Two-mode Bose-Einstein Condensate Research Articles

Noise-dissipation Correlated Dynamics of a Double-well Bose-Einstein Condensate-reservoir System

Abstract: In this work, we study the dissipative dynamics of a double-well Bose-Einstein condensate (BEC) out-coupled to reservoir at each side of its trap. The sub-system comprises of a simple Bose-Hubbard model, where the interplay of atom-tunneling current and inter-particle interaction are the main quantum features. The contact with two separate heat baths causes dissipation and drives the system into a non-equilibrium state. The system is well described by the Generalized Quantum Heisenberg-Langevin equation. We considered two Markovian dissipative BEC systems based on (i) the mean-field model (MF), where the internal noise has been averaged out and (ii) the noise-correlated model (FDT). Physical quantities, such as population imbalance, coherence and entanglement of the system, are computed for the models. The two-mode BEC phases, such as the quantum tunneling state and the macroscopic quantum-trapping state, evolved into complicated dynamics by controlling the non-linear interaction and dissipation strengths. We found that many important quantum features produced by the noise-correlated FDT model are not captured by the mean-field model. Keywords: Double-well BEC, Dissipation, Noise, Markovian, Non-Markovian, Fixed points. PACS: 03.75 Lm, 03.65 Yz, 03.75 Gg, 05.

Schrödinger cat state formation in small bosonic Josephson junctions at finite temperatures and dissipation

In this work, we consider the feasibility of Schrödinger cat (SC) and states formation by a convenient bosonic Josephson junction system in two-mode approximation. Starting with purely quantum description of two-mode Bose–Einstein condensate we investigate the effective potential approach that provides an accurate analytical description for the system with a large number of particles. We show that in the zero temperature limit SC states result from a quantum phase transition that occurs when the nonlinear strength becomes comparable with the Josephson coupling parameter. The Wigner function approach demonstrates the growth of the SC state halves separation and formation of -like states (a Fock state superposition) with the particle number increase. We examine the possibility to attain the SC state at finite temperatures and a weak dissipation leading to appearing of some critical temperature; it defines the second-order phase transition from classical activation process to the SC state formation through the quantum tunneling phenomenon. Numerical estimations demonstrate that the critical temperature is sufficiently below the temperature of atomic condensation. The results obtained may be useful for experimental observation of SC states with small condensate Josephson junctions.

Many-body Bell inequalities for bosonic qubits

Since John Bell formulated his paramount inequality for a pair of spin-1/2 particles, quantum mechanics has been confronted with the postulates of local realism with various equivalent configurations. Current technology, with its advanced manipulation and detection methods, allows to extend the Bell tests to more complex structures. The aim of this work is to analyze a set of Bell inequalities suitable for a possibly broad family of many-body systems with the focus on bosonic qubits. We develop a method that allows for a step-by-step study of the many-body Bell correlations, for instance among atoms forming a two-mode Bose-Einstein condensate or between photons obtained from the parametric-down conversion. The presented approach is valid both for cases of fixed and non-fixed number of particles, hence it allows for a thorough analysis of quantum correlations in a variety of many-body systems.

Non-linear Dynamics of a Double-Well Bose–Einstein Condensate–Reservoirs System

We investigate the dynamics of a Bose–Einstein condensate (BEC) confined within a symmetric double-well trap out-coupled to reservoirs. Our model is based on the quantum Heisenberg–Langevin approach. Dissipation in the system is induced by the interaction of condensate bosons with the reservoir which determines the Markovian and non-Markovian operational dynamics. These distinct dynamics correspond directly to the choice of function used in the dissipation kernel of the quantum Langevin equation. The main focus of this work is to study non-Markovian dynamics of the system proliferated by the use of Ornstein–Uhlenbeck (OU) memory function. Comparison is made with the Markovian dynamics induced by the use of δ-memory-less function. We have calculated numerically the time evolution of population imbalance and tunneling current of the system and found dissipation destroys the regular pattern of the two-mode BEC phases. Comparison between dissipative models shows significant differences in their dynamics, particularly in the strongly interacting case.

Criteria to detect macroscopic quantum coherence, macroscopic quantum entanglement, and an Einstein-Podolsky-Rosen paradox for macroscopic superposition states

