Abstract We discuss the multispecies time-dependent restricted-active-space (RAS) self-consistent-field theory, an ab initio wavefunction-based theory for mixtures of ultracold atomic and molecular gases. We present the general theory, based on the time-dependent variational principle, and derive the equations of motion. The theory captures in a time-dependent setting, via the specification of the RAS scheme, different levels of approximation from the mean-field to the full-configuration interaction approach. To assess its accuracy and to illustrate its ability to identify correlation effects at successive approximation levels, we apply the theory to compute the ground state energy of a Bose–Bose mixture interacting through a harmonic potential, for which the exact ground state energy is known analytically. We focus on the case of an ideal Bose gas interacting with a few impurities. The intra-species interaction between the impurities is relatively strong compared to the inter-species interaction between the impurities and the ideal non-interacting Bose gas. For this system, we find that an accurate description of the ground state necessitates the possibility of the theory to account for few-particle excitations out of the condensed phase; a situation well accounted for by the present RAS theory for mixtures; and not within reach for approaches not incorporating orbital-restriction schemes.
Abstract We discuss the multispecies time-dependent restricted-active-space (RAS) self-consistent-field theory, an ab initio wavefunction-based theory for mixtures of ultracold atomic and molecular gases. We present the general theory, based on the time-dependent variational principle, and derive the equations of motion. The theory captures in a time-dependent setting, via the specification of the RAS scheme, different levels of approximation from the mean-field to the full-configuration interaction approach. To assess its accuracy and to illustrate its ability to identify correlation effects at successive approximation levels, we apply the theory to compute the ground state energy of a Bose–Bose mixture interacting through a harmonic potential, for which the exact ground state energy is known analytically. We focus on the case of an ideal Bose gas interacting with a few impurities. The intra-species interaction between the impurities is relatively strong compared to the inter-species interaction between the impurities and the ideal non-interacting Bose gas. For this system, we find that an accurate description of the ground state necessitates the possibility of the theory to account for few-particle excitations out of the condensed phase; a situation well accounted for by the present RAS theory for mixtures; and not within reach for approaches not incorporating orbital-restriction schemes.
We develop an ab initio time-dependent wavefunction based theory for the description of a many-body system of cold interacting bosons. Like the multi-configurational time-dependent Hartree method for bosons (MCTDHB), the theory is based on a configurational interaction Ansatz for the many-body wavefunction with time-dependent self-consistent-field orbitals. The theory generalizes the MCTDHB method by incorporating restrictions on the active space of the orbital excitations. The restrictions are specified based on the physical situation at hand. The equations of motion of this time-dependent restricted-active-space self-consistent-field (TD-RASSCF) theory are derived. The similarity between the formal development of the theory for bosons and fermions is discussed. The restrictions on the active space allow the theory to be evaluated under conditions where other wavefunction based methods due to exponential scaling in the numerical effort cannot, and to clearly identify the excitations that are important for an accurate description, significantly beyond the mean-field approach. For ground state calculations we find it to be important to allow a few particles to have the freedom to move in many orbitals, an insight facilitated by the flexibility of the restricted-active-space Ansatz . Moreover, we find that a high accuracy can be obtained by including only even excitations in the many-body self-consistent-field wavefunction. Time-dependent simulations of harmonically trapped bosons subject to a quenching of their noncontact interaction, show failure of the mean-field Gross-Pitaevskii approach within a fraction of a harmonic oscillation period. The TD-RASSCF theory remains accurate at much reduced computational cost compared to the MCTDHB method. Exploring the effect of changes of the restricted-active-space allows us to identify that even self-consistent-field excitations are mainly responsible for the accuracy of the method.
We study a Bose–Fermi mixture within the framework of the mean-field theory, including three possible regimes for the fermionic species: fully polarized, BCS, and unitarity. Starting from the 3D description and using the variational approximation (VA), we derive one-dimensional and two-dimensional systems of equations, under the corresponding confining potentials. This method produces a pair of nonlinear Schrödinger equations coupled to algebraic equations for the transverse widths of the confined state. The equations incorporate interactions between atoms of the same species and between the species, assuming that the latter can be manipulated by means of the Feshbach resonance. As an application, we explore spatial density correlations in the ground state (GS) between the species, concluding that they strongly depend on the sign and strength of the inter-species interaction. Also studied are the dynamics of the mixture in a vicinity of the GS and the corresponding spatiotemporal inter-species correlation. The correlations are strongly affected by the fermionic component, featuring the greatest variation in the unitary regime. Results produced by the VA are verified by comparison with full numerical solutions.
