Single Coherent State Research Articles

Identical damped harmonic oscillators described by coherent states

Some aspects of quantum damped harmonic oscillator (DHO) obeying a Markovian master equation are considered in the absence of thermal noise. The continuity equation is derived and Bohmian trajectories are constructed. As a solution of the master equation, we take a single coherent state and compute analytically the relative entropy of coherence, [Formula: see text], in the energy, position and momentum bases. Although [Formula: see text] is constant in both the position and the momentum bases, it is a decreasing function of time in the energy basis becoming zero at long times, revealing its role as the preferred basis. Then, quantum coherence is computed for a superposition of two coherent states, a cat state, and also a superposition of two cat states in the energy basis as a function of separation, in the complex plane, between the two superposed states. It is seen that the quantum coherence increases with this separation. Furthermore, quantum coherence of superposition is compared to that of decomposed states in the superposition. Finally, by considering a system of two noninteracting DHOs, the effect of quantum statistics is studied on the coherence of reduced single-particle states, the joint detection probability and the mean square separation of particles. Our computations show that the single-particle coherence for antisymmetric states is always less than that of symmetric ones. Furthermore, boson anti-bunching and fermion bunching is seen in this open system. This behavior of bosons is the matter-wave analogue of photon anti-bunching seen in a modified Hanbury Brown–Twiss (HBT) interferometer.

Partially synchronized states in an ensemble of chemo-mechanical oscillators

Partially synchronized (clustered) states are defined as coexisting coherent (synchronized) and incoherent (unsynchronized) domains in an ensemble of interacting oscillators. We report these clustered states in experiments involving an ensemble of sixteen mercury beating heart (MBH) oscillators. These oscillators interact via resistors and are subjected to two different network schemes: 1) All to all and 2) Nonlocal. For the all to all network, the coupling strengths were inhomogeneously distributed, whereas for the nonlocal network scenario, each oscillator was coupled, with an identical coupling strength, with four of its nearest neighbors in either direction. For both of these network schemes, partially synchronized states results into grouping of these oscillators, wherein some oscillators are synchronized and rest are unsynchronized. For all to all network, the partially synchronized states are observed, for the intermediate inhomogeneities, when subjected to the power law and the ‘U’ shape profiles of coupling strengths. Irrespective of the coupling profile chosen, low inhomogeneities in the coupling strengths leaves all the oscillators in a single coherent state whereas for the high inhomogeneities scenarios oscillators are located in the incoherent domain. In comparison, for the nonlocal network partially synchronized states emerge when the coupling constant is appropriately chosen. The experimental results for both these network scenarios have been analyzed using the redox time series (chemical activity) and the time evolution of the normalized areas for the mercury drop (mechanical activity). The existence of partially synchronized states in the experiments was verified using different diagnostic tools such as time series plot, space–time plot and average frequency.

Quantum Nonadiabatic Cloning of Entangled Coherent States

We propose a systematic approach to the basis set extension for nonadiabatic dynamics of entangled combination of nuclear coherent states (CSs) evolving according to the time-dependent variational principle (TDVP). The TDVP provides a rigorous framework for fully quantum nonadiabatic dynamics of closed systems; however, the quality of results strongly depends on available basis functions. Starting with a single nuclear CS replicated vertically on all electronic states, our approach clones this function when replicas of the CS on different electronic states experience increasingly different forces. Created clones move away from each other (decohere), extending the basis set. To determine a moment for cloning, we introduce generalized forces based on derivatives that maximally contribute to a variation of the total quantum action and thus account for entanglement of all basis functions.

Hybrid entanglement concentration assisted with single coherent state**Project supported by the National Natural Science Foundation of China (Grant Nos. 11474168 and 61401222), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20151502), the Qing Lan Project in Jiangsu Province, China, the Natural Science Foundation of Jiangsu Higher Education Institutions, China (Grant No. 15KJA120002), and the Priority Academic Development Program

Hybrid entangled state (HES) is a new type of entanglement, which combines the advantages of an entangled polarization state and an entangled coherent state. HES is widely discussed in the applications of quantum communication and computation. In this paper, we propose three entanglement concentration protocols (ECPs) for Bell-type HES, W-type HES, and cluster-type HES, respectively. After performing these ECPs, we can obtain the maximally entangled HES with some success probability. All the ECPs exploit the single coherent state to complete the concentration. These protocols are based on the linear optics, which are feasible in future experiments.

