A charged particle in a homogeneous magnetic field accelerated by a time periodic Aharonov-Bohm flux
We consider a nonrelativistic quantum charged particle moving
on a plane under the influence of a uniform magnetic field and driven
by a periodically time-dependent Aharonov-Bohm flux. We observe an
acceleration effect in the case when the Aharonov-Bohm flux depends
on time as a sinusoidal function whose frequency is in resonance with
the cyclotron frequency. In particular, the energy of the particle
increases linearly for large times. An explicit formula for the acceleration
rate is derived with the aid of the quantum averaging method, and
then it is checked against a numerical solution with a very good agreement.
Keywords: electron-cyclotron resonance, Aharonov-Bohm flux, quantum averaging method, acceleration rate
Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Kolejní 2, 160 00 Praha, Czech Republic
Department of Mathematics, Faculty of Nuclear Science, Czech Technical University in Prague, Trojanova 13, 120 00 Praha, Czech Republic
The problem of acceleration in physical systems driven by time-periodic external forces, both in classical and quantum mechanics, has a rather long history though the results in the latter case are much less complete. One of the most prominent examples which initiated a lot of efforts in this field is the so called Fermi accelerator. On the basis of a theory due to Fermi to explain the acceleration of cosmic rays  Ulam formulated a mathematical model describing a massive particle bouncing between two infinitely heavy walls while one of the walls is oscillating . A thorough analysis finally did not fully confirm the expectations, however [28, 19, 20]. The model has also been reformulated in the framework of quantum mechanics .
More models of this sort have been studied in detail so far but we just mention one of them, the so called electron cyclotron resonance. One readily finds that electrons in a uniform magnetic field can gain energy from a microwave electric field whose frequency is equal to the electron cyclotron frequency. Because of an unlimited energy increase the relativistic effects cannot be neglected in a complete analysis. But even the relativistic model admits a quite explicit characterization of the resonant solution for a transverse circularly polarized electromagnetic wave propagating along the uniform magnetic field . In experimental arrangements the heated electrons are confined in a magnetic mirror field. Consequently, as they move along a flux tube of the mirror field they are exposed to the resonance heating only in a restricted region [24, 10, 12]. This acceleration mechanism is widely used in plasma physics.
Here we wish to discuss, on the quantum level, a model sharing some features with the preceding one. We again consider a charged particle placed in a uniform magnetic field. In our model the situation is simplified, however, in the sense that the particle is confined to a plane perpendicular to the magnetic field. Instead of a transverse electromagnetic wave propagating along the uniform field we apply, as an external force, an oscillating Aharonov-Bohm flux. The frequency of oscillations again coincides with the cyclotron frequency or, more generally, it may be an integer multiple of .
The Aharonov-Bohm effect itself received a tremendous attention as a genuinely quantum phenomenon , and almost all its possible aspects have been studied in the time-independent case. For example, a careful analysis can be found in . On the other hand, the time-dependent case represents an essentially more difficult mathematical problem and it has been treated so far only marginally in a few papers [18, 1, 3, 5].
The model we propose has already been studied in the framework of classical mechanics . It turns out that a resonance acceleration again exists but it has some remarkable new features if compared to the standard electron cyclotron resonance. If is an integer multiple of , then the classical trajectory eventually reaches an asymptotic domain where it resembles a spiral whose circles pass very closely to the singular flux line and, at the same time, their radii expand with the rate as approaches infinity. The particle moves along the circles approximately with frequency while its energy increases linearly with time. Denoting by the energy depending on time, an important characteristic of the dynamics is the acceleration rate which is computed in  and is given by the formula
Here is a real number which is expressible in terms of some asymptotic parameters of the trajectory.
The purpose of the current paper is to demonstrate that one can derive a formula analogous to (1) also in the framework of quantum mechanics. To this end and because of complexity of the problem, we restrict ourselves to the case when the AB flux depends on time as a sinusoidal function. To this system we apply the quantum averaging method getting this way an approximate time evolution for which we observe a resonance effect whose principal characterization is again a linear increase of energy.
