Characterization, properties and dynamics of squeezed states: single mode and multimode systems

Abstract

Squeezed states of the quantized electromagnetic field, as well as the squeezing phenomenon in the more general quantum mechanical context, have been topics of enormous interest recently. In addition to a very large number of theoretical studies, many experimental efforts at the production and detection of squeezed radiation have also been made. With the advent of the laser, photon shot noise has become a dominating limiting factor in more and more applications. The reason for this is that the field fluctuations in an ideal laser beam are a direct consequence of the quantum nature of light. The squeezed states of light were first studied to explore their properties as generalized minimum uncertainty states, which held significant potential for improving free-space optical communications and applications in ultrasensitive detection systems. Subsequently, squeezed states of light have been produced by four-wave mixing in atomic vapors and optical fibers, and by employing optical parametric oscillators. Sub-Poissonian light has been produced in resonance fluorescence experiments and by parametric down-conversion. In this thesis, we study squeezed states from several points of view: ways to define squeezing and set up criteria with suitable invariance properties; methods to geometrically depict the squeezing property when it is present, and to trace its dynamical evolution; effects of coupling to a dissipative environment in various regimes; the effect of dissipation on superposition at the quantum mechanical level when applied to squeezed states; characteristic signatures of squeezing such as the specific features of photon number distributions; new and global ways to define squeezing for multimode systems based on physically reasonable invariance requirements; and a study of the delocalization-localization transition in a two-level system coupled to a phonon bath, using a displaced squeezed state variational wave function for the phonons. Throughout, we base our discussion of squeezing, for single-mode as well as multimode systems, on the properties of variance matrices or second-order moments of any given quantum mechanical state. We also supplement analytic work by extensive numerical studies wherever necessary. In Chapter III, we study a state that is not a pure Gaussian but a linear superposition of two Gaussian states at the wavefunction level. The Wigner distribution of such a state consists of the sum of three terms, two of which correspond to the Wigner functions of each state independently and the third to the interference term. We have considered a particular superposition of two squeezed coherent states that can be generated by evolving an initial squeezed vacuum state under the influence of an anharmonic oscillator Hamiltonian. The effect of dissipation on such an initial density matrix has been investigated. We have looked at the effect of dissipation on this initial state in the Wigner distribution language. We have also shown the corresponding situation in the Minkowski space picture. We find that with time the interference fringes created by the superposition disappear and that even very small dissipative losses are enough to destroy quantum coherence. The loss of quantum coherence in the density matrix language or in the Wigner distribution language corresponds to the vanishing of the “spatial” components of the Minkowski space vector. We have also looked at the effect of dissipation on the photon number distribution of this superposition of squeezed states. The oscillations in the photon number distribution disappear due to dissipative effects. In Chapter IV, we study the photon number distribution of a squeezed state of a single-mode radiation field. We compare and contrast the U(1)-invariant squeezing criterion with the more restrictive usual or naive criterion, with the help of suggestive geometric representations. The U(1) invariance of the photon number distribution in a squeezed coherent state, with arbitrary complex squeeze and displacement parameters, is explicitly demonstrated analytically. The behavior of the photon number distribution for a representative value of the displacement and various values of the squeeze parameter is numerically investigated. A new kind of giant oscillation riding as an envelope over more rapid oscillations in this distribution is demonstrated. While oscillations per se have been observed earlier in the literature, this overriding oscillation effect is completely new and is revealed only by careful numerical work. Up to this point, we have dealt only with single-mode systems. In Chapter V, we develop a U(n)-invariant squeezing criterion, the natural extension of the U(1)-invariant squeezing criterion for the single-mode case to the multimode case. Physical reasons why such invariance is quite natural have been discussed. The first question that we have studied here is the complete set of necessary and sufficient conditions that must be imposed on a symmetric positive matrix for it to qualify to be the variance matrix of some physically realizable state. Now, given a bona fide variance matrix, we have given the algorithm by which it is possible to decide whether it is squeezed or not. We find it very interesting that the implementation of the U(n)-invariant squeezing criterion also reduces to that of finding out whether the least eigenvalue of the variance matrix is less than that prescribed by the standard shot noise limit. This is a nontrivial result. Finally, we have investigated the implications of this squeezing criterion in the context of one two-mode squeezed state well known in the literature and called the pair coherent state. In the last problem studied in this thesis, we turn from squeezing of radiation to an application of this phenomenon in the context of condensed matter physics. This is taken up in Chapter VI. Here we have looked at the symmetry-breaking transition in a two-level system coupled to a phonon bath. In this model, we have examined a particle tunneling between two equivalent minima. The two-level system is coupled linearly to the phonon coordinate. Both the coherent state and the displaced squeezed state had been proposed as variational wave functions for the ground state of the phonons. In the case of the displaced squeezed state, only the parameter controlling the amount of squeezing was treated as a variational parameter. We propose the displaced squeezed state as a variational wave function for the ground state treating both the parameters—one controlling the amount of displacement and the other controlling the amount of squeezing-as variational parameters. We have investigated the zero-temperature phase diagram for the delocalization-localization transition of the tunneling particle interacting with the phonon bath. We have considered both the cases of (i) ohmic dissipation and (ii) atomic tunneling in solids, for the spectral density of the bath. We have compared our results with known existing approximate treatments. The phase diagram is found to be modified on using the displaced squeezed state variational wave function. We observe that the “knee” found previously in the phase boundary curves of the coherent and squeezed state variational wave functions changes and appears now as a convex-down function instead of as a convex-up function.