ОПТИКА И СПЕКТРОСКОПИЯ, 2012, том 113, № 6, с. 694- 700
ФИЗИЧЕСКАЯ ^^^^^^^^^^^^^^ ОПТИКА
УДК 537.633
TOMOGRAPHIC PROBABILITY REPRESENTATION FOR STATES OF CHARGE MOVING IN VARYING FIELD1
© 2012 г. V. I. Man'ko*, **, E. D. Zhebrak*
* Moscow Institute of Physics and Technology, Moscow, Russia ** Lebedev Physical Institute of the Russian Academy of Sciences, Moscow, Russia Received July 22, 2011; in final form, May 16, 2012
The coherent and Fock states of a charge moving in varying homogeneous magnetic field are studied in the tomographic probability representation of quantum mechanics. The states are expressed in terms of quantum tomograms. The coherent states tomograms are shown to be described by normal distributions with varying dispersions and means. The Fock state tomograms are given in the form of probability distributions described by multivariable Hermite polynomials with time-dependent arguments.
1. INTRODUCTION
The classical-like packets [1] for charge moving in constant homogeneous magnetic field were shown [2] to be the coherent states with constant dispersions and means moving along the classical cyclotron trajectory and the coherent state wave functions were also constructed for the charge moving in varying uniform magnetic field [3—5]. In [6] the coherent states were used to obtain the Landau diamagnetism by new method.
The probability representation of quantum mechanics was suggested recently [7] for the case of continuous variables like position and momentum, and in [8, 9] for discrete variables like spin. Tomographic probability distributions for position and spin variables are related to wave function by means of invertible integral transforms and can be used to describe quantum states for all systems.
Generalized coherent states in [10] were introduced to describe the quasiclassical motion of electrons in a microwave and in a homogeneous magnetic field. In [11] the coherent states were considered for the quantum particle on a circle. The obtained results were applied in [12] to a problem of a charged particle moving in a uniform magnetic field. A special type of stationary states for a particle in a homogeneous magnetic field — so-called squeezed states were considered for example in [13, 14]. In [15] were introduced so-called trajectory coherent states corresponding to Gaussian wave packets with center of gravity moving along the classical trajectories of charged particles. Complete sets of trajectory coherent states were derived for Schrodinger, Klein-Gordon, and Dirac equations in [16—18]. In the purpose of study the rotational motion of a self-bound system of particles, such as a nucleus coherent states of the rotation groups are constructed by several authors (see [19]). The coher-
1 Работа представлена на Седьмом семинаре Клышко.
ent states construction algorithms were proposed in [2] (for systems with quadratic Hamiltonians with discrete spectra), in [20] where the approach introduced in [2] was generalized to systems with continuous spectra and in [21] (considering a charge in electromagnetic field).
The problem of charges moving in magnetic field has different applications. In [22, 23] the tunneling in quantum dots is considered in terms of Kondo effect. In [24] formation of a coherent Kondo — singlet state and stabilization of a spin liquid in a Kondo lattice is considered. In [25] were introduced two-scale magnetic Wannier orbitals of electrons in a von Neumann lattice. Transport properties of two-dimensional electron gases under high perpendicular magnetic fields were studied in [26]. In [27, 28] correspondingly was introduced a description of charged electron-hole complexes in magnetic fields and considered a system of two electrons and one hole in a strong magnetic field, i.e. a negatively charged magnetoexciton. In the paper [29] was studied the ground Landau level of the 2D Frohlich polaron in a magnetic field and in [30] the polarization properties of an atomic gas in a coherent state were presented.
In our previous work [31] the coherent states of the charge moving in constant magnetic field were studied in the tomographic probability representation. The aim of this work is to extend the consideration of probability representation of the problem and to find to-mographic distributions (tomograms) describing the quantum states of charged particle moving in magnetic field to the case of time-varying fields. Our goal is to obtain the tomograms for both coherent states and Fock states of the charged particle in explicit form.
We remind the discussion of quantum state concept which is the basic concept of quantum theory. In conventional formulation of quantum mechanics the notion of system state is associated with wave function y (x) [32] or density matrix [33, 34]. These complex
functions y (x) or p (x, x') differ essentially from the notion of a system state in classical statistical mechanics where the nonnegative probability density f (q, p) on the system phase space is identified with the state concept. In view of this from the very beginning of quantum mechanics some attempts to get a formulation of quantum mechanics where the notion of system state is similar to classical probability density f (q,P) were made [35—38].
