Squeezed Number States Research Articles

Nonclassicality of squeezed number states

Nonclassicality of squeezed number states

Stochastic approach to evolution of a quantum system interacting with environment in squeezed number state

We determine filtering and master equations for a quantum system interacting with wave packet of light in a continuous-mode squeezed number state. We formulate the problem of conditional evolution of a quantum system making use of model of repeated interactions and measurements. In this approach, the quantum system undergoes a sequence of interactions with an environment defined by a chain of harmonic oscillators. We assume that the environment is prepared in an entangled state being a discrete analogue of a continuous-mode squeezed number state. We present a derivation of a discrete stochastic dynamics that depends on the results of measurement performed on the field after its interaction with the system. In this paper, we consider a photon counting measurement scheme. By taking a continuous time limit, we finally obtain differential stochastic equations for the system. Analytical formulae for quantum trajectories and exclusive probability densities that allow to fully characterize the statistics of photons in the output field are given.

Squeezed Number State Representation of the Inflaton and Particle Production in the FRW Universe

In this study, we use a single-mode squeezed thermal vacuum state formalism and examine the nature of a massive homogeneous scalar field, minimally coupled to the gravity in the framework of semiclassical gravity in the Bianchi type-I universe. We have obtained an estimate leading solution to the semiclassical Einstein equation for the Bianchi type-I universe shows, each scale factor in its respective direction obeys t^(2/3) power-law expansion. The mechanism of the nonclassical thermal cosmological particle production is also analyzed in the Bianchi type-I universe.

Sub-shot-noise-limited phase estimation via single-mode inputs.

In 1981, Caves pointed out that the phase sensitivity of a Mach-Zehnder interferometer with single-mode inputs is bounded by the shot-noise limit. The quantum Fisher information analysis shows that this statement holds true for the scenario where two antisymmetric phase shifts occur in two arms, but it is invalid for the scenario where an unknown phase is embedded in one of two arms. In this paper, we focus on the phase sensitivity directed against the latter scenario. The optimal single-mode input is discussed by analyzing common states, including displaced squeezed states, displaced number states, squeezed number states, Schrödinger cat states and completely mixed states. We find that the best choice is a squeezed vacuum state and show the specific measurement scheme which is capable of saturating the corresponding phase sensitivity limit. In addition, we study the effects of several realistic factors-anti-squeezing noise, photon loss and dark counts-on the phase sensitivity. Our results suggest that sub-shot-noise-limited phase sensitivity is attainable with low noise or loss, which paves the way for practical metrology.

Modeling displaced squeezed number states in waveguide arrays

We present an exact analytical solution for a one-dimensional zigzag waveguide array with first and second neighbor interactions. It is found that this waveguide system acts as a classical analog to the displaced squeezed number states. The exact solution was compared directly with the numerical solution showing a perfect agreement. The implication of a linear index of refraction changing as a function of the site number is also studied; in this case, we show that the first neighbor interaction strongly influences the periodicity of Bloch oscillations.

Quantum interferometry via a coherent state mixed with a squeezed number state**Project supported by the National Natural Science Foundation of China (Grant No. 11404040) and the Qing Lan Project of the Higher Educations of Jiangsu Province of China.

We theoretically investigate the phase sensitivity with parity measurement on a Mach–Zehnder interferometer with a coherent state combined with a squeezed number state. Within a constraint on the total mean photon number, we find, via parity measurement, that the mixing of a coherent state and squeezed number state can give better phase sensitivity than mixing a coherent state and squeezed vacuum state when the phase shift deviates from the optimal phase φ =0. In addition, we show that the classical Fisher information for parity measurement saturates the quantum Fisher information when the phase shift approaches to zero. Thus, the quantum Cramér–Rao bound can be reached via the parity measurement in the case of φ =0.

Non-Gaussian swapping of entangled resources

We investigate the continuous-variable entanglement swapping protocol in a non-Gaussian setting, with non- Gaussian states employed either as entangled inputs and/or as swapping resources. The quality of the swapping protocol is assessed in terms of the teleportation fidelity achievable when using the swapped states as shared entangled resources in a teleportation protocol. We thus introduce a two-step cascaded quantum communication scheme that includes a swapping protocol followed by a teleportation protocol. The swapping protocol is fed by a general class of tunable non-Gaussian states, the squeezed Bell states, which, by means of controllable free parameters, allows for a continuous morphing from Gaussian twin beams up to maximally non-Gaussian squeezed number states. In the realistic instance, taking into account the effects of losses and imperfections, we show that as the input two-mode squeezing increases, optimized non-Gaussian swapping resources allow for a monotonically increasing enhancement of the fidelity compared to the corresponding Gaussian setting. This result implies that the use of non-Gaussian resources is necessary to guarantee the success of continuous-variable entanglement swapping in the presence of decoherence.

