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Quantum measurements without Schrödinger cat states

We report and give an alternative derivation of some results on a model for a quantum measurement studied in [1]. The measured microscopic system is coupled to the position of a macroscopic pointer, which itself interacts with its environment via its momentum. The entanglement between the system and the pointer produced by their mutual interaction is simultaneous with the decoherence of distinct pointer readings resulting from leakage of information to the environment. After a discussion on the various time scales in the model we calculate the matrix elements of the system-pointer density operator between eigenstates of the measured observable with distinct eigenvalues. In general, the decay with time of these coherences is neither exponential nor gaussian. We determine the decoherence (decay) time in terms of the strength of the system-pointer and pointer-environment couplings. This decoherence time does not depend upon the details of the pointer-bath coupling as soon as it is smaller than the bath correlation time (non-Markov regime). In contrast, in the Markov regime it depends strongly on whether this coupling is Ohmic or super-Ohmic.


Introduction
The issue of the interpretation and theoretical description of measurements on quantum systems has been the subject of intense debates since the birth of quantum mechanics [2]. In the last three decades the major role played by environment-induced decoherence in measurement processes has been fully acknowledged thanks to the works of Zeh, Zurek and others (see [3,4] and references therein). Various concrete models of a quantum system interacting with a measurement apparatus coupled to its environment have been investigated; some of them are inspired from statistical physics [5]. A renewal of interest for quantum detection and decoherence came in the last decade with new developments in quantum information. It is desirable to understand better the relation between quantum and classical information and how one can convert one into another. Moreover, a good control over all sources of decoherence is required to proceed quantum information. On the experimental side, measurements can be used both to extract information on quantum states and to monitor quantum systems (quantum trajectories [6], Zeno effect [7]). Experimental data are now available for the decoherence time in microwave cavities [8] and in solid state devices like superconducting tunnel junction nanocircuits [9,10].
A measurement on a quantum system consists in letting this system (called "object" in the following) interact with a measurement apparatus in such a way that some information about the object's state is transfered to the apparatus. It has been already recognised by Bohr (see e.g. [2]) that even though the composite (object and apparatus) system has to be described by the quantum theory, some part of the apparatus (called the "pointer" in the following) must be capable of a classical behaviour. In an ideal measurement, the interaction builds up a one-to-one correspondence between the eigenvalues s of the measured observable S (supposed here to have a discrete and non-degenerate spectrum) and some macroscopically distinguishable pointer states. These states may for instance correspond to macroscopically distinct positions of the pointer. As is well-known [2][3][4] a quantum measurement involves both the establishment of the aforementioned object-apparatus correlations and some "superselection rules" destroying coherences between the pointer states. In most previous discussions in the literature these two processes are treated separately. A first step consists in establishing quantum correlations between the object and the pointer by means of an appropriate unitary evolution. This evolution results from an interaction and produces an object-pointer entangled state. If the object is initially in a superposition of eigenstates |s of S, the unitary evolution builds up a Schrödinger cat state. This state is a superposition of separable object-pointer states given by tensor products of the eigenstates |s with some pointer states having, say, sharply-defined positions depending on s. Macroscopic distances separate pointer locations corresponding to distinct s's. This Schrödinger cat state is taken as the initial state for a second dynamical process called decoherence, which leads to superselection rules. During this second step the quantum correlations between the object and apparatus are transformed into classical correlations, i.e., the superposition of the object-pointer product states is transformed into a statistical mixture of the same states.
For such a sequential treatment to make physical sense, the time duration of the entanglement process must be short compared with the decoherence time after which coherences in the superposition disappear. However, it is known that the latter time is extremely short. The sequential treatment of the entanglement and decoherence can then be difficult to achieve in practice. The purpose of this paper is to study a model in which the two above steps are performed simultaneously. If, in contrast to the situation described above, the decoherence time is the shorter time, macroscopically distinguishable superpositions are transformed into statistical mixtures faster than entanglement can create them. The whole measurement is then performed without producing a Schrödinger cat state as an intermediate step.
Our model is a three-partite system consisting in the microscopic object to be measured, a pointer composed by a single degree of freedom of the macroscopic apparatus, singled out by its strong coupling to the object, and a bath consisting of all other degrees of freedom of the apparatus. A pointer-bath coupling is responsible for decoherence. In an ideal measurement, the measured observable S in the Heisenberg picture is weakly changed during the measurement. This can be the case provided that, on the one hand, the object-pointer interaction Hamiltonian producing the aforementioned entanglement nearly commutes with S and, on the other hand, the time scale for significant evolution of S under the object Hamiltonian is much larger than the time duration of the measurement. In particular, this time scale must be large compared with the decoherence time t dec . Furthermore, t dec is also much smaller than the classical time scale characterising the changes in position of the pointer under its proper Hamiltonian. As already acknowledged in [11] in the case of a system-bath coupling, explicit formulas for t dec can be derived on the basis of this separation of time scales, with no further assumptions about the relation between t dec and the bath correlation time T B . This allows us to treat the pointer-bath coupling in a non-Markov regime t dec T B as well as in the Markov regime t dec T B . Moreover, we need not to assume that the bath is composed of harmonic oscillators linearly coupled to the pointer, a supposition often made in previous studies. A third noteworthy generalisation with respect to previous works concerns the object-apparatus initial state. While it is appropriate to assume that the object and apparatus are uncorrelated initially, we shall consider two types of initial states for the apparatus in which the pointer and bath are either uncorrelated (i.e., in a product state) or in thermal equilibrium (i.e., in an entangled state) with each other.
