PHYSICAL REVIEW A 72, 012102 ͑2005͒
Decoherence time in self-induced decoherence
Mario Castagnino* CONICET-UNR-UBA, Institutos de Física de Rosario y de Astronomía y Física del Espacio, Casilla de Correos 67, Sucursal 28, 1428, Buenos Aires, Argentina
Olimpia Lombardi CONICET-IEC, Universidad Nacional de Quilmes, Rivadavia 2328, 6° Derecha, 1034, Buenos Aires, Argentina ͑Received 18 February 2005; published 1 July 2005͒
A general method for obtaining the decoherence time in self-induced decoherence is presented. In particular, it is shown that such a time can be computed from the poles of the resolvent or of the initial conditions in the complex extension of the Hamiltonian’s spectrum. Several decoherence times are estimated: 10−13–10−15 s for microscopic systems, and 10−37–10−39 s for macroscopic bodies. For the particular case of a thermal bath, the order of magnitude of our results agrees with that obtained by the einselection ͑environment-induced decoher- ence͒ approach.
DOI: 10.1103/PhysRevA.72.012102 PACS number͑s͒: 03.65.Yz
I. INTRODUCTION one issue which has been often taken for granted is looming big, as a foundation of the whole decoherence problem. It is The phenomenon of decoherence is usually considered as the question of what the ‘system’ is which plays such a cru- a relevant element for understanding how the classical world cial role in all the discussions of the emergent classicality. emerges from an underlying quantum realm. In a first period, This question was raised earlier but the progress to date has decoherence was explained as the result of the destructive been slow at best” ͑Ref. ͓16͔ p. 122͒. interference of the off-diagonal elements of a density matrix ͑iii͒ In the einselection approach, the definition of the ba- ͑see Refs. ͓1,2͔͒; however, this line of research was aban- sis where the system becomes classical, i.e., the pointer ba- doned due to technical difficulties derived from the formal- sis, relies on the “predictability sieve” which would produce ism used to describe the process. As a consequence, decoher- the set of the most stable states. But this definition seems ence began to be conceived as produced by the interaction very difficult to implement, at least in a generic case. In fact, between a system and its environment. This approach gave the basis vectors are only good candidates for reasonable ͓ ͔ rise to the environment-induced decoherence ͑EID͒ program, stable states 17 . based on the works of Zeh ͓3–5͔ and later developed by As the result of these and other difficulties, a number of ͓ ͔ alternative accounts of decoherence have been proposed Zurek and co-workers 6–12 . Although many relevant re- ͓ ͔ sults have been obtained by means of environment-induced 18–24 . In a series of papers ͓13,25–38͔ we have returned to the decoherence, this approach still involves certain unsolved initial idea of the destructive interference of the off-diagonal problems ͑see Refs. ͓13,14͔͒: ͑ ͒ terms of the density matrix, but now on the basis of the van i Einselection is based on the decomposition of the sys- Hove formalism ͓39–43͔. We have called this new approach tem into a relevant part, the proper system, and an irrelevant “self-induced decoherence” ͑SID͒ because, from this view- part, the environment. This decomposition is not always pos- point, decoherence is not produced by the interaction be- sible, as in the case of the universe. In fact, Zurek himself … tween a system and its environment, but results from the own considers the criticism: “ the Universe as a whole is still a dynamics of the whole quantum system governed by a single entity with no ‘outside’ environment, and therefore Hamiltonian with continuous spectrum. The aim of this pa- any resolution involving its division is unacceptable” ͑Ref. ͓ ͔ ͒ per is to present a general method for obtaining the decoher- 15 ,p.181. The same problem appears in any closed sys- ence time in self-induced decoherence. In particular, we will tem ͑and therefore with no interaction with an environment͒ … show that such a time can be computed from the poles of the that becomes classical. In fact, if “ the existence of emer- resolvent or of the initial conditions in the complex extension gent classically will be always accompanied by other mani- of the Hamiltonian’s spectrum. The general formalism has festations of openness such a dissipation of energy into the ͓ ͔ ͑ ͓ ͔ ͒ been developed in Refs. 44,45 , but in the context of a dis- environment” Ref. 11 ,p.6, the problem is to explain why cussion about the nature and properties of Gamow vectors. many systems behave classically maintaining their energy Here we will adopt that formalism for the computation of the constant. ͑ ͒ decoherence time. ii The einselection approach does not provide a clear Two comments are in order: definition of the “cut” between the proper system and its Comment 1. The time evolution of the decaying off- environment. In fact, as Zurek himself admits, “In particular, diagonal terms of the density matrix has three periods. ͑i͒ A short initial ͑nonexponential͒ Zeno period, where the time derivative is zero for t=0 ͓46͔. ͑ii͒ Then, a long exponential *Email address: [email protected] decaying period with a factor e−t/tD. This is the only period
