S. S. Mizrahi

Quantum mechanics in the Gaussian wave-packet phase space representation

We present here some results related to an alternative mapping formulation (to the well-known Wigner-Weyl transform) which makes use of the Gaussian Wave Packet or coherent states representation |pq↩. The pair p-q which labels this state defines a phase-space in which the abstract operators P, Q, of momentum and position, are represented as differential operators. In the mapped expression of an operator A(P, Q), the quantum effects appear when they are absent in the corresponding Wigner-Weyl transform.

Dissipative mass-accreting quantum oscillator

We begin by revisiting the so-called Caldirola - Kanai Hamiltonian and discuss its inherent ambiguity: does it represent a dissipative harmonic oscillator (HO) subject to a friction force, or does it describe an HO with a time-dependent mass (TDM)? Although classically both descriptions do coexist, in the quantum domain the solution of the Schrodinger equation (or Heisenberg equations of motion) with a TDM does not present inconsistencies, however, the dissipative Hamiltonian shows violation of the Heisenberg uncertainty principle. This violation is avoided by introducing a stochastic force in the equations of motion, which will take care of the fluctuations due to the environment. Once the distinction between the dissipative and amplifying Hamiltonian is made clear, we consider the problem of the quantum TDM HO subject to dissipation, showing that both phenomena may be merged and described by a single Hamiltonian, the amplifying - dissipative Hamiltonian. We obtain the solutions of the Heisenberg equations of motion for the canonical momentum and position; next, we specialize on the weak damping limit and analyse the effects of the amplifying - dissipative process on the mean values of the physical variables.

Energy spectrum, potential and inertia functions of a generalized f-oscillator

We consider a generalized four-parameter q-algebra AA† − qγA†A = qαN+β, [N, A] = −A, [N, A†] = A†, associating with operators A and A† the nonlinear f-oscillator operators, defined in terms of the usual harmonic oscillator operators as A ≡ af(N) and A† ≡ f*(N)a† (where a and a† are operators of the Weyl–Heisenberg algebra and N = a†a). The function f(N) is determined from the commutation relations. We write the Hamiltonian for the free f-oscillator and obtain its energy spectrum. Besides, expressing the Hamiltonian in terms of coordinate and momentum, we determine the potential and inertia functions (coordinate-dependent mass) and analyse their behaviour by varying the parameters.

Creating quanta with an 'annihilation' operator

The asymmetric nature of the boson ‘destruction’ operator ˆ a and its ‘creation’ partner ˆ a † is made apparent by applying them to a quantum state |ψ� different from the Fock state |n� .W e show that it is possible to increase (by many times or by any quantity) the mean number of quanta in the new ‘photonsubtracted’ state ˆ a|ψ� .M oreover, for certain ‘hyper-Poissonian’ states |ψ� the mean number of quanta in the (normalized) state ˆ a|ψ� can be much greater than in the ‘photon-added’ state ˆ a † |ψ� .T he explanation of this ‘paradox’ is given and some examples elucidating the meaning of Mandel’s q-parameter and the exponential phase operators are considered.

The geometrical phase: An approach through the use of invariants

Abstract An approach for the exact calculation of the geometrical and dynamical phases, by using the method of invariants of Lewis and Riesenfeld, is presented. The invariant parameter space, in which the parameters evolve dynamically, is introduced and the general scheme for constructing the invariants is discussed. Two examples are considered: (a) The spin- 1 2 particle in a magnetic field and (b) the forced harmonic oscillator for which we evaluate the interference effects, due to the geometrical phase only.

Quantum mechanics in the Gaussian wave-packet phase space representation II: Dynamics

The Heisenberg and Liouville dynamical equations are mapped using the Wave-Packet Phase Space Representation. A semiclassical perturbative expansion is introduced - the Quasi-Causal Approximation - for the Green function and an expression for transition probabilities is derived up to the first order.

Covariance entanglement measure for two-mode continuous variable systems

Abstract We introduce the measure of entanglement of continuous variable quantum systems, expressed in terms of the cross-covariances of the quadrature components or the annihilation/creation operators. Considering the examples of finite and infinite superpositions of coherent and Fock states, we compare the new parameter with the conventional entropic measures of entanglement and with the “linear entropy of entanglement”.

Generation of circular states and Fock states in a trapped ion

We propose three schemes to engineer a circular state (superposition of the harmonic oscillator coherent states on a circle in phase space) for the centre-of-mass motion of a trapped ion. We analyse the necessary duration of each laser pulse for constructing such states, and calculate the probability of obtaining the subtle superposition. We also show that it is possible to engineer Fock number states as a result of the interference effects in phase space of the coherent states superposition.

Pseudo-diffusion equation and information entropy of squeezed-coherent states

As the squeezed-coherent states are labeled by 3-tuple, the momentum, the coordinate and the squeeze parameter, a phase space probability distribution function associated with a generic wavefunction can be introduced. We verify that this probability function is the solution of a partial differential equation, the pseudo-diffusion equation, hence permitting to extend the concept of information entropy, as introduced by Wehrl for the coherent states, to the squeezed-coherent states. It is shown that the entropy functionals can be used as a measure of quantum correlations between the phase space variables, for a given wavefunction; we also present several properties and discuss the use of the entropic inequality as a concept that complements the Heisenberg uncertainty relationship.

Non-classical properties of even circular states

Here we investigate some general properties of the so-called even circular state (ECS) produced in a cavity, consisting of a superposition of N coherent states |αk� (k = 1, 2 ,...,N) with the same |αk|. Several special states may emerge from this kind of superposition: in particular, when 1 � (e|αk| 2 /N) N � 4 N a Fock state with N quanta is produced, whereas when (e|αk| 2 /N) N � 1 one gets a vacuum state. We analyse the atomic scattering the ECSs produce when two-level atoms go through the cavity and also the non-classical depth of these states.