A Fokker-Planck equation for the Wigner function evolution in a noisy Kerr medium ($χ^{(3)}$ non-linearity) is presented. We numerically solved this equation taking a coherent state as an initial condition. The dissipation effects are discussed. We provide examples of quantum interference, sub-Planck phase space structures, and Gaussian versus non-Gaussian dynamical evolution of the state. The results also apply to the description of a nanomechanical resonator with an intrinsic Duffing nonlinearity.
By introducing the thermo entangled state representation, we convert the calculation of Wigner function (WF) of density operator to an overlap between "two pure" states in a two-mode enlarged Fock space. Furthermore, we derive a new WF evolution formula of any initial state in self-Kerr Medium with photon loss and find that the photon number distribution for any initial state is independent of the coupling factor with Kerr Medium, where the number state is not affected by the Kerr nonlinearity and evolves into a density operator of binomial distribution.
The output field from a continuously driven linear parametric oscillator may exhibit considerably more squeezing than the intracavity field. Inspired by this fact, we explore the nonclassical features of the steady-state output field of a driven nonlinear Kerr parametric oscillator using a temporal wave packet mode description. Utilizing a new numerical method, we have access to the density matrix of arbitrary wave packet modes. Remarkably, we find that even though the steady-state cavity field is always characterized by a positive Wigner function, the output may exhibit Wigner negativity, dep…
The creation of quantum coherences requires a system to be anharmonic. The simplest such continuous 1D quantum system is the Kerr oscillator. It has a number of interesting symmetries we derive. Its quantum dynamics is best studied in phase space, using Wigner's distribution W and the associated Wigner phase space current J. Expressions for the continuity equation governing its time evolution are derived in terms of J and it is shown that J for Kerr oscillators follows circles in phase space. Using J we also show that the evolution's classical shear in phase space is quantum suppressed by an e…
Motivated by quantum experiments with nanomechanical systems, the evolution of a Kerr oscillator with focus on creation of states with a negative Wigner function is investigated. Using the phase space formalism, results are presented that demonstrate an asymptotic behavior in the large squeezing regime for the negativity of a squeezed vacuum state under unitary evolution. The analysis and model are extended to squeezed vacuum states of open systems, adding the decoherence effects of damping and dephasing. To increase experimental relevance, the regime of strong damping is considered. These eff…
A scalar Wigner distribution function for describing polarized light is proposed in analogy with the treatment of spin variables in quantum kinetic theory. The formalism is applied to the propagation of circularly polarized light in nonlinear Kerr media and an extended phase space evolution equation is derived along with invariant quantities. We further consider modulation instability as well as the extension to partially coherent fields.
We propose a feasible scheme for generation of strongly non-Gaussian states using the cross-Kerr nonlinearity. The resultant states are highly non-classical states of electromagnetic field and exhibit negativity of their Wigner function, sub-Poissonian photon statistics, and amplitude squeezing. Furthermore, the Wigner function has a distinctly pronounced ``banana'' or ``crescent'' shape specific for the Kerr-type interactions, which so far was not demonstrated experimentally. We show that creating and detecting such states should be possible with the present technology using electromagnetical…
We investigate non-classical effects such as fractional revivals, squeezing and higher-order squeezing of photon-added coherent states propagating through a Kerr-like medium.The Wigner functions corresponding to these states at the instants of fractional revivals are obtained, and the extent of non-classicality quantified.
We present exact results for the steady-state density matrix of a general class of driven-dissipative systems consisting of a nonlinear Kerr resonator in the presence of both coherent (one-photon) and parametric (two-photon) driving and dissipation. Thanks to the analytical solution, obtained via the complex P-representation formalism, we are able to explore any regime, including photon blockade, multiphoton resonant effects, and a mesoscopic regime with large photon density and quantum correlations. We show how the interplay between one- and two-photon driving provides a way to control the mu…
We derive a continuity equation for the evolution of the SU(2) Wigner function under nonlinear Kerr evolution. We give explicit expressions for the resulting quantum Wigner current, and discuss the appearance of the classical limit. We show that the global structure of the quantum current significantly differs from the classical one, which is clearly reflected in the form of the corresponding stagnation lines.
We show that the Truncated Wigner Approximation developed in the flat phase-space is mapped into a Lindblad-type evolution with an indefinite metric in the space of linear operators. As a result, the classically evolved Wigner function corresponds to a non-positive operator $\hat{R}(t)$, which does not describe a physical state. The rate of appearance of negative eigenvalues of $\hat{R}(t)$ can be efficiently estimated. The short-time dynamics of the Kerr and second harmonic generation Hamiltonains are discussed.
