定义了与热浴相互作用的压缩系统的熵S和不确定性函数，并利用影响泛函理论中约化密度矩阵的演化过程研究了它们随时间的变化.作为例子，我们计算了暴胀宇宙中两个精确可解压缩系统：一个倒谐振子和一个标量场模的熵。对于弱耦合到浴的反向振荡器，在高温和低温下，$S\to r $，其中r是挤压参数。在高温下的de Sitter情况下，$S\to (1-c)r$，其中$c = γ_0/H$，$γ_0$是与浴的耦合，H是哈勃常数。这三种情况证实了以前的结果，这些结果是基于计算熵的更特别的规定。但在低温下，de Sitter熵$S\to (1/2-c)r$明显不同。从更严格的方法获得的这一结果表明，通常被传统方法忽略的因素，即，环境的性质以及系统与环境之间的耦合强度是重要的。
英文摘要：
We define the entropy S and uncertainty function of a squeezed system interacting with a thermal bath, and study how they change in time by following the evolution of the reduced density matrix in the influence functional formalism. As examples, we calculate the entropy of two exactly solvable squeezed systems: an inverted harmonic oscillator and a scalar field mode evolving in an inflationary universe. For the inverted oscillator with weak coupling to the bath, at both high and low temperatures, $S\to r $, where r is the squeeze parameter. In the de Sitter case, at high temperatures, $S\to (1-c)r$ where $c = γ_0/H$, $γ_0$ being the coupling to the bath and H the Hubble constant. These three cases confirm previous results based on more ad hoc prescriptions for calculating entropy. But at low temperatures, the de Sitter entropy $S\to (1/2-c)r$ is noticeably different. This result, obtained from a more rigorous approach, shows that factors usually ignored by the conventional approaches, i.e., the nature of the environment and the coupling strength betwen the system and the environment, are important.
We derive a formula for the entanglement entropy of squeezed states on a lattice in terms of the complex structure J. The analysis involves the identification of squeezed states with group-theoretical coherent states of the symplectic group and the relation between the coset Sp(2N,R)/Isot(J_0) and the space of complex structures. We present two applications of the new formula: (i) we derive the area law for the ground state of a scalar field on a generic lattice in the limit of small speed of sound, (ii) we compute the rate of growth of the entanglement entropy in the presence of an instabilit…
Minimum uncertainty states and squeezed states from the sum uncertainty relation
Indian Institute of Science Education and Research, Pune（印度科学教育研究院，浦那）；Department of Physics, Institute of Science, BHU（贝拿勒斯印度教大学理学院物理系）；Indian Institute of Science Education and Research, Berhampur（印度科学教育研究院，贝尔汉普尔）
In a recent work, Kryuchkov, Suslov and Vega-Guzman [20013 J. Phys. B: At. Mol. Opt. Phys. 46 104007] described a multi-parameter family of minimum-uncertainty states satisfying the time-dependent Schrodinger equation for the harmonic oscillator. We show how a different parametrization and a proper geometrical insight reduces a complicated set of equations to two simple circular motions.
We analyze the entropy production and the maximal extractable work from a squeezed thermal reservoir. The nonequilibrium quantum nature of the reservoir induces an entropy transfer with a coherent contribution while modifying its thermal part, allowing work extraction from a single reservoir, as well as great improvements in power and efficiency for quantum heat engines. Introducing a modified quantum Otto cycle, our approach fully characterizes operational regimes forbidden in the standard case, such as refrigeration and work extraction at the same time, accompanied by efficiencies equal to u…
We study the entropy dynamics of a dephasing model, where a two-level system (TLS) is coupled with a squeezed thermal bath via non-demolition interaction. This model is exactly solvable, and the time dependent states of both the TLS and its bath can be obtained exactly. Based on these states, we calculate the entropy dynamics of both the TLS and the bath, and find that the dephasing rate of the system relies on the squeezing phase of the bath. In zero temperature and high temperature limits, we prove that both the system and bath entropy increases monotonically. Moreover, we find that the deph…
Density contrasts in the universe are governed by scalar cosmological perturbations which, when expressed in terms of gauge-invariant variables, contain a classical component from scalar metric perturbations and a quantum component from inflaton field fluctuations. It has long been known that the effect of cosmological expansion on a quantum field amounts to squeezing. Thus the entropy of cosmological perturbations can be studied by treating them in the framework of squeezed quantum systems. Entropy of a free quantum field is a seemingly simple yet subtle issue. In this paper, as different fro…
The squeezed states are states of minimum uncertainty, but unlike the coherent states, in which the uncertainty in the position and the momentum are the same, these allow to reduce the uncertainty, either in the position or in the momentum, while maintaining the principle of uncertainty in its minimum. It seems that this property of the squeezed states would allow you to get the position eigenstates as a limit case of them, doing null the uncertainty in the position and infinite at the momentum. However, there are two equivalent ways to define the squeezed states, which lead to different expre…