Let us now be more specific. We consider a quantum point particle of mass and charge moving on the plane in the presence of a homogeneous magnetic field of magnitude . For definiteness, all constants , , are supposed to be positive. Assume further that the particle is driven by an Aharonov-Bohm magnetic flux concentrated along a line intersecting the plane in the origin and whose strength is oscillating with frequency .
In the time-independent case, the Hamiltonian corresponding to a homogeneous magnetic field and a constant Aharonov-Bohm flux of magnitude reads
where are polar coordinates on the plane, and the Hilbert space in question is . Making use of the rotational symmetry of the model we restrict ourselves to a fixed eigenspace of the angular momentum with an eigenvalue , . Put
Then this restriction leads to the radial Hamiltonian
in . Without loss of generality, we can assume that (note that is a constant). The boundary conditions at the origin are chosen to be the regular ones (then is the so called Friedrichs self-adjoint extension of the symmetric operator defined on compactly supported smooth functions). Let us note that if , then more general boundary conditions are admissible  but here we confine ourselves to the above standard choice.
be the cyclotron frequency. The operator has a simple discrete spectrum, the eigenvalues are
with the corresponding normalized eigenfunctions
are the normalization constants and are the generalized Laguerre polynomials.
Thus our main goal is to study the time evolution governed by the periodically time-dependent Hamiltonian where
and is a -periodic continuously differentiable function, is a frequency and is a small parameter. This means that the Aharonov-Bohm flux is supposed to depend on time as
Without loss of generality one can assume that
As discussed in , for the values the domain of in fact depends on , and this feature makes the discussion from the mathematical point of view a bit more complicated. Nevertheless, the time evolution is still guaranteed to exist.
2 The Floquet operator and the quasienergy
Let be the propagator (evolution operator) associated with ; it is known to exist . An important characteristic of the dynamical properties of the system is the time evolution over a period which is described by the Floquet (monodromy) operator , with . We are primarily interested in the asymptotic behavior of the mean value of energy
for an initial condition as tends to infinity while focusing on the resonant case when
A basic tool in the study of time-dependent quantum systems is the quasienergy operator
acting in the so called extended Hilbert space which is, in our case,
The time derivative is taken with the periodic boundary conditions. This approach, very similar to that usually applied in classical mechanics, makes it possible to pass from a time-dependent system to an autonomous one. The price to be paid for it is that one has to work with more complex operators on the extended Hilbert space.
An important property of the quasienergy consists in its close relationship to the Floquet operator [11, 27]. In more detail, if is an eigenfunction or a generalized eigenfunction of , , which also implies that , then the wavefunction solves the Schrödinger equation with the initial condition . It follows that . Thus from the spectral decomposition of the quasienergy one can deduce the spectral decomposition of the Floquet operator.
be the unperturbed quasienergy operator. Its complete set of normalized eigenfunctions is
(here stands for nonnegative integers, the wave functions are defined in (4)), with the corresponding eigenvalues . Thus has a pure point spectrum which is in the resonant case (7) infinitely degenerated.
To take into account these degeneracies we perform the following transformation of indices. Denote by and the integer and the fractional part of a real number , respectively, i.e. , and . Furthermore, let
be the remainder in division of an integer by . The transformation of indices is a one-to-one map of onto itself sending to , with
Using the new indices we put
Then the vectors , , form an orthonormal basis in the extended Hilbert space . For a fixed integer let be the orthogonal projection onto the subspace in spanned by the vectors , . Then
Furthermore, using the basis one can identify with the Hilbert space . In particular, partial differential operators in the variables and like the quasienergy are identified in this way with matrix operators. In the sequel we denote matrix operators by bold uppercase letters.
3 The quantum averaging method
The full quasienergy operator depends on the small parameter . Let us write as a formal power series, . In our case,
and . The ultimate goal of the quantum averaging method in the case of resonances is a unitary transformation resulting in a partial (block-wise) diagonalization of . Thus one seeks a skew-Hermitian operator so that commutes with which is the same as saying that it commutes with all projections . This goal is achievable in principle through an infinite recurrence which in practice should be interrupted at some step. Here we shall be content with the first order approximation.
