to the set{Ai,Bi, . . . : i = 1, . . . ,N}.If, for a given i, Ai commutes withH , it does notevolve in time, being a conserved observable, andthe above argument forH applies straightforwardly.Instead, if, generally,[Ai,H ] = 0 we have to study thetime evolution of these observables by the Heisenbergequations of motion. By analogy withH we introducethe observableA = ∑N

i=1Ai (the same can be donewith any other set of operators belonging to the givenset), then (here and in the followingh̄= 1)

(9)A(t)= eiHtAe−iH t .

With this definition it is not difficult to obtain, with thesame state initial state,

(10)〈A(t)〉 =NA(t)

being

(11)A(t)= 1

N

N∑

k=1

〈φk|eiHktAke−iHkt |φk〉,

and

(12)[�A(t)]2 = ⟨A(t)2

⟩ − 〈A(t)〉2 =N�A(t)2

M. Frasca / Physics Letters A 308 (2003) 135–139 137

being

(13)

�A(t)2 = 1

N

N∑

k=1

[〈φk|eiHktA2ke

−iHkt |φk〉

− 〈φk|eiHktAke−iHkt |φk〉2],

proving finally the theorem as we have the fluctuationproportional to

√N and the mean value proportional

to N . We can recognize from this result a strictsimilarity with statistical mechanics as it should beexpected from the start (see [5]).✷

Once we have such a set of observables, one mayask if these operators can indeed produce decoherence.This is the content of the next theorem:

Theorem 2 (Decoherence).An ensemble of N nonin-teracting quantum systems, having a set of operators{Ai,Bi, . . . : i = 1, . . . ,N}, from which one can derivea set of observables behaving classically, and stronglyinteracting with a quantum system through a Hamil-tonian having forms like V0 ⊗ ∑N

i=1Ai , can producedecoherence if properly initially prepared.

Proof. By “strongly interacting” we mean that theHamiltonian of theN noninteracting quantum systemscan be neglected, and perturbation theory can beapplied. We want to use a theorem for strong couplingproved in Ref. [6]. In fact, the Hamiltonian of thissystem can be written, choosing as a observable∑N

i=1Ai acting in the Hilbert space of the bath,

(14)HSB =HS +N∑

i=1

Hi + V0 ⊗N∑

i=1

Ai

being HS the Hamiltonian of the quantum system,andV0 an operator, acting in the Hilbert space of thesystem, coupling the quantum system to the bath ofN

noninteracting systems. So, if we assume the couplingbetween the system and the bath to be very large, wecan apply the theorem of Ref. [6] to the Hamiltonianin the interaction picture in the system’s variables

(15)HI = eiHStV0e−iHSt

N∑

i=1

Ai,

stating that the strong coupling approximation is givenby

(16)

|ψ(t)〉 ≈∑

n

eiγ̇nt e−iNavnt |vn〉〈vn|ψS(0)〉N∏

i=1

|χi〉

being|ψS(0)〉 the initial state of the quantum system,

(17)V0|vn〉 = vn|vn〉,assuming a discrete spectrum and

(18)γ̇n = 〈vn|HS |vn〉.The initial state of the bath is chosen in such a wayto haveAi |χi〉 = ai |χi〉 and beinga = ∑N

i=1 ai/N aconstant.

The state (16) has a quite interesting aspect as thephases of the oscillating exponentials, already at verysmall energy, can have a time scale of the order ofthe Planck time in the thermodynamic limit, makingsenseless the possibility to observe such oscillationson the corresponding probabilities. This means, math-ematically, that such probability oscillations are aver-aged away [7]. Indeed, for the density matrix of thesystem one has

ρS(t)=∑

n

∣∣〈vn|ψS(0)〉∣∣2 |vn〉〈vn|

+∑

m=nei[γ̇m−γ̇n]t e−iNa[vm−vn]t

(19)× 〈vm|ψS(0)〉〈ψS(0)|vn〉|vm〉〈vn|with the interference terms being averaged away ona maximum time scaleτM = 1/(Namin[vm − vn])being min[vm − vn] the minimal energy differencebetween the eigenavalues ofV0. This time is reallysmall already at energies of order of eV forNbecoming very large and generally comparable withthe Planck time. As a final comment about thistheorem, we point out that use has been made of astrong coupling between a bath and a system. Thiskind of coupling is not generally common but can berealized in ion traps and could turn out to be used inquantum computation.✷

Finally, we prove a general theorem in measuretheory in quantum mechanics, stating that

138 M. Frasca / Physics Letters A 308 (2003) 135–139

Theorem 3 (Measure).If the operator V0, stronglycoupled to a quantum system with an observable of theensemble of N noninteracting systems, is linear in thegenerators of coherent states, Schrödinger cat statesare washed out in the leading order of the coupling inthe thermodynamic limit.

Proof. We assume, initially, that the system has thefollowing Hamiltonian, neglecting the bath contribu-tion at the leading order in the strong coupling expan-sion,

(20)HF = ωa†a + (γ a† + γ ∗a

) ⊗N∑

i=1

Ai

so that, we haveV0 given by a linear combination ofgenerators of coherent states [8]. Besides, the initialstate is taken to be a superposition state as

(21)|ψ(0)〉 =N(∣∣αeiφ

⟩ + ∣∣αe−iφ⟩) N∏

i=1

|χi〉

beingN a normalization factor and|αe±iφ〉 coherentstates as to have a Schrödinger cat state [9]. At theleading order, we can write the unitary evolutionoperator as [10],

(22)UF (t)= eiξ(t)e−iωa†at exp[β̂(t)a† − β̂(t)∗a

]

being

(23)ξ̂ (t)=(∑N

i=1Ai)2|γ |2

ω2

(ωt − sin(ωt)

)

and

(24)β̂(t)=(∑N

i=1Ai)γ

ω

(1− eiωt

).

So, it is straightforward to obtain the wave function as

|ψ(t)〉 ≈UF (t)|ψ(0)〉= eiξ(t)N

(eiφ1(t)

∣∣β(t)e−iωt + αeiφ−iωt ⟩

+ eiφ2(t)∣∣β(t)e−iωt + αe−iφ−iωt ⟩)

(25)×N∏

i=1

|χi〉

being now

(26)ξ(t)= N2|γ |2ω2

(ωt − sin(ωt)

),

and

(27)β(t)= Nγ

ω

(1− eiωt

)

and

(28)φ1(t)= −i α2

[β(t)e−iφ − β∗(t)eiφ

],

(29)φ2(t)= −i α2

[β(t)eiφ − β∗(t)e−iφ

].

The state (26), in the thermodynamic limitN → ∞,reduces to a pure coherent state for the system,|β(t)e−iωt 〉, proving the main assertion of the theo-rem. The Schrödinger cat state is washed away in thethermodynamic limit.

To keep this Letter shorter, we omit to prove thatthe system of Theorem 3 undergoes true decoherence.This can be seen from the interference term of theWigner function that, in the thermodynamic limit, dis-plays very rapid oscillations on a time scale of thePlanck time or smaller, being not physical. We can ap-ply the theory of divergent series [11] to assume in thesense of Abel or Euler that limN→∞ cos(Nf (t))= 0and limN→∞ sin(Nf (t))= 0 and all boils down to anaverage in time. So, no interference term can be ac-tually observed and true classical behavior emerges.This argument is similar to the one applied in the proofof Theorem 2. ✷

In conclusion, we have generalized our approach tothe study of quantum mechanics in the thermodynamiclimit, given in [4], to a large class of quantum systems,proving its wide applicability.

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