A solution of this problem was also suggested by Pauli [39] who proposed a conjecture that the quantum state complex wave function can be reconstructed if one knows both the probability density of position
(x )|2 and probability density of momentum |\jz (p)|2. Though this conjecture was shown to be incorrect (see, e.g. [40]) the generalization of this suggestion which now is called probability representation of quantum mechanics or tomographic probability representation solved this problem [7]. One can find such probability distribution w (X, p, v) of position X, depending on extra real parameters ^ and v that the density matrix or wave function can be obtained from this distribution by means of an integral transform. In our work we apply this new description of quantum state to the problem of charge moving in varying fields.
The paper is organized as follows. In Section 2 a short review of tomographic probability representation for systems with one and two degrees of freedom is presented.
In Section 3 the construction of coherent states and its tomograms for charged particle moving in constant magnetic field by means of creation and annihilation operators is reminded ([3, 41]). In Section 4 the Gaussian states of quantum charge including coherent states are considered in tomographic probability representation of quantum mechanics. In Section 5 some conclusions and perspectives are presented.
2. QUANTUM TOMOGRAMS
We construct the tomograms of the coherent states of a charge moving in varying fields and the tomo-grams identified with the quantum states determined by time-dependent integrals of motion which are initial energy and initial orbital momentum possessing the discrete time-independent eigenvalues.
The quantum tomography is based on the map of density operator p of the state | y) onto tomographic probability distribution w (X, p, v) called symplectic tomogram determined by the relation:
ws (X, p, v) = Tr p -8 (X -pq -vp) • (1)
Here, we consider a system with one degree of freedom and X, p, v are reals, q and p are position and momentum operators, respectively.
The tomogram is nonnegative probability distribution of random variable X which is position in rotated and rescaled reference frame in the phase-space. The parameters of the reference frame are labeled by reals p and v. The inverse of (1) reads
p = -L Jw (X, p, v)i( Wvp)dXdpdv. (2)
The tomogram has homogeneity property w (XX, Xp, Xv) = j1 w (X, p, v) •
(3)
For normalized states p the tomographic probability distribution is normalized, i.e.
Jw (X, p, v)dX = 1. (4)
In case of p = cos 0, v = sin 8 the tomogram
wop (X, 0) = w (X, cos 0, sin 0) (5)
is called optical tomogram.
The symplectic tomogram of pure state with the wave function y (y) is determined by the formula [42]:
w (X, p, v) =
1
2n|v
J¥ (y)
2 iX —y —y
dy
(7)
which is related to fractional Fourier transform of the wave function. For the several degrees of freedom the formulas (1)—(7) are given as simple direct product of the transforms presented for one-degree freedom. For example for two degrees of freedom the symplectic tomogram w (X1, pb v1, X2, p2, v 2) is determined by the fractional Fourier transform of the wave function V (y1, y2) and it reads
w (Xi, pb Vi, X2, ^2, V2) = JV (yi, y 2 )
1
4n2 |viv 2I
<>i y 2 ^2 „ 2 iXi y iX2 y
"—y1 +"-y2 -y1-y2
2vi 2v2 V] v2 1 1
e 12 dy1dy
(8)
The optical tomogram wop (X1, X2,01,02) is given by Eq. (8) where p1 = cos 01, p2 = cos 02, v1 = sin 01, v2 = = sin 02. If one has two states p1 and p2 the fidelity providing transition probability P12 = P21 between these states reads
P12 =
= Tr P1P2 = fw1 ((1, pb V1, X2, p2, V2)x
4n J
x w2(Y1, pb v1, Y2, p2, V2) x
x ei(X1 -Y1+X2-Y)dX 1dY1d^1dv1dp2dV 2.
(9)
We can use this formula to study the transition probabilities between the energy levels of the charge excited by time-varying field.
2
3. COHERENT STATES OF CHARGED PARTICLE MOVING IN CONSTANT MAGNETIC FIELD
We shall consider a particle of mass m = 1 and charge e = 1 moving in a constant magnetic field with a vector potential
MAN'KO, ZHEBRAK where
A = 1 [H x r].
2 J
(10)
Let the cyclotron frequency ® = 1. The Hamilto-nian for such a system is
H = I
(Px - Ax)2 + (Py - Ay)2
h = c = e = 1. (11)
In order to construct the coherent and excited states of the charge moving in the constant magnetic field let's introduce the following operators:
(Px + iPy ) + 1(y - ix)
A =
V2
1,
(12)
(Py + iPx ) + - (x - iy)
B =
V2
The following commutation relations hold:
[A,A+] = [B,B+] = 1, [A,B~\ = [A,B+] = 0.
(13)
For oppositely charged particles, the lowering and raising operators A, B and AB + change their places. For simplicity we suppose e > 0.
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