Rethinking Approach and Avoidance in Achievement Contexts: The Perspective of Dynamical Systems

An integrative model of approach and avoidance goals in achievement contexts is proposed based on the concepts and principles of nonlinear dynamical systems. These goals are conceptualized as self-organizing systems in which information relevant to competence expectancies and perceptions of benefit and threat for the self is integrated with respect to either a single coherent state ( static integration) or 2 conflicting coherent states ( dynamic integration). Mathematical equations depict the emergence of approach and avoidance attractor dynamics from the proposed integration process. The model specifies the conditions under which states of goal involvement will display patterns of resistance (stability), oscillation between approach and avoidance (instability), or reversal (nonlinear change). As a result, this model of approach and avoidance goals equips previous dynamical models of self-regulation with greater social–cognitive substance. It also provides a parsimonious account of achievement motivation and resolves lingering inconsistencies and uncertainties in the achievement goal literature. Moreover, the dynamical systems approach, on which the model is based, provides a heuristic framework for generating new predictions regarding the ebb and flow of approach and avoidance in achievement contexts.

Phase-space interference of states optically truncated by quantum scissors: Generation of distinct superpositions of qudit coherent states by displacement of vacuum

Conventional Glauber coherent states (CS) can be defined in several equivalent ways, e.g., by displacing the vacuum or, explicitly, by their infinite Poissonian expansion in Fock states. It is well known that these definitions become inequivalent if applied to finite $d$-level systems (qudits). We present a comparative Wigner-function description of the qudit CS defined (i) by the action of the truncated displacement operator on the vacuum and (ii) by the Poissonian expansion in Fock states of the Glauber CS truncated at $(d\ensuremath{-}1)$-photon Fock state. These states can be generated from a classical light by its optical truncation using nonlinear and linear quantum scissors devices, respectively. We show a surprising effect that a macroscopically distinguishable superposition of two qudit CS (according to both definitions) can be generated with high fidelity by displacing the vacuum in the qudit Hilbert space. If the qudit dimension $d$ is even (odd), then the superposition state contains Fock states with only odd (even) photon numbers, which can be referred to as the odd (even) qudit CS or Schr\"odinger's cat state. This phenomenon can be interpreted as an interference of a single CS with its reflection from the highest-energy Fock state of the Hilbert space, as clearly seen via phase-space interference of the Wigner function. We also analyze nonclassical properties of the qudit CS including their photon-number statistics and nonclassical volume of the Wigner function, which is a quantitative parameter of nonclassicality (quantumness) of states. Finally, we study optical tomograms, which can be directly measured in the homodyne detection of the analyzed qudit cat states and enable the complete reconstructions of their Wigner functions.

Measurement- and comparison-based sizes of Schrödinger cat states of light

We extend several measurement-based definitions of effective "cat-size" to coherent state superpositions with branches composed of either single coherent states or tensor products of coherent states. These effective cat-size measures depend on determining the maximal quantum distinguishability of certain states associated with the superposition state: e.g., in one measure, the maximal distinguishability of the branches of the superposition is considered as in quantum binary decision theory; in another measure, the maximal distinguishability of the initial superposition and its image after a one-parameter evolution generated by a local Hermitian operator is of interest. The cat-size scaling with the number of modes and mode intensity (i.e., photon number) is compared to the scaling derived directly from the Wigner function of the superposition and to that estimated experimentally from decoherence. We also apply earlier comparison-based methods for determining macroscopic superposition size that require a reference GHZ state. The case of a hierarchical Schr\"{o}dinger cat state with branches composed of smaller superpositions is also analyzed from a measurement-based perspective.

Mean-field theory of baryonic matter for QCD in the largeNcand heavy quark mass limits

We discuss theoretical issues pertaining to baryonic matter in the combined heavy-quark and large ${N}_{c}$ limits of QCD. Witten's classic argument that baryons and interacting systems of baryons can be described in a mean-field approximation with each of the quarks moving in an average potential due to the remaining quarks is heuristic. It is important to justify this heuristic description for the case of baryonic matter since systems of interacting baryons are intrinsically more complicated than single baryons due to the possibility of hidden color states---states in which the subsystems making up the entire baryon crystal are not color-singlet nucleons but rather colorful states coupled together to make a color-singlet state. In this work, we provide a formal justification of this heuristic prescription. In order to do this, we start by taking the heavy quark limit, thus effectively reducing the problem to a many-body quantum mechanical system. This problem can be formulated in terms of integrals over coherent states, which for this problem are simple Slater determinants. We show that for the many-body problem, the support region for these integrals becomes narrow at large ${N}_{c}$, yielding an energy which is well approximated by a single coherent state---that is a mean-field description. Corrections to the energy are of relative order $1/{N}_{c}$. While hidden color states are present in the exact state of the heavy quark system, they only influence the interaction energy below leading order in $1/{N}_{c}$.

