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The Quantization process for the Driven Quantum Well - Perturbative Expansion and the Classical Limit

We consider the quantum mechanical behavior of a driven particle in an infinite 1D potential well. We show that the time dependent perturbation series is induced by the delicate non-trivial properties of the momentum operator in this case, namely, its non-self-adjointness. Using this expansion, we calculate the first order contribution to the cross section and the energy gain, and discuss their classical limit. In this limit the one-period energy gain converges to its classical analog - the classical local (momentum space) diffusion coefficient. Both the classical and quantum mechanical results are compared with numerical simulations.


I. INTRODUCTION
Modern semiconductor technology has enabled the fabrication of 1D quantum wells [1].
Such a quantum well is fabricated by varying the alloy composition in a compound semiconductor like Al x Ga 1−x As along one dimension. Conduction electrons in such structures experience an arbitrarily shaped effective potential in the growth direction while remaining free in the perpendicular plane. Quantum wells are typically 200 − 300 meV deep with level spacing ∆E between several meV and 150 meV. These systems are of special interest since they can be treated in means of pure quantum mechanical considerations while they are still experimentally accessible.
Recently, there has been interest in the behavior of such systems under the influence of an electromagnetic field [2][3][4][5][6][7][8][9][10]. The quantum well structure can be considered as an analogue of an 1D atom, and thus a study of the driven well may help us learn about the interaction of atoms and high-field electromagnetic radiation. In the region where the electric field is strong with respect to the level spacing of the well, one obtains a system in which to study non-perturbative effects in light-matter interaction. Quantum wells with parameters in the region of interest in this study are well in the range of experimental capabilities. In fact, currently there are some experiments carried out at UCSB with the Free Electron Laser (FEL) in which an intense monochromatic far-infrared radiation is applied to a quantum well [8]. In particular, some authors have studied in the recent years the quantum dynamics of a particle in an infinite potential well driven by a spatially constant ac electric field [7,8].
These studies analyzed the energy absorption characteristics as a function of the driving force amplitude and frequency. A non-perturbative resonance was found for a special value of the driving amplitude, corresponding to the replacement of the states localized in phase space by more extended states. This resonance was predicted by avoided crossings in the quasienergy spectrum of the Floquet operator. Other mesoscopic applications of this model have been also considered [10].
The problem of a driven particle in an infinite potential well is of interest in the field of quantum 'chaology' [11] as well. The study of the effect of quantization on the properties of classically chaotic systems has attracted increasing attention in the recent years [12][13][14][15]. It was argued long ago that quantum chaos is strictly impossible in autonomous finite quantum systems which exhibits a discrete energy spectrum [16]. Moreover, a peculiar effects of quantum suppression of classical chaotic diffusion in the dynamics of the kicked [17,18] and two-sided kicked [19,20] rotator has been shown to exist. Since our system is classically chaotic, the study of its quantum properties may enrich the search for the fingerprints of classical chaos in the quantized system. A WKB approach was employed to this system, and information about the quasienergy levels and the quasistates is available in the classical limit.
In this paper we consider the quantum mechanical behavior of the above model, i.e., a driven particle in an infinite potential well. We discuss the proper way to apply a perturbative expansion to the system, and explain in detail the origin of the failure of the more naive approach. Our treatment applies also strong fields in the high frequency limit. We now turn to a concise qualitative description of our approach.
Applying a guage transformation, one can show that the Hamiltonian can be written as a function of the operators P and P 2 alone with time-dependent coefficients. Thus, it seems that the Hamiltonian commutes with itself for different times, and one may simply integrate over the time to get the evolution operator. The resulting evolution operator for a complete period (and a number of periods as well) behaves as one which follows from an integrable Hamiltonian. Moreover, for a properly chosen period, i.e., a proper choice of the start (and end) point, the complete period behavior is exactly the same as that of the unperturbed well.
