1D Harmonic Trap Research Articles

Control of Correlation Using Confinement in Case of Quantum System

AbstractThis article investigates the behavior of a Moshinsky atom in a 1D harmonic trap. Focus is given on the theoretical foundations of confinement and its impact on the correlation between particles in the Moshinsky atom. The investigation begins by illustrating the (de)localization of the probability density function using Shannon entropy. The basics of correlation and interpretation of correlation using tools such as mutual information and statistical correlation coefficients and how these can be quantified are discussed. Then the concept of confinement is explored. The impact of interaction strength and confinement on Shannon entropy, statistical correlation coefficients, and mutual information is investigated. How interaction strength and confinement can be used to induce correlations between previously uncorrelated particles, as well as how they can be used to suppress correlations between previously correlated particles is discussed. Their implications for quantum information processing and quantum simulation are discussed. In conclusion, confinement is a powerful tool for controlling correlations in quantum systems, and its impact on correlation can be understood through theoretical models. The importance of experimental studies in this field, which provide insights into the behavior of quantum systems under confinement and pave the way for future applications in quantum technology is also emphasized.

A microscopic study of spin-polarized 40K atoms in one dimension using the static fluctuation approximation (SFA)

The thermodynamic and magnetic properties of a spin-mixture of 40K atoms in a 1D harmonic trap, in the presence of an external magnetic field H, are studied using the static fluctuation approximation (SFA). The two-body interaction used is the contact potential. The thermodynamic properties include the chemical potential, energy per particle, specific heat capacity, and entropy per particle. The magnetic properties comprise the magnetization per particle and susceptibility per particle. A phase transition is observed for the repulsive interactions around 0.35TF and 0.5TF in the channels {–9/2, –7/2} and {–9/2, –5/2}, respectively. A peak appears in the entropy and susceptibility at effective scattering length |a1d| ≈ 0.16 μm. For the attractive interactions, three different behaviors depending on H are found in the low-temperature limit. The first behavior is when H < Hc, where Hc is the critical magnetic field, and typical peaks emerge. The second behavior is at H = Hc, and a magnetic phase transition occurs. The third behavior is when H > Hc and the system becomes a partially polarized phase. Our results are consistent with the reported results obtained by the local density approximation, thermal Bethe ansatz method, mean-field approximation, and quantum transfer matrix method.

Two distinguishable fermions entanglement generation and fermionization

We propose a procedure for entanglement generation which could be implemented in current experiments performed with two distinguishable interacting fermions confined in a 1D harmonic trap [Phys. Rev. Lett. 108 , 075303 (2012)]. Our results indicate that entangled states could be obtained by preparing a non-interacting non-entangled product state of the trap and then properly controlling the scattering length by means of the magnetic field. Entanglement could be generated when the prepared non-entangled initial state undergoes a turn on and off interaction procedure in which the state is brought into the interacting regime and then back to the absence of interaction point. Entanglement could also be generated when reaching the strongly repulsive regime by tuning the scattering length near the confinement-induced resonance (CIR), i.e. when the two distinguishable fermions are fermionized. We give exact upper bounds for the entanglement and show that maximally entangled states could be produced. Moreover, our results suggest that the maximum entanglement could be reached by repeatedly turning on and off the interaction or by repeatedly crossing the CIR, being the latter the most convenient procedure because it ensures the obtention of maximally entangled states with a higher probability. The turn on and off procedure, the fermionization of the states, and the CIR crossing constitute three basic experimentally realizable main steps upon which several paths for entanglement generation could be diagrammed. • Entanglement generation in a two distinguishable interacting fermion system. • Entangled states are obtained tuning the interaction strength with a magnetic field. • Entanglement could be generated with three basic experimental main steps. • Several paths for entanglement generation could be diagrammed. • Maximum entanglement could be generated by repeatedly crossing the resonance.

Constructing tensor network influence functionals for general quantum dynamics.

