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arXiv:1407.7678 · Journal of Physics Conference Series

Many-body excitation spectra of trapped bosons with general interaction by linear response

Ofir E. Alon

Published Mar 18, 2015• 11 citations
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Abstract

The linear-response theory of the multiconfigurational time-dependent Hartree for bosons method (LR-MCTDHB) for computing many-body excitations of trapped Bose-Einstein condensates [Phys. Rev. A 88 , 023606 (2013); J. Chem. Phys. 140 , 034108 (2014)] is implemented, for the first time, for systems with general interparticle interaction. This allows us to investigate the many-body excitation spectrum of interacting bosons with, for instance, a long-range interaction. Illustrative numerical examples for repulsive and attractive bosons are provided. The LR-MCTDHB theory is capable of identifying all excitations, including the excitations which are not unraveled within Bogoliubov-de Gennes equations. The theory is herewith benchmarked against the exactly-solvable one-dimensional harmonic-interaction model. As a complementary result, using a complex transformation, we represent LR-MCTDHB in a compact block-diagonal form, opening up thereby an avenue for treating larger many-body systems. We expect the LR-MCTDHB theory and its implementation for general interparticle interaction to provide a proved probe into the many-body excitations involved in the out-ofequilibrium dynamics of trapped interacting bosons.

DOI
10.1088/1742-6596/594/1/012039

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The linear-response theory of the multiconfigurational time-dependent Hartree for bosons method (LR-MCTDHB) for computing many-body excitations of trapped Bose-Einstein condensates [Phys. Rev. A 88 , 023606 (2013); J. Chem. Phys. 140 , 034108 (2014)] is implemented, for the first time, for systems with general interparticle interaction. This allows us to investigate the many-body excitation spectrum of interacting bosons with, for instance, a long-range interaction. Illustrative numerical examples for repulsive and attractive bosons are provided. The LR-MCTDHB theory is capable of identifying all excitations, including the excitations which are not unraveled within Bogoliubov-de Gennes equations. The theory is herewith benchmarked against the exactly-solvable one-dimensional harmonic-interaction model. As a complementary result, using a complex transformation, we represent LR-MCTDHB in a compact block-diagonal form, opening up thereby an avenue for treating larger many-body systems. We expect the LR-MCTDHB theory and its implementation for general interparticle interaction to provide a proved probe into the many-body excitations involved in the out-ofequilibrium dynamics of trapped interacting bosons.
We derive a general linear-response many-body theory capable of computing excitation spectra of trapped interacting bosonic systems, e.g., depleted and fragmented Bose-Einstein condensates (BECs). To obtain the linear-response equations we linearize the multiconfigurational time-dependent Hartree for bosons (MCTDHB) method, which provides a self-consistent description of many-boson systems in terms of orbitals and a state vector (configurations), and is in principle numerically exact. The derived linear-response many-body theory, which we term LR-MCTDHB, is applicable to systems with interaction potentials of general form. For the special case of a $\ensuremath{\delta}$ interaction potential we show explicitly that the response matrix has a very appealing bilinear form, composed of separate blocks of submatrices originating from contributions of the orbitals, the state vector (configurations), and off-diagonal mixing terms. We further give expressions for the response weights and density response. We introduce the notion of the type of excitations, useful in the study of the physical properties of the equations. From the numerical implementation of the LR-MCTDHB equations and solution of the underlying eigenvalue problem, we obtain excitations beyond available theories of excitation spectra, such as the Bogoliubov--de Gennes (BdG) equations. The derived theory is first applied to study BECs in a one-dimensional harmonic potential. The LR-MCTDHB method contains the BdG excitations and, also, predicts a plethora of additional many-body excitations which are out of the realm of standard linear response. In particular, our theory describes the exact energy of the higher harmonic of the first (dipole) excitation not contained in the BdG theory. We next study a BEC in a very shallow one-dimensional double-well potential. We find with LR-MCTDHB low-lying excitations which are not accounted for by BdG, even though the BEC has only little fragmentation and, hence, the BdG theory is expected to be valid. The convergence of the LR-MCTDHB theory is assessed by systematically comparing the excitation spectra computed at several different levels of theory.
This review explores the dynamics and the low-energy excitation spectra of Bose-Einstein condensates (BECs) of interacting bosons in external potential traps putting particular emphasis on the emerging many-body effects beyond mean-field descriptions. To do so, methods have to be used that, in principle, can provide numerically exact results for both the dynamics and the excitation spectra in a systematic manner. Numerically exact results for the dynamics are presented employing the well-established multicongurational time-dependent Hartree for bosons (MCTDHB) method. The respective excitation spectra are calculated utilizing the more recently introduced linear-response theory atop it (LR-MCTDHB). The latter theory gives rise to an, in general, non-hermitian eigenvalue problem. The theory and its newly developed implementation are described in detail and benchmarked towards the exactly-solvable harmonic-interaction model. Several applications to BECs in one- and two-dimensional potential traps are discussed. With respect to dynamics, it is shown that both the out-of-equilibrium tunneling dynamics and the dynamics of trapped vortices are of many-body nature. Furthermore, many-body effects in the excitation spectra are presented for BECs in different trap geometries. It is demonstrated that even for essentially-condensed systems, the spectrum of the lowest-in-energy excitations computed at the many-body level can differ substantially from the standard mean-field description. In general, it is shown that bosons carrying angular momentum are more sensitive to many-body effects than bosons without. These effects are present in both the dynamics and the excitation spectrum.