1050-2947/2005/72͑1͒/012102͑9͒/$23.00012102-1 ©2005 The American Physical Society M. CASTAGNINO AND O. LOMBARDI PHYSICAL REVIEW A 72, 012102 ͑2005͒ studied in this paper because it allows us to define an expo- equation, and that this time can be obtained in terms of the nential decoherence characteristic time tD by means of the poles of the involved functions. In Sec. IV we find the origin poles method. ͑iii͒ A final Khalfin period ͓47͔, where the of these poles in the resolvent of the Hamiltonian or in the evolution has a long powers series tail that essentially de- initial conditions. In Secs. V and VI we estimate the deco- pends on the interaction. herence time in microscopic systems and in macroscopic These three periods are studied in Refs. ͓27,48–50͔ with bodies, respectively. Section VII is devoted to study the case the same methods used in the present paper ͑see also the of a thermal bath, where the decoherence time is estimated coincidence of our results with those of Ref. ͓51͔͒. These and compared with the corresponding results obtained by the 1 three periods are ubiquitous in decaying processes. einselection approach. After the conclusions, we include two Comment 2. The models studied by EID and SID are quite appendixes: in Appendix A we introduce a necessary math- different. In the typical models treated by EID, the proper ematical remark, and in Appendix B we sketch a model with system interacts with an environment, the evolution spec- two characteristic times: decoherence and relaxation. trum is usually discrete, only decoherence in the proper sys- tem is considered since the environment is traced away, and II. SELF-INDUCED DECOHERENCE normally there is dissipation of energy from the system to the environment; this means that EID is a dissipative decoher- In this section we will present the formalism of self- ence theory. On the contrary, SID models are closed systems induced decoherence by means of a very simple case. We and therefore there is no dissipation, the evolution spectrum refer the reader to our previous papers; in particular, for more is continuous, and the decoherence of the whole system is complex cases, see Ref. ͓30͔, and for a conceptual discussion considered: SID is a nondissipative decoherence theory. Un- about the physical meaning of self-induced decoherence, see fortunately, the comparative study between EID and SID is Ref. ͓13͔. only in a first stage in Ref. ͓53͔ and, in certain sense, in the Let us consider a quantum system endowed with a Hamil- present paper. More generally, there is no comparative study tonian with continuous spectrum, ͑ ͓ ͔ ͒ between dissipative theories like Refs. 3–5 , and nondissi- ϱ pative ones ͓54–57͔.2 Therefore the comparison between the H = ͵ ͉͉͗͘d. ͑1͒ decoherence times obtained by EID and SID is a comparison 0 of data resulting from different models. As a consequence, we will try to find obviously not an exact coincidence, but A generic observable reads only a certain similarity in orders of magnitude. Neverthe- ϱ ϱ less, to the extent that EID is an accepted theory ͑albeit the O = ͵ ͵ O˜ ͑,Ј͉͒͗͘Ј͉ddЈ, ͑2͒ problems listed above͒, this comparison is interesting and 0 0 necessary, and it is the best we can do while we wait for where O͑,Ј͒ is any kernel or distribution. Since the alge- more model research and more experimental data. The paper is organized as follows. In Sec. II we briefly bra of these observables is too large for our purposes, we review the formal basis of self-induced decoherence. In Sec. only consider the van Hove operators such that III we show that the decoherence time can be computed as O˜ ͑,Ј͒ = O͑͒␦͑ − Ј͒ + O͑,Ј͒, ͑3͒ the characteristic decaying time of the expectation value where O͑,Ј͒ is a regular function. Then, 1In some cases where an approximate analytical solution can be ϱ ϱ ϱ obtained by means of the EID approach, these periods can be easily O = ͵ O͉͉͑͒͗͘d + ͵ ͵ O͑,Ј͉͒͗͘Ј͉ddЈ, identified. For instance, in Ref. ͓52͔, when the distribution of the 0 0 0 interaction couplings is Lorentzian, the time evolution is exponen- ͑4͒ tial, but when these couplings are constant, an approximately Gaussian evolution is obtained. In the Gaussian regime, the initial where the first term of the right-hand side ͑rhs͒ is the singu- period can be conceived as a Zeno period where the time derivative lar term OS, and the second term is the regular term OR. is zero for t=0; after the first inflection point, the evolution can be These observables belong to an algebra Aˆ , that we will call approximated by an exponential; and at infinity, the Gaussian evo- Aˆ lution and all its derivatives vanish, so we find a final Khalfin con- van Hove algebra: they are the self-adjoint operators of stat zero tail ͑see also Fig. 29 of Ref. ͓48͔ as an illustration͒. and