This result is physically unacceptable. It is well known [21,22] that the quantum dynamics of a system corresponds to its classical dynamics for time periods which scale (at least) like ln(1/h). This result can be rigorously proved using the path-integral formulation of the expansion around the WKB limit. Accordingly, since the classical driven particle experiences a chaotic motion, one should expect the quantum dynamics to show some fea-tures that resembles this chaotic (diffusive) behavior in the short time limit (although the long time limit is charaterized by the energy-space localization which is a purely quantum mechanical effect). Nevertheless, if the quantum dynamics is determined by the integrated Hamiltonian, which is exactly equal to the one obtained from an integrable system, namely, the unperturbed well, it must also describe in this short time limit the classical dynamics related to this integrable Hamiltonian. However, the classical dynamics of these two are off course qualitatively differenet; for example, while the first diffuses over the whole stochastic layer, the second is energy-preserving. Note that the above described derivation is exact for any value ofh eff (the effectiveh). Thus one may chooseh to be arbitrarily small, such that the correspondence time diverges. One may therefore conclude that the result of this derivation contradicts the well-established 'log-time' correspondence principle.
It turns out that the origin for this apparent contradiction is found in the delicate nontrivial properties of the P operator, when defined on a finite interval. We discuss some of the manifestations of this behavior, and explain in detail its implication on our derivation.
It is shown that the seemingly trivial relation [P, P 2 ] = 0 does not hold for a particle in an infinite potentail well, and therefore our guaged Hamiltonian does not commute with itself at different times. An explanation to this non-trivial behavior is given in terms of the self-adjoint extension procedure relevant to our model. This unexpected non-commutativity may be of importance to the analysis of many other finite quantum systems. In particular, each finite system under the influence of an electric or magnetic fields may be influenced by this property.
In order to study the effect of this non-commutativity of P and P 2 over the dynamics, we use the explicit form of the operator [P, P 2 ] in the energy basis, and employ the timedependent perturbation theory to the interaction representation of the guaged Hamiltonian, which takes into account an infinite number of commutators [P 2 , [P 2 , [· · · , [P 2 , P ] · · ·]]], and obtain the first correction to the exact periodicity implied by the previous false derivation.
Our expansion is appropriate even for cases in which E L = λL, the (maximal) potential difference between the ends of the well, is much larger then E 0 , the spacing of the energy levels in the unperturbed well. The matrix elements of the perturbation are not small and in fact, the reason that the usual time-dependent perturbation theory does not break down is the approximate periodicity implied by the preceding wrong argument. We thus obtain an expression for the integral cross section, i.e., the probability to leave the initial state after one period of the perturbation. The high frequency limit of the frequency dependence of this cross section is O(ω −4 ), i.e. much faster then the usual O(ω −2 ) asymptotics, which characterizes the response in the high frequency limit.
The first-order treatment yields an infinite series for the escape probability -the probability not to return to the initial state after a complete period, and the energy gain. We discuss the asymptotics of these serieses in the various regimes. In particular, we study the behavior in the classical limit, i.e.,h eff → 0 while keeping the energy fixed. We show how the approximate periodicity breaks down in this limit, thus furnishing the transition to the classically chaotic behavior. The classical limit of the energy gain is then compared to the corresponding classical quantity. We find that, surprisingly, the indirect calculation of this quantity, i.e., evaluating the quantum result and then considering the classical limit, is much more efficient then a direct calculation based on the classical equations of motion.
The first order results also imply the exsistence of an antiresonance phenomenon, namely, a sharp anti-peak in the escape probability and the energy gain for some values ofh eff .
This phenomenon was discussed in detail recently [23], and it was shown that it persists for higher orders of perturbation theory as well. This was tested numerically, and it was found that the antiresonance behavior is seen even for high values of β the perturbation strength parameter, even up to β = 2. Since the full discussion was given recently elsewhere, we present here only a short descriptive consideration of this effect.
The rest of this paper is organized as follows: In section II we present the naive 'exact' solution of the model in terms of the transverse guage. The surprising result is that the effects of the ac field are cancelled out over a complete period. We discuss the implications of this result. In section III we explain the origin of the error in this derivation in terms of the algebraic properties of the operators involved with respect to their self-adjoint extension.