We describe a six-parameter family of the minimum-uncertainty squeezed states for the harmonic oscillator in nonrelativistic quantum mechanics. They are derived by the action of corresponding maximal kinematical invariance group on the standard ground state solution. We show that the product of the variances attains the required minimum value 1/4 only at the instances that one variance is a minimum and the other is a maximum, when the squeezing of one of the variances occurs. The generalized coherent states are explicitly constructed and their Wigner function is studied. The overlap coefficien…
We provide an explicit construction for Gazeau-Klauder coherent states related to non-Hermitian Hamiltonians with discrete bounded below and nondegenerate eigenspectrum. The underlying spacetime structure is taken to be of a noncommutative type with associated uncertainty relations implying minimal lengths. The uncertainty relations for the constructed states are shown to be saturated in a Hermitian as well as a non-Hermitian setting for a perturbed harmonic oscillator. The computed value of the Mandel parameter dictates that the coherent wavepackets are assembled according to sub-Poissonian s…
We provide an explicit construction of entangled states in a noncommutative space with nonclassical states, particularly with the squeezed states. Noncommutative systems are found to be more entangled than the usual quantum mechanical systems. The noncommutative parameter provides an additional degree of freedom in the construction by which one can raise the entanglement of the noncommutative systems to fairly higher values beyond the usual systems. Despite of having classical-like behaviour, coherent states in noncommutative space produce little amount of entanglement and therefore they posse…
In parametric systems, squeezed states of radiation can be generated via extra work done by external sources. This eventually increases the entropy of the system despite the fact that squeezing is reversible. We investigate the entropy increase due to squeezing and show that it is quadratic in the squeezing rate and may become important in the repeated operation of tunable oscillators (quantum buses) used to connect qubits in various proposed schemes for quantum computing.
The Morse potential is relatively closed to the harmonic oscillator quantum system. Thus, following the idea used for the latter, we study the possibility of creating entanglement using squeezed coherent states of the Morse potential as an input field of a beam splitter. We measure the entanglement with the linear entropy for two types of such states and we study the dependence with the coherence and squeezing parameters. The new results are linked with observations made on probability densities and uncertainty relations of those states. The dynamical evolution of the linear entropy is also ex…
According to quantum theory the interactions between physical systems are quantized. As a direct consequence, measurement sensitivities are fundamentally limited by quantization noise, or just `quantum noise' in short. Furthermore, Heisenberg's Uncertainty Principle demands measurement back-action for some observables of a system if they are measured repeatedly. In both respects, squeezed states are of high interest since they show a `squeezed' uncertainty, which can be used to improve the sensitivity of measurement devices beyond the usual quantum noise limits including those impacted by quan…
We study the evolution of the hybrid entangled squeezed states of the qubit-oscillator system in the strong coupling domain. Following the adiabatic approximation we obtain the reduced density matrices of the qubit and the oscillator degrees of freedom. The oscillator reduced density matrix is utilized to calculate the quasiprobability distributions such as the Sudarshan-Glauber diagonal P -representation, the Wigner W -distribution, and the nonnegative Husimi Q-function. The negativity associated with the W -distribution acts as a measure of the nonclassicality of the state. The existence of …
The Generalized Uncertainty Principle (GUP) is a modification of Heisenberg's Principle predicted by several theories of Quantum Gravity. It consists of a modified commutator between position and momentum. In this work we compute potentially observable effects that GUP implies for the harmonic oscillator, coherent and squeezed states in Quantum Mechanics. In particular, we rigorously analyze the GUP-perturbed harmonic oscillator Hamiltonian, defining new operators that act as ladder operators on the perturbed states. We use these operators to define the new coherent and squeezed states. We com…
Using the f-deformed oscillator formalism, we introduce two types of squeezed coherent states for a Morse potential system (Morse-like squeezed coherent states) through the following definitions: i) as approximate eigenstates of a linear combination of f-deformed ladder operators and ii) as deformed photon-subtracted coherent states. For the states thus constructed we analyze their statistical properties, their uncertainty relations, and their temporal evolution on phase space.
We study the sensitivity and resolution of phase measurement in a Mach-Zehnder interferometer with two-mode squeezed vacuum (<n> photons on average). We show that super-resolution and sub-Heisenberg sensitivity is obtained with parity detection. In particular, in our setup, dependence of the signal on the phase evolves <n> times faster than in traditional schemes, and uncertainty in the phase estimation is better than 1/<n>.
