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arXiv:1704.05840

Squeezed Fourier Meets Toeplitz Algebras

Bogdan Mielnik, Jesús Fuentes

Published Apr 19, 2017• 0 citations
arXivHTMLPDF

Abstract

We look for new steps on the dynamical operations that may squeeze simultaneously some families of quantum states, independently of their initial shape, induced by softly acting external fields which might produce the squeezing of the canonical observables $q,p$ of charged particles. Also, we present some exactly solvable cases of the problem which appear in the symmetric evolution intervals permitting to find explicitly the time dependence of the external fields needed to generate the required evolution operators. Curiously, our results are interrelated with a simple, non--trivial, anti--commuting algebra of Toeplitz which describes the problem more easily than the frequently used Ermakov--Milne invariants.

DOI
10.48550/arxiv.1704.05840

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We look for new steps on the dynamical operations that may squeeze simultaneously some families of quantum states, independently of their initial shape, induced by softly acting external fields which might produce the squeezing of the canonical observables q,pq,pq,p of charged particles. Also, we present some exactly solvable cases of the problem which appear in the symmetric evolution intervals permitting to find explicitly the time dependence of the external fields needed to generate the required evolution operators. Curiously, our results are interrelated with a simple, non--trivial, anti--commuting algebra of Toeplitz which describes the problem more easily than the frequently used Ermakov--Milne invariants.
In the studies of the squeezing it is customary to focus more attention on the particular squeezed states and their evolution than on the dynamical operations that could squeeze simultaneously some wider families of quantum states, independently of their initial shape. We look for new steps in this direction, carried out by softly acting external fields which might produce the squeezing of the canonical observables q,pq,pq,p of charged particles. The works on these problems collect so many valuable results that the question is whether something more is indeed something more in our knowledge. Yet we decided to present some exactly solvable cases of the problem which appear in the symmetric evolution intervals permitting to find explicitly the time dependence of the external fields needed to generate the required evolution operators. Curiously, our results are interrelated with a simple anti-commuting algebra of Toeplitz which describes the problem more easily than the frequently used Ermakov--Milne invariants. Some trending topics, as well as some fundamental problems in quantum theory, might also be involved.
We explore a class of quantum control operations based on a wide family of harmonic magnetic fields that vary softly in time. Depending on the magnetic field amplitudes taking part, these control operations can produce either squeezing or loop (orbit) effects, and even parametric resonances, on the canonical variables. For these purposes we focus our attention on the evolution of observables whose dynamical picture is ascribed to a quadratic Hamiltonian that depends explicitly on time. In the first part of this work we survey such operations in terms of biharmonic magnetic fields. The dynamical analysis is simplified using a stability diagram in the amplitude space, where the points of each region will characterise a specific control operation. We discuss how the evolution loop effects are formed by fuzzy (non-commutative) trajectories that can be closed or open, in the latter case, even hiding some features that can be used to manipulate the operational time. In the second part, we generalise the case of biharmonic fields and translate the discussion to the case of polyharmonic fields. Using elementary properties of the Toeplitz matrices, we can derive exact solutions of the problem in a symmetric evolution interval, leading to the temporal profile of those magnetic fields suitable to achieve specific control operations. Some of the resulting fuzzy orbits can be destroyed by the influence of external forces, while others simply remain stable.
Quantum squeezing, an intriguing phenomenon that amplifies the uncertainty of one variable while diminishing that of its conjugate, may be studied as a time-dependent process, with exact solutions frequently derived from frameworks grounded in adiabatic invariants. Remarkably, we reveal that exact solutions can be ascertained in the presence of time-variant elastic forces, eschewing dependence on invariants or frozen eigenstate formalism. Delving into these solutions as an inverse problem unveils their direct connection to the design of elastic fields, responsible for inducing squeezing transformations onto canonical variables. Of particular note is that the dynamic transformations under investigation belong to a class of gentle quantum operations, distinguished by their delicate manipulation of particles, thereby circumventing the abrupt energy surges commonplace in conventional control protocols.
In this paper, we consider quantum multidimensional problems solvable by using the second quantization method. A multidimensional generalization of the Bogolyubov factorization formula, which is an important particular case of the Campbell-Baker-Hausdorff formula, is established. The inner product of multidimensional squeezed states is calculated explicitly; this relationship justifies a general construction of orthonormal systems generated by linear combinations of squeezed states. A correctly defined path integral representation is derived for solutions of the Cauchy problem for the Schrödinger equation describing the dynamics of a charged particle in the superposition of orthogonal constant ( E,H )-fields and a periodic electric field. We show that the evolution of squeezed states runs over compact one-dimensional matrix-valued orbits of squeezed components of the solution, and the evolution of coherent shifts is a random Markov jump process which depends on the periodic component of the potential.
We study the time evolution of a class of exactly solvable time-dependent quantum systems with a time-dependent Hamiltonian given by a linear combination of SU(1,1) and SU(2) generators with the help of the invariant Hermitian operator. The exact common solutions of the Schrödinger equations for both the SU(1,1) and SU(2) systems are obtained in terms of eigenstates of the invariant operator. The adiabatic and non-adiabatic Berry phases are calculated with the exact solutions. Moreover, we derive an explicit time-evolution operator which is used to investigate the time-dependent two-photon squeezing states and SU(2) squeezing states. The squeezing properties of the time-dependent SU(1,1) coherent states are also discussed.
In this Paper we present an approach to Quantum Mechanical Canonical Transformations. Our main result is that Time Dependent Quantum Canonical Transformations can always be cast in the form of Squeezing Operators. We revise the main properties of these operators in regard to its Lie group properties, how two of them can be combined to yield another operator of the same class and how can also be decomposed and fragmented. In the second part of the paper we show how this procedure works extremely well for the Time Dependent Quantum Harmonic Oscillator. The issue of the systematic construction of Quantum Canonical Transformations is also discussed along the lines of Dirac, Wigner and Schwinger ideas and to the more recent work by Lee. The main conclusion is that the Classical Phase Space Transformation can be maintained in the operator formalism but the construction of the Quantum Canonical Transformation is not clearly related to the Classical Generating Function of a Classical Canonical Transformation. We propose the road of Squeezing Operators rather than the old one attached to Quantum Operators constructed under the guideline of the exponential of the Classical Generating Function.