We investigate the quantum properties for the codirectional three-mode Kerr nonlinear coupler. We investigate single-, two- and three-mode quadrature squeezing, Wigner function and purity. We prove that this device can provide richer nonclassical effects than those produced by the conventional coupler, i.e. the two-mode Kerr coupler. We show that it can provide squeezing and the quadrature squeezing exhibiting leaf-revival-collapse phenomenon in dependence on the values of the interaction parameters. In contrast to the conventional Kerr coupler two different forms of cat states can be simultan…
Non-Gaussian states represent a powerful resource for quantum information protocols in the continuous variables regime. Cat states, in particular, have been produced in the motional degree of freedom of trapped ions by controlled displacements dependent on the ionic internal state. An alternative method harnesses the Kerr nonlinearity naturally existent in this kind of system. We present detailed calculations confirming its feasibility for typical experimental conditions. Additionally, this method permits the generation of complex non-Gaussian states with negative Wigner functions. Especially,…
In this paper, we investigate the single mode quantum properties of the codirectional Kerr nonlinear coupler when the frequency mismatch is involved and a condition for an exact solution of equations of motion is fulfilled. Particularly, we investigate quadrature and principal squeezing, Wigner function, quadrature distribution, phase distribution and phase variance. We show that the quadrature squeezing and the phase variance can exhibit collapse-revival and collapse-revival-subrevival phenomena, respectively, based on the values of the detuning parameter. Furthermore, we analytically demonst…
In this paper we consider multimode multiphoton Jaynes-Cummings model, which consists of a two-level atom, initially prepared in an excited atomic state, interacting with $N$ modes of electromagnetic field prepared in general pure quantum states. For this system we show that under certain conditions the evolution of the Wigner function at the phase space origin provides direct information on the corresponding atomic inversion. This relation is also valid even if the system includes Kerr-like nonlinearity, Stark shift effect, different types of the initial atomic state as well as moving atom. F…
We present the reconstruction of the Wigner function of a classical phase-sensitive state, a pulsed coherent state, by measurements of the distributions of detected-photons of the state displaced by a coherent probe field. By using a hybrid photodetector operated above its photon-resolving regime, we obtained both the statistics at different values of the probe field and the values of the probe phase required to reconstruct the Wigner function.
We propose a Wigner function based parameter that can be used as an indicator of quantum chaos. This parameter is defined as "entropy" from the time-dependence of "non-classicallity" proposed in \cite{KZ04}. We perform our considerations for the system of damped nonlinear (Kerr-like) oscillator excited by a series of ultra-short external pulses.
We propose a scheme to prepare optical Schrödinger-cat states in a traveling wave setting. Two states are similarly prepared via the self-Kerr effect and after mixing them, one mode is measured by homodyne detection. In the other mode a superposition of coherent states is conditionally prepared. The advantage of the scheme is that assuming a small Kerr effect one can prepare with high probability one from a set of Schrödinger-cat states. The measured value of the quadrature provides the information, which state from the set is actually prepared.
Few-photon optomechanical effects are not only important physical evidences for understanding the radiation-pressure interaction between photons and mechanical oscillation, but also have wide potential applications in modern quantum technology. Here we study the few-photon optomechanical effects including photon blockade and generation of the Schrödinger cat states under the assistance of a cross-Kerr interaction, which is an inherent interaction accompanied the optomechanical coupling in a generalized optomechanical system. By exactly diagonalizing the generalized optomechanical Hamiltonian a…
Quantum states can be stabilized in the presence of intrinsic and environmental losses by either applying active feedback conditioned on an ancillary system or through reservoir engineering. Reservoir engineering maintains a desired quantum state through a combination of drives and designed entropy evacuation. We propose and implement a quantum reservoir engineering protocol that stabilizes Fock states in a microwave cavity. This protocol is realized with a circuit quantum electrodynamics platform where a Josephson junction provides direct, nonlinear coupling between two superconducting wavegu…
Exact nonlinear stationary solutions of the one-dimensional Wigner and Wigner-Poisson equations in the terms of the Wigner functions that depend not only on the energy but also on position are presented. In this way, the Bernstein-Greene-Kruskal modes of the classical plasma are adapted for the quantum formalism in the phase space. The solutions are constructed for the case of a quartic oscillator potential, as well as for the self-consistent Wigner-Poisson case. Conditions for well-behaved physically meaningful equilibrium Wigner functions are discussed.
