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Geometrical phase for a three-dimensional anisotropic quantum well

A three-dimensional anisotropic quantum well placed in an adiabatically precessing uniform magnetic field is considered and an explicit formula for the Berry phase is obtained. To get the Berry phase, a purely algebraic algorithm of reducing a quadratic Hamiltonian to the canonical form via symplectic transformations of the phase space is presented.


I. INTRODUCTION
In the classical work [1], M.Berry has shown that if an evolution of a physical system is cyclic and adiabatic the geometric phase (so-called Berry phase) is accumulated on the wave function in addition to the usual dynamical phase. The Berry phase has a purely geometric nature and reflects geometric properties of the parameter space of the system. B.Simon [2] has given an interpretation of the Berry phase as due to holonomy in a line bundle over the parameter space.
The space of values of the magnetic induction as a parameter space for the considered system is frequently dealt with. As a rule, for a magnetic field, the Berry phase is studied in the two cases: (1) a potential well is transported along a closed loop in the usual configuration space [3,4,5]; (2) spin particles interact with a time-depended magnetic field [6,7,8]. The most part of theoretical studies is devoted to the evolution of the spin in a magnetic field; in this case explicit formulae are easy to obtain using finite-dimensional algebra. However, the Berry phase can arise in the case of a spin-less particle interacting with a precessing magnetic field [9].
The aim of this paper is to obtain an explicit formula for the Berry phase in the case of a three-dimensional anisotropic quantum well placed in a precessing magnetic field. The classical Hamiltonian of such a system is a quadratic form of momenta and positions. The considered problem has attracted attention of physicists since the 1980's. In particular, M.Berry [10] has calculated the geometric phase for a one-dimensional quadratic potential with variable coefficients. As for now, the most general result for quadratic Hamiltonians is contained in [9] where the following simple case has been studied: a quadratic potential has an axial symmetry with respect to the z-axis and a magnetic field precesses around z-axis. These assumptions reduce the dimension of the problem and significantly simplify calculations. We investigate the most general problem in the case in which the frequencies of the parabolic potential are mutually different and the magnetic field is arbitrarily directed.
The significant peculiarity of a quadratic quantum Hamiltonian lies in the fact that the Feynman integrals for their propagators can be calculated explicitly. What is more, the problem of calculating these integrals is reduced to finding the canonical form for a classical counterpart of the Hamiltonian via symplectic transformations of the phase space; in our case this form is a sum of squares of momenta and positions). Finding the canonical form is a standard (but as a rule very cumbersome) problem of linear algebra. For convenience of readers, in the next section we present an algorithm of finding the classical (symplectic) and quantum (unitary) transformations that reduce the corresponding Hamiltonian to the canonical form.
It is important to stress that we obtain our results using only simple methods of linear algebra. It is strange enough that in the similar situations, more complicated methods are used in the physical literature; in particular, the method of the Bogolubov canonical transformations [11] or other indirect methods. Note another advantage of the method we use. To find various physical quantities one needs the matrix elements of the corresponding operators. However, a direct calculation of matrix elements can be a complicated problem. Our method allows to resolve this problem using only simple calculations from linear algebra. As an example we can mention the papers [12] and [13] where the hybrid and hybrid-phonon resonances for an asymmetric quantum well in a magnetic field have been studied by means of this approach.

II. PRELIMINARIES
The advantage of a description of mechanical systems with the help of the Hamilton formalism is conditioned by the following main reason: a choice of the generalized coordinates is restricted by no conditions (any one of quantities defining the state of a system may be the generalized coordinates.) What is more it is possible to choice such a class of transformations of phase coordinates that includes the transformation of the all initial phase coordinates p i , q i to the new ones P i , Q i (here i = 1, 2...n; n is the dimension of the configuration space) because momenta and positions play the same role in Hamilton equations Q = Q(p, q); P = P (p, q). (1) However there is a restriction. The transformation (1) is permissible (canonical) if and only if it conserves the Poisson brackets. The equivalent criterion for the transformation (1) to be canonical is the equality to the exact differential the 1-form pdq − P dQ. This 1-form is the exact differential of the generating function of the canonical transformation (1).
Note that if we know the generating function we can always find the functions determing the transformation (1).
Through the paper we use the notation for the standard symplectic product in the phase space R 2n = R n q × R n p ; here x j = (p j , q j ), (j = 1, 2).