Let us introduce the block-wise diagonal part of an operator in as
Thus surely commutes with . The off-diagonal part is then defined as . Developing formally in one has and
After switching on the perturbation, any unperturbed eigenvalue gives rise to a perturbed spectrum which, in the first order approximation, equals the spectrum of the operator restricted to the subspace . If the degeneracy of is infinite then the character of the perturbed spectrum may be arbitrary, depending on the properties of . The corresponding perturbed (generalized) eigenvectors span a subspace which is the range of the orthogonal projection
is the reduced resolvent of taken at the isolated eigenvalue . Thus the first order averaging method is in fact nothing but the standard quantum perturbation method in the first order but accomplished on the extended Hilbert space simultaneously for all eigenvalues of (compare to [17, Chp. II§2]).
Our strategy in the remainder of the paper is based on replacing the true quasienergy by its first order approximation
and, consequently, is replaced by an approximate Floquet operator associated with . To determine the approximate Floquet operator one has to solve the spectral problem for . To this end, as already pointed out above, one can employ the orthonormal basis in order to identify operators in with infinite matrices indexed by .
Let denote the standard basis in , and denote the standard basis in . It is convenient to write as the tensor product of Hilbert spaces which also means identification of the standard basis in with the set of vectors .
where is the matrix operator in with the entries
To compute the matrix entries of one observes that formally (see (2))
stands for the th Fourier coefficient of . Recall that, by the assumption (6), . Moreover, for one has , hence
In  it is derived that, for ,
Note that , as defined in (9), is –periodic in the integer variable , and so is the matrix , i.e. . Moreover, since one also has (see (11)). For an integer , , let be the closed subspace in the original Hilbert space spanned by the vectors , . Then decomposes into the orthogonal sum
and from the relationship between and , as recalled in Section 2, it follows that every subspace is invariant with respect to .
In the example which we study in more detail in the following section (for a sinusoidal function ), the matrix operators have purely absolutely continuous spectra. For the sake of simplicity of the notation let us confine ourselves to this case. For a fixed index , , suppose that all generalized eigenvectors and eigenvalues of are parametrized by a parameter . Let us call them and , respectively, i.e.
The generalized eigenvectors are supposed to be normalized to the function, i.e.
which in fact means that as a function in the variables and is a kernel of a unitary mapping between the Hilbert spaces and . Thus the spectral decomposition of reads:
and, for all ,
To get a correct approximation in the first order of the propagator one further has to take into account the transformation which is inverse to that generated by . First observe that is a multiplication operator on the Hilbert space in the following sense. Let be the unitary operator on acting as
An operator on commutes with if and only if there exists a one-parameter -periodic family of operators on such that . Notice that
With this equality, it is obvious from (14) that if commutes with then the same is true for . Furthermore, as one can see from (12), commutes with , and from (15) one infers that commutes with as well. Hence there exists a one-parameter -periodic family of skew-Hermitian operators on such that
Next notice that a transformation of the quasienergy operator of the form , where again is a -periodic family of skew-Hermitian operators on , implies a transformation of the associated propagators according to the rule
Hence the correct approximation of the Floquet operator reads
Let us note, however, that one has, for and ,
where . If the commutator happens to be bounded then it does not contribute to the acceleration rate.
4 A sinusoidally time-dependent AB flux
In the remainder of the paper we discuss the example when . The goal of the current section is to provide more details on the spectral decomposition of the averaged quasienergy derived in (16). Naturally, rather than directly with the quasienergy we shall deal with its matrix, as given in (17) and (18).
We still assume that is fixed. For this choice of , an immediate evaluation of formula (22) gives
where . Thus one has
where is the Jacobi (tridiagonal) matrix with zero diagonal,
and with the positive entries
and is the unitary diagonal matrix with the diagonal .
This is an elementary fact that the spectrum of is simple since any eigenvector or generalized eigenvector is unambiguously determined by its first entry. Moreover, one readily observes that the matrices and are unitarily equivalent, and so the spectrum of is symmetric with respect to the origin.