Quantum theory of self-phase and cross-phase modulation of ultrashort light pulses in the Kerr medium is developed with taking into account the response time of an electronic nonlinearity. The correspondent algebra of time-dependent Bose-operators is elaborated. It is established that the spectral region of the pulse, where the quadrature fluctuations level is lower than the shot-noise one, depends on the value of the nonlinear phase shift, the intensity of another pulse, and the relaxation time of the nonlinearity. It is shown that the frequency of the pulse spectrum at which the suppression …
We derive a cross-Kerr type effective Hamiltonian for the four-level atom interacting with three electromagnetic fields in the N-configuration. When the atom has relaxed into the ground state a cross-Kerr nonlinearity arises between two weak probe fields. As a development on earlier work we show in general that the atom will also display a linear and self-Kerr response. However, if certain resonance conditions are satisfied then the linear and self-Kerr interactions will vanish. The electrical susceptibilities of the probe transitions are also explored and it is shown that a large, pure cross-…
We study Kerr nonlinear resonators (KNR) driven by a continuous wave field in quantum regimes where strong Kerr interactions give rise to selective resonant excitations of oscillatory modes. We use an exact quantum theory of KNR in the framework of the Fokker-Planck equation without any quantum state truncation or perturbation procedure. This approach allows non-perturbative consideration of KNR for various quantum operational regimes including cascaded processes between oscillatory states. We focus on understanding of multi-photon non-resonant and selective resonant excitations of introcavity…
This paper derives Langevin equations for an optical cavity where the dominant nonlinearity arises from free-carrier dispersion. We define a generalized Wigner function, compute a Fokker-Planck equation that approximates the master equation, and convert this to a system of stochastic differential equations. These equations are similar to the Wigner equations for an optical Kerr cavity, but have additional noise terms due to the incoherent carrier excitation and decay processes. We use these equations to simulate a phase-sensitive amplifier and latch and compare the results to a Kerr model.
Dissipative Kerr solitons arising from parametric gain in ring microresonators are usually described within a classical mean-field framework. Here, we develop a quantum-mechanical model of dissipative Kerr solitons in terms of the truncated Wigner method, which accounts for quantum effects to lowest order. We show that the soliton experiences a finite lifetime due to quantum fluctuations originating from losses. Reading the results in terms of the theory of open quantum systems, allows to estimate the Liouvillian spectrum of the system. It is characterized by a set of eigenvalues with finite i…
We present two complete maximally hyperentangled state analysis protocols for photons entangled in the polarization and spatial-mode degrees of freedom. The first protocol is a hyperentangled Bell state analysis scheme for two photons and the second is a hyperentangled Greenberger-Horne- Zeilinger (GHZ) state analysis scheme for three photons. In each scheme, a set of mutually orthogonal hyperentangled basis states are completely and deterministically discriminated with the aid of cross-Kerr nonlinearities and linear optics. We also generalize the schemes to unambiguously analyze the N-photon …
We identify and discuss nonlinear phase noise arising in Kerr self-phase modulation of a coherent light pulse propagating through an attenuating medium with third-order nonlinearity in a dispersion-free setting. This phenomenon, accompanying the standard unitary Kerr transformation of the optical field, is described with high accuracy as Gaussian phase diffusion with parameters given by closed expressions in terms of the system properties. We show that the irreversibility of the nonlinear phase noise ultimately limits the ability to transmit classical information in the phase variable over a l…
In a recent paper, Żukowski and Markiewicz showed that Wigner's Friend (and, by extension, Schrödinger's Cat) can be eliminated as physical possibilities on purely logical grounds. I validate this result and demonstrate the source of the contradiction in a simple experiment in which a scientist S attempts to measure the position of object $|O\rangle = |A\rangle_S + |B\rangle_S$ by using measuring device M chosen so that $|A\rangle_M \approx |A\rangle_S$ and $|B\rangle_M \approx |B\rangle_S$. I assume that the measurement occurs by quantum amplification without collapse, in which M can entangle…
Frauchiger and Renner have recently claimed to prove that "Single-world interpretations of quantum theory cannot be self-consistent". This is contradicted by a construction due to Bell, inspired by Bohmian mechanics, which shows that any quantum system can be modelled in such a way that there is only one "world" at any time, but the predictions of quantum theory are reproduced. This Bell-Bohmian theory is applied to the experiment proposed by Frauchiger and Renner, and their argument is critically examined. It is concluded that it is their version of "standard quantum theory", incorporating st…
The consistent theory of formation of pulsed squeezed states as a result of self-action of ultrashort light pulse in the medium with relaxation Kerr nonlinearity has been developed. A simple method to form the ultrashort light pulse with sub-Poissonian photon statistics is analyzed too.
