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Quantum Monte Carlo Simulations of Ultracold Bosons in a Double Well

Joel Corbo Joel Corbo

Published Jan 1, 2013• 1 citation
OpenAlex

Abstract

We present path integral ground state (PIGS) quantum Monte Carlo calculations for the ground state (T = 0) properties of repulsively interacting bosons in a three-dimensional external double well potential over a range of interaction strengths and potential parameters. We focus our calculations on ground state number statistics and the one-body density matrix in order to understand the level of squeezing and fragmentation that the system exhibits as a function of interaction strength, and we compare our PIGS results to both a standard and an improved two-mode model and a recently-proposed eight-mode model. In general, the various models agree with the PIGS simulations for weak interactions, but the full quantum Monte Carlo treatment is required to correctly predict the amount of squeezing and fragmentation exhibited for strong interactions. One novel and somewhat surprising result from our simulations involves the relationship between squeezing and interaction strength: rather than a monotonic relationship between these quantities, we find that for certain double well barrier heights the squeezing increases as a function of interaction strength until it reaches a maximum, after which it decreases again. We also see a similar relationship between fragmentation and interaction strength. We propose novel physical mechanisms to account for this behavior.

Related papers

Number-squeezed and fragmented states of strongly interacting bosons in a double well

Joel Corbo Joel Corbo, Jonathan L. DuBois, K. Birgitta Whaley

We present a systematic study of the phenomena of number squeezing and fragmentation for a repulsive Bose-Einstein condensate (BEC) in a three-dimensional double-well potential over a range of interaction strengths and barrier heights, including geometries that exhibit appreciable overlap in the one-body wave functions localized in the left and right wells. We compute the properties of the condensate with numerically exact, full-dimensional path-integral ground-state (PIGS) quantum Monte Carlo simulations and compare with results obtained from using two- and eight-mode truncated basis models. The truncated basis models are found to agree with the numerically exact PIGS simulations for weak interactions, but fail to correctly predict the amount of number squeezing and fragmentation exhibited by the PIGS simulations for strong interactions. We find that both number squeezing and fragmentation of the BEC show nonmonotonic behavior at large values of interaction strength $a$. The number squeezing shows a universal scaling with the product of number of particles and interaction strength ($Na$), but no such universal behavior is found for fragmentation. Detailed analysis shows that the introduction of repulsive interactions not only suppresses number fluctuations to enhance number squeezing, but can also enhance delocalization across wells and tunneling between wells, each of which may suppress number squeezing. This results in a dynamical competition whose resolution shows a complex dependence on all three physical parameters defining the system: interaction strength, number of particles, and barrier height.

arXiv:1404.2250•Physical Review A•Nov 22, 2017•3 citations

Squeezed and fragmented states of strongly interacting bosons in a double well. Part I: Theoretical models

Joel Christopher Corbo, Jonathan L. DuBois, K. Birgitta Whaley

We present a systematic study of the phenomena of squeezing and fragmentation for a BoseEinstein condensate (BEC) in a three dimensional double well potential over a range of interaction strengths and barrier heights. We analyze the properties of the condensate in the context of both a standard and an improved two-mode model and a recently-proposed eight-mode model. One fundamental dierence between this and previous two-mode work is that we explicitly examine well geometries that exhibit appreciable overlap in the one-body wavefunctions localized in the left and right wells rather than assuming that they are nearly isolated from each other. One particularly surprising observation that we have made in our study of these models is the existence of an interaction strength that maximizes squeezing, which is in stark contrast to the predictions of the two-mode model with no overlap between the wells for which squeezing monotonically increases with interaction strength. We explain this and other novel results through a careful study of the ground state properties produced by various terms in the model Hamiltonians and the relationship between the strength of these terms and the system parameters, such as well geometry and atomic interaction strength.

•Apr 8, 2014•0 citations

Ground-State Properties of Charged Bosons Confined in a One-Dimensional Harmonic Double-Well Trap: Diffusion Monte Carlo Calculations

Jing Jin, Tang Yi

The diffusion Monte Carlo method is applied to study the ground-state properties of charged bosons in one dimension confined in a harmonic double-well trap. The particles interact repulsively through a Coulombic 1/ r potential. Numerical results show that the well separation has significant influence on the ground-state properties of the system. When the interaction of the system is weak, ground-state energy decreases with the increasing well separation and has a minimal value. If the well separation increases continually, the ground-state energy increases and approaches to a constant gradually. This effect will be abatable in the strong interacting system. In addition, by calculating the density of the systems for different interaction strengths with various well separations, we find that the density increases abnormally when the well separation is large at the centre of the system.