belong to a space of operators Oˆ . The basis of Aˆ is 2It is interesting to note that, since in SID the whole system ͕͉͒,͉,Ј͖͒, where ͉͒=͉͉͗͘ and ͉,Ј͒=͉͗͘Ј͉. ͑ ͒ proper system plus environment reaches an equilibrium decohered Let us now consider the space OˆЈ, that is, the dual of ͑ state, all the parts of the whole system in particular, the proper Oˆ ͕͉͑ ͑ Ј͉͖ system͒ also reach a final equilibrium state. Therefore the proper space with basis , , . A generic state belonging system has a final of equilibrium whose eigenbasis can be ob- to OˆЈ reads tained: the proper system decoheres in that basis. This means that, ϱ ϱ ϱ when the conditions for the application of SID are satisfied in the = ͵ ͉͑͒͑d + ͵ ͵ ͑,Ј͒͑,Ј͉ddЈ ͑5͒ ͑ ͓ ͔͒ whole system see Ref. 30 , the SID approach should explain the 0 0 0 decoherence of the proper system as treated by the EID approach. Of course, this is only a preliminar suggestion that we will develop and must satisfy the usual constraints: ͑͒ must be real and ͐ϱ͑͒ elsewhere. positive and 0 d =1. We also introduce the require-
012102-2 DECOHERENCE TIME IN SELF-INDUCED DECOHERENCE PHYSICAL REVIEW A 72, 012102 ͑2005͒
ment that ͑,Ј͒ be a regular function. Again, the first term self-induced decoherence can be applied to closed systems as ͑ ͒ ͓ ͔ of the rhs of Eq. 5 is the singular term S, and the second the universe 34 , the problem of the “cut” between the ͓ ͔ term is the regular term R. proper system and its environment is absent 13 , and the The expectation value of the observable O in the state pointer basis is perfectly defined ͑e.g., the energy eigenbasis results from the action of the functional on the operator O, in the above example͓͒28͔. Nevertheless, three points need ͉͑O͒, to be emphasized: ͑ ͒ ϱ ϱ ϱ i In this approach, the coarse graining necessary to turn ͗ ͘ ͵ ͑͒ ͑͒ ͵ ͵ ͑ Ј͒ ͑ Ј͒ Ј the usual unitary time evolutions of quantum mechanics into O = O d + , O , d d , 0 0 0 nonunitary time evolutions is not explicit in the formalism. Aˆ ͑6͒ However, the choice of a particular algebra , the van Hove algebra, among all possible algebras and the systematic use ͑͒ ͑͒ ͗ ͘ ͑ ͉͒ where functions and O are such that the first integral of expectation values O ͑t͒ =( t O) play the role of a is well defined. The time evolution of this last equation reads coarse graining. In fact, we can define the projector ⌸=͉O͒ ϱ ͑ ͉ ͉ ͒Aˆ ͑ ͉ ͒ ͑͑ ͉͒ 0 , with O and 0 O =1, that projects t as ͗O͑͘ ͒ = ͵ ͑͒O͑͒d ͑͑ ͉͒⌸ ͗ ͘ ͑ ͉ t t = O ͑t͒ ˆ 0 and allows us to translate our results in 0 the language of projectors: in this case, from Eq. ͑10͒ we will ϱ ϱ obtain lim →ϱ͑͑t͉͒⌸=͑ ͉⌸, where ͑ ͉ is diagonal. This ͵ ͵ ͑ Ј͒ ͑ Ј͒ i͓͑−Ј͒/ប͔t Ј t * * + , O , e d d . projection is what breaks the unitarity of the primitive evo- 0 0 lution. A detailed discussion on this point can be found in ͑7͒ Refs. ͓13,38͔. ͑ ͒ Since the first term of the rhs is time-constant and the second ii As a consequence of the Riemann-Lebesgue theorem, →ϱ is a function of time, we will call them constant term and full decoherence occurs at t . However, as in any expo- fluctuating term, respectively. nential decaying process, there is a characteristic decaying The Riemann-Lebesgue theorem, which mathematically time that can be considered as the time at which, in practice, expresses the phenomenon of destructive interference, states the decaying is approximately completed. In the next sec- that, if f͑͒L , tions we will compute the decoherence time of a self-induced 1 decoherence process as the characteristic decaying time of the fluctuating term in the expression of ͗O͑͘ ͒ of Eq. ͑7͒. lim ͵ df͑͒eit =0. ͑8͒ t t→ϱ ͑iii͒ In the model presented here, full decoherence is ob- tained for t→ϱ. However, as explained in Ref. ͓30͔, if the ͑ Ј͒ ͑ Ј͒ ͑ ͒ Therefore, if the function , O , of Eq. 7 is L1 total Hamiltonian has more than one discrete eigenvalue in variable = − Ј, we can apply the Riemann-Lebesgue nonoverlapping with the continuous spectrum, there is no theorem, decoherence in the cross terms of the discrete eigenvalues. ϱ ϱ ͑ ͉͒ ͵ ͵ ͑ Ј͒ ͑ Ј͒ i͓͑−Ј͒/ប͔t lim„ R t OR… = lim , O , e t→ϱ t→ϱ 0 0 III. POLES IN THE EXPECTATION VALUE EQUATION ϫddЈ =0. ͑9͒ In order to study and compute the decoherence time, we As a consequence, will use the standard theory of analytical continuation in ϱ scattering quantum theory ͑see, e.g., Refs. ͓58,59͔͒ and its ͗ ͘ ͑ ͉͒ ͑͒ ͑͒ ͑ ͒ lim O ͑t͒ = lim„ t O… = ͵ O d . 10 extension to the Liouville–von Neumann space ͑see, e.g., t→ϱ t→ϱ 0 Refs. ͓26,44͔͒. By means of this theory, we can compute the This last equation can be expressed as a weak limit, decoherence time in terms of the poles corresponding to the ͗ ͘ ͓ ϱ functions involved in the fluctuating term of O ͑t͒ see Eq. ͑͑ ͉͒ ͑ ͉ ͵ ͉͑͒͑ ͑ ͒ ͑7͔͒: W − lim t = * = d , 11 →ϱ t 0 ϱ ϱ ͓͑ Ј͒ ប͔ 3 ͑ ͉͒ ͵ ͵ ͑ Ј͒ ͑ Ј͒ i − / t Ј ͑ ͉ R t OR = , O , e d d . where * has only singular-diagonal terms. „ … 0 0 It is not difficult to see that this approach to decoherence avoids the problems of the einselection program. In fact, ͑12͒ In this equation, we can introduce the following change of 3Here we are working in the Schrödinger picture. However, ana- variables ͑see Appendix A͒: log results can be obtained in the Heisenberg picture: 1 ϱ = ͑ + Ј͒, = − Ј, ddЈ = Jdd = dd. 2 ͗ ͑ ͒͘ ͉ ͑ ͒ ͵ ͑͒ ͑͒ ͉͑ ͒ lim O t = lim O t = O d = O* . →ϱ →ϱ„ … t t 0 ͑13͒ Then,