Section IV presents a systematic perturbative expansion around the limit in which the naive argument holds. This derivation is based on the interaction representation of the P operator. We apply time-dependent perturbative methods to calculate the first correction to the apparent exact periodicity. Asymptotic analysis of these perturbative rsults are discussed, with particular emphasis on the classical limit. Numerical results are presented, which verify the above perturbative derivation and test its range of applicability. Section V describes in short the anti-resonance phenomenon, and a conclusion is given in section VI.

The Hamiltonian considered is
where f (t) is a periodic function whose period is T = 2π/ω. The wave-function is imposed to satisfy the boundary conditions of vanishing of the wave-function at the ends of the well x = 0;x = L. This model has been investigated classically for the case f (t) = cos(ωt) [2,3,5], and has been found to be chaotic. Using dimensionless form, i.e., it can be shown that the classical behavior of this model depends on a single parameter β = eE/(mω 2 L). However, the quantum evolution depends also on an additional parameter, the effectiveh,h eff =h/(mωL 2 ). Using the dimensionless form the Hamiltonian reads out to be where the position representation of the scaled momentum operator is P = −ih ef f d dx . We now transform to the transverse gauge, in which the Hamiltonian takes the form The lower limit of the integral is t = 0 since we choose the field to be turned on at t = 0, and thus the vector potential should vanish at t = 0. At this point, one may naively use the following argument: since the hamoltoniam (4) commutes with itself at different times, the Schrödinger equation may be trivially integrated over τ , yielding the evolution operator where The diagonalization problem for this effective Hamiltonian is rather trivial since for each τ it is just the Hamiltonian of a particle in an infinite well with a guage term, whose eigenfunctions are of the form sin(kx) exp(iγ(τ )x).
A basic tool for analyzing the chaotic features of classical systems is the Poincare section [12,14]. For a time-periodic Hamiltonian system with one degree of freedom, the Poincare surface of section is just a strobe plot, that is one plots (x, p) once every period τ . Such a plot for the driven well system is given in Figure particular, if the evolution of a system is periodic, the strobe plot will give a one-dimensional curve. On the other hand if the evolution is chaotic, the trajectory will spread over a two-dimensional region. Accordingly, most of the research interest in the field of quantum chaos was directed to the study of the Floquet operator, which determines the quantum time evolution of periodic systems for times which are integer in the driving force period units.
Apparently, one sees that for our model, namely, the driven one-dimensional potential well, the explicit form of the Floquet operator can be easily obtained from Eq. (6) to be One ontains that the stoboscopic behavior of the system is rather trivial, i.e., equivalent to that of a particle in an infinite well subjected to a constant electric field. Moreover for an appropriate choice of the function f (ωt), e.g. f (τ ) = cos(τ ) the guage term vanishes, and the effective Hamiltonian has no intraction dependence.
The above result is completely unacceptable physically, since it contradicts the correspondence principle for arbitrary short times (with respect toh). In particular, although the classical particle in the well reaches (in general) a completely different point in phase space after one period of the perturbation, we have just shown that the quantum particle develops into the state it should have been developed in the absence of the perturbation.
Thus, according to the correspondence principle, forh → 0 the quantum mechanical state should correspond to two different classical states, i.e., those points in phase space to which the system arrives in the presence and in the absence of the driving force. This ambiguity manifests the apparent breakdown of the correspondence principle (at least) for times of order of the period, no matter how small ish. We therefore turn now to discuss the origin of the failure of our derivation.

III. ALGEBRAIC PROPERTIES OF THE P OPERATOR
In order to find the problematic point in the above arguments, we state explicitly the assumptions made above. Apart from simple algebra, we used two basic relations, i.e., The first assumption was used to perform the guage transformation, and the second in the statement that the guaged Hamiltonian (4) commutes with itself for different τ s.
It seems that these two relations are well established. The first is the basic principle of quantum mechanics, whose failure leads to the breakdown of the whole method of canonical quantization in the system. The second assumption is just a manifestation of the associativity of operators multiplication. However, this second assumption turns out to be false as we now explain in detail.