We describe an iterative formalism to compute influence functionals that describe the general quantum dynamics of a subsystem beyond the assumption of linear coupling to a quadratic bath. We use a space-time tensor network representation of the influence functional and investigate its approximability in terms of its bond dimension and time-like entanglement in the tensor network description. We study two numerical models, the spin-boson model and a model of interacting hard-core bosons in a 1D harmonic trap. We find that the influence functional and the intermediates involved in its construction can be efficiently approximated by low bond dimension tensor networks in certain dynamical regimes, which allows the quantum dynamics to be accurately computed for longer times than with direct time evolution methods. However, as one iteratively integrates out the bath, the correlations in the influence functional can first increase before decreasing, indicating that the final compressibility of the influence functional is achieved via non-trivial cancellation.

Multicritical Edge Statistics for the Momenta of Fermions in Nonharmonic Traps.

We compute the joint statistics of the momenta p_{i} of N noninteracting fermions in a trap, near the Fermi edge, with a particular focus on the largest one p_{max}. For a 1D harmonic trap, momenta and positions play a symmetric role, and hence the joint statistics of momenta are identical to that of the positions. In particular, p_{max}, as x_{max}, is distributed according to the Tracy-Widom distribution. Here we show that novel "momentum edge statistics" emerge when the curvature of the potential vanishes, i.e., for "flat traps" near their minimum, with V(x)∼x^{2n} and n>1. These are based on generalizations of the Airy kernel that we obtain explicitly. The fluctuations of p_{max} are governed by new universal distributions determined from the nth member of the second Painlevé hierarchy of nonlinear differential equations, with connections to multicritical random matrix models. Finite temperature extensions and possible experimental signatures in cold atoms are discussed.

Correlations of occupation numbers in the canonical ensemble and application to a Bose-Einstein condensate in a one-dimensional harmonic trap

We study statistical properties of $N$ non-interacting identical bosons or fermions in the canonical ensemble. We derive several general representations for the $p$-point correlation function of occupation numbers $\overline{n_1\cdots n_p}$. We demonstrate that it can be expressed as a ratio of two $p\times p$ determinants involving the (canonical) mean occupations $\overline{n_1}$, ..., $\overline{n_p}$, which can themselves be conveniently expressed in terms of the $k$-body partition functions (with $k\leq N$). We draw some connection with the theory of symmetric functions, and obtain an expression of the correlation function in terms of Schur functions. Our findings are illustrated by revisiting the problem of Bose-Einstein condensation in a 1D harmonic trap, for which we get analytical results. We get the moments of the occupation numbers and the correlation between ground state and excited state occupancies. In the temperature regime dominated by quantum correlations, the distribution of the ground state occupancy is shown to be a truncated Gumbel law. The Gumbel law, describing extreme value statistics, is obtained when the temperature is much smaller than the Bose-Einstein temperature.

Scattering of particles in the presence of harmonic confinement perturbed by a complex absorbing potential

A system of two interacting atoms confined in 1D harmonic trap and perturbed by an absorbing boundary potential is studied using the Lippmann–Schwinger formalism. The atom–atom interaction potential was considered as a nonlocal separable model. The perturbed absorbing boundary potential was also assumed in the form of Scarf II complex absorbing potential. The model is used for the study of 1D optical lattices that support the trapping of a pair atom within a unit cell. Moreover, it allows to describe the scattering particles in a tight smooth trapping surface and to analyze the bound and resonance states. The analytical expressions for wavefunctions and transition matrix as well as the absorption probabilities are calculated. A demonstration of how the complex absorbing potential affecting the bound states and resonances of particles confined in a harmonic trap is described.

Spectral properties and breathing dynamics of a few-body Bose–Bose mixture in a 1D harmonic trap

We investigate a few-body mixture of two bosonic components, each consisting of two particles confined in a quasi one-dimensional harmonic trap. By means of exact diagonalization with a correlated basis approach we obtain the low-energy spectrum and eigenstates for the whole range of repulsive intra- and inter-component interaction strengths. We analyse the eigenvalues as a function of the inter-component coupling, covering hereby all the limiting regimes, and characterize the behaviour in-between these regimes by exploiting the symmetries of the Hamiltonian. Provided with this knowledge we study the breathing dynamics in the linear-response regime by slightly quenching the trap frequency symmetrically for both components. Depending on the choice of interactions strengths, we identify 1 to 3 monopole modes besides the breathing mode of the centre of mass coordinate. For the uncoupled mixture each monopole mode corresponds to the breathing oscillation of a specific relative coordinate. Increasing the inter-component coupling first leads to multi-mode oscillations in each relative coordinate, which turn into single-mode oscillations of the same frequency in the composite-fermionization regime.