012102-3 M. CASTAGNINO AND O. LOMBARDI PHYSICAL REVIEW A 72, 012102 ͑2005͒
ϱ 2 tonian H , and call ͕͉E͖͘ the eigenbasis of H , where 0ഛE ͑ ͉͒ Ј͑ ͒ Ј͑ ͒ i͑/ប͒t ͑ ͒ 0 0 „ R t OR… = ͵ d ͵ d , O , e , 14 Ͻϱ. Decoherence is due to the vanishing of the fluctuating 0 −2 ͗ ͘ ͓ ͑ ͔͒ term of O ͑t͒ see Eq. 12 , where Ј͑,͒=͑,Ј͒, OЈ͑,͒=O͑,Ј͒ and the new ϱ ϱ limits of the integrals are due to the fact that , Јജ0. Now ͑ ͉͒ ͵ ͵ ͑ Ј͒ ͑ Ј͒ i͓͑E−EЈ͒/ប͔t lim„ R t OR… = lim E,E O E,E e we promote the real variable to a complex variable Z;ifthe t→ϱ t→ϱ 0 0 function Ј͑Z,͒OЈ͑Z,͒ has no poles in the upper Z half ϫdEdEЈ =0, ͑18͒ plane, we obtain ϱ where ͑E,EЈ͒=͗E͉͉EЈ͘ are the coordinates of the state, and ͑ ͉͒ ͵ Ј͑ ͒ Ј͑ ͒ i͑Z/ប͒t O͑E,EЈ͒=͗E͉O͉EЈ͘ are the coordinates of the considered ob- „ R t OR… = ͵ d dz Z, O Z, e , 0 C͑−2,2͒ servable, both in the basis ͕͉E͖͘. Calling, as above, ͑15͒ 1 = ͑E + EЈ͒, = E − EЈ where C͑−2,2͒ is any curve that goes from −2 to 2 by 0 2 0 the upper complex half plane. If the function ͑ ͒ ͑ ͒ ␥ the limit reads Ј Z, OЈ Z, has, say, a pole at Z0 = ˜ +i in the upper ϱ half plane, we can, as usual, decompose C͑−2,2͒ 2 0 ⌫͑ ͒ഫ ͑t͉͒O ͵ d ͵ d ͑ ͒ = −2 ,2 CZ , where CZ is a residue circle around the lim„ R R… = 0 lim 0 0 + 0/2, 0 − 0/2 0 0 t→ϱ t→ϱ ⌫͑ ͒ 0 −2 0 pole Z0 and −2 ,2 is the remaining “background” ͑ ប͒ ϫ ͑ ͒ i 0/ t ͑ ͒ curve. If, as usual, we neglect the background, only the fac- O 0 + 0/2, 0 − 0/2 e . 19 tor ei͑Z/ប͒t becomes relevant at the pole Z ; this factor reads 0 Proceeding in the same way as in the previous section, we ͑ ប͒ ͓͑ ␥͒ ប͔ ͑ ប͒ ͑␥ ប͒ ei Z0/ t = ei ˜ +i / t = ei ˜ / te− / t, ͑16͒ arrive at an expression similar to Eq. ͑15͒, −͑␥/ប͒t ϱ where e is a dumping factor appearing in the regular ͑ ͉͒ ͵ lim R t OR = d 0 fluctuating term of ͗O͑͘ ͒ =(͑t͉͒O). Therefore the decoher- „ … t t→ϱ 0 ence time can be computed as the characteristic decaying time of the process as ϫ ͵ ͑ ͒ lim dZ 0 + Z/2, 0 − Z/2 t→ϱ ͑ ͒ ប C −2 0,2 0 ͑ ͒ tD = . 17 ϫ ͑ ͒ i͑Z/ប͒t ͑ ͒ ␥ O 0 + Z/2, 0 − Z/2 e , 20 Let us note that, up to this point, we have worked in the where some poles could be found ͑see details in Refs. eigenbasis of the complete Hamiltonian H, that we will call ͓26,44͔͒. However, since this is a free period, there should be ͕͉͘+͖.4 no decoherence, i.e., the decoherence time should be infinite. Once the decoherence time has been computed, a single This means that we have to adjust our theory to this physical question remains: what is the origin of the pole in the prod- fact by asking the following conditions for the complex con- uct Ј͑Z,͒OЈ͑Z,͒? We will address this problem in the tinuation of the coordinates ͑E,EЈ͒=͗E͉͉EЈ͘, O͑E,EЈ͒ next section. =͗E͉O͉EЈ͘: ͑i͒ Condition 1: In the free eigenbasis ͕͉E͖͘, the coordi- ͑ ͒ nates of the observable O, O 0 +Z/2, 0 −Z/2 , have no IV. ORIGIN OF THE POLES poles in the upper half plane. This is a natural requirement In the physical evolutions we are interested on, there are because, if not, the observable O would introduce by itself a “free” periods with no decoherence ͑e.g., very long ones like finite decoherence time for any state and any evolution the “in” and “out” periods used to modelize a scattering pro- Hamiltonian, which is certainly not a physical situation. ͑ ͒ ͕͉ ͖͘ cess, or short periods but long enough to fix the initial con- ii Condition 2: In the free eigenbasis E , the coordi- ͑ ͒ ditions for an “interaction” period͒ and interaction periods nates of the state , 0 +Z/2, 0 −Z/2 , have no poles in 5 where decoherence occurs. On this basis, we will first con- the upper half plane. sider in detail the free period governed by a free Hamiltonian If these two conditions are satisfied, the function ͑ ͒ ͑ ͒ ͑ ͒ H0 case 1 , and then the interaction period governed by a 0 +Z/2, 0 −Z/2 O 0 +Z/2, 0 −Z/2 will have no poles ͑ ͒ “perturbed” Hamiltonian H=H0 +V case 2 . in the upper half plane. In this case, the decoherence time is Case 1. Let us consider the free case with a free Hamil- infinite and decoherence is only nominal, as one would have expected in a free evolving situation. Case 2. Once we have “calibrated” our theory in order to 4 ͕͉͘±͖ Strictly speaking, we will find a double basis correspond- satisfy the physical condition according to which a free ing to a decaying process and a growing process, respectively. But since we are only interested in the former one, we will use only ͕͉͘+͖ and work in the lower half plane, trying to find poles in the 5Nevertheless, in Sec. VII we will see that, in some cases, there second sheet. In Sec. II, we have simply called ͉͘ the eigenvectors are poles in the free period, e.g., in the case of the evolution of a ͉͘+ ͓see Eq. ͑1͔͒. system with a thermal bath.