Non-Gaussianity dynamics of two-mode squeezed number states subject to different types of noise based on cumulant theory**Project supported by the Natural Science Foundation of Hunan Province, China (Grant No. 2017JJ2214) and the Key Project Foundation of the Education Department of Hunan Province, China (Grant No. 14A114.

We provide a measure to characterize the non-Gaussianity of phase-space function of bosonic quantum states based on the cumulant theory. We study the non-Gaussianity dynamics of two-mode squeezed number states by analyzing the phase-averaged kurtosis for two different models of decoherence: amplitude damping model and phase damping model. For the amplitude damping model, the non-Gaussianity is very fragile and completely vanishes at a finite time. For the phase damping model, such states exhibit rich non-Gaussian characters. In particular, we obtain a transition time that such states can transform from sub-Gaussianity into super-Gaussianity during the evolution. Finally, we compare our measure with the existing measures of non-Gaussianity under the independent dephasing environment.

Quantum state engineering via coherent-state superpositions in traveling optical fields

We propose two experimental schemes for producing coherent-state superpositions which approximate different nonclassical states conditionally in traveling optical fields. Although these setups are constructed of a small number of linear optical elements and homodyne measurements, they can be used to generate various photon number superpositions in which the number of constituent states can be higher than the number of measurements in the schemes. We determine numerically the parameters to achieve maximal fidelity of the preparation for a large variety of nonclassical states, such as amplitude squeezed states, squeezed number states, binomial states and various photon number superpositions. The proposed setups can generate these states with high fidelities and with success probabilities that can be promising for practical applications.

Squeezing effects applied in nonclassical superposition states for quantum nanoelectronic circuits

Quantum characteristics of a driven series RLC nanoelectronic circuit whose capacitance varies with time are studied using an invariant operator method together with a unitary transformation approach. In particular, squeezing effects and nonclassical properties of a superposition state composed of two displaced squeezed number states of equal amplitude, but 180° out of phase, are investigated in detail. We applied our developments to a solvable specific case obtained from a suitable choice of time-dependent parameters. The pattern of mechanical oscillation of the amount of charges stored in the capacitor, which are initially displaced, has exhibited more or less distortion due to the influence of the time-varying parameters of the system. We have analyzed squeezing effects of the system from diverse different angles and such effects are illustrated for better understanding. It has been confirmed that the degree of squeezing is not constant, but varies with time depending on specific situations. We have found that quantum interference occurs whenever the two components of the superposition meet together during the time evolution of the probability density. This outcome signifies the appearance of nonclassical features of the system. Nonclassicality of dynamical systems can be a potential resource necessary for realizing quantum information technique. Indeed, such nonclassical features of superposition states are expected to play a key role in upcoming information science which has attracted renewed attention recently.

Universality in uncertainty relations for a quantum particle

A general theory of preparational uncertainty relations for a quantum particle in one spatial dimension is developed. We derive conditions which determine whether a given smooth function of the particle’s variances and its covariance is bounded from below. Whenever a global minimum exists, an uncertainty relation has been obtained. The squeezed number states of a harmonic oscillator are found to be universal: no other pure or mixed states will saturate any such relation. Geometrically, we identify a convex uncertainty region in the space of second moments which is bounded by the inequality derived by Robertson and Schrödinger. Our approach provides a unified perspective on existing uncertainty relations for a single continuous variable, and it leads to new inequalities for second moments which can be checked experimentally.

A generalized Jaynes-Cummings model: The relativistic parametric amplifier and a single trapped ion

We introduce a generalization of the Jaynes-Cummings model and study some of its properties. We obtain the energy spectrum and eigenfunctions of this model by using the tilting transformation and the squeezed number states of the one-dimensional harmonic oscillator. As physical applications, we connect this new model to two important and novelty problems: the relativistic parametric amplifier and the quantum simulation of a single trapped ion.

Two mode mechanical non-Gaussian squeezed number state in a two-membrane optomechanical system

We consider an optomechanical system with two membranes when a bichromatic laser field with red-sideband and blue-sideband frequencies is applied in the single photon strong coupling regime. It is shown that using the mode selecting method and under the Lamb–Dicke approximation, motion of membranes can evolve to single or two mode squeezed number states. By considering the environmental effect, a Wigner function is plotted for understanding the conditions that lead to the generation of non-Gaussian states. The results show that, in this system, initial states of membranes are important to generation of non-Gaussian mechanical squeezed number states.

Time evolution of a squeezed chaotic field in an amplitude damping channel when used as a generating field for a squeezed number state**Project supported by the National Natural Science Foundation of China (Grant No. 10574647), the Natural Science Foundation of Shandong Province, China (Grant No. Y2008A16), and the University Experimental Technology Foundation of Shandong Province of China (Grant No. S04W138).