The paper is organised as follows. The model is introduced in Section 2. We discuss the separation of the time scales and a subsequent simplification of the object-pointer dynamics in Section 3. Sections 4 and 5 contain a separate study of the two dynamical processes producing the entanglement of the object with the pointer and the loss of coherences between well-separated pointer readings. Section 6 provides a short summary of these two sections. We derive and discuss our results for a separable pointer-bath initial state in Section 7. In the next Section, we report without proofs the results derived in [1] for a pointer and bath initially entangled and in thermal equilibrium. Our conclusions are drawn in Section 9. Appendix A contains technical proofs of some statements made in Section 7.

The model
As in much of the previous literature on quantum measurement [3][4][5], we shall deal with a threepartite system: the object of measurement is some microscopic system (S); a single-degree-of-freedom macroscopic pointer (P) will allow read-outs; finally, a bath (B) with many (N 1) degrees of freedom serves to decohere distinct pointer readings. We shall have to deal with the following dynamical variables: for the object S, the observable S to be measured; for P the position X and the momentum P ; and for B, a certain coupling agent B. The pointer is coupled to S and B via the Hamiltonians where is a coupling constant and α a positive integer. The object-pointer coupling H PS is chosen so as to (i) not change the object observable S to be measured (i.e., [H PS , S] = 0); (ii) be capable of shifting the pointer position by an amount proportional to (the eigenvalues of) S, in such a way that each eigenvalue s of S becomes entangled with a specific pointer reading; (iii) be a strong coupling ( is large), so that different eigenvalues s = s eventually become associated with pointer readings separated by large distances. The pointer-bath interaction, in turn, is chosen for most efficient decoherence of distinct pointer positions [11]. Depending on the value of α, nonlinear (α > 1) as well as linear (α = 1) couplings will be considered. The free evolutions of S, P and B are generated by respective Hamiltonians H S , H P and H B . We do not have to specify H S explicitly. The pointer Hamiltonian H P = P 2 /2M + V (X) has a potential V (x) with a local minimum at x = 0, so that V (0) = 0 and V (0) > 0; moreover, the potential barriers at the left and right of this minimum have a height V 0 satisfying V 0 k B T , where T is the temperature and k B the Boltzmann constant. An example of such potential is sketched in Fig. 1. A confining potential with a single minimum at x = 0 provides another example, for which V 0 = ∞. The Hamiltonian of the full system S + P + B is We now proceed to describe the initial states of the object S and the apparatus P + B allowed for. It is appropriate to require initial statistical independence between S and P + B. The initial density operator ρ S of the object may represent a pure or a mixed state. Two types of initial conditions for the apparatus will be considered. The first one, to be referred to as partial equilibrium, is a product state in which P has some density operator ρ P and B is at thermal equilibrium with the Gibbs density operator ρ (eq) For this first initial state all three subsystems are statistically independent. In the second (more realistic) initial state, the apparatus P + B is in thermal equilibrium according to the density operator ρ (eq) . For a pointer potential as sketched in Fig.1, this thermal equilibrium must be understood as a local equilibrium around x = 0. It can be achieved by first preparing P in some state localised near x = 0 at a time t = −t in , with t in larger than the relaxation rate but small with respect to the tunnelling escape rate, and then letting P interact with B until t = 0. The two initial states of S + P + B are We further specify the partial-equilibrium state (3) by requiring that the probability density x|ρ P |x to find the pointer at position x has a single peak of width ∆x = ∆ centred at x = 0. A momentum uncertainty ∆p = 2π /λ defines a second length scale λ. A macroscopic pointer has both ∆ and λ negligibly small against any macroscopic read-out scale ∆ class , x Figure 1. Sketch of a candidate for the pointer potential. The height V 0 of the potential barriers around x = 0 and the width of the potential wall, W ≈ (V 0 /V (0)) 1/2 , are much larger than k B T and the thermal fluctuation ∆ th = (k B T /V (0)) 1/2 . The unstable parts of V (x) at the left and right of the barriers serves to amplify the separation between the peaks in the pointer density produced by its interaction with the object. Such an amplification can happen after the object-pointer coupling is turned off at a time t int > W/( δs) (see Section 6).
where the first inequality is the uncertainty principle. We shall also require that which means that the state ρ P is not highly squeezed in momentum nor in position. As a concrete example we may consider a Gaussian pointer density matrix, If P is initially in a pure state then tr P ρ 2 P = dx dx x|ρ P |x 2 = 1, which implies that this state has the minimum uncertainty product ∆x∆p = /2, i.e., λ = 4π∆. The Gaussian density (7) also arises if P is in a Gibbs state ρ (eq) P = Z −1 P e −βH P at high enough temperature. To see this, we note that the pointer observables X and P evolve noticeably under the Hamiltonian H P on a classical time scale T P , which is much larger than all other (quantum) time scales in the model. In particular, T P is much larger than the thermal time, T P β. As a result, the matrix elements x|ρ (eq) P |x of ρ (eq) P can be approximated by Z −1 P e −β(V (x)+V (x ))/2 e −2π 2 (x−x ) 2 /λ 2 th , wherein λ th = 2π (β/M ) 1/2 is the thermal de Broglie wavelength. The reader may recognise in this expression the short-time behaviour of the quantum propagator x|e −itH P / |x for t = −i β (see e.g. [12]). Since the potential V (x) has a local minimum at x = 0, it can be approximated near the origin by a quadratic potential, V (x) V (0) + x 2 V (0)/2. Therefore, for small x and x , x|ρ (eq) P |x has the Gaussian form (7) with ∆ = ∆ th = (βV (0)) −1/2 and λ λ th . It is important to bear in mind the separation of length scales λ th ∆ th ∆ class . Inasmuch as the pointer classical time scale T P may be defined as T P = (M/V (0)) 1/2 , the fact that λ th is much smaller than ∆ th is equivalent to T P β. To fix ideas, for T P = 1 s, M = 1 g, ∆ class = 1 cm and a temperature of 1 K, the above-mentioned length scales differ by more than eight orders of magnitude. Hence (5) and (6) are well satisfied if ρ P = ρ (eq) P . All of these illustrations, including the Gaussian (7), are meant to give an intuitive picture. What we shall need in actual fact are the quasi-classical nature of the pointer initial state, as implied by (5) and (6), together with the single-peak character of the initial density of pointer positions.