The Rényi entanglement entropy is calculated exactly for mode-partitioned isolated systems such as the two-mode squeezed state and the multi-mode Silbey-Harris polaron ansatz state. Effective thermodynamic descriptions of the correlated partitions are constructed to present quantum information theory concepts in the language of thermodynamics. Boltzmann weights are obtained from the entanglement spectrum by deriving the exact relationship between an effective temperature and the physical entanglement parameters. The partition function of the resulting effective thermal theory can be obtained d…
Spin-squeezing is a well-established "quantum technology", where well-designed correlations in an ensemble of two-level systems reduce the statistical uncertainty of spectroscopic experiments. This paper reviews some important advances in the field, with emphasis on the author's contributions concerning in particular the fundamental limits imposed by decoherence. Building on the material presented in the first part, new ideas and some promising developments are outlined in the last section.
We consider a finite-time quantum Otto heat engine that consists of two isochoric (thermal-contact) process, where the system is alternatively coupled to a hot squeezed and a cold thermal reservoir, and two unitary driven strokes, where the system is isolated from these two baths and its von Neumann entropy keeps constant. Both quantum inner friction and coherence are generated along the driven stroke and coherence cannot be fully erased after the finite-time hot isochore. Using full counting statistics, we present the probability distribution functions of heat injection and total work per cyc…
With the successes of the Laser Interferometer Gravitational-wave Observatory, we anticipate increased interest in working with squeezed states in the undergraduate and graduate quantum-mechanics classroom. Because squeezed-coherent states are minimum uncertainty states, their wavefunctions in position and momentum space must be Gaussians. But this result is rarely discussed in treatments of squeezed states in quantum textbooks or quantum optics textbooks. In this work, we show three different ways to construct the wavefunction for squeezed-coherent states: (i) a differential equation-based ap…
The hydrodynamic interpretation of quantum mechanics treats a system of particles in an effective manner. In this work, we investigate squeezed coherent states within the hydrodynamic interpretation. The Hamiltonian operator in question is time dependent, n-dimensional and in quadratic order. We start by deriving a phase space Wigner probability distribution and an associated equilibrium entropy for the squeezed coherent states. Then, we decompose the joint phase space distribution into two portions: a marginal position distribution and a momentum distribution that is conditioned on the post-s…
Using squeezed states it is possible to surpass the standard quantum limit of measurement uncertainty by reducing the measurement uncertainty of one property at the expense of another complementary property. Squeezed states were first demonstrated in optical fields and later with ensembles of pseudo spin-1/2 atoms using non-linear atom-light interactions. Recently, collisional interactions in ultracold atomic gases have been used to generate a large degree of quadrature spin squeezing in two-component Bose condensates. For pseudo spin-1/2 systems, the complementary properties are the different…
We report a scheme for generation of high-order vortex states using two-mode photon-number squeezed states, generated via the non-linear process of Spontaneous Parametric Down Conversion. By applying a parametric rotation in quadrature space $(X,Y)$, using a $ϕ$ converter, the Gaussian quadrature profile of the photon-number squeezed input state can be mapped into a superposition of Laguerre-Gauss modes with $N$ vortices or singularities, for an input state containing $N$ photons, thus mapping photon-number fluctuations to interference effects in quadrature space. Our scheme has the potential …
We explore the effects of quantum mechanical squeezing on the nonequilibrium fluctuations of bosonic transport between two squeezed harmonic reservoirs and a two level system. A standard full counting statistics technique based on a quantum master equation is employed. We derive a nonzero thermodynamic affinity under equal temperature setting of the two squeezed reservoirs. The odd cumulants are shown to be independent of squeezing under symmetric conditions, whereas the even cumulants depend nonlinearly on the squeezing parameters. The odd and even cumulants saturate at two different but uniq…
We perform state tomography of an itinerant squeezed state of the microwave field prepared by a Josephson parametric amplifier (JPA). We use a second JPA as a pre-amplifier to improve the quantum efficiency of the field quadrature measurement (QM) from 2% to 36 +/- 4%. Without correcting for the detection inefficiency we observe a minimum quadrature variance which is 69 +/- 8% of the variance of the vacuum. We reconstruct the state's density matrix by a maximum likelihood method and infer that the squeezed state has a minimum variance less than 40% of the vacuum, with uncertainty mostly caused…
We define an EPR-like uncertainty by using the Duan et al.'s inequality which gives a necessary and sufficient condition for the separability of any two-mode Gaussian state. We show that for a given amount of entanglement, the uncertainty is minimized by pure two-mode squeezed states. Using this fact, we write the optimal pure-state decomposition and derive a compact form for the entanglement of formation of two-mode Gaussian states. For illustration purposes, we consider symmetric and squeezed thermal states as special cases and evaluate their entanglement of formation explicitly. For the sym…
Many non-classicality indicators are used to measure quantum effects of different systems. Kenfack's and Sadeghi's non-classicality indicators are introduced in terms of the amount of Wigner function's negativities and interferences in phase space quantum mechanics, respectively. They are, effectively, applied for some real distribution functions. In this paper, we compared these non-classicality indicators for the entangled state of photonic number states in the Wigner, Husimi and Rivier representations. It is shown that for a two-level entangled state, Sadeghi's indicator has more benefits w…
