测不准关系和最小不确定态

利用力学量算符的厄密性和希尔伯特状态矢量模的非负性，重新给出了关于测不准关系的数 学证明.简单的证明过程不仅揭示了测不准关系某些经常被忽视的特征，而且还可以直接给 出最小不确定态的充要条件.在此基础上，我们提出了对易子为非零常数的任意一对力学量 的最小不确定态问题，并且采用玻色型产生和湮没算符给出了它们的压缩态的明显表达式. 关键词： 海森堡测不准原理 最小不确定态 压缩态     

Coherent states and uncertainty relations

The most general form of the minimality-preserving Hamiltonians is found and the uncertainty products ΔqΔp are calculated for some quadratic quantum systems     

Multimode uncertainty relations and separability of continuous variable states

A multimode uncertainty relation (generalizing the Robertson-Schr?dinger relation) is derived as a necessary constraint on the second moments of n pairs of canonical operators. In turn, necessary conditions for the separability of multimode continuous variable states under (m+n)-mode bipartitions are derived from the uncertainty relation. These conditions are proven to be necessary and sufficient for (1+n)-mode Gaussian states and for (m+n)-mode bisymmetric Gaussian states.     

混合态的不确定关系与压缩效应

在关于混合态的海森堡不确定关系的基础上，研究了纯态和混合态的最小不确定性和压缩效应.虽然最小不确定态必定是纯态，但在某些并非最小不确定态的纯态或混合态中，依然可 以实现力学量不确定度的压缩.还给出了普通统计学的不确定关系，它们不涉及量子相干性却与量子力学的海森堡不确定关系具有相似的数学结构. 关键词： 混合态 最小不确定性 压缩效应     

Minimization of Heisenberg uncertainty relations with quasi-classical path-coherent states

The problem of minimizing Heisenberg uncertainty relations with path-coherent states is investigated. Minimizing conditions on the Hamiltonian and the path are derived. It is shown that path-coherent states minimize the uncertainty relations only when a special Cauchy problem is posed for the Schrödinger equation.Tomsk Polytechnic Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 49–53, October, 1994.     

Squeezed states and quantum chaos

We examine the dynamics of a wave packet that initially corresponds to a coherent state in the model of a quantum rotator excited by a periodic sequence of kicks. This model is the main model of quantum chaos and allows for a transition from regular behavior to chaotic in the classical limit. By doing a numerical experiment we study the generation of squeezed states in quasiclassical conditions and in a time interval when quantum-classical correspondence is well-defined. We find that the degree of squeezing depends on the degree of local instability in the system and increases with the Chirikov classical stochasticity parameter. We also discuss the dependence of the degree of squeezing on the initial width of the packet, the problem of stability and observability of squeezed states in the transition to quantum chaos, and the dynamics of disintegration of wave packets in quantum chaos. Zh. éksp. Teor. Fiz. 113, 111–127 (January 1998)     

Extension of the uncertainty relations to the mixed states

In this paper we have extended the uncertainty relations to mixed states for some special cases of quantum oscillators, relations which are calculated for this kind of states by means of the earlier established virial and Hellmann-Feynman theorems. The lower bound of these uncertainty relations is estimated for a few exactly solvable potentials, too.     

Squeezed states and Helmholtz spectra

The “classical interpretation” of the wave function ψ(x) reveals an interesting operational aspect of the Helmholtz spectra. It is shown that the traditional Sturm-Liouville problem contains the simplest key to predict the squeezing effect for charged particle states.     

Squeezed states of electromagnetic field

Lebedev Physics Institute, Academy of Sciences of the USSR, Moscow. Translated from Preprint No. 232 of the Lebedev Physics Institute, Academy of Sciences of the USSR, Moscow, 1988.     

Squeezed states in two-phonon devices

New variables called two-mode quadrature-phase amplitudes are used in order to analyze squeezing in two-phonon devices. The Fokker-Planck equation obtained for two-mode quasiprobability distributions presents a diffusion matrix independent of the non-linear coupling and with non-negative eigenvalues. Assuming that the initial state is a two-mode coherent state, the uncertainties in the two-mode quadrature-phase amplitudes have been obtained, and the dependence of squeezing in the non-linear coupling, the pumping phase and the modulation frequency has been studied.     

Squeezed states in direct and coherent detection

With reference to squeezed states, we analyse the noise performances of the basic detection schemes in fibre telecommunications and interferometry. We have developed a semiclassical approach that allows us to translate the quantum mechanical formalism into synthetic expressions and rules only involving directly measurable physical quantities. As a result, we find that in the telecommunication field squeezed states do not allow significant improvements in system performances: in practice, the signal-to-noise ratio cannot be increased in standard detection schemes, both direct and coherent, by squeezing either the signal (because of propagation losses) or by the local oscillator. On the contrary, a significant improvement can be obtained in interferometric measurements if squeezed vacuum is coupled into the normally unused beamsplitter port.     

New uncertainty relations for tomographic entropy: application to squeezed states and solitons

Using the tomographic probability distribution (symplectic tomogram) describing the quantum state (instead of the wave function or density matrix) and properties of recently introduced tomographic entropy associated with the probability distribution, the new uncertainty relation for the tomographic entropy is obtained. Examples of the entropic uncertainty relation for squeezed states and solitons of the Bose-Einstein condensate are considered.     

Phase Transitions of Thermal Squeezed states

By using thermo field dynamics, we studied the phase transitions of thermal squeezed states. Two critical points are obtained in certain cases. They correspond to the transitions from thermal squeezed state to pure squeezed state and to thermal state respectively.     

Squeezed states in the Jaynes-Cummings model

It is shown that squeezed states of the electromagnetic field are obtained in the Jaynes-Cummings model of an atomic transition coupled to a single mode, quantized electromagnetic field.     

Generalized uncertainty relations

The standard uncertainty relations (UR) in quantum mechanics are typically used for unbounded operators (like the canonical pair). This implies the need for the control of the domain problems. On the other hand, the use of (possibly bounded) functions of basic observables usually leads to more complex and less readily interpretable relations. In addition, UR may turn trivial for certain states if the commutator of observables is not proportional to a positive operator. In this letter we consider a generalization of standard UR resulting from the use of two, instead of one, vector states. The possibility to link these states to each other in various ways adds additional flexibility to UR, which may compensate some of the above-mentioned drawbacks. We discuss applications of the general scheme, leading not only to technical improvements, but also to interesting new insight.     

Higher-order squeezing of quantum electromagnetic fields and higher-order uncertainty relations in two-mode squeezed states

It is found that the two-mode output quantum electromagnetic field in two-mode squeezed states exhibits higherorder squeezing to all even orders, and the degree of higher-order squeezing is greater than that of the second-order.The higher-order squeezed parameter and squeezed limit due to the modulation frequency are investigated. The smaller the modulation frequency is, the stronger the degree of higher-order squeezing becomes. Furthermore, the higher-order uncertainty relations in two-mode squeezed states are presented for the first time. The product of higher-order noise moments is related to even order number N and the squeeze factor r.     

Entropic uncertainty relations

New entropic uncertainty relations for angle-angular momentum and position-momentum, derived recently by Partovi, are related to older relations of a similar type, which were proved by Bialynicki-Birula and Mycielski. Significantly improved lower bounds are obtained in both cases.