We will study in what follows the dynamics of the reduced state of S + P (object and pointer). This state is defined by a density operator ρ PS (t) obtained by tracing out the bath degrees of freedom in the state of S + P + B, Here and in what follows tr j refers to the partial trace over the Hilbert space of j = S, P or B.
One may associate with the time evolution of X(t), B(t) and S(t) four distinct time scales. The time scale T P = (M/V (0)) 1/2 has been already introduced above; it is the time scale for significant evolution of X(t) (or, equivalently, of P (t) = M d X/dt ) when the pointer is in the initial state ρ P . If ρ P is a pure state, its energy uncertainty ∆E is given by Heisenberg's uncertainty relation ∆E ≈ 2π /T P . In fact, ∆E 2 = tr P (H P 2 ρ P ) − tr P (H P ρ P ) 2 is found by means of (7) to be equal to (2π ) 4 /(2λ 4 M 2 )+V (0) 2 ∆ 4 /2− 2 V (0)/(4M ); one can check that the three terms in this expression are of the order of 2 /T P 2 by combining λ = 4π∆ and (6). The bath correlation time T B is defined with the help of the bath auto-correlation function For simplicity, it is assumed here and in the whole paper that Because the bath has many degrees of freedom, h(t) decays (often exponentially) to zero as |t| goes to infinity. We define T B (respectively t B ) as the largest (smallest) time constant characterising the variations of h. More precisely, h(t) 0 whenever |t| T B and h(t) h(0) whenever |t| t B . Note that with B in thermal equilibrium, the thermal time β figures among the decay rates of h and thus The time scale T S is defined in an analogous way as t B , so as to signal significant variation of the mean value tr S ( S(t)ρ S ) of S(t) and of the object correlation functions such as tr S ( S(t)Sρ S ). Let us stress that T S can be larger than the inverse of the typical object Bohr frequency (E − E )/ (here E and E denote two neighbouring eigenvalues of H S ). For instance, if S is (or commutes with) the energy H S , then T S = ∞.

Separation of time scales
As stressed in the Introduction, during the time span of an ideal measurement the measured observable S(t) does not evolve noticeably under the "free" Hamiltonian H S . The same conclusion holds for the pointer observables X(t) and P (t), which evolve noticeably on the classical time scale T P . It is thus legitimate to assume that the object and pointer are put in contact during a time t int much shorter than both T S and T P . As far as we are aware, this natural separation of time scales in a quantum measurement has not been so far fully exploited in previous works. We will see below that it allows for a great simplification of the dynamics implemented by the Hamiltonian (2). The density matrix (8) can then be determined at time t int without relying on an allegedly short bath correlation time T B t int (Markov approximation) and without making a specific choice for the pointer-bath interaction (among such choices, the quantum Brownian motion and the spin-boson model are widely discussed in the literature, see [4,18,19] and references therein). In fact, we will see in this Section that for t int T S , T P the Hamiltonians H S and H P can be dropped out in the total Hamiltonian (2) provided that the initial conditions (3) or (4) are modified appropriately.
For times t T S , T P , the object-apparatus evolution operator can be approximated by This can be established by expressing this evolution operator in the interaction picture with respect to H 0 = H S + H P . By (1) and (2), wherein T denotes the time-ordering and S(τ ), P (τ ) and X(τ ) are given by (9). Note that for |t| T S , T P these operators are almost constant in time between τ = 0 and τ = t. Hence one may replace them in (13) by S, P and X, respectively. In other words, the right-hand side of (13) can be approximated by exp{−it(H B + H PS + H PB )/ }. Multiplying both members of (13) by e −itH 0 / , one obtains (12).
Let us substitute t by −t in (12), take the adjoint and replace the result into (8). One then finds that the impact of H S and H P on the dynamics of ρ PS (t) can be fully accounted for at times t T S , T P by modifying the initial state of S + P + B according to After this slippage of the initial condition, one makes a negligible error in (8) by dropping out H S and H P in the total Hamiltonian (2). Thanks to the quasi-classical nature of the pointer initial state ρ P in (3), this state is not modified by the substitution (14). This can be justified as follows if ρ P is a pure state. Let |E k be the eigenvectors of H P with eigenenergies E k . We set ρ (0) Indeed, if |E k − E l | is of the order of (or smaller than) the energy uncertainty ∆E ≈ 2π /T P then the phase factor in the second member of (15) is close to unity when t T P ; otherwise E k |ρ P |E l 0. One can also show [1] that ρ (0) P (t) ρ P at times t T P if ρ P is a gaussian mixed state given by (7) and, for the initial condition (4), it holds e −itH P / ρ (eq) PB provided that t, β T P . The situation is different for the object initial state ρ S : then cannot be approximated by ρ S . For instance, for S = H S it has been argued above that T S = ∞ and s|ρ   S (t)|s has to approximate tr(Sρ S ) = s s s|ρ S |s in this limit. It follows from the above discussion that it is well justified to set (3) and (4) by the time-dependent density operator (16).

