Approximate formulas for expectation values using coherent states

... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . For time-independent Hamiltonians, the dynamics of quantum expectation values of observables in coherent states ĀT can be easily represented as an integral formula involving forward and backward propagators K±. In the semiclassical regime, an approximate formula Āsc T can be constructed via the replacement of K± by their semiclassical versions, followed by a consistent integration procedure.Alternatively, one can keep the original propagators and rewrite the integral formula for ĀT as a truncated series expansion, thus introducing a new approximate formula Āse T . Yet a third approximation Ācl T can be derived by use of a classical statistical approach based on the Liouville equation and Gaussian probability distributions. In the present paper, we develop these three approximate formulas for expectation values, apply them to simple systems, and evaluate their accuracy. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subject Index A02, A60


Introduction
Semiclassical techniques applied to quantum dynamics are quite ubiquitous. Usually, their cornerstone is the propagator, written as a function of classical trajectories, an idea that was initially developed by van Vleck [1] and later improved by Gutzwiller [2]. Such a semiclassical propagator is an approximation for its quantum partner in the position representation and is widely known by the scientific community, including the nonspecialized part of it. Arguing, in a parallel manner, that coherent-states are more appropriate for semiclassical studies because of their natural connection with classical phase-space, and also considering that they have a simpler extension when, e.g., spin degrees of freedom are included, approximations for the quantum propagator in the coherent-state representation have been derived in the literature (Refs. [3][4][5][6][7][8]). These results have been shown to be very useful for the description of time-dependent phenomena, with direct application to propagation of wave-packets in a number of physical systems (Refs. [9][10][11][12][13][14][15]).
In this context, one should also mention other relevant expressions that are closely related to the coherent-state semiclassical propagator, namely, its initial value representations (Refs. [16][17][18][19][20][21][22][23][24][25]). Their most important advantage is that, while the propagator depends on classical trajectories with mixed (initial-and final-time) boundary conditions, the initial value representations avoid the so-called root search problem by means of an integration over classical trajectories determined exclusively by initial conditions.
Although this brief survey illustrates how important coherent-state-based semiclassical approaches are in describing dynamical behavior, there are only a few works using this machinery to deal with expectation values (Refs. [26][27][28][29][30][31][32][33]). In particular, the methods used in all these articles essentially involve concepts related to initial value representations. As expectation values depend not only on the time-evolution operator, but also on its conjugate, backward (in time) trajectories naturally emerge from the formalism. The quantity of interest is therefore expressed essentially as the sum of contributions of combined forward-backward trajectories. In addition, it is important to mention other alternative semiclassical approaches in phase-space that also address the same issue. In, e.g., Refs. [34,35], using the Weyl-Wigner formalism, the authors have written expectation values of operators as an -series expansion and then evaluated the contribution of its first terms. In another approach (Ref. [36]), a proper association between the Ehrenfest theorem and Heller's thawed Gaussian approximation was established in order to define average trajectories.
We point out that none of the above works on semiclassical expectation values explicitly identify a backward semiclassical coherent-state propagator in their methods. In the present paper, we employ both the backward and the forward semiclassical propagators to study quantum averages. We follow, to some extent, the mathematical framework put forward recently for the study of entanglement in the semiclassical regime (Refs. [15,37]). More precisely, our goal is to develop approximations for the dynamics of expectation values of arbitrary ordered operatorsÂ 0 ≡ A 0 (â † ,â), assuming that the initial state of the (one-dimensional) system of interest is a coherent state |z 0 . The expression ordered operators means that, in each term of the polynomial function A 0 , operatorsâ † are always disposed to the left side ofâ. Under the action of a time-independent HamiltonianĤ , the mean valueĀ with T the evolution time, can be written in terms of the forward (ξ = +1) and backward (ξ = −1) coherent-state quantum propagators according toĀ where Here the HamiltonianĤ ≡ H (â † ,â) refers to one-dimensional systems, but extensions of the present formalism to greater degrees of freedom can be straightforwardly derived. To find the integral expression (3), we used the coherent-state completeness relation Throughout the paper, we will use the following definitions for the annihilation and creation operators:

Semiclassical coherent-state propagator
The semiclassical formula for both forward (ξ = +1) and backward (ξ = −1) quantum propagators in the coherent-state representation (2), as introduced in Refs. [15,37], is written as where the sum runs over classical trajectories, which, in terms of the auxiliary variables u and v, are solutions of the Hamilton equations The semiclassical Hamiltonian is defined byH (v, u), withH (z * , z) ≡ z|Ĥ |z . In addition, trajectories contributing to Eq. (6) must satisfy the boundary conditions where single (double) prime refers to initial (final) time. Before proceeding, some comments about the nature of the contributing trajectories are opportune. First, notice that Eqs. (7) and (8) completely define the trajectories to be considered in the sum of Eq. (6). Also, the independent variables of the problem are z * η , z μ , and T . From them, classical trajectories are evaluated, which, used in Eq. (6), determine the value of K ξ . More than one trajectory may exist for a given set (z * η , z μ , T ), which explains the existence of the sum in K ξ . Because in general u(t) and v(t) are not complex conjugates of each other, these solutions are said to be complex A. L. Foggiatto et al. trajectories. In some particular instances in which u(t) = v * (t), these solutions constitute real trajectories, since we can conveniently change variables in such a way that they would live in a real phase-space. Contrarily, when u(t) = v * (t), we can consider either a real phase-space whose dimension is twice the original one, or a complex phase-space. We also point out that u and v have no ξ index attached because the context in general implies no doubt about which propagator is being used. In case of potential confusion, their differences will be explicitly indicated.
The complex action S ξ = S ξ (z * η , z μ , T ) and the function G ξ = G ξ (z * η , z μ , T ) depend on the classical trajectories according to The elements , which account for the normalization, and˜ , are given by It should be emphasized that K ξ explicitly depends on z η and z * μ only by means of . That is, the nonnormalized semiclassical coherent-state propagator is a function of z * η and z μ only. Lastly, we write the prefactor P ξ as which concludes the description of all terms appearing in formula (6). Differentiating S ξ , we obtain a result that will prove important for the present work: It also follows that ∂S ξ /∂T = −ξH (v , u ) = −ξH (v , u ). Equation (13), in particular, allows us to write second derivatives of S ξ in terms of elements of the stability matrix M, which is defined as where δu and δv are arbitrarily small initial displacements around the classical trajectory, while δu and δv represent their propagation until the final time T . It can be proved (Ref. [15]) that which is clearly more appropriate for numerical studies. For completeness and future convenience, we write down other relations involving second derivatives of S ξ : For the harmonic oscillator and free particle Hamiltonians, the results obtained from Eq. (6) are completely equivalent to those achieved via full quantum mechanical calculation. In what follows, we will briefly review the semiclassical propagators for these systems. More interestingly, we will discuss an application involving the quartic oscillator, in which case the results do not perfectly agree with the exact calculation. This study not only illustrates the modus operandi of Eq. (6), but also paves the way for the next calculations concerning expectation values.

Free particle
Using the definitions (5), one may write the free-particle Hamilton operator aŝ The corresponding semiclassical Hamiltonian is simplỹ The equations of motion (7) then become Although the trajectories contributing to both K ± are governed by the same equations of motion, their boundary conditions are distinct in general. As can be straightforwardly checked, for a given set (z * η , z μ , T ) only one trajectory contributes to K + , while another one contributes to K − . The sum appearing in Eq. (6), in this case, consists of only one term. Using these results, we find that where α ≡ iω 0 T /2. This formula coincides with the exact one.

Harmonic oscillator
For the harmonic oscillator, the quantum and the semiclassical Hamiltonians are respectively given byĤ where the oscillation frequency is chosen to be the same as the parameter ω 0 of the creation and annihilation operators (see Eq. (5)). Equations of motion (7) reduce to which can be easily solved. By imposing boundary conditions (8), we again find only one trajectory contributing to each propagator. It follows that As well as for the free particle model, the semiclassical propagator is equivalent to the exact one. In fact, for these two systems, this agreement could be anticipated since the Hamiltonians are at most A. L. Foggiatto et al. quadratic. Given that the expansions involved in the deduction of Eq. (6) are polynomials of second degree, the procedure turns out to be exact rather than approximate. The complete correspondence between the exact propagator K ξ and the semiclassical one K ξ is not verified when nonlinear terms appear in the Hamiltonian, which is the case for the quartic model introduced next.