Recall that a linear transformation
is canonical (i.e., preserves the Poisson brackets) if and only if the matrix L is symplectic: It is convenient to write the matrix L in the form The following commutation relations between components of the matrix L follow immediately from the fact that the matrix L is symplectic where E is the unit matrix.
Denote by S the generating function for the transformation L (this means by definition that dS = pdq − P dQ); we will suppose that S is a function of p and P . Then In this case the generating function of the canonical transformation (4) is expressed in terms of L i by the following formula (we assume that |L 2 | = 0) According to the Dirac-Fock theorem [14] the classical transformation (4) corresponds to a quantum transformation U : L 2 (R n P ) → L 2 (R n p ) that is to a unitary operator related to L as follows The operator U can be represented as an integral operator with the kernel where the normalization constant C = ((2π ) 3 | det L 2 |) −1/2 . Then using the unitary of the operator U we can express the initial wave function of the quantum Hamiltonian in terms of the final wave functions by the following formula here ψ(p) is the wave function of the original Hamiltonian andψ(P ) is the wave function of the unitary equivalent Hamiltonian.

III. DIAGONALIZATION OF QUADRATIC HAMILTONIANS
The purpose of this section is to demonstrate an algorithm of reducing the quadratic Hamil- whereH is a symmetric matrix of order 2n × 2n, to the canonical form. This form must contain only squares of the corresponding momenta and positions.
Then Hamilton equations corresponding to our quadratic Hamiltonian (14) can be written in the formẋ = IHx, here I is the symplectic unity At first we find the family (λ i ) 1≤i≤2n of eigenvalues the matrix IH, where I is the symplectic and the family (f i ) 1≤i≤2n of the corresponding eigenvectors. For this purpose we use the following simple lemma Lemma. The characteristic polynomial det(IH −λE) has only real coefficients and contains only terms of even degree.
It follows from the Lemma that if λ i is an eigenvalue of the matrix IH then −λ i and λ * i are eigenvalues of this matrix too. Taking into account this assertion we can arrange λ i in a finite sequence of the form where ν i , ω k > 0; l, m ≥ 0; l + m = n. Now we transform the basis (f i ) into a symplectic one. Let (b i ) 1≤i≤2n be the following sequence of vectors from R 2n : It is clear that the vectors (b i ) 1≤i≤2n form a basis of the space R 2n . These vectors satisfy the skew-orthogonality conditions Denote form a symplectic basis of the space R 2n . Using coordinates with respect to this basis, we reduce the Hamiltonian H to the form where ε i = sgn c i . Now we define the canonical basis of the quadratic formH by the relations With respect to this basis, the Hamiltonian H has the canonical form The transition matrix from the initial phase coordinates p, q to the final ones P, Q can be written in the following form where each vector a i is written in the corresponding column.
IV. DIAGONALIZATION OF THE HAMILTONIAN OF A 3D ANISOTROPIC OS-

CILLATOR WITH A MAGNETIC FIELD
Let us consider the Hamiltonian of a 3D anisotropic harmonic oscillator with a magnetic field B arbitrarily directed with respect to the potential symmetry axes Here A is the vector potential of the magnetic field, c is the light velocity, Ω i (i = x, y, z) are the characteristic frequencies of the parabolic potential, m, e, and p are the mass, the charge, and the momentum of the considered particle.
The results of Sec. III show that Hamiltonian (25) is unitary equivalent to a quadratic Hamiltonian without magnetic field but with new frequencies named "hybrid frequencies".
The purpose of this section is to reduce Hamiltonian (25) to the canonical form using the algorithm developed in the preceding section; here P, Q are the new phase coordinates, ω i (i = 1, 2, 3) are the hybrid frequencies. It is convenient to choose the following gauge for the vector potential A Below we use the notations