In our case,
Hence is rather close to the “free” Jacobi matrix for which for all . The spectral problem for is readily solvable explicitly (see below). It turns out that the spectral properties of are close to those of as well , see also . In particular, it is known that the singular continuous spectrum of is empty, the essential spectrum coincides with the absolutely continuous spectrum and equals the interval . Furthermore, there are no embedded eigenvalues, i.e. if is an eigenvalue of then .
Splitting into the sum of the upper triangular and the lower triangular part, one notes that . In our example, for all and so and, consequently, the spectrum of is contained in the interval . This means that the only possible eigenvalues of are . But one can exclude even this possibility. In fact, suppose that , with and . Then
(while putting ). Summing this equality for , and using that , one finds that for . Hence for all , and so is not square summable. Thus one can summarize that the spectrum of is simple, purely absolutely continuous and equals .
Let us parametrize the spectrum of by a continuous parameter , , so that
is a point from the spectrum and is the corresponding normalized generalized eigenvector with components , (here we drop the index at and in order to simplify the notation). The asymptotic behavior of the components is known [14, 6]; one has
for . Here is a normalization constant and is a phase which depends on the initial conditions imposed on the sequence (the initial condition is simply ) but the asymptotic methods employed in the cited articles do not provide an explicit value for it. In the limit case the generalized eigenvectors are known explicitly, namely
for all . Hence .
The generalized eigenvectors are supposed to be normalized so that
For , one can use the equality
which is valid for and where the symbol indicates the regularization of a nonintegrable singularity in the sense of the principal value. The normalization is an immediate consequence of this identity.
For general , the contribution to the function should come from the most singular and, at the same time, the leading term in the asymptotic expansion of , as given in (31). This time, when investigating the singularity near the diagonal in the scalar product of two generalized eigenvectors, one is lead to considering the sum
where is a real constant. Using the Lerch function one has for (see [9, § 9.55]),
From here one deduces that, for any real ,
where is a regular distribution, i.e. a locally integrable function. Hence in the general case, too, the normalization constant is given by
As already mentioned, the phase in the asymptotic solution (31) remains undetermined. But we remark that a bit more can be said about the behavior of the phase near the spectral point (the center of the spectrum) which corresponds to the value of the parameter . More precisely, one can compute the derivative . Though this result is not directly used in the sequel it represents an additional information about generalized eigenfunctions of . We briefly indicate basic steps of the computation in Appendix.
5 The acceleration rate
In the case when the commutator occurring in (26) can be shown to be bounded. This implies that instead of the approximate Floquet operator , as given in (25), one can work directly with defined in (24) when deriving a formula for the acceleration rate. On the other hand, one should not forget about the transformation of the initial state, i.e. has to be replaced by , see (26).
Of course, the parallels to the diagonal determined by can be explicitly evaluated as well but for our purposes it is sufficient to know that they are bounded. In [3, Lemma 6] it is shown that the matrix operator in with the entries
for and otherwise is bounded. Thus to verify the boundedness of the commutator it suffices to show that the difference of matrices (33) and (34) has a finite operator norm. This can be readily done, for example, with the aid of the following estimate for the norm of a Hermitian matrix operator [17, § I.4.3],
To proceed further, we again fix an integer , . Suppose one is given a function . Recalling (23) we put
In what follows, we drop the index and, whenever convenient, write simply instead of . Using (24), one has, for ,
Note that is an orthonormal basis in and so
Hence, in view of (3), the leading contribution to the acceleration rate comes from the expression
Furthermore, restricting this sum to an arbitrarily large but finite number of summands results in an expression which is uniformly bounded in . This justifies replacement of by the leading asymptotic term, as given in (31) (with ). Hence the leading contribution to the acceleration rate is expressible as
Estimating the acceleration rate we can restrict ourselves to a sufficiently small but fixed neighborhood of the diagonal with a radius . Thus we arrive at the expression
Further we carry out the differentiation, as indicated in the integrand, and get rid of the terms which are not proportional to or which are non-singular. Moreover, we use the substitution . Thus we obtain the expression