Entanglement of the object and pointer
Before studying the dynamics generated by the total Hamiltonian (2), it is instructive to discuss what happens if one disregards the free dynamics of S, P and B as well as the pointer-bath interaction, so that H = H PS = SP . In such a situation the bath can be ignored. The initial product state of S and P evolves into an entangled state in which each eigenstate |s of S is tied up with a given pointer position depending on s. Recalling that P is the generator of space translations we have x is the joint eigenstate of S and X with eigenvalues s and x, normalised as s, x|s , x = δ ss δ(x − x ). Hence an initial product state of S + P becomes at time t It is now well to put forth a specification: Throughout the present paper we assume for simplicity that S has a discrete and non-degenerate spectrum. Moreover, if the Hilbert space of S has infinite dimension we restrict ourselves to initial states of the object satisfying s|ρ S |s = 0 if s or s belong to a part of the spectrum containing arbitrarily close eigenvalues (near an accumulation point).
In the state (17), the diagonal object operator s|ρ S |s |s s| is multiplied by the initial pointer density matrix ρ P shifted by t s in position space, as given by (18) with s = s. The interaction has thus tied up each eigenstate |s of S with a pointer state which have position x t s with uncertainty ∆ and momentum p 0 with uncertainty 2π /λ. The probability density to find P at position x given that S is in the state |s s| has a single peak at x = st with a width ∆. The peaks corresponding to different values of s are separated by at least the distance t δs, where δs is the minimum of |s − s | over all pairs (s, s ) of non-degenerate eigenvalues such that s|ρ S |s = 0. In order to be able to infer the value of s from the position of the pointer, one must wait until all peaks are well resolved. That resolvability begins at the entanglement time At that time, the reduced pointer density operator ρ P (t) = tr S (ρ PS (t)) has a Wigner function represented in Fig.2. Much later yet, the separation between the peaks reaches a macroscopic value ∆ class at the time allowing for a "reading" of the result by a classical observer. The entanglement in ρ PS (t) comes from the presence of the off-diagonal object operators s|ρ S |s |s s | in (17). These operators are multiplied by the (non self-adjoint) pointer operators (18), the matrix elements of which have a peak at (x, x ) = ( st, s t) with a height (2π∆ 2 ) −1/2 and a width dominated by ∆ in both x and x -directions of the (x, x )-plane. Therefore, the modulus of the matrix elements s, x|ρ PS (t)|s , x for fixed s = s reaches its maximal value when x = st and x = st. For those values of x and x , is time-independent and proportional to s|ρ S |s . Hence all coherences between different eigenstates of S present in the initial state of the object are still alive, no matter how large the time t is. At times t t class , ρ PS (t) resembles a Schrödinger cat state, i.e., has nonzero matrix elements between macroscopically distinguishable pointer position eigenstates. For such an object-pointer state, no classical probabilistic interpretation is possible: one cannot assign a probability to the pointer being located e.g. in the vicinity of x = ts, henceforth implying that S has the value s. In a quantum measurement, the entanglement process must be completed by a decoherence process suppressing the coherences (21). We describe such a decoherence process in the next Section.

Decoherence and disentanglement of the object and pointer
Let us now turn to the dynamics generated by H = H PB only, momentarily disregarding all other terms in the Hamiltonian (2). The coupling between P and B allows for decoherence: As shown in [11] for a similar model, a quantum superposition of coherent states of P with well-separated peaks in position evolves under H PB to a statistical mixture of these coherent states. If S and P are entangled, this also modifies the state of S. We recall in this Section the fundamental role played by decoherence in quantum measurements (for more details, see [3][4][5]). Assume object and pointer at time t 0 entangled, with ρ PS = ρ ent PS given by (17); the time t 0 should be chosen larger than t ent , possibly as large as the classical time scale introduced above, . If s = s , x t 0 s, x t 0 s and t 0 t class , this decay brought about by the pointer-bath coupling requires a time span t dec (x, x ) t dec (t 0 s, t 0 s ) much smaller than the dissipation time scale on which the pointer-bath coupling can irreversibly change the pointer position. Let us define the decoherence time t dec as the largest of the times t dec (t 0 s, t 0 s ) for all distinct eigenvalues s, s such that s|ρ S |s = 0. For t − t 0 t dec , the object-pointer state has shed all terms s = s in the double sum in the density operator (17), ρ (s) wherein it has been assumed that t dec t − t 0 t B , T S , T P (so that H B , H S and H P can be neglected) and t ent t 0 ≈ t class . While ρ PS (t) is not (and actually never can become) strictly diagonal in the position basis of the pointer, the matrix elements of the pointer state (23) almost vanish if |x − x | is larger than either the uncertainty λ (see (7)) or the decoherence length √ 2 /(α ∆ α−1 (t − t 0 ) B 2 ) (see the second factor inside the integral in (23)).
It is worth noting that the object-pointer states appearing in the sum over s in (22) are product states; ρ PS (t) is a statistical mixture of these states with probabilities p s = s|ρ S |s . Hence the decoherence disentangles S and P. This implies that in the time regime indicated after (23), S and P can be given independent states ρ S (t) and ρ P (t), The object S is in one of the eigenstates |s with probability p s , in agreement with von Neumann's postulate. The pointer P is in the quasi-classical state ρ (ii) the separation between the peaks of the distinguished pointer states ρ (s) P (t) reaches a macroscopic value ∆ class ; this occurs at time t 0 t class , see (20).