The self-Kerr interaction is an optical nonlinearity that produces a phase shift proportional to the square of the number of photons in the field. At present, many proposals use nonlinearities to generate photon-photon interactions. For propagating fields these interactions result in undesirable features such as spectral correlation between the photons. Here, we engineer a discrete network composed of cross-Kerr interaction regions to simulate a self-Kerr medium. The medium has effective long-range interactions implemented in a physically local way. We compute the one- and two-photon S matrice…
By using the localized character of canonical coherent states, we give a straightforward derivation of the Bargmann integral representation of Wigner function (W). A non-integral representation is presented in terms of a quadratic form V*FV, where F is a self-adjoint matrix whose entries are tabulated functions and V is a vector depending in a simple recursive way on the derivatives of the Bargmann function. Such a representation may be of use in numerical computations. We discuss a relation involving the geometry of Wigner function and the spacial uncertainty of the coherent state basis we us…
Technological applications of many-body structures that emerge in gated devices under minimal control are largely unexplored. Here we show how emergent Wigner crystals in a semiconductor quantum wire can facilitate a pivotal requirement for a scalable quantum computer, namely transmitting quantum information encoded in spins faithfully over a distance of micrometers. The fidelity of the transmission is remarkably high, faster than the relevant decohering effects, independent of the details of the spatial charge configuration in the wire, and realizable in dilution refrigerator temperatures. Th…
We investigate the nonlinear refraction induced by Rydberg excitons in $\mathrm{Cu_2O}$. Using a high-precision interferometry imaging technique that spatially resolves the nonlinear phase shift, we observe significant shifts at extremely low laser intensity near each exciton resonance. From this, we derive the nonlinear index $\mathrm{n_2}$, present the $\mathrm{n_2}$ spectrum for $n\geq 5$ and report large $\mathrm{n_2}$ values of order $10^{-3}$ mm$^2$/mW. Moreover, we observe a rapid saturation of the Kerr nonlinearity and find that the saturation intensity $\mathrm{I_{sat}}$ decreases as …
A Kerr-nonlinear parametric oscillator (KPO) can generate a quantum superposition of two oscillating states, known as a Schrödinger cat state, via quantum adiabatic evolution, and can be used as a qubit for gate-based quantum computing and quantum annealing. In this work, we investigate complex dynamics, i.e., chaos, in two coupled nondissipative KPOs at a few-photon level. After showing that a classical model for this system is nonintegrable and consequently exhibits chaotic behavior, we provide quantum counterparts for the classical results, which are quantum versions of the Poincaré surface…
The engineering of Kerr interactions has great potential for quantum information processing applications in multipartite quantum systems and for investigation of many-body physics in a complex cavity-qubit network. We study how coupling multiple different types of superconducting qubits to the same cavity modes can be used to modify the self- and cross-Kerr effects acting on the cavities and demonstrate that this type of architecture could be of significant benefit for quantum technologies. Using both analytical perturbation theory results and numerical simulations, we first show that coupling…
The equation of motion for the reduced Wigner function of a system coupled to an external quantum system is presented for the specific case when the external quantum system can be modeled as a set of harmonic oscillators. The result is derived from the Wigner function formulation of the Feynman-Vernon influence functional theory. It is shown how the true self-energy for the equation of motion is connected with the influence functional for the path integral. Explicit expressions are derived in terms of the bare Wigner propagator. Finally, we show under which approximations the resulting equatio…
Photons are ideal carriers for quantum information as they can have a long coherence time and can be transmitted over long distances. These properties are a consequence of their weak interactions within a nearly linear medium. To create and manipulate nonclassical states of light, however, one requires a strong, nonlinear interaction at the single photon level. One approach to generate suitable interactions is to couple photons to atoms, as in the strong coupling regime of cavity QED systems. In these systems, however, one only indirectly controls the quantum state of the light by manipulating…
We present a unified theoretical study of the bright solitons governed by self-focusing and defocusing nonlinear Schrodinger (NLS) equations with generalized parity-time (PT)-symmetric Scarff II potentials. Particularly, a PT-symmetric k-wavenumber Scarff II potential and a multi-well Scarff II potential are considered, respectively. For the k-wavenumber Scarff II potential, the parameter space can be divided into different regions, corresponding to unbroken and broken PT-symmetry and the bright solitons for self-focusing and defocusing Kerr nonlinearities. For the multi-well Scarff II potenti…
Dirac's wave equation for a point electron in the topologically nontrivial maximal analytically extended electromagnetic Kerr--Newman spacetime is studied in a zero-gravity limit; here, "zero-gravity" means $G\to 0$, where $G$ is Newton's constant of universal gravitation. The following results are obtained: the formal Dirac Hamiltonian on the static spacelike slices is essentially self-adjoint; the spectrum of the self-adjoint extension is symmetric about zero, featuring a continuum with a gap about zero that, under two smallness conditions, contains a point spectrum. Some of our results exte…