Optimal Power and Efficiency of Quantum Thermoacoustic Micro-cycle Working in 1D Harmonic Trap

Thermoacoustic engines (including heat engines and refrigerators) are energy conversion devices without moving part. They have great potential in aviation, new energy utilization, power technology, refrigerating and cryogenics. The thermoacoustic parcels, which compose the working fluid of a thermoacoustic engine, oscillate within the sound channel with a temperature gradient. The thermodynamic foundation of a thermoacoustic engine is the thermoacoustic micro-cycle (TAMC). In this paper, the theory of quantum mechanics is applied to the study of the actual thermoacoustic micro-cycle for the first time. A quantum mechanics model of the TAMC working in a 1D harmonic trap, which is named as a quantum thermoacoustic micro-cycle (QTAMC), is established. The QTAMC is composed of two constant force processes connected by two straight line processes. Analytic expressions of the power output and the efficiency for QTAMC have been derived. The effects of the trap width and the temperature amplitude on the power output and the thermal efficiency have been discussed. Some optimal characteristic curves of power output versus efficiency are plotted, and then the optimization region of QTAMC is given in this paper. The results obtained here not only enrich the thermoacoustic theory but also expand the application of quantum thermodynamics.

Fermionized Dipolar Bosons Trapped in a Harmonic Trap

We explore entanglement properties of systems of identical dipolar bosons confined in a 1D harmonic trap by using explicitly correlated Jastrow-type wavefunctions. Results for the linear entropy in dependence on the dimensionless coupling and the number of particles are provided and discussed.

A model study on a pair of trapped particles interacting with an arbitrary effective range

We study the effects of the effective range of interaction on the eigenvalues and eigenstates of two particles confined in a three-dimensional (3D) isotropic as well as one- or quasi-one dimensional harmonic (1D) traps. For this we employ model potentials which mimic finite-range s-wave interactions over a wide range of s-wave scattering length as including the unitarity limits . Our results show that when the range is larger than the 3D or 1D harmonic oscillator length scale, the eigenvalues and eigenstates are nearly similar to those of noninteracting two particles in the 3D or 1D trap, respectively. In case of 3D, we find that when the range goes to zero, the results of contact potential as derived by Busch et al (1998 Foundations of Physics 28 549) are reproduced. However, in the case of 1D, such reproducibility does not occur as the range goes to zero. We have calculated the eigenvalues and eigenstates in a 1D harmonic trap taking one dimensional finite-range model potential. We have also calculated the bound state properties of two particles confined in a highly anisotropic quasi-1D trap taking three-dimensional finite-range model potential, and examined whether these quasi-1D results approach towards 1D ones as the aspect ratio η of the radial to axial frequency of the trap increases. We find that if the range is very small compared to the axial size of the trap, then one can reach 1D regime for . However, for a large range, one can almost get 1D results for smaller values of η. This study will be important for the exploration of two-body or many body physics of trapped ultracold atoms interacting with narrow Feshbach resonance for which the effective range can be large.

Quantum breathing dynamics of ultracold bosons in one-dimensional harmonic traps: Unraveling the pathway from few- to many-body systems

Following a `bottom-up approach' in understanding many-particle effects and dynamics we provide a systematic ab initio study of the dependence of the breathing dynamics of ultracold bosons in a 1D harmonic trap on the number of bosons ranging from few to many. To this end, we employ the Multi-Layer Multi-Configuration Time-Dependent Hartree method for Bosons (ML-MCTDHB) which has been developed very recently [S. Kr\"onke, L. Cao, O. Vendrell and P. Schmelcher. {\it New J. Phys.} {\bf 15}, 063018 (2013)]. The beating behavior for two bosons is found numerically and consequently explained by an analytical approach. Drawing on this, we show how to compute the complete breathing mode spectrum in this case. We examine how the two-mode breathing behavior of two bosons evolves to the single-frequency behavior of the many-particle limit when adding more particles. In the limit of many particles, we numerically study the dependence of the breathing mode frequency on both the interaction strength as well as on the particle number. We provide an estimate for the parameter region in which Gross-Pitaevskii theory is well applicable.