Chinese Physics Letters•Aug 23, 2007•1 citation

Effective three-body interactions for bosons in a double-well confinement

Jacek Dobrzyniecki, Xikun Li, Anne E. B. Nielsen +1 more

When describing the low-energy physics of bosons in a double-well potential with a high barrier between the wells and sufficiently weak atom-atom interactions, one can, to a good approximation, ignore the high-energy states and thereby obtain an effective two-mode model. Here we show that the regime in which the two-mode model is valid can be extended by adding an on-site three-body interaction term and a three-body interaction-induced tunneling term to the two-mode Hamiltonian. These terms effectively account for virtual transitions to the higher-energy states. We determine appropriate strengths of the three-body terms by an optimization of the minimal value of the wave-function overlap within a certain time window. Considering different initial states with three or four atoms, we find that the resulting model accurately captures the dynamics of the system for parameters where the two-mode model without the three-body terms is poor. We also investigate the dependence of the strengths of the three-body terms on the barrier height and the atom-atom interaction strength. The optimal three-body interaction strengths depend on the initial state of the system.

arXiv:1711.05100•Physical Review A•Jan 12, 2018•16 citations

Quantum Monte Carlo study of strongly interacting bosonic one-dimensional systems in periodic potentials

Krešimir Dželalija, Leandra Vranješ Markić

We present diffusion Monte Carlo (DMC) and path-integral Monte Carlo (PIMC) calculations of a one-dimensional Bose system with realistic interparticle interactions in a periodic external potential. Our main aim is to test the predictions of the Luttinger liquid (LL) theory, in particular with respect to the superfluid-Mott insulator transition at both zero and finite temperatures, in the predicted robust and fragile superfluid regimes. For that purpose, we present our results of the superfluid fraction ${\ensuremath{\rho}}_{s}/{\ensuremath{\rho}}_{0}$, the one-body density matrix, the two-body correlation functions, and the static structure factor. The DMC and PIMC results in the limit of very low temperature for ${\ensuremath{\rho}}_{s}/{\ensuremath{\rho}}_{0}$ agree, but the LL model for scaling ${\ensuremath{\rho}}_{s}/{\ensuremath{\rho}}_{0}$ does not fit the data well. The critical depth of the periodic potential is close to the values obtained for ultracold gases with different models of interaction, but with the same value of the bare LL parameter, demonstrating the universality of LL description. Algebraic decay of correlation functions is observed in the superfluid regime and exponential decay in the Mott-insulator one, as well as in all regimes at finite temperature for distances larger than a characteristic length.

arXiv:2001.07163•Physical Review A•Jan 22, 2020•0 citations

Bose-Einstein Condensation in a Double-Well System

Jesus Caloca

The two-site version of the Bose-Hubbard model can be used as a simplified model of N bosons confined to two quantum wells. We study the behavior of low-temperature Bose-Einstein condensates as a function of the strength of the repulsive interactions between the bosons and of the size of the quantum-mechanical tunneling between the quantum wells. The ground-state of weakly interacting bosons tends to be one in which almost all particles are in the same quantum-mechanical state (‘Bose-Einstein Condensate’, or ‘BEC’). It is found that even very small amounts of interwell tunneling lead to a BEC, but that zero tunneling leads to two separate BECs, one in each well (‘fragmentation’). We study systematically the transition between a single BEC comprised of bosons in both wells, and a fragmented condensate consisting of two BECs, one for each well. Our analysis will use matrix computational methods to solve the Schrödinger equation of the two-well boson system, and obtain the energies and wavefunctions as eigenvalues and eigenvectors of the Hamiltonian matrix. We will compare the results obtained with previous analyses, and against analytical approximations developed for the limiting case of a large number of bosons (large N).