012102-4 DECOHERENCE TIME IN SELF-INDUCED DECOHERENCE PHYSICAL REVIEW A 72, 012102 ͑2005͒ evolving system does not decohere, we will consider the pro- 1 ͉͗z͘+ = ͉͗z͘ + ͉͗ V͉z͘. ͑27͒ cess in the interaction period. The total Hamiltonian now z − H reads ϱ ϱ ϱ If we suppose, as before, that V is a well behaved vector ͵ ͉͉͗͘ ͵ ͵ ͑ ͉͒͘ value function, then even if ͉͗͘ has no poles ͑as required H = H0 + V = d + V , ͒ ͉͗͘+ 0 0 0 in case 1 , has a pole at z0, coming from the resolvent +͗ ͉͘ ͗ ͉͘ ϫ͗Ј͉ddЈ, ͑21͒ term. The same applies to z : even if z has no poles, +͗z͉͘ gets the poles of the resolvent. The argument can be ͕͉͖͘ ͑ where is the eigenbasis of H0 here we have replaced E extended to states and observables: even if ͗z͉͉zЈ͘ has no with to emphasize that now this is an interaction period͒. poles, +͗z͉͉zЈ͘+ has poles, and even if ͗z͉O͉zЈ͘ has no poles, Let us consider the resolvent, namely, the complex value op- +͗z͉O͉zЈ͘+ has poles. ͑ ͓ ͔͒ erator see Ref. 60 As a consequence, if the initial condition of the interac- R͑z͒ = ͑z − H͒−1, ͑22͒ tion period is given by the states and operators of a previous free period which, as shown in case 1, have no poles in the ͓ ͑ ͒ i.e., the analytical continuation to the lower second sheet of eigenbasis of H0 precisely, the continuations of E,EЈ =͗E͉͉EЈ͘ and O͑E,EЈ͒=͗E͉O͉EЈ͘ have no poles͔, then −1 R͑͒ = ͑ + i0−H͒ . ͑23͒ ͑z,zЈ͒=͗z͉͉zЈ͘ and O͑z,zЈ͒=͗z͉O͉zЈ͘ have no poles but +͗ ͉͉ Ј͘+ +͗ ͉ ͉ Ј͘+ The poles of R͑z͒ are known as the poles of the resolvent, z z and z O z do have poles that produce the ͑ ͒ 8 and they coincide with those of the S matrix. In fact, for the dumping factor of Eq. 16 . These results are presented in ͓ ͔ Hamiltonian ͑21͒, the S-matrix coefficients read ͑see Ref. great detail in Ref. 45 , where the dumping factor appears in ͑ ͒ ͓61͔͒ Eq. 70 , precisely, ϱ 1 ͑ ͒ ͑ ͉͒ ͵ ͑ ͉⌽ ͒͑˜⌽ ͉ ͒ i ¯z0−z0 t͑ ͉⌽ ͒͑˜⌽ ͉ ͒ S͑͒ =1−2i͉͗V͉͘ −2i͉͗V V͉͘. „ t O… = d 0 O + e 0 00 00 O + i0−H 0 ͑24͒ ͑ Ј͒ ͵ Ј i ¯z0−z t͑ ͉⌽ ͒͑˜⌽ ͉ ͒ + dz e 0 0zЈ 0zЈ O Then, if V is well behaved, i.e., the analytical continuation of ⌫ V͉͘ is a vector value analytical function ͓60͔, functions 6 ͑ ͒ R͑͒ and S͑͒ have the same poles. ͵ i z−z0 t͑ ͉⌽ ͒͑˜⌽ ͉ ͒ + dze 0 z0 z0 O As before, for the sake of simplicity we will assume that ¯⌫ ͑ ͒ 7 R z has just one pole z0, and we will only consider pure ͵ Ј͵ i͑z−zЈ͒t͑ ͉⌽ ͒͑˜⌽ ͉ ͒ ͑ ͒ states. On this basis we will show that, if the continuation to + dz dze 0 zzЈ zzЈ O . 28 the lower second sheet of ͉͗͘ has no poles, the continua- ⌫ ¯⌫ ͉͗͘+ tion to the lower second sheet of gets the pole z0 of ͑ ͒ Independently of the precise meaning of each symbol ͑which function R z . In fact, from the Lippmann-Schwinger equa- ͓ ͔͒ ͑ can be found in Ref. 45 , it is quite clear that the term tion we know that there are two eigenbases for H see Ref. ͑ ͒ i ¯z0−z0 t͑ ͉⌽ ͒͑⌽˜ ͉ ͒ iZ0t͑ ͉⌽ ͒͑⌽˜ ͉ ͒ ͓61͔͒, e 0 00 00 O =e 0 00 00 O represents the main contribution to decoherence, that is, the pole con- ± 1 tribution. The first term is the “constant term” and the last ͉͘ = ͉͘ + V͉͘, ͑25͒ 9 ± i0−H three terms are background terms. Furthermore, it can be proved that a sufficient condition for the coefficient where ±i0 symbolizes the analytical continuation to the ͑ ͉⌽ ͒͑⌽˜ ͉O͒ be well defined is that ͑z,zЈ͒=͗z͉͉zЈ͘ and ͑ ͒ 0 00 00 lower upper half plane in the second sheet. As explained, O͑z,zЈ͒=͗z͉O͉zЈ͘ have no poles. we will consider only the decaying case and therefore we will only use ͕͉͘+͖. Then, 8Moreover, if conditions 1 or 2 of case 1 are not satisfied, 1 ͑z,zЈ͒=͗z͉͉zЈ͘ and O͑z,zЈ͒=͗z͉O͉zЈ͘ may have poles, as it will be ͉͗͘+ = ͉͗͘ + ͉͗ V͉͑͘26͒ + i0−H shown in Sec. VII. Nevertheless, in a certain sense more poles are welcomed because what we are essentially trying to prove is that and we can make the analytical continuation of this equation decoherence time is very small. 9 to the lower second sheet, All this computation can also be made by using the Laplace transform ͑see Ref. ͓62͔͒