Two interacting fermions in a one-dimensional harmonic trap: Matching the Bethe ansatz and variational approaches

In this work, combining the Bethe ansatz approach with the variational principle, we calculate the ground state energy of the relative motion of a system of two fermions with spin up and down interacting via a delta-function potential in a 1D harmonic trap. Our results show good agreement with the analytical solution of the problem, and provide a starting point for the investigation of more complex few-body systems where no exact theoretical solution is available.

Quantum Otto engine of a two-level atom with single-mode fields

We establish a quantum Otto engine (QOE) of a two-level atom, which is confined in a one-dimensional (1D) harmonic trap and is coupled to single-mode radiation fields. Besides two adiabatic processes, the QOE cycle consists of two isochoric processes, along one of which the two-level atom as the working substance interacts with a single-mode radiation field. Based on the semigroup approach, we derive the time for completing any adiabatic process and then present a performance analysis of the heat engine model. Furthermore, we generalize the results to the performance optimization for a QOE of a single two-level atom trapped in a 1D power-law potential. Our result shows that the efficiency at maximum power output is dependent on the trap exponent θ but is independent of the energy spectrum index σ.

Quantum criticality and universal scaling of strongly attractive spin-imbalanced Fermi gases in a one-dimensional harmonic trap

We investigate thermodynamics and quantum criticality of strongly attractive Fermi gases confined in a one-dimensional (1D) harmonic trap. Finite temperature density profiles, entropy, compressibility and susceptibility of the trapped gas are studied using analytic results for the thermodynamics within the local density approximation. We demonstrate that current experiments are capable of measuring universal Tomonaga-Luttinger liquid physics and quantum criticality of 1D strongly interacting Fermi gases. The results provide insights on recent measurements of key features of the phase diagram of a spin-imbalanced atomic Fermi gas [Liao et al., Nature 467, 567 (2010)] and point to further study of quantum critical phenomena in ultracold atomic Fermi gases.

Bose-Einstein condensation in dark power-law laser traps

We investigate theoretically an original route to achieve Bose-Einstein condensation using dark power-law laser traps. We propose to create such traps with two crossing blue-detuned Laguerre-Gaussian optical beams. Controlling their azimuthal order $\ensuremath{\ell}$ allows for the exploration of a multitude of power-law trapping situations in one, two, and three dimensions, ranging from the usual harmonic trap to an almost square-well potential, in which a quasihomogeneous Bose gas can be formed. The usual cigar-shaped and disk-shaped Bose-Einstein condensates obtained in a 1D or 2D harmonic trap take the generic form of a ``finger'' or of a ``hockey puck'' in such Laguerre-Gaussian traps. In addition, for a fixed atom number, higher transition temperatures are obtained in such configurations when compared with a harmonic trap of the same volume. This effect, which results in a substantial acceleration of the condensation dynamics, requires a better but still reasonable focusing of the Laguerre-Gaussian beams.

Spin ½ Fermions in 1D Harmonic Trap with Repulsive Delta Function Interparticle Interaction

Following the general theory developed in our earlier papers I [Chin. Phys. Lett. 26 (2009) 120506] and II [Chin. Phys. Lett. 27 (2010) 020506] for Bosons, we extend the investigation to trapped spin ½ Fermions.

Spinless Bosons in a 1D Harmonic Trap with Repulsive Delta Function Interparticle Interaction II: Numerical Solutions

Following the general theory developed in paper I [Chin. Phys. Lett. 26 (2009) 120506] we present here the numerical results for an N spinless boson system in a 1D harmonic trap with repulsive delta function interparticle interaction as N → ∞, with √N/g = fixed.

Ground State Energy for Fermions in a 1D Harmonic Trap with Delta Function Interaction

Conjectures are made for the ground state energy of a large spin 1/2 Fermion system trapped in a 1D harmonic trap with delta function interaction. States with different spin J are separately studied. The Thomas–Fermi method is used as an effective test for the conjecture.

Spinless Bosons in a 1D Harmonic Trap with Repulsive Delta Function Interparticle Interaction I: General Theory

In these two papers, we solve the N body 1D harmonically trapped spinless Boson problem with repulsive δ function interaction in the limit N → ∞. The general theory is given in paper I and the numerical solutions will be given in paper II.