According to classical theory, a system with two or more macroscopically distinct states available to it is in one of those states at all times. Quantum mechanics gives a different interpretation where the system can be in a superposition of such states. In this paper, we derive criteria in the form of inequalities to detect this effect, referred to as mesoscopic quantum coherence, where the states are (at least) mesoscopically distinct. Such criteria are also signatures of a mesoscopic Schr\odinger cat paradox. We extend the treatment to consider definitions and criteria for mesoscopically entangled states, which are a subset of the states exhibiting mesoscopic quantum coherence. It is shown how the criteria (referred to as type I criteria) can be applied to detect the mesoscopic entanglement of systems prepared in NOON states, Greenberger-Horne-Zeilinger states, and entangled cat states involving coherent states. Here, a larger system $C$ is entangled with a second system $S$, which can be small. The proposed definition of mesoscopic (macroscopic) entanglement involves the Schmidt decomposition, and can be applied to systems in a superposition of many states, only some pairs of which are mesoscopically (macroscopically) distinguishable. We prove that a higher-order Hillery-Zubairy entanglement criterion will detect mesoscopic entanglement of this type, and use this to demonstrate the mesoscopic entanglement of the ground state of a two-mode Bose-Einstein condensate. Finally, we explain how a subset of the type I criteria (called type II criteria) are EPR steering inequalities allowing realization of a mesoscopic or macroscopic version of the Einstein-Podolsky-Rosen paradox for macroscopic superposition states. Where the two systems $C$ and $S$ are spatially separated, we then use results from the literature to point out it is not possible to complete quantum mechanics using hidden-variable theories compatible with the assumption of locality between the two systems. The violation of certain Bell inequalities is sufficient to prove this result, which gives a stricter signature of the Schr\odinger cat paradox and of mesoscopic quantum coherence. We call such signatures type III criteria.

Generation of various classes of entangled states in a two-mode Bose–Einstein condensate under the influence of interatom collisions

In this paper, we generate some new classes of entangled states of a bimodal Bose–Einstein condensate (BEC), a pair of tunnel-coupled BEC, in the presence of two- and three-body elastic as well as mode-exchange collisions. The Hamiltonian of the considered system is very complicated, moreover, it can be fortunately transformed into a simple form using a two-mode displacement operator. After introducing the general form of the time evolved state, various classes of entangled states are generated. Indeed, the influence of different orders of tunneling strengths on the generated entangled states has been studied. Depending on the tunneling strength constants, two-, three- and four-partite entangled states are generated, all of which are superposition states of macroscopic number of BEC atoms. Considering three-particle collision dramatically changes the generated entangled states. Moreover, in particular cases, the resulted states are non-entangled. Also, we show that tunneling and collisional interactions can be manipulated to generate a pair of atomic entangled coherent states (quasi-Bell states). In addition, it is observed that the degree of entanglement for two-partite entangled states can be tuned via the number of BEC atoms, i.e. the corresponding concurrences tend to their maximum value by increasing the atoms in both modes of system.

Population imbalance, macroscopic tunneling and intermodal entanglement of two-mode Bose–Einstein condensate under the influence of dissipation process

In this paper, we study the dissipative dynamics of a system which is composed of two atomic Bose–Einstein condensates (BECs) interacting through the Josephson-like coupling. To model a more realistic physical system, the inevitable dissipation effect is considered via the interaction between the system and its environment, in particular, a thermal reservoir. In this respect, after introducing a proper Hamiltonian for the model, we analytically solve the corresponding quantum Heisenberg–Langevin equations and then obtain explicit analytical expressions for population imbalance, macroscopic tunneling current and intermodal entanglement. Generally, the dynamics of the system is very sensitive to the chosen values of tunneling coupling strength, initial population as well as the characteristics of interaction between the system and its reservoirs. Also, the time evolution of the above-mentioned physical quantities shows oscillatory decaying behavior where the frequency of oscillations depends on the strength of tunneling interaction between the two subsystems. The oscillatory pattern of population imbalance and tunneling current is more regular in comparison with the intermodal entanglement. Although the system is always separable for low initial population, we show that it tends to an entangled state as its initial population is increased. In particular, the amount and the time interval of the entanglement can be effectively controlled via the dissipation parameter. Also, to get an insight into the effect of nonlinear interaction on the behavior of dynamical evolution of the considered system, we numerically investigate the population imbalance in the absence and presence of such interactions. A qualitative comparison shows that the previous theoretical works and numerical simulations confirm our obtained results.

Dissipative evolution of two-mode Bose–Einstein condensate in the presence of nonlinear interactions: Heisenberg operator approach

In this paper we study the qualitative dynamical evolution of atomic population imbalance and tunneling current between two Bose–Einstein condensate (BEC) interacting with each other in the presence of the intra- and inter-species collisions while the Rabi coupling between them is switched on, too. In particular, we consider the influence of dissipation process in our model to get more realized physical results. In this regard, to solve the Hamiltonian of the system, the Heisenberg operator method is used, by which the analytical solutions of the related differential equations are obtained. The population imbalance between the two BECs and the respected atomic tunneling current are then explicitly deduced for two initial state, i.e., number and coherent states in different conditions. The analytical as well as numerical results confirm the dependence of the mentioned quantities on the chosen initial state and the total number of atoms in the system which is in consistence with experimental observations. As is observed, a new kind of macroscopic quantum self-trapping (MQST) effect is also occurred in the population imbalance at certain conditions. Two distinct types of collapse-revival phenomena, depending on the relative strength of tunneling and collisional interactions, can be seen in the behavior of atomic population imbalance and tunneling current. Interestingly, collapse-revival phenomenon is disappeared in the intermediate regime and a new oscillatory step-like pattern is observed in the dynamics of the system.