6 1 1 If V is not well behaved, some new poles may appear. ͑ ͒ ͑ ͒ 7 ͉͑ ͒͘ exp − iLt = ͵ exp − izt dz z0 is the pole of the t evolution, Z0 =¯z0 −z0 is the pole of the 2i L − z ͑ ͒ C t evolution. The minus sign in the −z0 produces the change from the lower half plane for the decaying processes in the ͉͑t͒͘ evolu- with the same result. In this case, all the conditions required in this tion to the upper half plane for these processes in the ͑t͒ evolution. section are also needed.
012102-5 M. CASTAGNINO AND O. LOMBARDI PHYSICAL REVIEW A 72, 012102 ͑2005͒
V. DECOHERENCE TIME FOR MICROSCOPIC SYSTEMS N N N N ͑ ͒ H = ͚ Hi + ͚ Vi + ͚ Vij + ͚ Vijk + ¯ , 33 In these three final sections we will present some esti- i=1 i=1 i,j=1 i,j,k=1 mates of decoherence time, in order to show that self- induced decoherence can account for already known results where the Hi are the free Hamiltonians of the microscopic and opens the way to more detailed models. particles, the Vi are the averages of the interactions between As already explained, if ␥ is the imaginary part of the pole each particle and the remaining particles, the Vij are the two ͑or of the pole closer to the real axis͒, the decoherence time particle interactions, and so forth. The eigenvectors of H are ͉ ͘ is ,x1 ,...,xN−1 , such that ប H͉,x , ...,x ͘ = ͉,x , ...,x ͘, ͑34͒ t = ͑29͒ 1 N−1 1 N−1 D ␥ where is the total energy and x1 ,...,xN−1 are the remaining because, as we have said, the characteristic decaying time of labels necessary to define the eigenstate. Since decoherence ͗ ͘ ͑ ͉͒ the fluctuating term of O ͑t͒ =( t O) is the decoherence occurs in variable , we can ignore the rest of the labels for time tD. our argument. Disregarding for simplicity the Vij,Vijk,..., The decoherence time can be estimated in particular cases and considering all the Vi equal, we obtain ͓ ͔ like, e.g., the Friedrich model studied in Refs. 26,44 , where N we obtain ͑ ͒ V = ͚ Vi = NVi. 35 ប i=1 ͑ ͒ tD = 2 30 1 2͉V⍀͉ Then, the characteristic energy in Eq. ͑30͒ is now
being V⍀ the interaction function. It is clear that, if the inter- ប t →ϱ D1 ͑ ͒ action vanishes, tD . In turn, if the characteristic energy tD = = . 36 2 NV N 2͉V⍀͉ ϳV is, say, 1 eV ͑a natural energy scale for quantum i atomic interactions, see, e.g., Ref. ͓63͔͒, the decoherence Let us suppose again that all the V are of the order of 1 eV. time is ϳ10−15 s.10 i If we consider a macroscopic object of 1 mol, where N This means that, when the theory is calibrated in such a =1024, the decoherence time results, way that the free period does not lead to decoherence, a generic system does decohere ͑and, in general, very fast͒ in −39 tD =10 s, the interaction period. It is interesting to remark that the method of Ref. ͓45͔ was a very tiny time indeed. This decoherence time is so close to compared with the usual methods of nuclear physics Plank time 10−43 s that it should be considered more as an ͓͑ ͔͒ 208 ͑ ͒ 64–66 in the case of a Pb 2d5/2 proton state in a illustration than as a physical result. Nevertheless, it shows Woods-Saxon potential, including spin-orbit interaction with that decoherence is fantastically fast in macroscopic bodies. ͓ ͔ ͑ ͑ ͒ parameters as in Ref. 67 , with an excellent agreement see Let us compare Eq. 36 with the low temperature tdec of ͓ ͔͒ ␥ −1 Fig. 3 of Ref. 45 . In this case, =10 MeV and tD Ref. ͓11͔,p.51, ϳ10−20 s. ⌬x 2 ␥−1ͩ ͪ tdec = 0 VI. DECOHERENCE TIME OF MACROSCOPIC BODIES 2L0 ͉͘ where we have also a interaction factor term ␥−1ϳt and a A pure state of a macroscopic object can be consid- 0 D1 ͉ ͘ ͑⌬ ͒2 ϳ ered as the tensor product of N states