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Quantum Energy Expectation in Periodic Time-Dependent Hamiltonians via Green Functions
Let be the Floquet operator of a time periodic Hamiltonian . For each positive and discrete observable (which we call a probe energy), we derive a formula for the Laplace time average of its expectation value up to time in terms of its eigenvalues and Green functions at the circle of radius . Some simple applications are provided which support its usefulness.
Consider a general periodically driven quantum hamiltonian system with period acting in a separable Hilbert space , and let denote its Floquet operator, so that if is the initial state (at time zero) of the system, then is this state at time . Typically, the unperturbed hamiltonian is assumed to have purely point spectrum so that the same is true for . What happens when is perturbed by ? A natural physical question is if the expectation values of the unperturbed energy remain bounded when . This question is formulated based on many physical models, in particular on the Fermi accelerator in which a particle can acquire unbounded energy from collisions with a heavy periodically moving wall. Here quantum mechanics is considered and, more precisely, if is finite or not, where , the domain of .
Motivated by models with hamiltonians as above , one is suggested to probe quantum (in)stability through the behavior of an “abstract energy operator" which we call a probe operator and will be represented by a positive, unbounded, self-adjoint operator and with discrete spectrum, , such that if for all , then, for each , the expectation value is finite. It is convenient to write if does not belong to .
We say the system is A-dynamically stable when is a bounded function of time , and A-dynamically unstable otherwise (usually we say just (un)stable). If the function is not bounded, one can ask about its asymptotic behavior, that is, how does behave as goes to infinity? Usually this is a very difficult question and sometimes the temporal average of is considered, as we will do in this work.
Quantum systems governed by a time periodic hamiltonian have their dynamical stability often characterized in terms of the spectral properties of the corresponding Floquet operator. As in the autonomous case, the presence of continuous spectrum is a signature of unstable quantum systems; this is a rigorous consequence of the famous RAGE theorem , firstly proved for the autonomous case and then for time-periodic hamiltonians [2, 3]. At first sight a Floquet operator with purely point spectrum would imply stability, but one should be alerted by examples with purely point spectrum and dynamically unstable [4–6] in the autonomous case and, recently, also a time-periodic model with energy instability  was found.
Dynamical stability of time-dependent systems was studied, for example, in references [2, 8–19]. In  it was proved that the applicability of the KAM method gives a uniform bound at the expectation value of the energy for a class of time-periodic hamiltonians considered in .
For hamiltonians , not necessarily periodic, with a positive self-adjoint operator whose spectrum consists of separated bands such that , upper bounds of the type were obtained in  if the gaps grow like , with , and if is strongly with . The proof is based on adiabatic techniques that require smooth time dependence and therefore do not apply to kicked systems. In [11, 13] upper bounds complementary to those of  described above are obtained.
In [2, 8, 9, 15] stability results are obtained through topological properties of the orbits for , while in [16–19] lower bounds for averages of the type are obtained for periodic hamiltonians through dimensional properties of the spectral measure associated with and (the exponent depends on the measure ).
In this work we study (in)stability of periodic time-dependent systems. As for tight-binding models (see  and references therein) we consider the Laplace-like average of , that is, where is a probe energy, is an element of , and is the Floquet operator. The main technical reason for working with this expression for the time average is that it can be written in terms of (see Theorem 2.3) the eigenvalues of , that is, , and the matrix elements of the resolvent operator (with ) with respect to the orthonormal basis of the Hilbert space (here denotes the identity operator). Lemma 2.2 relates the long run of Laplace-like average with the usual Cesàro average. In Section 2 we shall prove this abstract results and present some applications in Section 3.
Since our main results are for temporal Laplace averages of expectation values of probe energies (see Section 2), in practice we will think of (in)stability in terms of (un)boundedness of such averages. Note that unbounded Laplace averages imply unboundedness of expectation values of probe energies themselves.
2. Average Energy and Green Functions
Consider a time-dependent hamiltonian with for all , acting in the separable Hilbert space . Suppose the existence of the propagators , so that the Floquet operator is at our disposal. Let be a probe energy and as in the introduction. Also, is an orthonormal basis of .
The main interest is in the study of the expectation values, herein defined by as function of time . Another quantity of interest is the time dependence of the moment of this probe energy, which takes values in and is defined by Our first remark is the equivalence of both concepts (under certain circumstances).
Proposition 2.1. If for all , then This holds, in particular, if is invariant under the time evolution and .
Proof. Since , one has , for all , and so which is the stated result.
We introduce the temporal Laplace average of (see also the appendix) by the following function of , which also takes values in : Under certain conditions, the next result shows that the upper and lower growth exponents of this average, that is, roughly they are the best exponents so that for large there exist with and the corresponding exponents for the temporal Cesàro average are closely related; this follows at once by Lemma 2.2, which perhaps could be improved to get equality also between lower exponents. Note that, although not indicated, these exponents depend on the initial condition .
Lemma 2.2. If is a nonnegative sequence and for some and , then and , where
Proof. Note that for we have , and so
On the other hand, for each , denoting by the smallest integer larger or equal to , one has Now, for large enough . Thus Therefore, for each and large enough Since as , it follows that As was arbitrary, .
Recall that the Green functions associated with the operators at and , are defined by the matrices elements of the resolvent operator along the orthonormal basis , that is, Note that is always well defined since for that resolvent operator is bounded. Theorem 2.3 is the main reason for considering the temporal averages . It presents a formula that translates the Laplace average of wavepackets at time into an integral of the Green functions over “energies” in the circle of radius in the complex plane (centered at the origin). As grows, the integration region approaches the unit circle where the spectrum of lives and takes singular values, so that (hopefully) -(in)stability can be quantitatively detected.