Quartic oscillator
The last model to be studied is the quartic oscillator, whose semiclassical regime has already been discussed in Refs. [14,45] with different approaches. Its Hamiltonian readŝ with c = 1/4 and ζ a parameter whose dimension is inverse energy. The semiclassical Hamiltonian to be used in the calculation is now given bỹ where we introduceω ≡ 2ζ ω 2 0 . The equations of motion read which clearly render uv constant. Then, we find where A ± and B ± are arbitrary constants to be determined by the imposition of boundary conditions (8). In contrast to the last two examples, here one may find more than one trajectory for a given set (z * η , z μ , T ). To check this, we use Eq. (8) to conclude that A + = z μ and B − = z * η . However, finding B + and A − is more involved. They should satisfy which are transcendental equations with many solutions, in general. Writing the numerical values of A − (B + ) obtained from the first (second) equation as a n (b * n ), where each solution is labeled by n, we find that the trajectories contributing to Eq. (6) are formally given by u (n) These expressions can be inserted in the functions S ξ and G ξ . On the other hand, writing the prefactor P ξ explicitly in terms of the trajectories is not straightforward. Through Eq. (28), we may consider that b * n = b * n (z * η , z μ ) and a n = a n (z * η , z μ ), so that their differentiation leads to Using the last equation, the prefactor can be easily found by means of Eq. (12). Therefore, where γ + n ≡ z μ b * n and γ − n ≡ a n z * η . Before reporting the exact calculation, it is instructive to analyze the last expression in the limit T → 0. In the extremal case T = 0, we have only one solution for Eq. (28), which implies γ ± n = z μ z * η , so that Eq. (31) clearly becomes z η |z μ , as expected. For finite but small T , if we keep only terms of order T in Eq. (28), we find again just one solution for the transcendental equation, which implies γ This result can be shown to agree with the exact result. Then we can be assured that K qo ξ is satisfactorily accurate for sufficiently short times.
Concerning the exact calculation, we can write K qo ξ as the series which cannot be resummed in general. In particular, by setting T =T ≡ 2πj/ω, with j integer, we can show that the last expression yields the periodicity relation |K Clearly, numerical calculation is needed for a more precise comparison between K qo ξ and K qo ξ .
However, in what follows we will see that this step is not mandatory for the study of expectation values, which are the quantities of interest in the present paper.