i of microscopic macroscopic coefficient x/2L0 1/N. We can see a co- H particles, belonging to Hilbert spaces i, respectively, incidence between the orders of magnitude of our result and of the standard theory’s result, in the spirit of the Comment 2 N N ͉͘ ͉ ͘ H ͑ ͒ of the Introduction. = i i 31 i=1 i=1 VII. INITIAL THERMAL BATH or, more generally, if the microscopic states are mixed, i L H H i = i i, then the state of the macroscopic object will Finally, we will consider the case of a system with a ther- be mal bath, following the formalism of Ref. ͓27͔, Sec. IV.B. N N For the model considered in that section, an oscillator in a L ͑ ͒ ͑ ͒ = i i. 32 thermal bath, the fluctuating term of Eq. 7 is i=1 i=1 ͓͑ ͒ ប͔ ͵ i p− pЈ t/ Ј ͑ ͒ Therefore the generic total Hamiltonian reads OppЈ¯ppЈe dpdp , 37
10 ͉͑ † ͒ ͑ ͒ ͓ ͔ This is, of course, a general case: ␥ is usually of the order of where ¯ppЈ= ApApЈ , given in Eq. 37 of Ref. 27 ,isthe magnitude of the characteristic interaction energy V. initial condition of the oscillator and thermal bath,
012102-6 DECOHERENCE TIME IN SELF-INDUCED DECOHERENCE PHYSICAL REVIEW A 72, 012102 ͑2005͒
͗ ͘ 2 n 0VpVpЈ S ML ͉͑A†A ͒ = ͑p͒␦3͑p − pЈ͒ + N = = ͑45͒ p pЈ ͑ ͒ ͑ ͒ ប⌼ + p − pЈ S1 ͑ ͒ and the decoherence time reads VpVpЈ p + ͑ ͒͑ − − i0͒ ប2 − pЈ pЈ p ϳ ⌼ ͑ ͒ tD . 46 ͑ ͒ ML2kT VpVpЈ p VpVpЈ + + ͑ ͒͑ ͒ ͑ ͒ ͑ ͒ ⌼ ␥−1 + p p − pЈ + i0 + p − pЈ In the particular case that = 0 , L=L0 and M =M are the characteristic time, length, and mass of the model of Sec. dkV2͑k͒ ͓ ͔ ϫ ͵ k , 4.1 of Ref. 11 , when we introduce the de Broglie length ͑ ͒͑ ͒͑ ͒ ប ͱ ϳប ͱ − pЈ pЈ − p − i0 p − pЈ + i0 DB= / 3MkT / MkT, we obtain ͑38͒ 2 ϳ ␥−1ͩ DBͪ ͑ ͒ tD 0 , 47 where ͑p͒ is given in Eq. ͑40͒ of that paper, L0 1 namely, Eq. ͑4.10͒ of Ref. ͓11͔ which represents the deco- ͑ ͒ ͑ ͒ p =  39 herence time for the model of Sec. 4.1. This shows, again in e p −1 the spirit of Comment 2 of the Introduction, the agreement and where =1/kT. Again, independently of the precise between the orders of magnitude of the decoherence times meaning of each term ͑which can be found in Ref. ͓27͔͒,itis obtained by the SID and the EID approaches. ͑ ͒ obvious that the p factor has a pole in the p complex plane and therefore the initial condition is not pole free. Then, defining as before VIII. CONCLUSION In a series of previous papers we have developed an ap- = p − pЈ,2= p + pЈ, p = + , etc., 2 proach to decoherence that avoids the drawbacks of the ein- selection approach. In the present paper we have shown that ͑ ͒ 40 our formalism supplies a precise method for computing the the terms of Eq. ͑37͒ have the form decoherence time, and that the results obtained with such a method are physically meaningful. 1 ͵ ei͑t/ប͒dd, ͑41͒ However, we are aware of the fact that a great number of ¯ e͓+͑/2͔͒ −1 results have been obtained in the context of the einselection program when compared with the cases treated by means of where the dots symbolize other factors coming from the in- ͑ ͒ the self-induced approach. Therefore our future work has to teraction which may have poles not considered here . When be directed to enlarge the set of applications of our theory. we introduce the complex variable z=+i␥, the poles of the ͑ ͒ We consider that this task will be worth the effort to the factor p turn out to be located where the following equa- extent that decoherence is a key element in the explanation tion is satisfied: of the emergence of classicality from the quantum world. ␥ ␥ e͓+͑z/2͔͒ = e͓+͑/2͔͒ͩcos + i sin ͪ =1. ͑42͒ 2kt 2kt APPENDIX A: HARTOG THEOREM Then, the poles are located in the coordinates A necessary mathematical remark is in order: from the =−2, ␥ =4nkT, ͑43͒ Hartog theorem, we know that the realm of analytical func- n n tions of more than one complex variable is