Theorem 2.3. Assume that for all . Then
Before the proof of this theorem, we underline that this formula, that is, the expression on the right-hand side of (2.15), is a sum of positive terms and so it is well defined for all if we let it take values in hence, in principle it can happen that this formula is finite even for vectors not in the domain of , where . The general case, that is, , can then be gathered in the following inequality: so that lower bound estimates for this formula always imply lower bound estimates for the Laplace average.
Proof of Theorem 2.3. First note that, by hypothesis, for each . Denote by the spectral measure of associated with the pair and by the Fourier transform . By the spectral theorem for unitary operators For each let be the sequence Since and is a unitary operator, it follows that and also with . Therefore and so From such relation it follows that which is exactly the stated result.
Theorem 2.3 clearly remains true if the eigenvalues of have finite multiplicity. In this case, for each consider the corresponding orthonormal eigenvectors , and one obtains with as before.
In case the initial condition is , put . Thus, and so . Hence and by denoting one concludes.
In Section 3 we discuss some Floquet operators that are known in literature and analyze their Green functions through the equation
This section is devoted to some applications of the formula obtained in Theorem 2.3. In general it is not trivial to get expressions and/or bounds for the Green functions of Floquet operators, and so one of the main goals of the applications that follow is to illustrate how to approach the method we have just proposed.
3.1. Time-Independent Hamiltonians
As a first example and illustration of the formula proposed in Theorem 2.3, we consider the special case of autonomous hamiltonians. In this case for all , and we assume that is a positive, unbounded, self-adjoint operator and with simple discrete spectrum, , so that is an orthonormal basis of and with . For we can consider as our abstract energy operator , so that its eigenvalues are (since and have the same eigenfunctions, we are justified in using the notation for the eigenfunctions of ). We take (time ) and for Since is invariant under the time evolution , then for and we have
Thus we need to calculate the integral . Let be the closed path in given by with , and , then As and , is the unique pole in the interior of . Thus, by using residues, and is independent of .
3.2. Lower-Bounded Green Functions
As a first theoretical application we get dynamical instability from some lower bounds of the Green functions. See  for a similar result in the one-dimensional discrete Schrödinger operators context; there, a relation to transfer matrices allows interesting applications to nontrivial models, what is not available in the unitary setting yet (and it constitutes of an important open problem). As before, denote the increasing sequence of positive eigenvalues of the abstract energy operator , the ones we use to probe (in)stability.
Let denotes the integer part of a real number, and indicates Lebesgue measure.
Theorem 3.1. Suppose that there exist and such that for each large enough there exists a nonempty Borel set such that holds for all with (the -neighborhood of ). Let , then for large enough such that , one has Moreover, if , , then
The above theorem becomes appealing when the exponent of is greater than zero and instability is obtained, for instance, when in case . However, up to now we have not yet been able to find explicit estimates in models of interest; in any event, we think that the future applications will be useful, and so we point out some speculations. First, note that it applies even if the set is a single point! Nevertheless, we expect that Theorem 3.1 will be applied to models whose Floquet operators have some kind of “fractal spectrum” (usually singular continuous or uniformly Hölder continuous spectral measures), and, somehow, should be related to dimensional properties of those spectra; indeed, this was our first motivation for the derivation of this result, and, in our opinion, such applications are among the most interesting open problems left here.
3.3. Rank-One Kicked Perturbations
Now consider with as in Section 3.1, with eigenvectors and the corresponding eigenvalues; where and is a normalized cyclic vector for , in the sense that and the closed subspace spanned by equals . Let In this case (see [22–24]) with and . Note that , , and so for , also belongs to ; a simple iteration process shows that for all , and we are justified in using the formula in Theorem 2.3 to estimate Laplace averages.
We are interested in . As , it follows that belongs to the Hilbert space, and so one can write Note that , and we have By the relation and (3.19) it follows that that is, and we get the equations Thus
For the trivial case or, equivalently, , one has and . In this case the analysis of is reduced to as calculated in Section 3.1. Thus for large , as expected.
Returning to the general case , note that So By denoting by (3.24) we finally obtain the relations
3.3.1. A Harmonic Oscillator
Now we present an application of the above relations to a kicked harmonic oscillator with natural frequency equals to 1; we will write .
Proposition 3.2. Let be a harmonic oscillator hamiltonian with appropriate parameters so that its eigenvalues are integers , , and as aforementioned. Then for any and cyclic vector for , there exists so that, for large enough, where is the harmonic oscillator ground state. Hence one has -dynamical stability.
Proof. We use the above notation; note that , and Theorem 2.3 can be applied. In this case we have and so Now we evaluate . For and , , where , , , and ; only and are poles in the interior of . By residue, for , and for and evaluating the integrals we obtain and after inserting this in the expression of the average energy we get Therefore, for large there is a constant so that This completes the proof.
For harmonic oscillators with eigenvalues , , the evaluations of the resulting integrals are more intricate and were not carried out.
3.4. Kicked Perturbations by a in ()
3.4.1. Kicked Linear Rotor
Consider where , , and . The Hilbert space is ; this model was considered in [12, 25, 26] and references therein. The Floquet operator is Denote , , and to be the eigenvectors of whose eigenvalues are the square of integers ; all eigenvalues have multiplicity 2 (the corresponding eigenvectors are and ), except for the null eigenvalue which is simple.
Consider the case ; then and so Denote . It follows that For fixed denote . If , , one has and by residues Hence
The analytical evaluation of these integrals is not a simple task. As an illustration, consider the particular potential ; since by Cauchy's integral formula