The simplest way to realize the failure of the relation [P, P 2 ] = 0 is through the matrix representation of these operators in the energy basis of the unperturbed well. The matrix elements of these matrices are given by (P 2 ) mn =h 2 eff π 2 m 2 δ mn (9) 4mn m 2 −n 2 m + n odd 0 m + n even (10) One clearly sees that these matrices do not commute. In fact the commutator gives In terms of this matrix representation, the reason for the un-associativity of the matrix multiplication is that this multiplication involves summations which do not absolutely converge, and thus the sum depends on the order of summation. However, since one expects operators to be associative, the question still holds whether this strange result really manifests an operator property (and if so -why ?), or is only an artifact of the particular basis used.
First, we note that the P operator possesses some delicate properties when restricted to a finite interval. Physically, due to the uncertainty principleand the finite size of the well, an exact measurement of the momentum is impossible, and thus there is no eigenstate of P in the Hilbert space as can be seen also by applying the appropriate boundary conditions to the differential eigenvalue problem. Moreover, it can be easily seen that since P is canonically conjugate to X, it generates spatial translations. Thus its domain is restricted to functions which do not leave the well for infinitesimal translations. A well known criterion for the commutativity of two operators is the exsistence of a basis in which the two of them are diagonal. As we have seen, P has an empty spectrum and thus cannot be diagonalized in any basis. In particular it is not diagonal in the basis in which P 2 is. Thus they do not commute.
The reason for this un-associativity is as follows. The operator P 2 in the Hamiltonian is not really defined as P · P but rather as the self-adjoint extension of this form. P itself is not essentially-self-adjoint operator when restricted to a finite interval [24], and has therefore no self-adjoint extension [25]. Thus, although the "bare" differential operators P and P 2 (i.e., the operators defined only the (twice) differentiable functions whose derivative vanishes on the boundaries, without the self-adjoint extension) commute on their common dense domain, after the procedure of self-adjoint extension, the domain of P 2 is much larger then that of P . If there had been such an extension for the P operator, its square would have been the extension of the P 2 operator (uniqueness of the extension). However, since due to the above (both mathematical and physical) reasons, P can not be extended to a self-adjoint form, the "bare" P does not commute with the extended P 2 .
In the next section we study the perturbative expansion with respect to the terms obtained due to the uncommutativity.

IV. PERTURBATIVE APPROACH
The starting point for the perurbative expansion is the dimensionless form of the guaged where P here is the dimensionless momentum operator, whose matix elements in the energy representation are (P 2 ) mn =h 2 eff m 2 π 2 δ mn (13) We now take into account the fact that P and P 2 do not commute, and treat the P term as a time-dependent perturbation to the Hamiltonian H 0 = P 2 2 . We use the interaction picture in which where α =h eff π 2 /2. The latter equation can be derived simply by taking the matrix element of the whole expression in the middle of the above equation, and applying the exponential operators to the bra and ket states (These are eigenstates of the exponential operators).
However, another way to look at this equation is as a sum over the commutator series as follows It is therfore clear that in the perturbative approach one contain an infinite number of commutators in the derivation, and thus the problems described in the previous sections are avoided.
We demonstrate the approach for the calculation of the escape probability, i.e. the probability to leave the initial state m after one period of the perturbation, to second order in β. The amplitude for a transition to the n state is given by The escape probability (integral cross section) is therefore given by In the following, we estimate this sum for the various relevant regimes.

A. Low-Energy Regime
We first consider the regime in which the sum is appropriately estimated by taking the continuous limit (this regime is going to be defined precisely later), i.e., replacing the sum by the integral √ α where z = αm 2 , and x was defined through x = √ αn. We now consider two different subregimes. For z << 1, one may estimate the integral by its value for z = 0, which is I = 4.663.... The condition for replacing the sum by the integral is that the spacing of the summation points, i.e., 2 √ α is sufficiently small compared to the scale of changes in the function f (x). In this subregime this condition is equivalent to √ α << 1. This, together with the condition z << 1 implies E = 2zα/π 2 << 1. Thus the regime in which the integral approach is appropriate is the low energy regime. However, where z << 1, the condition √ α = √ z/m << 1 holds trivially since m, the quantum number is a positive integer m ≥ 1.
Thus the limit z << 1 is relevant only to the low-energy regime.