much more in- where is an integer number. For =1 we obtain the deco- ͓ ͔ herence time volved than that of just one variable. In Refs. 26,44,45 ,we have considered the analytical continuation of two variables, ប z and zЈ, in products like f ͑z͒ f ͑zЈ͒; this suggests that we t = ͑44͒ 1 2 D 4kT should have worked with the theory of analytical functions of two complex variables. However, this is not the case be- which, for room temperature, Tϳ102 K, gives t =10−13 s D1 cause the only relevant variable for the problems considered for a single-particle system, and t =t /N=10−37 s for a D D1 in the quoted papers is the difference Z=z−zЈ that appears in mol-particle system. the evolution factor ei͑z−zЈ͒t; then, we can always introduce a ប Moreover, if S1 = is the characteristic action for a par- change of variables, ticle system,11 and S=ML2 /⌼ is the action of a macroscopic system ͑where M, L, and ⌼ are the characteristic mass, z + zЈ =2, z − zЈ = Z, length, and time͒, the particle number can be estimated as ប ͑ ͒ q=p=1 we obtain S1 =QP= . ii In a model with spherical sym- 11 E.g., ͑i͒ In the harmonic oscillator, the unidimensional coordi- metry we have LzY͑,͒=mបY͑,͒; then, by making m=1 we 1/2 1/2 ប nates are q=͑M/ប͒ Q and p=͑1/Mប͒ P; then, by making obtain S1 = Lz = , and so forth.
012102-7 M. CASTAGNINO AND O. LOMBARDI PHYSICAL REVIEW A 72, 012102 ͑2005͒
Z Z ϱ ϱ ϱ Ј ͑ ͒ ͑1͒ ͵ ͉͉͗͘ ͵ ͵ ͑1͒͑ Ј͉͒͘ z = + , z = − , A1 H = H0 + V1 = d + V , 2 2 0 0 0 ͑ ͒ ͑ ͒ ͑ ͒ and functions f z,zЈ , like the product f1 z f2 zЈ , have to be ϫ͗Ј͉ddЈ. ͑B2͒ considered as functions f͑z,zЈ͒= f͓+͑Z/2͒,−͑Z/2͔͒ ͕͉͘+ ͖ where ZC; but since in all the cases we have taken R, If we change the basis to ͑1͒ , we obtain f͑z,zЈ͒ is really a function of only one complex variable Z. 1 For instance, in Eq. ͑71͒ of Ref. ͓45͔, the symbol ͉͘+ = ͉͘ + V͑1͉͒͘. ͑B3͒ ͑1͒ cont→ cont → should be understood as + i0−H ¯z0 Ј z0 Since the interaction V͑1͒ is macroscopic, the dumping of the cont ͑ ͒→ ͑␥ ͒cont ͑ ͒→ ͑␥ ͒ = cont→cont→ ␥, + /2 ˜ +i /2 − /2 ˜ −i /2 ˜ i off-diagonal terms has a characteristic time of 10−37–10−39 s. ͑A2͒ Therefore, after an initial period much larger than this mag- where is a real number. In this equation, only the second nitude, the state can be considered nearly diagonal for all continuation of the rhs is an analytical continuation in the practical purposes. However, the state is not yet in complete equilibrium, complex plane, being the first one a simple change of a real ͑2͒ variable, from to ˜ . because the interaction V is always present: now it be- These considerations show that Refs. ͓26,44,45͔ and, in comes relevant. Then, after the initial period we can consider general, the method presented in this paper, are free from the total Hamiltonian as Hartog’s objection. ϱ ϱ ϱ ͑1͒ ͵ ͉͘+ ͉͗+ ͵ ͵ ͑2͒Ј͑ Ј͒ H = H + V2 = ͑1͒ ͑1͒d + V , APPENDIX B: TWO-TIMES EVOLUTION 0 0 0 ϫ͉͘+ ͗Ј͉+ Ј Let us now generalize Eq. ͑21͒ by adding two interac- ͑1͒ ͑1͒d d , tions, ͑2͒Ј͑ Ј͒ ͑2͒͑ Ј͒ ͕͉͘+ ͖ where V , is V , in the new basis ͑1͒ . ϱ ϱ ϱ When we make a final change of basis ͵ ͉͉͗͘ ͵ ͵ ͓ ͑1͒͑ Ј͒ ͑2͒ H = H0 + V = d + V , + V 0 0 0 + + 1 ͑2͒ + ͉͘ = ͉͑͘ ͒ + V ͉͑͘ ͒ ϫ͑,Ј͔͉͒͗͘Ј͉ddЈ, ͑B1͒ 1 + i0−H 1 ͑ ͒ ͑ ͒ where V 1 represents a macroscopic interaction and V 2 rep- we can compute the characteristic time in this case. But now ͑ ͒ ͑ ͒ resents a microscopic interaction: V 1 ͑,Ј͒ӷV 2 ͑,Ј͒. this time may be of the order of 1 s, that is, the relaxation As a consequence, in a first step we can neglect V͑2͒͑,Ј͒ time of a macroscopic body. In this way we can describe a and repeat what was said in Sec. IV. Then, we can begin with two-times process, with a extremely short decoherence time considering a Hamiltonian and a long relaxation time.
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