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.
1. Introduction
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 [1], 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 [7] was found.
Dynamical stability of time-dependent systems was studied, for example, in references [2, 8–19]. In [14] 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 [20].
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 [10] 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 [10] 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 [21] 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 [1], 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
Hence .
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.
Lemma 2.4.
In Section 3 we discuss some Floquet operators that are known in literature and analyze their Green functions through the equation
3. Applications
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 .
Therefore by (3.2) it follows that
Since , for large it is found that
with (for )
Then we conclude that the function
is bounded for , which is (of course) an expected result (see Proposition 2.1).
3.2. Lower-Bounded Green Functions
As a first theoretical application we get dynamical instability from some lower bounds of the Green functions. See [21] 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
Proof. By the formula in Theorem 2.3, or its more general version (2.16),
and we have used that . If , then
The proof is complete.
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
and by residue theorem
it is found that
Therefore, by (2.16) it follows that for any
and we conclude that (see the appendix)
and also that the sequence is unbounded. This behavior is expected since the spectrum of is absolutely continuous in this case [25], but here we got the result explicitly without passing through spectral arguments, although in a rather involved way; indeed, a much simpler derivation is possible by direct calculating and the corresponding expectation values.
For with integer , similar results are obtained, that is
and so
Therefore, we have the following lower bound for the Laplace average:
(see appendix). The same is valid if with denoting any negative integer number.
3.4.2. Power-Kicked Systems
Due to the difficulty in evaluating the integrals in (3.47), in order to estimate in some situations we take an alternative way.
Consider the Kicked models in with Floquet operator
corresponding to the hamiltonian
with as before and for some . Let be the Fourier transform. Then and
where is represented by a diagonal matrix whose elements are
and is represented by a matrix whose elements are
where . Denote ; so
Put ; then
and using (3.62) we obtain
Thus, for each ,
so that
Tridiagonal Case
In order to deal with the above equations, we try to simplify them by supposing that is such that if . Then, for each fixed (3.66) becomes
and is tridiagonal and has the structure
where .
Now, a tridiagonal unitary operator on is either unitarily equivalent to a (bilateral) shift operator or an infinite direct sum of and unitary matrices, as shown in [27, Lemma 3.1]. For proving this result it was only used that is unitary and , where is the canonical basis of , that is,
It then follows that for all
Applying these relations to we obtain the following.
(i)If , then and . (ii)If , then and . (ii) If , then and .
The next step is to investigate these cases. If , it reduces to the autonomous case previously considered.
The cases and are similar, so we only discuss that . For fixed, (3.67) takes the form
and so we can write in terms of and for all . More precisely
moreover, for in (3.71) we obtain , and so for and
and there exists so that
Therefore, by (2.16), for one has
so that, by the discussion at the end of the appendix,
and is unbounded. Hence we have instability.
Pentadiagonal Case
Suppose now that is such that if . Then for each fixed, equation (3.66) becomes
and is pentadiagonal and has a structure similar to the corresponding operator in the previous case, just adding the elements whose distance to the diagonal is . The elements in the new upper diagonal are , and those in the new lower diagonal are .
For not repeating the tridiagonal case we suppose that either or is different from zero. If is a pentadiagonal unitary operator in , that is, one gets the matrix representation
From this we obtain the following relations, for each ,
Suppose that . The case is similar. Then by the above relations we obtain and so (3.77) becomes
For one gets and analogously to the previous case
with and . Since for
we obtain
hence
and is unbounded.
N-Diagonal Case
If satisfies for , we suppose that either or is different from zero. In case , by unitarity and the structure of we obtain that , thus (3.66) becomes, for each ,
and so
with , . Moreover, for
Similarly to the previous cases we conclude that
Therefore we can state the following result.