Quantum Entanglement in Two Mode Bose-Einstein Condensate

We investigate a general model of a two-mode Bose-Einstein condensate (BEC) coupled via Josephson tunneling. By the bosonization method, we analytically and numerically calculate the entanglement parameter. It shows that the better entanglement can be achieved by increasing the number of particles and decreasing the the coupling strength for Josephson tunneling.

Weak measurements and quantum weak values for NOON states

Quantum weak values arise when the mean outcome of a weak measurement made on certain preselected and postselected quantum systems goes beyond the eigenvalue range for a quantum observable. Here, we propose how to determine quantum weak values for superpositions of states with a macroscopically or mesoscopically distinct mode number, that might be realized as two-mode Bose-Einstein condensate or photonic NOON states. Specifically, we give a model for a weak measurement of the Schwinger spin of a two-mode NOON state, for arbitrary $N$. The weak measurement arises from a nondestructive measurement of the two-mode occupation number difference, which for atomic NOON states might be realized via phase contrast imaging and the ac Stark effect using an optical meter prepared in a coherent state. The meter-system coupling results in an entangled cat-state. By subsequently evolving the system under the action of a nonlinear Josephson Hamiltonian, we show how postselection leads to quantum weak values, for arbitrary $N$. Since the weak measurement can be shown to be minimally invasive, the weak values provide a useful strategy for a Leggett-Garg test of $N$-scopic realism.

Transition dynamics of a bright soliton in a binary Bose–Einstein condensate

Transition dynamics have been studied widely in two-mode Bose–Einstein condensate systems with linear coupling effects. Nonlinear interaction between atoms would induce a nonlinear Josephson oscillation, which admits an oscillation form different from the classical Josephson oscillation. Similar extensions for the Rosen–Zener transition and the Landau–Zener transition also suggest that a strong nonlinear strength would induce nonlinear-type transition dynamics, which is in sharp contrast to the linear process. Interestingly, here we show a standard Josephson oscillation that has a constant linear coupling strength, no matter how great the nonlinear interaction strength is. This characteristic is in sharp contrast to the nonlinear Josephson oscillation reported before in nonlinear coupled systems. By manipulating the linear coupling strength with the RF field, we demonstrate that the Rosen–Zener transition particles, with an invariant distribution profile, can be managed well under exponential and periodic forms. The nonlinear interaction strength is found to have no effect on the transition rate between atoms in the two components for the case where the inter- and intra-atomic interaction strengths are equal. Furthermore, we test the robustness of the transition dynamics against parameter deviations and noises in the initial condition. These results are helpful for controllable quantum state preparations.

Squeezing and entanglement in two-mode Bose–Einstein condensates

We investigate the influence of one-body losses on the dynamics of squeezing and entanglement in two-mode Bose–Einstein condensates. We show that one-body losses play an important role in the dynamical process of generating squeezing and quantum entanglement. The stronger one-body losses induce smaller squeezing and lesser entanglement, but maintain in a longer time interval.

Phase Dissipation of an Open Two-Mode Bose–Einstein Condensate**Supported by the National Natural Science Foundation of China under Grant No 11374197, the Program for Changjiang Scholars and Innovative Research Team in Universities of China under Grant No IRT13076, and the Hundred Talent Program of Shanxi Province.

We study the dynamics of a two-mode Bose–Hubbard model with phase dissipation, based on the master equation. An analytical solution is presented with nonzero asymmetry and phase noise. The effects of asymmetry and phase noise play a contrasting role in the dynamics. The asymmetry makes the oscillation fast, while phase noise enlarges the period. The conditions for the cases of fast decay and oscillation are presented. As a possible application, the dynamical evolution of the population for cold atomic gases with synthetic gauge interaction, which can be understood as two-mode dynamics in momentum space, is predicted.

Probing two-particle exchange processes in two-mode Bose-Einstein condensates

We study the fidelity decay and its freeze for an initial coherent state of two-mode Bose-Einstein condensates in the Fock regime considering a Bose-Hubbard model that includes two-particle tunneling terms. By using linear-response theory we find scaling properties of the fidelity as a function of the particle number that prove the existence of two-particle mode-exchange when a non-degeneracy condition is fulfilled. Tuning the energy difference of the two modes serves to distinguish the presence of two-particle mode-exchange terms through the appearance of certain singularities. Numerical results confirm our findings. Experimental verification of our findings could improve cold atom interferometry.