In this notation
i.e., H is a quadratic form of p and q with the matrix To find the eigenvalues of the matrix IH we solve the equation det(IH − λE) = 0, which is easily reduced to the form where µ = mλ. Since for any n×n matrix A, B, C, we can obtain the hybrid frequencies from the following sixth-order algebraic equation [15] ( where ω ic = eB i /mc are the components of the cyclotron frequency and λ ± i = ±iω i . Note that equation (32) always has three different real negative solutions with respect to λ 2 . Now our purpose is to find the eigenvectors of the matrix IH associated with the eigenvalues λ ± i . Using elementary matrix transformations one can reduce the matrix M = IH − λE to the form The eigenvectors of (33) are obtained from the system (34) Solving this system we get the coordinates of the eigenvectors g(µ) associated with the eigen-values µ = mλ ± i of IH: Consider the vectors Then we have for the coordinates of f i the following expressions. If i = 1, 2, 3, then If i = 4, 5, 6, then for i = 1, 2, 3; and M i = M i−3 for i = 4, 5, 6. It is easy to show that the following basis is where we put ω i = ω i−3 for i = 4, 5, 6.
With respect to this basis the Hamiltonian H has the form herep i , andq i are new phase coordinates.
To reduce this Hamiltonian to the canonical form (26) we change the variables once again Finally, the transition matrix L from the initial coordinates to the coordinates P i , Q i has the components where a ij are defined as follows: if i = 1, 2, 3 then As a result we have obtained an explicit formula for the transition matrix L.
Note that the quantum counterpart of classical Hamiltonian (21) is the quantum harmonic oscillator with the eigenvalues E n 1 n 2 n 3 = ω 1 n 1 + 1 2 + ω 2 n 1 + 1 2 where n 1 , n 2 , n 3 = 0, 1, ... In the P -representation, the corresponding eigenfunctions arẽ Here χ n is the n-th oscillator function where H n is the corresponding Hermite polynomial.

V. CALCULATING THE BERRY PHASE
The purpose of this section is to obtain an explicit formula for the Berry phase of a 3D anisotropic parabolic quantum well placed in a precessing magnetic field. We deal with the case when a considered quantum state undergoes an adiabatic evolution in the space of the values of the magnetic induction B in such a way that eigenvalues (41) remain nondegenerate.
Let ψ n be the wave function of the original state (here n = (n 1 , n 2 , n 3 ) is the set of the quantum numbers). If the system undergoes an evolution in question, then the wave function obtains the geometric phase where C is a closed path B(t) in the parameter space such that B(T ) = B(0), and V n (B) is so-called Berry vector potential given by Since the Berry phase is determined by V n (B), it is sufficient to calculate only the vector potential. In our case the magnetic field doesn't vanish and hence the parameter space is R 3 \{0}, which is topologically non-trivial, and we can expect that the Berry phase is nontrivial, too.
To find the scalar product in (44) one needs eigenfunctions of the initial Hamiltonian. However, a direct calculation of the wave functions to obtain the Berry phase is a complicated computational problem. To calculate the Berry phase, we suggest to simplify the calculations using the method of linear canonical transformation of the phase space developed in the preceding sections. In this case we can use formula (11) to express the initial wave functions ψ n (P; B) in terms of the final ones (42): Taking into account thatψ n (P; B) is real-valued we can calculate ∇ B ψ n (p; B): In what follows we denote ∇ B simply by ∇.
By changing the variables L −1 2 p/ = y; dp = | det L 2 | 3 dy, we perform the integration with respect to p in the first term of (47) and get exp i (S(p, P ′ ) − S(p, P)) dp = Hence the first term in (47) is real-valued and doesn't contribute in the Berry phase. Therefore, we need to calculate only the second term in (47).
Finally we have Equation (60) is the main result of our paper. This formula extends a result of Berry [10] to the 3D case and contains as a particular case some results from [9].