•Jan 1, 2014•0 citations
We present path integral ground state (PIGS) quantum Monte Carlo calculations for the ground state (T = 0) properties of repulsively interacting bosons in a three-dimensional external double well potential over a range of interaction strengths and potential parameters. We focus our calculations on ground state number statistics and the one-body density matrix in order to understand the level of squeezing and fragmentation that the system exhibits as a function of interaction strength, and we compare our PIGS results to both a standard and an improved two-mode model and a recently-proposed eight-mode model. In general, the various models agree with the PIGS simulations for weak interactions, but the full quantum Monte Carlo treatment is required to correctly predict the amount of squeezing and fragmentation exhibited for strong interactions. One novel and somewhat surprising result from our simulations involves the relationship between squeezing and interaction strength: rather than a monotonic relationship between these quantities, we find that for certain double well barrier heights the squeezing increases as a function of interaction strength until it reaches a maximum, after which it decreases again. We also see a similar relationship between fragmentation and interaction strength. We propose novel physical mechanisms to account for this behavior.
We present a systematic study of the phenomena of number squeezing and fragmentation for a repulsive Bose-Einstein condensate (BEC) in a three-dimensional double-well potential over a range of interaction strengths and barrier heights, including geometries that exhibit appreciable overlap in the one-body wave functions localized in the left and right wells. We compute the properties of the condensate with numerically exact, full-dimensional path-integral ground-state (PIGS) quantum Monte Carlo simulations and compare with results obtained from using two- and eight-mode truncated basis models. The truncated basis models are found to agree with the numerically exact PIGS simulations for weak interactions, but fail to correctly predict the amount of number squeezing and fragmentation exhibited by the PIGS simulations for strong interactions. We find that both number squeezing and fragmentation of the BEC show nonmonotonic behavior at large values of interaction strength aaa. The number squeezing shows a universal scaling with the product of number of particles and interaction strength (NaNaNa), but no such universal behavior is found for fragmentation. Detailed analysis shows that the introduction of repulsive interactions not only suppresses number fluctuations to enhance number squeezing, but can also enhance delocalization across wells and tunneling between wells, each of which may suppress number squeezing. This results in a dynamical competition whose resolution shows a complex dependence on all three physical parameters defining the system: interaction strength, number of particles, and barrier height.
We present a systematic study of the phenomena of squeezing and fragmentation for a BoseEinstein condensate (BEC) in a three dimensional double well potential over a range of interaction strengths and barrier heights. We analyze the properties of the condensate in the context of both a standard and an improved two-mode model and a recently-proposed eight-mode model. One fundamental dierence between this and previous two-mode work is that we explicitly examine well geometries that exhibit appreciable overlap in the one-body wavefunctions localized in the left and right wells rather than assuming that they are nearly isolated from each other. One particularly surprising observation that we have made in our study of these models is the existence of an interaction strength that maximizes squeezing, which is in stark contrast to the predictions of the two-mode model with no overlap between the wells for which squeezing monotonically increases with interaction strength. We explain this and other novel results through a careful study of the ground state properties produced by various terms in the model Hamiltonians and the relationship between the strength of these terms and the system parameters, such as well geometry and atomic interaction strength.
The diffusion Monte Carlo method is applied to study the ground-state properties of charged bosons in one dimension confined in a harmonic double-well trap. The particles interact repulsively through a Coulombic 1/ r potential. Numerical results show that the well separation has significant influence on the ground-state properties of the system. When the interaction of the system is weak, ground-state energy decreases with the increasing well separation and has a minimal value. If the well separation increases continually, the ground-state energy increases and approaches to a constant gradually. This effect will be abatable in the strong interacting system. In addition, by calculating the density of the systems for different interaction strengths with various well separations, we find that the density increases abnormally when the well separation is large at the centre of the system.
When describing the low-energy physics of bosons in a double-well potential with a high barrier between the wells and sufficiently weak atom-atom interactions, one can, to a good approximation, ignore the high-energy states and thereby obtain an effective two-mode model. Here we show that the regime in which the two-mode model is valid can be extended by adding an on-site three-body interaction term and a three-body interaction-induced tunneling term to the two-mode Hamiltonian. These terms effectively account for virtual transitions to the higher-energy states. We determine appropriate strengths of the three-body terms by an optimization of the minimal value of the wave-function overlap within a certain time window. Considering different initial states with three or four atoms, we find that the resulting model accurately captures the dynamics of the system for parameters where the two-mode model without the three-body terms is poor. We also investigate the dependence of the strengths of the three-body terms on the barrier height and the atom-atom interaction strength. The optimal three-body interaction strengths depend on the initial state of the system.
We present diffusion Monte Carlo (DMC) and path-integral Monte Carlo (PIMC) calculations of a one-dimensional Bose system with realistic interparticle interactions in a periodic external potential. Our main aim is to test the predictions of the Luttinger liquid (LL) theory, in particular with respect to the superfluid-Mott insulator transition at both zero and finite temperatures, in the predicted robust and fragile superfluid regimes. For that purpose, we present our results of the superfluid fraction ${\ensuremath{\rho}}_{s}/{\ensuremath{\rho}}_{0}$, the one-body density matrix, the two-body correlation functions, and the static structure factor. The DMC and PIMC results in the limit of very low temperature for ${\ensuremath{\rho}}_{s}/{\ensuremath{\rho}}_{0}$ agree, but the LL model for scaling ${\ensuremath{\rho}}_{s}/{\ensuremath{\rho}}_{0}$ does not fit the data well. The critical depth of the periodic potential is close to the values obtained for ultracold gases with different models of interaction, but with the same value of the bare LL parameter, demonstrating the universality of LL description. Algebraic decay of correlation functions is observed in the superfluid regime and exponential decay in the Mott-insulator one, as well as in all regimes at finite temperature for distances larger than a characteristic length.
The two-site version of the Bose-Hubbard model can be used as a simplified model of N bosons confined to two quantum wells. We study the behavior of low-temperature Bose-Einstein condensates as a function of the strength of the repulsive interactions between the bosons and of the size of the quantum-mechanical tunneling between the quantum wells. The ground-state of weakly interacting bosons tends to be one in which almost all particles are in the same quantum-mechanical state (‘Bose-Einstein Condensate’, or ‘BEC’). It is found that even very small amounts of interwell tunneling lead to a BEC, but that zero tunneling leads to two separate BECs, one in each well (‘fragmentation’). We study systematically the transition between a single BEC comprised of bosons in both wells, and a fragmented condensate consisting of two BECs, one for each well. Our analysis will use matrix computational methods to solve the Schrödinger equation of the two-well boson system, and obtain the energies and wavefunctions as eigenvalues and eigenvectors of the Hamiltonian matrix. We will compare the results obtained with previous analyses, and against analytical approximations developed for the limiting case of a large number of bosons (large N).