Only for t 0 t class can a classical observer infer a measured value s by looking at the position of the pointer. Such a "reading" of the pointer, while still a physical process in principle perturbing P, surely cannot blur the distinction of the peaks. Rather, the pointer will behave classically under a reading, i.e., it will not noticeably react. It is clear on (20) that (ii) can hardly arise unless the object-pointer coupling constant is very large (otherwise t class would be larger than the time duration of the measurement). This is related to the well-known amplification problem in quantum measurements [13]. One may, however, consider a different situation than that described in Section 4, in which the interaction H PS is switched off at some time t int < t class . The pointer potential V (x) must then be non-confining, with two potential barriers separated by a distance W as in Fig.1. If t int is larger than W/( δs), the pointer dynamics allows for a subsequent amplification of the inter-peak distance. Assuming also that ∆ W δs T P , one has t int ≈ W/( δs) T P . In this situation, the small quantum system S must be able to perturb the pointer strongly enough in order to produce in it a "mesoscopic change" (i.e., a distance W between the peaks in its density), instead of a macroscopic change as required in Section 4. In particular, if ρ P is a Gibbs state with position uncertainty ∆ th = (k B T /V (0)) 1/2 , this arises when the height V 0 ≈ W 2 V (0) of the two potential barriers satisfies k B T V 0 M ( δs) 2 (recall that T P 2 = M/V (0)). Then condition (ii) will be fulfilled at time t 0 ≈ T P . If moreover V 0 V (0)( δs T S ) 2 , t int T S , T P and the simplification of the Hamiltonian (2) discussed in Subsection 3.2 is valid up to time t int .
7. Partial-equilibrium apparatus initial state As was stressed above, we are interested in a quantitative description of a quantum measurement when, unlike in the situation described in Sections 4 and 5, S and P evolve under the simultaneous action of H PS and H PB . Furthermore, we will no longer assume that the decoherence time is small compared with the bath time t B ; as we shall see, the bath Hamiltonian H B then plays an important role as well. We study in this Section the object-pointer dynamics for an apparatus initially in the partial-equilibrium initial state (3).

Derivation of the result
Our starting point is the time-dependent Redfield equation (also known as the time-convolutionless master equation) [14,15] for the object-pointer density matrix. It reads where h(τ ) is the bath correlation function (10). In (25) we have neglected H S and H P . As we have seen in Subsection 3.2, this is justified at times t T P , T S provided that ρ (0) S (t) is substituted to the object initial state ρ S at the end of the calculation. As a matter of fact, the Redfield equation is valid in a perturbative regime with respect to the pointer-bath interaction. However, the result to be obtained below can be derived differently [1] by using a non-perturbative approach which only relies on the assumption t T S , T P and on a property of the bath coupling agent (namely, B is a sum of operators B ν acting on single degrees of freedom of the bath). In virtue of (1) and of the formula Hence the matrix elements of ρ PS in the eigenbasis of S and X satisfy at time t The solution of (27) with the initial condition s, x|ρ PS (0)|s , x = s|ρ wherein x s (t) and x s (t) are given by (18) and the function f (x, x , t) satisfies As is well-known [16], the real and imaginary parts of the bath correlation function (10)    where φ t is a certain real phase (depending on t, x, x , s and s ) which we do not specify here since it is irrelevant for decoherence, and (32) Note that (31) is valid for times t T S , T P only. Due to both the initial statistical independence and our special choice of the interactions, the density matrix of S + P retains at such times a remarkably simple product structure. The first factor in (31) accounts for free evolution of the object initial state ρ S , as generated by H S , see (16). It is equal to p s = s|ρ S |s if s = s and t T S (Subsection 3.2). The second factor is nothing but the matrix elements of the pointer operators (18). Here, the Hamiltonian H P does not show up because of our assumption t T P and our choice of a quasi-classical initial state ρ P . Most important is now the third factor in (31); it accounts for decoherence, i.e., for the suppression of coherences with respect to pointer displacements associated with different eigenvalues s = s . We proceed to the discussion of that "decoherence factor".
The exponent D t appearing in (32) has the following properties:

The non-negativity (a) is a consequence of the fact that the correlation function h(t) and its real part
h(t) are of positive type, i.e., they have nonnegative Fourier transforms h(ω) and ( h)(ω). Actually, one may rewrite (32) as where we have used ( h)(ω) = ( h)(−ω). Property (c) is easily checked by a change of the time integration variable in (33). Let us recall from Section 4 that the dynamics generated by H PS maps the object-pointer coordinate (x, s) into (x s (t), s) after time t and, similarly, (x , s ) is mapped into (x s (t), s ). Hence one may interpret (c) as the invariance of D t under time reversal, i.e., under t → −t and the exchange of the initial and final coordinates.

Decoherence time
As it was stressed in Section 4, the object-pointer matrix elements (31) for x s (t) = x s (t) = 0, are of particular importance for decoherence in a quantum measurement. The main difference between (34) and (21) lies in the presence of the damping factor exp{−D peak t (s, s )} given by Note that if α = 2, 4, . . . is even then D peak t (s, −s) = 0. This means that, if moreover the spectrum of S is symmetric with respect to s = 0, the coherences (34) for s = −s are not damped. This comes from the symmetry x ↔ −x of the coupling Hamiltonian H PB in (1), which allows for the existence of decoherence-free subspaces [17]. Due to these long-living coherences, P + S fails to reach (at least within a time span t T P , T S ) the statistical mixture required to be able to give a classical result to the measurement. We therefore exclude such a situation in the discussion below. In more precise words, we assume that |s α − s α |/|s − s| α is bounded below by a constant c α > 0 of the order of unity, for all pairs (s, s ) of eigenvalues in the spectrum of S such that s|ρ S |s = 0.