Quantum formula
To infer the quality of the approximate formulas to be developed in this work, we will make direct comparisons with the exact result. Thus, we start this section by explicitly stating what we regard as the exact quantum mean value. From now on, contrary to the procedure implemented in Sect. 2, the starting point is a classical Hamiltonian and an arbitrary function, whose quantum counterparts are derived via the following quantization rule.
Consider a canonically conjugated pair (x, y), with quantum counterpart (x,ŷ) and commutation relation [x,ŷ] resulting in a complex number. The quantum operator corresponding to a classical function x n y m is given by Qx ,ŷ [x nŷm ], where the quantizer is a differential operator that acts onx nŷm without changing the ordering. As an example, consider the quantization of q 2 p. According to the above rule, the corresponding quantum operator is given by Qq ,p [q 2p ] =q 2p − i q. It can be checked that this Hermitian operator corresponds to the ordered expression of the symmetrized version 1 3 q 2p +qpq +pq 2 of q 2 p. (The interested reader is referred to Ref. [46] for further details and examples related to this quantization rule.) LetÃ(q, p) be a function that does not depend explicitly on time. Because our approach is concerned with coherent states, it is convenient to work with the variables (z * , z) instead of (q, p). We then use the parametrization with the numerical choice mω 0 = 1, to establish the relation Assume that the classical function can be expanded as for a nm = a * mn . Applying the quantizer Qâ † ,â to the above expression giveŝ Using the Heisenberg operators (â T ,â † , we finally write the exact formula for the quantum expectation value as The quantum Hamiltonian is obtained through the same quantization process. We apply the quantizer Qâ † ,â to H(z * , z) = n,m h nm (z * ) n z m to obtain We are now ready to introduce the main contribution of this work. Next, we present a classical counterpart for the quantum expectation value (39), followed by the derivation of its two distinct approximate formulas.  which is a Gaussian probability distribution centered at (q 0 , p 0 ). This choice, along with the identification z 0 = z(q 0 , p 0 ), is intended to make the link with the coherent states. Also for this reason, the widths q,p are chosen to be numerically equal to √ . The classical statistical mean value of A(q, p) is then defined as the ensemble averagē
This function clearly includes information about the classical trajectory that departs from (q, p) and reaches (Q, P) after a time T . The last integral, which can be derived from the fact that the Hamiltonian flow preserves the phase-space volume, i.e., dq dp = dQ dP, can be expressed in terms of a series representation (see Appendix A for details). By direct application of the identity (A.5), we obtain our classical mean value formulā Using the parametrization (35), we may introduce which allows us to express Eq. (43) as where z 0 = z(q 0 , p 0 ). Notice that A c , in analogy withÃ c , takes into account the equations of motion of the problem. Also, we learn from the last equation that classical averages via Gaussian probability distributions depend exclusively on the trajectory followed by its central point (z * 0 , z 0 ). It is worth noticing that the resultĀ cl T can be derived through a conceptually different scheme. Take the initial function (37). Purely classical evolution leads to A c (z * , z, T ) = n,m a nm (T )(z * ) n z m , which, by action of Qâ † ,â , is quantized asÂ c (T ) = n,m a nm (T ) exp 1 2 ∂â † ∂â (â † ) nâm . Since this operation preserves normal ordering, we can directly compute z 0 |Â c (T )|z 0 , which turns out to fully reproduce Eq. (45). Thus, we see that the classical statistical mean value and the exact quantum result are obtained, respectively, by the procedures That is, evolving in time and then quantizing is just an approximation of quantizing and then evolving in time. This remark put in evidence a central difference between the classical statistical approximation and the quantum result.

Saddle point method
Our proposal here is the evaluation of the expectation value using the same kind of approximation involved in the derivation of the semiclassical propagators K ξ (Refs. [7,15] (6), to obtain where In the last integral, for simplicity we omitted the sum over the trajectories contributing to K − , and similarly to K + . Also, we renamed the original integration variables z 2 and z * 1 as u − and v + , respectively, because of their clear interpretation in this framework. In addition, to establish a connection with the results of the previous section, in particular with the quantization process adopted, we should rewrite A 0 (z * 2 , z 1 ) as Integral (46) essentially means that, for each point (z * 2 , u − , z 1 , v + ) of the integration variables, forward and backward semiclassical propagators should be evaluated, so that its integrand can be calculated. Then, the result should be numerically summed over all integration points, implying in the semiclassical expectation valueĀ sc T . However, this is an extensive procedure that can be avoided by realizing that integral (46) can be approximated by the steepest descent method (Ref. [41]). Actually, as the semiclassical propagators themselves are built by the same method, their inherent inaccuracy makes exact and approximate integrations equivalent. As we will see in the following, the condition for the critical point of integral (46) couples trajectories of both propagators in such a way that the final result becomes a function of only the real trajectory.
The first step of the steepest descent method consists of finding the saddle pointr = (z * 2 ,ū − ,z 1 ,v + ), which defines the critical trajectories to be considered in the evaluation of integral (46). Assuming as usual (Ref. [7]) that derivatives of the prefactors can be disregarded rendersr to be the solution of In addition, disregarding derivatives of G ± (Ref. [7]) and applying Eq. (13) we get These four equations imply thatū − =ū + =z 1 andv − =v + =z * 2 , meaning that the final phasespace points of both critical trajectories should be identical. Because they are governed by the same equations of motion, the two trajectories turn out to be one and the same. Since the initial conditions are constrained toū + = z 0 andv − = z * 0 , the only way to satisfy all these conditions is by considering 10/25 a real trajectory, namely, the one defined byū + =ū − = z 0 andv − =v + = z * 0 . Proceeding with the method, we expand the integral up to second order aroundr to find where we have defined δ 2 F ≡ δr T · Q · δr, with δr T ≡ (δz * 2 , δu − , δv + , δz 1 ) and The integral appearing in Eq. (50) is a Gaussian integral whose result is | det Q| −1/2 exp (iσ ), where σ represents the phase of the integral. Finally, using Eq. (16) and M uu M vv − M uv M vu = 1, which expresses volume conservation of phase-space flow, we conclude that det Q = (M vvMuu ) −1 , so that which, by Eq. (47), can be finally written as Naturally, the question arises of whether one may expressĀ sc T in terms of the initial conditionsv = z * 0 andū = z 0 . By writingū =ū (z * 0 , z 0 , T ) andv =v (z * 0 , z 0 , T ) and assuming their inverse, we findĀ sc T = exp where Ac(z * 0 , z 0 , T ) ≡ A(v (z * 0 , z 0 , T ),v (z * 0 , z 0 , T )) may be different from A c (z * 0 , z 0 , T ) (see Eq. (44)), since the former is governed by the semiclassical HamiltonianH , while the latter is governed by H. To make this difference explicit, we will keep usingv andū (z * and z ) for trajectories derived fromH (H). Equation (53) is our semiclassical formula for the expectation value (39). As well as for the classical average (45), only the (real) trajectory starting at the central point (z * 0 , z 0 ) contributes toĀ sc T . To some extent, Eq. (54) can be viewed as an expression of the Ehrenfest theorem. 1