We investigate how an optical squeezed chaotic field (SCF) evolves in an amplitude dissipation channel. We have used the integration within ordered product of operators technique to derive its evolution law. We also show that the density operator of SCF can be viewed as a generating field of the squeezed number state.

Tunable non-Gaussian resources for continuous-variable quantum technologies

We introduce and discuss a set of tunable two-mode states of continuous-variable systems, as well as an efficient scheme for their experimental generation. This novel class of tunable entangled resources is defined by a general ansatz depending on two experimentally adjustable parameters. It is very ample and flexible as it encompasses Gaussian as well as non-Gaussian states. The latter include, among others, known states such as squeezed number states and de-Gaussified photon-added and photon-subtracted squeezed states, the latter being the most efficient non-Gaussian resources currently available in the laboratory. Moreover, it contains the classes of squeezed Bell states and even more general non-Gaussian resources that can be optimized according to the specific quantum technological task that needs to be realized. The proposed experimental scheme exploits linear optical operations and photon detections performed on a pair of uncorrelated two--mode Gaussian squeezed states. The desired non-Gaussian state is then realized via ancillary squeezing and conditioning. Two independent, freely tunable experimental parameters can be exploited to generate different states and to optimize the performance in implementing a given quantum protocol. As a concrete instance, we analyze in detail the performance of different states considered as resources for the realization of quantum teleportation in realistic conditions. For the fidelity of teleportation of an unknown coherent state, we show that the resources associated to the optimized parameters outperform, in a significant range of experimental values, both Gaussian twin beams and photon-subtracted squeezed states.

Displacing, squeezing, and time evolution of quantum states for nanoelectronic circuits

The time behavior of DSN (displaced squeezed number state) for a two-dimensional electronic circuit composed of nanoscale elements is investigated using unitary transformation approach. The original Hamiltonian of the system is somewhat complicated. However, through unitary transformation, the Hamiltonian became very simple enough that we can easily treat it. By executing inverse transformation for the wave function obtained in the transformed system, we derived the exact wave function associated to the DSN in the original system. The time evolution of the DSN is described in detail, and its corresponding probability density is illustrated. We confirmed that the probability density oscillates with time like that of a classical state. There are two factors that drive the probability density to oscillate: One is the initial amplitude of complementary functions, and the other is the external power source. The oscillation associated with the initial amplitude gradually disappears with time due to the dissipation raised by resistances of the system. These analyses exactly coincide with those obtained from classical state. The characteristics of quantum fluctuations and uncertainty relations for charges and currents are also addressed.

Squeezed number state and squeezed thermal state: decoherence analysis and nonclassical properties in the laser process

We investigate how the squeezed thermal state (STS) and squeezed number state (SNS) evolve undergoing decoherence in the laser process. Remarkably, the initial SNS, an example of a pure state, evolves into a mixed state, which turns out to be a Laguerre polynomial of combination of creation and annihilation operators within normal ordering; however, the STS, a mixed state, still keeps squeezed and thermal. At long times, these fields lose their nonclassical nature and decay to a highly classical thermal field. The normally ordered density operators of such states in the laser channel lead to deriving the analytical time-evolution expressions of the Wigner functions (WFs). Their nonclassicality is investigated in reference to the time-evolution WFs, which indicates that both of the WFs decay to the same thermal WF as a result of decoherence when the decay time t or the cavity loss κ of the laser is large enough.

Photon-number distribution of two-mode squeezed thermal states by entangled state representation

Using the entangled state representation, we convert a two-mode squeezed number state to a Hermite polynomial excited squeezed vacuum state. We first analytically derive the photon number distribution of the two-mode squeezed thermal states. It is found that it is a Jacobi polynomial; a remarkable result. This result can be directly applied to obtaining the photon number distribution of non-Gaussian states generated by subtracting from (adding to) two-mode squeezed thermal states.

Nonclassicality and decoherence of photon-added squeezed thermal state in thermal environment

Theoretical analysis is given of nonclassicality and decoherence of the field states generated by adding any number of photons to the squeezed thermal state (STS). Based on the fact that the squeezed number state can be considered as a single-variable Hermite polynomial excited state, the compact expression of the normalization factor is derived, a Legendre polynomial. The nonclassicality is investigated by exploring the sub-Poissonian and negative Wigner function (WF). The results show that the WF of single photon-added STS (PASTS) always has negative values at the phase space center. The decoherence effect on PASTS is examined by the analytical expression of WF. It is found that a longer threshold value of decay time is included in single PASTS than in single-photon subtraction STS.

New Hermite-polynomial-operator identities and their application in quantum squeezing

By introducing the Hermite-polynomial-operator Hn(X), where X is the coordinate operator (or the quadrature operator in quantum optics theory), and combining the technique of integration within an ordered product of operators, we derive some new operator identities about quantum squeezing, which are useful for studying the squeezed number state.