This PhD thesis contains new results on the mathematical study of Bose-Einstein condensation and the main part of it is devoted to mixtures of condensates, i.e., systems composed of multiple bosonic species in interaction. We prove the validity of effective ground state theories for mixtures in the Gross-Pitaevskii and mean-field regime. We show that the ground state energy asymptotics, in the large-N limit, is captured by the minimum of a suitable one-body functional. Moreover, we prove that in the ground state all species exhibit Bose-Einstein condensation onto the minimizer of that functional. For mixtures in the mean-field regime, we provide a rigorous justification of Bogoliubov’s theory. This is done by computing the contribution to the ground state energy which is due to excited particles. We also prove a norm approximation for the ground state vector, in the Fock space norm. From the time-dependent viewpoint, we prove for the first time the validity of the effective equations that were previously known due to heuristic physical arguments, and that are confirmed by robust experimental evidence. Our results show that, for mixtures in the Gross-Pitaevskii and mean-field regime, the effective dynamics is governed by a system of non-linear Schrödinger equations, one for each species of the mixture. In the final part of the thesis we present additional results on problems and models related to the study on mixtures. We were able to derive the effective dynamics for spinor- and pseudo-spinor condensates. The equations that we obtain are precisely those of modern experiments with ultra-cold spin bosons. We also show that the mean-field model provide a time-dependent control of condensation that is very accurate for the typical duration times of experiments. A further result is the global well-posedness in the energy space of the singular Hartree equation. Last, we present new remarks on the adaptation of known techniques that one needs in order to prove the derivation of the magnetic Gross-Pitaevskii equation.
We study a Bose-Fermi mixture within the framework of the mean-field theory, including three possible regimes for the fermionic species: fully polarized, BCS, and unitarity. Starting from the 3D description and using the variational approximation (VA), we derive 1D and 2D systems of equations, under the corresponding confining potentials. This method produces a pair of nonlinear Schr\"{o}dinger (NLS) equations coupled to algebraic equations for the transverse widths of the confined state. The equations incorporate interactions between atoms of the same species and between the species, assuming that the latter can be manipulated by means of the Feshbach resonance (FR). As an application, we explore spatial density correlations in the ground state (GS) between the species, concluding that they strongly depend on the sign and strength of the inter-species interaction. Also studied are the dynamics of the mixture in a vicinity of the GS and the corresponding spatiotemporal inter-species correlation. The correlations are strongly affected by the fermionic component, featuring the greatest variation in the unitary regime. Results produced by the VA are verified by comparison with full numerical solutions.
A time-dependent multiconfigurational self-consistent field theory is presented to describe the many-body dynamics of a gas of identical bosonic atoms confined to an external trapping potential at zero temperature from first principles. A set of generalized evolution equations are developed, through the time-dependent variational principle, which account for the complete and self-consistent coupling between the expansion coefficients of each configuration and the underlying one-body wave functions within a restricted two state Fock space basis that includes the full effects of the condensate's mean field as well as atomic correlation. The resulting dynamical equations are a classical Hamiltonian system and, by construction, form a well-defined initial value problem. They are implemented in an efficient numerical algorithm. An example is presented, highlighting the generality of the theory, in which the ballistic expansion of a fragmented condensate ground state is compared to that of a macroscopic quantum superposition state, taken here to be a highly entangled number state, upon releasing the external trapping potential. Strikingly different many-body matter-wave dynamics emerge in each case, accentuating the role of both atomic correlation and mean-field effects in the two condensate states.
Abstract We systematically investigate and illustrate the complete ground-state phase diagram for a one-dimensional, three-species mixture of a few repulsively interacting bosons trapped harmonically. To numerically obtain the solutions to the many-body Schrödinger equation, we employ the improved Exact Diagonalization method (Anh-Tai et al 2023 SciPost Phys. 15 048), which is capable of treating strongly-correlated few-body systems from first principles in an efficiently truncated Hilbert space. We present our comprehensive results for all possible combinations of intra- and interspecies interactions in the extreme limits that are either the ideal limit (g = 0) or close to the hard-core limit ( g → ∞ ). These results show the emergence of unique ground-state properties related to correlations, coherence and spatial localization stemming from strongly repulsive interactions.