Semiclassical Husimi function
Before closing this section, it is instructive to make a brief digression on the application of the present formalism to the Husimi function. By considering in Eq. (1) the operatorÂ 0 as the operator h ≡ |z z|, the mean valueĀ T becomes the Husimi function 1 According to the Ehrenfest theorem m d 2 dT 2 q = − ∂qV (q) , the short-time dynamics of a particle of mass m subjected to an arbitrary potential V (q) is approximately governed by Newton's second law m d 2 dT 2 q ∼ = −∂ q V ( q ), whenever the initial state is sufficiently sharp. The bridge with our result (53) is made by setting A(q, p) = q, so that q (T ) ∼ =Ā sc For this particular case, our semiclassical formula (53) becomes completely useless because the semiclassical version of h(z, z 0 , T ) is simply |K + (z * , z 0 , T )| 2 , which is an expression much richer than Eq. (53), since it generally includes contributions from more than one trajectory, as illustrated by Refs. [9,47]. The natural question then arises about the possibility of writingĥ in the polynomial formÂ 0 = n,m c n,m (â † ) nâm , which a priori would imply the validity of Eq. (53) also for the semiclassical version of h(z, z 0 , T ). Actually, we can investigate this issue by expressing matrix elements ofĥ and A 0 in the usual number basis, where μ is the maximum between 0 and (k−j). If we now equate these terms, for any value of j and k, we can, in principle, recursively 2 find all coefficients c n,m for the polynomial form ofĥ. This suggests that |K + (z * , z 0 , T )| 2 as well as Eq. (53) can be directly used as semiclassical approximations of h(z, z 0 , T ). However, it is clear that the former is better than the latter, because |K + (z * , z 0 , T )| 2 admits contributions from more than one trajectory. Although the last paragraph seems to imply that Eq. (53) can be used to semiclassically approximatê h, we now argue that it is an inappropriate application. The point to be defended demonstrates that, for the operator |z z|, the term A 0 (z * 2 , z 1 ) z 2 |z 1 of Eq. (3) is indeed a separable function of z 1 and z 2 , as expressed by Eq. (53) rewritten as The semiclassical evaluation of this integral, via the saddle point method, clearly implies in critical trajectories, involved in each K ± , which are not coupled to each other (notice also that the term z 2 |z z|z 1 will have an influence on the definition of critical trajectories, given its particular exponential form). We remind that this coupling is the vital point to restrict the number of contributing trajectories to Eq. (53). By applying the saddle point method to the last integral, we will arrive at K − (z * 0 , z, T )K + (z * , z 0 , T ) = |K + (z * , z 0 , T )| 2 , which is manifestly different from Eq. (53), explaining why we classify this use of formulaĀ sc T as an inappropriate procedure. The important lesson to be learned from the treatment of the Husimi function concerns the inefficacy of Eq. (53) when A 0 is such that z 2 |Â 0 |z 1 is a separable function of z 1 and z 2 , which may possibly happen after resumming its respective series. Similar operators, such as d 2 z 2 π f (z, z * ) |z z|, deserve the same special treatment. Still, if f (z, z * ) contains specific exponential terms, it may influence the task of finding critical trajectories, making it even harder.