It is shown in Appendix A that for s α = s α , D peak t (s, s ) is an increasing convex function of time. This means that, quite generally, the graph of D peak t looks qualitatively like in Fig.3. We denote by t dec (s, s ) the time after which the coherences (34) for s = s are significantly damped. More precisely, this time is given by solving the implicit equation D peak t=t dec (s, s ) = 1, i.e., is the entanglement time (whose physical interpretation has been illustrated in Section 4), η is the (fluctuation of the) initial pointer-bath coupling energy in units of k B T , By inspection on (36), t dec (s, s ) depends on the object-pointer and pointer-bath coupling constants and η through the single parameter η 1/α . Recalling that (31-36) are valid with the proviso t T S , T P , one has to check a posteriori that the "free" evolutions of S and X, P are slow compared to t dec (s, s ), If moreover t dec (s, s ) ≥ t ent (s, s ) then at time t t dec (s, s ) the object-pointer coherences s, x|ρ PS (t)|s , x are vanishingly small for all values of (x, x ) . This comes from the single-peak character of the pointer initial state: As already pointed out in Section 5, for values of (x, x ) such that |x s (t)| ∆ or |x s (t)| ∆ the pointer coherences x s (t)|ρ P |x s (t) are small. On the other hand, if (x, x ) is close to (t s, t s ) with uncertainty ∆ and t t dec (s, s ) ≥ t ent (s, s ), i.e., if |x s (t)|, |x s (t)| ∆ t|s − s|, it follows from (32)  in (31) is close to exp{−D peak t } and is thus small. As a result, for t t dec (s, s ) the product of the pointer coherence and the decoherence factor in (31) is small for all (x, x ). It is worth to emphasise that if, unlike in the situation just described, t dec (s, s ) is smaller than t ent (s, s ) then some matrix elements s, x|ρ PS (t)|s , x may still be large at time t dec (s, s ) for some (x, x ) located within a distance ∆ from the centre ( ts, ts ) of the peak of the pointer coherence. In such a case the decoherence time must be defined as the time t at which the minimum of D t (x, x ; s, s ) over all values of (x, x ) is equal to 1. For α = 1 this minimum turns out to be given by the right-hand side of (35) excepted that the range [0, t] of both time integrals must be replaced by the symmetric range [−t/2, t/2]. It can be observed in Fig.3 that min x,x D t (x, x ; s, s ) can be significantly smaller than D peak t at times t T B , but it is possible to show that for t T B these two quantities differ by a numerical factor only. We refer the reader to [1] for more details.
If S has a discrete spectrum satisfying the requirements of Section 4, the decoherence time t dec of the measurement is the largest of the times t dec (s, s ) for all pairs of distinct eigenvalues (s, s ) such that s|ρ S |s = 0 (with the proviso t dec ≥ t ent = ∆/( δs), in the light of the discussion of the preceding paragraph). This amounts to replace |s α − s α | in (35) by its minimum value over all such pairs (s, s ) (recall that D peak t is an increasing function of time). For α = 1 this minimum value is by definition equal to δs (see Section 4); for α ≥ 2 it depends on the spectrum of S in a more subtle way. At times t t dec , the object-pointer state (31) is very close to the separable state (22). In other words, S and P are in the statistical mixture (24) with the probabilities p s = s|ρ S |s and with the pointer states ρ The initial superpositions of eigenstates |s have disappeared by indirect decoherence via the pointer. The pointer is in a statistical mixture of quasi-classical states having densities localised around x = t s with uncertainty ∆. The essence of quantum measurements lies in this loss of coherences: for indeed, as pointed out in Section 6 it is only when all object-pointer coherences for s = s are vanishingly small that a classical probability can be given for the result of the measurement. The measurement is accomplished under the supplementary condition that the distance between the nearest peaks of the pointer states (41) is macroscopic. This occurs at the time t class given by (20). Because all our calculations are valid with the proviso t T S , T P , one must assume for consistency that t class T S , T P , in addition to (40). A situation under which this condition on t class is not necessary has been discussed in Section 6.
It is worthwhile mentioning here that if t class t dec , the object and pointer are never in a Schrödinger cat state as in (17). This is because in the limit in question, decoherence subdues linear superpositions faster (see the third factor in (31)) than entanglement between P and S can produce them (second factor in (31)). Due to the simultaneous action of H PS and H PB , the whole measurement process directly produces the mixture of macroscopically distinct pointer states ρ (s) (t), i.e. without allowing for the intermediate appearance of superpositions of macroscopically distinguishable pointer states.
Our results (31-36) go beyond the so-called Markovian limit which would require T B t dec . This is an important point, since for sufficiently large decoherence takes place within the "non-Markovian" regime t T B . Explicit asymptotical results for t dec can now be drawn from the foregoing expressions, both for t dec t B and t dec T B . to t(x α − x α ) 2 (the fact that D t ∝ t is well-known [3]); the indicated time behaviour of D peak t then follows by replacing (x, x ) by ( ts, ts ). Since |h(τ )| ≤ h(0) = B 2 and h(τ ) 0 for τ T B (Subsection 3.1), the integrals ∞ 0 dτ h(τ ) and | ∞ 0 dτ τ h(τ )| are at most of the order of B 2 T B and B 2 T B 2 , respectively. If |s /s − 1| is not close to unity (so that c α (s, s ) in (39) is not very large) the factor inside the parenthesis in (46) is of the order of ( β/T B ) 2 or smaller. Thus, for coupling strength η β/T B the condition t ent (s, s ) ≤ t dec (s, s ) holds in the Markov regime for super-Ohmic baths. The situation is different for Ohmic baths: then by (44) the condition in question is violated even for small η if the entanglement time t ent (s, s ) is large enough compared with β. More precisely, still assuming that c α (s, s ) is of the order of unity in (44), t dec (s, s ) becomes smaller than t ent (s, s ) when For super-Ohmic baths, the decoherence time (46) decreases by increasing α for η β/T B and |s /s − 1| not close to unity, i.e., provided that t dec (s, s ) ≥ t ent (s, s ). Otherwise the reverse statement holds. For Ohmic baths and if c α (s, s ) ≈ 1 and t ent (s, s ) satisfies (47), we find that t dec (s, s ) is larger in the nonlinear case α > 1 than in the linear case α = 1 by a factor t dec (α>1) /t dec (α=1) of the order of (t ent /t dec (α=1) ) (2α−2)/(2α+1) ≥ 1. This means that nonlinear pointer-bath couplings become less efficient than a linear coupling as soon as t ent (s, s ) is large enough so as to fulfil (47). This is in striking contrast with the super-Ohmic case: then, for a fixed weak enough pointer-bath coupling strength η, nonlinear object-pointer couplings always win over a linear coupling in efficiency for decoherence.