For the z >> 1, one should notice that the integrand in (19) is a highly peaked function around the region x 2 = z ± 1 (See figure 3. The maximum of the function is obtained at x 2 = z and its value is 2π 2 z, while for x 2 = z ± 1 the integrand is π 2 4 (z ± 1), i.e., about a one fourth of the maximum. The decrease of the function away from this region is O((x 2 −z) −6 ), and thus one may neglect the contributions to the integral from any region other than the maximum. The width of the peak is therefore approximated by the distance between the two points x 2 = z ± 1, i.e. ∆ = 1/ √ z. Thus one obtains for the z >> 1 regime the estimate I ∼ 2π 2 √ z. Numerical integration shows that while the z dependence of the integral is as stated above, a more accurate value of the coefficient is The condition for the validity of the integral approximation in this case is that the spacing of the sum is less then the peak width, i.e. 2 √ α << 1/ √ z which implies E = 2zα/π 2 << 1.
Accordingly, the following results for the escape probabilities in the low energy limit are In figure 4 we show the results of a numerical summation of the 1st order series, compared to the two asymptotes. The transition between the two regimes is clearly seen.
The nature of the transition from the extreme quantum regime where the system is almost periodic (as implied by the wrong argument of Sec. II) to the classical regime where the dynamics turns out to be chaotic over larger and larger time scales can now be clearly seen.
As one approaches the classical limith eff → 0 the approximate periodicity of the system is lost, since the escape probability, i.e. the deviation from periodic behavior, diverges ash eff approaches zero. One may also define the time scale after which the periodicity is ruined as the inverse of the escape probability, i.e.
τ chaotic ∼ T /σ (22) and this time scale vanishes in the classical limit. It is interesting to note that while the exsistence of an upper limit (in time) to the correspondence principle to hold is well established and discussed in the recent years [21,22], the system under consideration exhibits the opposite type of behavior, i.e., a lower bound on the time in which correspondence holds.
We now apply similar arguments to the calculation of the energy change of the driven particle after one period. This quantity has a concrete classical meaning and thus this example may further clarify the nature of the classical limit. Moreover, the energy absorption of the quantum well is also an experimentally measurable quantity, and thus one may study experimentally the complete transition from the extreme quantum regime to the classical limit, by changing the parameters of the well such thath eff is in the appropriate regime. The arguments concerning the integral estimations are quite similar to those described above, and the results are Figure 5 shows the energy gain per period, as a function ofh eff , as obtained from a numerical summation of the appropriate 1st order sum. The two asymptotes are also indicated on the graph, and the agreement seems quite good.
One can clearly see that for a fixed energy E, ash eff → 0, z = E/h eff approaches infinity.
Thus the relevant regime to the classical limit is the second one, in which the quantity converges to the well-defined limit ∆E (classical) = 17.772β 2 √ E. It is interesting to note that we have thus obtained a classical prediction based on quantum calculations. This has to be appriciated considering the fact that the "quantum" calculation is tractable analytically, as we have just demonstrated. The functional form can be completely deduced theoretically, and even the accurate value of the coefficient is obtained through only one numerical integration. A direct "classical" calculation, i.e., one which is based on classical equations of motion is by far more complicated. As far as we know, the best way to calculate the local momentum-diffusion constant is using a Monte-Carlo method, which involved averaging over a large number of paths related to different initial conditions. In figure  is 17.77 ±.04. Note that for this degree of accuracy about 10 8 paths were considered for each case, and the whole calculation consumed many CPU hours. We therefore conclude that we have shown in this example not only that the classical limit is obtained correctly from the quantum calculations, but also that the latter may turn out to be by far more efficient.
We wish to note here about one more point. Considering the high frequency limit of this problem, it is clearly seen that while ω >> 1, bothh eff and β approach zero. Thus, the pertubative, classical-limit calculation holds and one may conclude that the high-frequncy asymptote of the absorption is O(ω −4 ) rather than the usual O(ω −2 ). This is another manifestation of the approximate periodicity of the system in the extreme quantum regime.