Gaussian-state quantum-illumination receivers for target detection

The signal half of an entangled twin beam, generated using spontaneous parametric downconversion, interrogates a region of space that is suspected of containing a target and has high loss and high (entanglement-breaking) background noise. A joint measurement is performed on the returned light and the idler beam that was retained at the transmitter. An optimal quantum receiver, whose implementation is not yet known, was shown to achieve 6 dB gain in the error-probability exponent relative to that achieved with a single coherent-state (classical) laser transmitter and the optimum receiver. We present two structured optical receivers that achieve up to 3 dB gain in the error exponent over that attained with the classical sensor. These are designs of quantum-optical sensors for target detection, which can be readily implemented in a proof-of-concept experiment, that appreciably outperform the best classical sensor in the low-signal-brightness, high-loss, and high-noise operating regime.

Decoherence and dissipation in a molecular system coupled to an environment: An application of semiclassical hybrid dynamics

Applying the recently developed semiclassical hybrid dynamics [Grossmann, J. Chem. Phys. 125, 014111 (2006)], we study the decay of interference patterns in the reduced density as well as of the purity in a Morse oscillator test system due to the interaction with a finite harmonic bath at zero temperature. In the case that the bath mimics a continuous Ohmic spectral density, in addition to the quantum classical transition induced by the interaction with the environment, we corroborate the existence of a blueshift due to the bath coupling, predicted by Pollak [Phys. Rev. A 33, 4244 (1986)]. Furthermore, the decoherence dynamics of cat states is confirmed to be faster than that of single coherent states and we show that for a resonant bath the dissipation leads to an increase in the decoherence rate as compared to the low frequency bath.

Nonlocal quantum macroscopic superposition in a high-thermal low-purity state.

Quantum state exchange between light and matter is an important ingredient for future quantum information networks as well as other applications. Photons are the fastest and simplest carriers of information for transmission but in general, it is difficult to localize and store photons, so usually one prefers choosing matter as quantum memory elements. Macroscopic superposition and nonlocal quantum interactions have received considerable interest for this purpose over recent years in fields ranging from quantum computers to cryptography, in addition to providing major insights into physical laws. However, these experiments are generally performed either with equipment or under conditions that are unrealistic for practical applications. Ideally, the two can be combined using conventional equipment and conditions to generate a "quantum teleportation"-like state, particularly with a very small amount of purity existing in an overall highly mixed thermal state (relatively low decoherence at high temperatures). In this study we used an experimental design to demonstrate these principles. We performed optical coherence tomography (OCT) using a thermal source at room temperatures of a specifically designed target in the sample arm. Here, position uncertainty (i.e., dispersion) was induced in the reference arm. In the sample arm (target) we placed two glass plates separated by a different medium while altering position uncertainty in the reference arm. This resulted in a chirped signal between the glass plate reflective surfaces in the combined interferogram. The chirping frequency, as measured by the fast Fourier transform (FFT), varies with the medium between the plates, which is a nonclassical phenomenon. These results are statistically significant and occur from a superposition between the glass surface and the medium with increasing position uncertainty, a true quantum-mechanical phenomenon produced by photon pressure from two-photon interference. The differences in chirping frequency with medium disappears when second-order correlations are removed by dual balanced detection, confirming the proposed mechanism. We demonstrated that increasing position uncertainty at one site leads to position uncertainty (quantum position probability amplitude) nonlocally via second-order correlations (two-photon probability amplitude) from a low coherence thermal source (low purity, high local entropy). The implications, first, are that the phenomenon cannot be explained through classical mechanisms but can be explained within the context of quantum mechanics, particularly relevant to the second-order correlations where controversy exists. More specifically, we provide the theoretical framework that these results indicate a nonlocal macroscopic superposition is occurring through a two-photon probability amplitude-induced increase in the target position probability amplitude uncertainty. In addition, as the experiments were performed with a classical source at room temperature, it supports both the quantum-mechanical properties of second-order correlations and that macroscopic superposition is obtainable in a target not in a single coherent state (mixed state). Future work will focus on generalizing the observations outside the current experimental design and creating embodiments that allow practical application of the phenomenon.