Finally, it is worth mentioning that in the Markov regime Ohmic baths win in efficiency over super-Ohmic baths. Actually, (t dec Ohm /t dec sup Ohm ) 2α+1 is equal up to a numerical factor of the order of unity to the product of Since the last factor must be small compared with 1 for consistency (recall that β ≤ T B ), it follows that t dec Ohm (s, s ) is smaller than t dec sup Ohm (s, s ).

Bath of harmonic oscillators linearly coupled to P
To study the transition between the limiting time regimes discussed in the two preceding Subsections, let us consider a bath of N 1 harmonic oscillators, H B = ν ω ν (b † ν b ν + 1/2), coupled to the pointer via a coupling agent B linear in each of its creation and annihilation operators b † ν and b ν , Here ω ν is the frequency and κ ν the coupling constant of the νth oscillator. We shall take the following specific choice for the power spectrum function: wherein m is an odd positive integer, γ > 0 and ω D is a cut-off frequency. The case m = 1 corresponds to the so-called Ohmic damping, whereas one speaks of super-Ohmic damping for m > 1.
For instance, m = d or d + 2 for a phonon bath in d dimensions, depending on the underlying symmetries [19]. As is well-known [18,19] for such a harmonic oscillator bath with a linear interaction the imaginary part of the bath function (10) is temperature-independent. Moreover, its   where we have expressed h(t) in terms of its Fourier transform (see (33)) and relied on (48). We did not write explicitly in (49) the dependence of τ ent and τ dec in (s, s ). The right-hand side of (49) is shown in Fig.3. We have computed numerically the integrals appearing in this right-hand side for various values of α, m and w D , so as to obtain τ dec as function of τ ent and η. The main results are shown in Fig.4 and 5 in a log-log scale. For fixed α and η, the plain curves representing τ dec in Fig.4 split by increasing τ ent into distinct branches corresponding to distinct m's, as predicted by (43) and (45). This splitting occurs when τ dec is in the transition region w −1 D τ dec 1. After this splitting τ dec is larger for larger m so that, in particular, a Ohmic bath (m = 1) has a lower decoherence time than a super-Ohmic bath (m = 3, 5 . . .) as stated above. For comparison, the power law behaviours found in Subsection 7.3 and 7.4 in the small time (τ dec w −1 D ) and Markov (τ dec 1) regimes are also shown in Fig.4 (broken lines). A remarkably good agreement between the exact and asymptotic behaviours of τ dec is obtained: the exact results are well approximated by their small-time behaviours (42) up to τ dec ≤ w −1 D and they are hardly distinguishable from the Markov approximation as soon as τ dec ≥ 1 (or τ dec ≥ 10 for the gray curve corresponding to η = 0.5). Our aforementioned statement that a nonlinear pointer-bath coupling is more efficient for decoherence than a linear one when τ ent is not too large (and even for arbitrarily large τ ent if m ≥ 3 and η 1) is well confirmed. Indeed, it is seen in Fig.4 and 5 that τ dec becomes significantly smaller when the value of α is increased from α = 1 to α = 3. If the broken lines showing ln(τ dec ) above the horizontal axis would be drawn further to the right in Fig.4, those lines corresponding to distinct α and fixed m = 1 would intersect; after this intersection (not shown in the figure) the reverse situation of higher values of α leading to higher values of τ dec occurs. In contrast, for m = 3, 5, . . . the broken lines associated with distinct α's never intersect (their are all parallel to the line ln(τ dec ) = ln(τ ent )); hence τ dec decreases with α and τ ent ≤ τ dec for all values of τ ent . We also emphasise that τ dec increases in Fig. 5 with the Debye cut-off frequency ω D . Even though the results in Fig.4 and 5 correspond to the simplifying choice of a bath of harmonic oscillators with power spectrum function (48), for more general baths these figures should give the correct qualitative picture.

Equilibrium apparatus initial state
We now turn to the initial state (4), i.e., to a pointer and bath initially entangled and locally in thermal equilibrium. Our result for this initial state looks quite similar to that for the initial state (3) discussed in Section 7. We shall here only state this result, postponing the details of its proof to [1]. Let us first introduce the pointer effective potential wherein γ 0 is given in terms of the imaginary part of the bath correlation function h(t) by We write Z P,eff = dx e −βV eff (x) for the partition function associated with V eff . It can be shown with the help of the KMS property (A2) and the positivity of ( h)(ω) that γ 0 ≥ 0. Considering e.g. a linear pointer-bath coupling, γ 0 / is the mean force per unit length exerted by the bath on the pointer. Note that V eff (x) is a non-confining potential if V (x) = o(x 2α ) at large distances. For instance, if P is a harmonic oscillator (V (x) ∝ x 2 for all x) and α > 1 then V eff (x) looks like in Fig. 1. This means that an initial pointer density localised around x = 0 will eventually spread over the whole real line via tunnel effect after the pointer-bath coupling (1) is switched on. Our result for the initial state (4) relies, in addition to t T P , T S , on the two following additional hypotheses: (a) the separation of time scales β T P or, equivalently, the separation of length scales λ th ∆ th (Section 2); (b) a weak enough pointer-bath coupling satisfying where η th is given by (38) with ∆ = ∆ th = (βV (0)) −1/2 . Condition (52) is motivated by the following requirement: The effective potential (50) must have a local minimum at x = 0 and the width and height of the potential well at the origin must be much larger than ∆ th and k B T , respectively (see Fig. 1). If this requirement is not fulfilled then, due to the instability induced by the coupling H PB , it is not possible to prepare a pointer-bath initial equilibrium-like state ρ (eq) PB with a pointer density x| tr B (ρ PB )|x having a single peak at the origin as discussed in Section 2. We exclude here such a possibility, henceforth also excluding the occurrence of symmetry breaking and phase transitions considered in [5]. It is shown in [1] that (52) implies the required stability of V eff (x).