B. High-Energy Regime
As we have shown above, in the high-energy regime only the limit z >> 1 should be considered. We can then maintain the picture of a highly peaked function f (x) as in the low-energy case. However, here the function values are taken in large steps compared to the peak width. Since the decay of f (x) away from the peak is extremely fast, we can approximate the sum by the contribution of the closest points, i.e., n = m ± 1. This implies (for simplicity we assume that the initial state m is not close to the ground state, m >> 1) The E dependence of σ (h eff fixed) is presented in figure 7, as obtained from summation of the first order series. The transintion between the three different regimes is clearly seen. A similar analysis is done for the energy absorption. Again, we calculate the classical limit of the quantum result and get This result is obtained only through the above analytical argument and much tedious but straight forward series expansions. This is to be compared to the standard, "direct", classical approach which calls for massive computer simulations.

V. ANTI-RESONANCE
The antiresonance phenomena in the quantum driven well have been discussed elsewhere.
We therefore just scetch the general arguments and main results.
The first order calculation indicated the possibility of an antiresonance to occur. Under the antiresonance conditions, a sharp anti-peak in the escape probability and the energy absorption is expected. The antiresonance is obtained when the nominator in Eq. (24) vanishes (See also fig. 7). It can be easily checked that this corresponds to the fact that the particle passes the well an integer number of times during one period of the perturbation. Recently, it was shown that this anti-resonance is maintained even in higher orders of perturbation theory and numerical integration of Scroedinger equation was carried out and showed that it is stable for β values as high as β = 2.
In order to complete the picture, we present here similar results for the energy absorption around the resonance. These are shown in figure 8. The numerical integration of Schroedinger equation was done using a quality-controlled Runge-Kutta method. This sharp peak (even for large β) should be easily detected experimentally.

VI. CONCLUSION
The ac-driven particle in an infinite potential well is considered. This system, which is classically chaotic presents a rich quantum behavior.
In the first sections we have shown that all the non-trivial properties of the system are related to the fact that in the infinite potential well the relation [P, P 2 ] = 0 does not hold. If it did, the evolution would become exactly periodic, which contradicts the chaotic behavior in the classical limit. It is really worth noting that here the classical limit depends upon some non-commutativity property, which seems to be a pure quantum-mechanical effect.
Apart from being crucial for the analysis of our system, the non-commutativity of P and P 2 may turn out to be of importance in many cases in which the response of quantum wells and other finite quantum systems to electric and magnetic fields. For example, it is interesting to check the influence of this uncommutativity on the Landau quantization of particles in a finite box. This problem is closely related to the investigation of the quantum Hall effect with realistic, i.e., non-periodic, boundary conditions.
In the second part of the paper, a perturbative expansion around the limit in which the system is periodic was developed. The various regimes were considered, and the analytical estimates for the different asymptotes were obtained. These were tested with respect to an exact summation. The classical limit was also discussed. It was shown how the approximate periodicity breaks down ash eff → 0. A classical limit for the absorption was obtained from the classical limit of the quantum calculations, and it was shown that it agrees with the classical result obtained from integrating the classical equations of motion. Moreover, the indirect calculation of this quantity, i.e., taking the classical limit of the quantum result, turns out to be much more efficient then the "pure" classical one. In fact, while the first is done analytically, and involves only one numerical integration for determining a numerical factor, the latter is done through a time-consuming and not very efficient Monte Carlo simulation. Another feature of this quantum system is the antiresonance phenomena for which the system evolution turns out to be almost periodic for some sharp values of the parameterh eff .
As stressed in the Introduction, this system, including all the above various regimes, is experimentally realizable, using quantum wells radiated by a laser. Different values of the width of the well and the laser frequency controll the parametersh eff and β such that the whole parameter space is accessible. The influence of perurbations caused by impurities and imperfections in the well, dissipation due to phonon excitation, finiteness or distortion of the well, motion in the plane of the well, and even electron-electron interactions were discussed in Ref. [8] and further publishings, and it is claimed that the one-particle, 1D, ideal system effects can be in fact realized. It is therefore our belief that this system may become the subject of intense experimental, as well as theoretical, study, as a prototype of driven, classically-chaotic mesoscopic systems.