Phase variance of squeezed vacuum states

We consider the problem of estimating the phase of squeezed vacuum states within a Bayesian framework. We derive bounds on the average Holevo variance for an arbitrary number $N$ of uncorrelated copies. We find that it scales with the mean photon number $n$, as dictated by the Heisenberg limit, i.e., as ${n}^{\ensuremath{-}2}$, only for $N>4$. For $N\ensuremath{\leqslant}4$ this fundamental scaling breaks down and it becomes ${n}^{\ensuremath{-}N∕2}$. Thus, a single squeezed vacuum state performs worse than a single coherent state with the same energy. We find the optimal splitting of a fixed given energy among various copies. We also compute the variance for repeated individual measurements (without classical communication or adaptivity) and find that the standard Heisenberg-limited scaling ${n}^{\ensuremath{-}2}$ is recovered for large samples.

Stabilizer quantum error correction with quantum bus computation

In this paper we investigate stabilizer quantum error correction codes using controlled phase rotations of strong coherent probe states. We explicitly describe two methods to measure the Pauli operators that generate the stabilizer group of a quantum code. First, we show how to measure a Pauli operator acting on physical qubits using a single coherent state with large average photon number, displacement operations, and photon detection. Second, we show how to measure the stabilizer operators fault-tolerantly by the deterministic preparation of coherent quantum superposition (``cat'') states along with one-bit teleportations between a qubitlike encoding of coherent states and physical qubits.

LATTICE EQUILIBRIUM THEORY AND SIZE EFFECTS FOR BOSONS IN A BOUNDED HARMONIC POTENTIAL

Effects of finite spatial size of boson assemblies in traps are studied in a self-consistent lattice theory by modeling the trap as a bounded harmonic potential of size R0. The thermodynamic quantities exhibit scaling and crossover from ideal gas behaviour at small (R0/a0) to that appropriate to an unbounded harmonic potential at large (R0/a0) with a crossover parameter [Formula: see text], a0 being the harmonic oscillator length, and τ denoting the dimensionless thermal energy. The numerical results obtained earlier by computing the energy levels of the bounded harmonic oscillator fit the general structure predicted by the theory very well. For a1>10, the spatial size effects are negligible but for a1<10 they become appreciable and experimentally measurable in suitably designed traps. At low temperatures the self consistent cell size is found to be about 2.5a0 implying that the condensate is essentially a single coherent state contained in the central cell.

Dynamical decoherence in a cavity with a large number of two-level atoms

We consider a large number of two-level atoms interacting with the mode of a cavity in the rotating-wave approximation (Tavis–Cummings model). We apply the Holstein–Primakoff transformation to study the model in the limit of the number of two-level atoms, all in their ground state, becoming very large. The unitary evolution that we obtain in this approximation is applied to a macroscopic superposition state showing that, when the coherent states forming the superposition are distant enough, then the state collapses on a single coherent state describing a classical radiation mode. This appears as a true dynamical effect that could be observed in experiments with cavities.

Sampling entropies and operational phase-space measurement. II. Detection of quantum coherences.

We use the operational phase-space distributions and sampling entropies developed in the preceding paper [V. Bu\ifmmode \check{z}\else \v{z}\fi{}ek, C. H. Keitel, and P. L. Knight, Phys. Rev. A 51, 2575 (1995)] to discuss the nature of quantum interference between components of superpositions of states. We show how the Wehrl entropy, a special case of the sampling entropy, is a useful discriminator between different kinds of superpositions and of statistical mixtures, and is determined essentially by the coherent-state content. Apart from interference terms, this content is given by the quantum uncertainty of a single coherent state and the classical contribution to the number of coherent states necessary to tile the dominant phase-space ``patch'' representing the quantum state of interest. We illustrate these ideas using nonclassical superpositions of coherent states, where interference modifies the phase-space distributions, and show how these features are sensitive to dissipation.

Two-Photon Correlations in a Mixture of Gaussian and Laser Light

Measurement of the two-photon time distribution in a mixture of light from a He-Ne laser and a Gaussian source of the same central frequency agrees well with the prediction of coherence theory when the laser is represented by a single coherent state. The peak in the distribution near zero time difference is made up of contributions due to the second-order interference between pairs of Gaussian photons and between laser and Gaussian photons.