Series representation
Now we develop a formal series expansion for the expectation valueĀ T . We start by writing Eq. (3) as where k ξ = e K ξ are nonnormalized quantum propagators corresponding to the exact versions of Eq. (11). Writing the integration variables as a function of the real and imaginary parts of z 1 and z 2 (see Eq. (35)) and defining we haveĀ We now employ again the results proved in Appendix A. Direct application of Eq. (A.5) allows us to rewrite Eq. (58) as which, in terms of the original variables, can be written as with A 0 (z * 2 , z 1 ) given by Eq. (47) and z = (z * 2 , z 2 , z * 1 , z 1 ). Observe that in writing Eq. (60b) we have artificially introduced the integer N to truncate the series and thus obtain an approximate formula. In fact, had we kept N → ∞, then we would have just an alternative representation for the exact expectation value, i.e., lim N →∞Ā se T =Ā T . It follows that the quality of the approximationĀ se T must proportionally increase with N .
Expression (60) can be further simplified by noticing that ∂ z 2 and ∂ z * 1 act only on the nonnormalized propagators k ± whereas ∂ z 1 and ∂ z * 2 act only on the term exp z * 2 z 1 A 0 (z * 2 , z 1 ). Then, by use of the identity [exp ∂ x ∂ y f (x) g(y)]| x=0 = f (∂ y ) g(y), one shows that Eq. (60) can be written as where it should be understood that exp ∂ z * 1 ∂ z 2 ∼ = N n=0 (∂ z * 1 ∂ z 2 ) n /n!, with a given truncation integer N . As for the result (60), here we have that lim N →∞Ā se T =Ā T .

Harmonic oscillator
Given the classical Hamiltonian H ho = p 2 2m + 1 2 mω 2 0 q 2 = ω 0 z * z, where Eq. (35) was used, we solve the corresponding equations of motion to find z = z 0 exp (−iω 0 T ) and its complex conjugate. Because ∂ z * 0 ∂ z 0 = ∂ z * ∂ z , we can write Eq. (45) as A cl T = exp 1 2 ∂ z * ∂ z A(z * , z ). As far as the quantum result is concerned, we first apply the prescription (40) to the classical Hamiltonian H ho to obtainĤ ho = ω 0 â †â + 1 2 . Given that the Heisenberg solution for the dynamics iŝ a T =â exp (−iω 0 T ), we obtain from Eq. (39) that For the semiclassical result, since H ho andH ho differ only by a constant term, we haveū = z and v = z * . Then, it follows from Eq. (53) thatĀ sc T =Ā T .
Finally, as far as the series approximation (60) is concerned, we have used the propagators (23) and performed numerical calculations for several values of and N . These studies have shown thatĀ se T succeeded in reproducingĀ T with arbitrarily high accuracy, for all quantities investigated, whenever the number N of terms in the series was taken to be sufficiently large, which roughly means that N should increase with −1 (in agreement with the criteria (A.8) discussed in Appendix A). That is, the smaller the numerical value of the greater N has to be in order for the series approximation to yield a good mimic of the quantum expectation value. The same behavior was observed for other systems, as we will illustrate later. To a certain extent, the good agreement among the results is not surprising, since expectation values in coherent states constitute a privileged framework for the harmonic oscillator dynamics. In any case, this was a necessary consistency test for our approach.