Quantifying the mesoscopic quantum coherence of approximate NOON states and spin-squeezed two-mode Bose-Einstein condensates

We examine how to signify and quantify the mesoscopic quantum coherence of approximate two-mode NOON states and spin-squeezed two-mode Bose-Einstein condensates (BEC). We identify two criteria that verify a nonzero quantum coherence between states with quantum number different by $n$. These criteria negate certain mixtures of quantum states, thereby signifying a generalised $n$-scopic Schrodinger cat-type paradox. The first criterion is the correlation $\langle\hat{a}^{\dagger n}\hat{b}^{n}\rangle\neq0$ (here $\hat{a}$ and $\hat{b}$ are the boson operators for each mode). The correlation manifests as interference fringes in $n$-particle detection probabilities and is also measurable via quadrature phase amplitude and spin squeezing measurements. Measurement of $\langle\hat{a}^{\dagger n}\hat{b}^{n}\rangle$ enables a quantification of the overall $n$-th order quantum coherence, thus providing an avenue for high efficiency verification of a high-fidelity photonic NOON states. The second criterion is based on a quantification of the measurable spin-squeezing parameter $\xi_{N}$. We apply the criteria to theoretical models of NOON states in lossy interferometers and double-well trapped BECs. By analysing existing BEC experiments, we demonstrate generalised atomic "kitten" states and atomic quantum coherence with $n\gtrapprox10$ atoms.

Coherence and Squeezing of Bose-Einstein Condensates in Double Wells

We investigate coherence and squeezing of a two-mode Bose-Einstein condensate trapped in a double-well potential. By analytically deriving the form of coherence and numerically calculating the squeezing parameter, we show that the coherence and the squeezing may be controlled by adjusting some parameters of the two-mode Bose-Einstein condensate.

Dynamic fragmentation in a quenched two-mode Bose–Einstein condensate

We investigate the fragmentation in a two-mode Bose–Einstein condensate with Josephson coupling. We explore how the fragmentation and entropy of the ground state depend on the intermode asymmetry and interparticle interactions. Owing to the interplay between the asymmetry and the interactions, a sequence of notches and plateaus in the fragmentation appears with the single-atom tunneling and interaction blockade, respectively. We then analyze the dynamical properties of the fragmentation in three typical quenches of the asymmetry: linear, sudden, and periodic quenches. In a linear quench, the final state is a fragmented state due to the sequential Landau–Zener tunneling, which can be analytically explained by applying the two-level Landau–Zener formula for each avoided level crossing. In a sudden quench, the fragmentation exhibits persistent fluctuations that sensitively depend on the interparticle interactions and intermode coupling. In a periodic quench, the fragmentation is modulated by the periodic driving, and a suitable modulation may allow one to control the fragmentation.

Effects of three-body collisions in a two-mode Bose-Einstein condensate

We study the effects of three-body collisions in the physical properties of a two-mode Bose-Einstein condensate. The model introduced here includes two-body and three-body elastic and mode-exchange collisions and can be solved analytically. We will use this fact to show that three-body interactions can produce drastic changes in the probability distribution of the ground state and the dynamics of the relative population. In particular, we find that three-body interactions under certain circumstances may inhibit the collapse of the relative population.

Dynamics of the double-well Bose–Einstein condensate coupled to a dual Markovian reservoirs system

The dynamics of atomic condensate loaded in a double-well symmetric traps coupled to distinct dual reservoirs is studied. The effect of damping on the population evolution through the exposure of traps to their respective reservoir has been analysed at various temperatures. Damping in our system is induced by the fluctuation of the reservoir in agreement with the fluctuation–dissipation theorem. We found damping due to strong reservoir correlation destroy the quantum tunnelling state and macroscopic quantum self-trap state of the closed two-mode Bose–Einstein condensate states at all finite temperatures. Alternatively switching the trap’s out-coupling damping rates shows significant change in the population dynamics particularly in the strongly interacting case. Dissipation coupled with strong on-site interaction in the traps exhibits even more interesting dynamics as the equilibrium temperature in system is increased.

Non-Markovian dynamics of a two-mode Bose–Einstein condensate

We study the dynamics of an atomic condensate loaded in two symmetric traps where one of the trap is coupled to a non-Markovian reservoir. The system is fully described by the quantum-Langevin equation subjecting to Ornstein–Uhlenbeck dissipation kernel. We consider Ohmic dissipation with Lorentzian cutoff in our calculation. In this model dissipation is induced by the reservoir fluctuation in agreement with the Fluctuation–Dissipation theorem. We use simple numerical method to calculate the non-Markovian population evolution of the system. We found that stronger dissipation and higher temperature significantly effecting the population evolution of the non-Markovian two-mode system. Surprising dynamics is observed when the tunneling between the traps was inhibited as the system reaches it equilibrium.