Theorem 3.3. For Kicked systems in L2 with
as in (3.56), one can obtain that is represented by the matrix with elements , where . If satisfies for and either or is different from zero, then , for some , and is unitarily equivalent to (the th power of ) where is the bilateral shift. Furthermore,
Proof. It is enough to prove that is unitarily equivalent to . Suppose that (the case for is similar); then by the above discussion we obtain
that is, where is the canonical basis of . Since , write . Let be the unitary operator defined by
where are elements in . If satisfies for all
it follows that . Equation (3.93) is satisfied taking, for example, and another obeying (3.93).
Although Theorem 3.3 gives a nice illustration of the potential applications of our expression for the Laplace average, since it is one of the few instances that such average can be explicitly estimated from below, again it can be derived by more direct methods and one can also conclude [25] that the spectrum of the corresponding Floquet operators is absolutely continuous.
4. Conclusions
Although most of our applications of Theorem 2.3 give expected results (sometimes known results that can be derived in simpler ways), we believe that that formula is interesting and has a potential to be applied to more sophisticated models as the Fermi accelerator. The difficulty is to get expressions or estimates for the Green functions, since calculating the resolvent of an operator is not always an easy task; sometimes we have the expressions for resolvent operators (e.g., for kicked systems), but the resulting integrals can be too involved. We have not tried any numerical approach to formula (2.15), which might be useful for some specific models.
In the case of one-dimensional discrete Schrödinger operators, where the hamiltonian is defined by
a bounded sequence, a similar formula can be handled in some cases by relating the resolvent to transfer matrices. Then adequate upper bounds of such transfer matrices, on some set of energies , result in lower estimates for the corresponding Green functions, and then transport properties are obtained for interesting models (see [21] and references therein).
In [27] a class of Floquet operators displaying a pentadiagonal structure was introduced; for these models there is a transfer matrix formalism. However, such transfer matrices are too complicated, and analytical estimates seem far from trivial.
Anyway, the technique here is quite general; it asks no particular regularity of the time-dependence and can be virtually applied to any time-periodic system as soon as the time evolution is well posed. As already said, the chief difficulty is related to suitable bounds of matrix elements of the resolvents of unitary (Floquet) operators, a task harder than we initially envisaged. Herein we put forward for consideration the challenge of getting additional applications for the formula (2.15) deduced for the Laplace averages, including an application of Theorem 3.1 to physical models. It is also worth mentioning the question left open in Lemma 2.2, that is, is it true that ?
Appendix
Laplace Transform of Sequences
Let be a sequence of positive real numbers. The Laplace transform of , denoted by , is the function defined by
for in a subset of . It will also be denoted by .
We say that the Laplace transform of exists if the series in (A.1) converges for some . For example, if , then the sum in (A.1) diverges for all .
Examples
(1) For the constant sequence it follows that
for . By using Taylor expansion, for small one finds that .
(2) Since , for , it follows that
for and as above. Thus, the Laplace transform of is
For small , .
A sequence of complex numbers is said to be exponential of order (real) if there exists so that , . That is, does not increase faster than as . If is exponential of order then
is convergent for any .
Let denote the set of positive sequences of exponential order . The Laplace transform satisfies
where is a positive number, and and are sequences in . Moreover, if and then and so for all , that is, . Thus is injective on .
The Laplace average (2.5) is related to the Laplace transform of by
If for all , then
for large enough. If then
for large . Hence, if grows like , then the same law holds for its average Laplace transform. We have a restricted converse, that is, if grows with a positive power of then, by Lemma 2.2, its Cesàro average is unbounded (with a rather similar behavior at large times) and so is . These properties are repeatedly used in the text.
One should be aware that there are special situations of unbounded positive sequences with bounded average Laplace transforms (so that ); an explicit example is and for . The same phenomenon is well known for Cesàro averages, and, by Lemma 2.2, such phenomena are connected.
Acknowledgments
The authors thank the anonymous referees for careful readings and suggestions. C. R. de Oliveira thanks the warm hospitality of PIMS (Vancouver) and the support by CAPES (Brazil).