We can now state our result for the object-pointer density operator when P and B are initially in the state (4). For α = 1, it is given at times t T S , T P by wherein D t is the same decoherence exponent as before, see (32), φ t is a real phase and The only difference between (53) and the formula (31) for the partial-equilibrium initial state lies in the replacement of the initial pointer density x|ρ P |x by the Gibbs-like density R 0 (x, x ). Interestingly, (53) entails the following result on the reduced pointer initial state ρ P = tr B (ρ PB ) Comparing (54) with the expression of x|ρ (eq) P |x at high-temperatures given in Section 2, we see that under the aforementioned conditions (a) and (b) the coupling between P and B can be fully accounted for by the effective potential (50). Furthermore, for a linear coupling α = 1, the matrix elements (55) can be approximated for small x and x by the Gaussian (7) with an almost unchanged uncertainty in momentum ∆p 2π /λ th and a renormalised uncertainty in position ∆ eff given by The analog of (34) for the equilibrium-apparatus initial state is ρ peak PS (t; s, s ) = s|ρ wherein D t and R 0 are given by (32) and (54), φ t is a real phase and It can be shown [1] that for α > 1, (52) entails g t (x, x ; s, s ) 2 (λ th /∆ th ) 2 D t (x, x ; s, s ). The denominator in (57) is thus close to unity if condition (a) holds and D t (x, x ; s, s ) 1. Therefore, in the interesting regime 0 ≤ t t dec (s, s ), (56) gives a good approximation of ρ peak PS (t; s, s ) for α = 2 as well. It is argued in [1] that this is still true for higher α's.
It follows from (53-57) that the coherences of ρ PS (t) for s = s decay with the same rate for the initial states (3) and (4), at least in the early time regime 0 ≤ t t dec . Furthermore, in view of (54) the whole discussion of Section 4 about the emergence of classically discernible peaks remains qualitatively valid. Hence our result for the equilibrium apparatus initial state goes a long way towards supporting the general conclusions about quantum measurement of Section 7. In particular, t dec is given by (42), (44) and (46) in the short-time and Markov limits and it behaves like in Fig.4 and 5 in the intermediate region between these two limits.

Conclusion
Let us summarise the main results of this paper. The model for a quantum measurement presented in Section 2 has five parameters: the object-pointer coupling constant , the thermal variance B 2 of the bath coupling agent, the temperature T = (k B β) −1 of the bath, the uncertainty ∆ in the initial pointer position (equal to ∆ eff ≈ ∆ th for the initial state (4)) and the exponent α associated with the nonlinearity of the pointer-bath Hamiltonian (1). One may construct out of the first four parameters two relevant dimensionless constants. The first one is the entanglement time τ ent = ∆( δs β) −1 in units of the thermal time. It measures the efficiency of the pointer-bath interaction (a coupling is efficient if τ ent is small). More precisely, it is the time after which distinct positions of the pointer corresponding to distinct eigenvalues s begin to be resolved. The second constant is the coupling energy η = B 2 1/2 ∆ α β in units of k B T , which measures the strength of the pointer-bath interaction. We have found that the joint state (8) of the object and pointer is close to the statistical mixture of separable states (22) at times t large compared with the decoherence time t dec = β τ dec given by This formula is valid provided that t dec T S , T P . We have omitted all numerical factors, which are given explicitly in (42), (44) and (46). Two distinct regimes ought to be identified in (59): in the interaction-dominated regime, the decoherence time is shorter than the characteristic time t B after which the bath correlation function h(t) in (10) differs significantly from its value B 2 at t = 0; in the opposite Markov regime, one must wait more than the bath correlation time T B , i.e., the largest decay time of h(t), to obtain the required statistical mixture (22). While t dec presents a universal behaviour in the interactiondominated regime (it depends on the bath through the parameter η only), in the Markov regime it is determined by the small-frequency behaviour of the Fourier transform of the imaginary part of h(t), ( h)(ω) ∼ −i γ ω m . Larger values of t dec are found for larger m's. Although the dependence of t dec on m is partly hidden in the proportionality factors in (59), a significant change of behaviour between m = 1 (Ohmic bath) and m > 1 (super-Ohmic bath) is manifest in this equation. In both time regimes, it turns out that the value of t dec depends strongly on the nonlinearity exponent α. Smaller decoherence times are obtained for larger α's excepted in the Markov regime if m > 1 and η β/T B or if m = 1 and η 2 τ ent β/T B , where the reverse statement holds. Moreover, with the same restrictions one has τ dec ≥ τ ent . Let us emphasise again that for a small τ ent , the time t dec needed for the transformation of linear superpositions into statistical mixtures can be so small that the whole measurement is performed without producing a Schrödinger cat state as an intermediate step. It would be interesting to study concrete models for the object and pointer involving projective measurements in this "no-cat" regime, in connection with recent experiments in Solid State Physics and with parametric oscillators in Quantum Optics.