We employ the multiconfigurational time-dependent Hartree for bosons (MCTDHB) method to study excited states of interacting Bose-Einstein condensates confined by harmonic and double-well trap potentials. Two approaches to access excitations, one static and the other dynamic, are investigated and contrasted. In static simulations the low-lying excitations are computed by utilizing a linear-response theory constructed on top of a static MCTDHB solution (LR-MCTDHB). Complimentarily, we propose two dynamic protocols that address excitations by propagating the MCTDHB wave function. In particular, we investigate dipolelike oscillations induced by shifting the origin of the confining potential and breathinglike excitations by quenching the frequency of a parabolic part of the trap. To contrast static predictions and dynamic results we compute the time evolution and regard the respective Fourier transform of several local and nonlocal observables. Namely, we study the expectation value of the position operator $\ensuremath{\langle}x(t)\ensuremath{\rangle}$, its variance Var[x(t)]\mathrm{Var}[x(t)]Var[x(t)], and a local density computed at selected positions. We find that the variance is the most sensitive and informative quantity: Along with excitations it contains information about deexcitations even in a linear regime of the induced dynamics. The dynamic protocols are found to access the many-body excitations predicted by the static LR-MCTDHB approach.
The exactly solvable quantum many-particle model with harmonic one- and two-particle interaction terms is extended to include time dependency. We show that when the external trap potential and interparticle interaction have a time dependency, the numerically exact solutions of the corresponding time-dependent many-boson Schr\"odinger equation are still available. We use these exact solutions to benchmark the recently developed multiconfigurational time-dependent Hartree method for bosons (MCTDHB) [Phys. Rev. Lett. 99, 030402 (2007); Phys. Rev. A 77, 033613 (2008)]. In particular, we benchmark the MCTDHB method for (i) the ground state; (ii) the breathing many-body dynamics activated by a quench scenario where the interparticle interaction strength is suddenly turned on to a finite value; (iii) the nonequilibrium dynamic for driven scenarios where both the trap- and interparticle-interaction potentials are time-dependent. Excellent convergence of the ground state and dynamics is demonstrated. The great relevance of the self-consistency and time adaptivity, which are the intrinsic features of the MCTDHB method, is demonstrated by contrasting the MCTDHB predictions and those obtained within the standard full configuration interaction method spanning a Fock space of the same size, but utilizing as one-particle basis set the fixed-shape eigenstates of the one-particle potential. Connections of the model's results to ultracold Bose-Einstein condensed systems are addressed.
The evolution of Bose-Einstein condensates is amply described by the time-dependent Gross-Pitaevskii mean-field theory which assumes all bosons to reside in a single time-dependent one-particle state throughout the propagation process. In this work, we go beyond mean field and develop an essentially exact many-body theory for the propagation of the time-dependent Schr\"odinger equation of NNN interacting identical bosons. In our theory, the time-dependent many-boson wave function is written as a sum of permanents assembled from orthogonal one-particle functions, or orbitals, where both the expansion coefficients and the permanents (orbitals) themselves are time-dependent and fully determined according to a standard time-dependent variational principle. By employing either the usual Lagrangian formulation or the Dirac-Frenkel variational principle we arrive at two sets of coupled equations of motion, one for the orbitals and one for the expansion coefficients. The first set comprises of first-order differential equations in time and nonlinear integrodifferential equations in position space, whereas the second set consists of first-order differential equations with time-dependent coefficients. We call our theory multiconfigurational time-dependent Hartree for bosons, or MCTDHB(M)\text{MCTDHB}(M)MCTDHB(M), where MMM specifies the number of time-dependent orbitals used to construct the permanents. Numerical implementation of the theory is reported and illustrative numerical examples of many-body dynamics of trapped Bose-Einstein condensates are provided and discussed. The convergence of the method with a growing number MMM of orbitals is demonstrated in a specific example of four interacting bosons in a double well.
In order to solve the many-boson Schr\"odinger equation we utilize the multiconfigurational time-dependent Hartree method for bosons (MCTDHB). To be able to attack larger systems and/or to propagate the solution for longer times, we implement a parallel version of the MCTDHB method, thereby realizing the recently proposed [Streltsov et al., Phys. Rev. A 81, 022124 (2010)] idea on how to construct efficiently the result of the action of the Hamiltonian on a bosonic state vector. As an illustrative example of its own interest, we study the real-space dynamics of repulsive bosonic systems made of N=12N=12N=12, 51, and 3003 bosons in triple-well periodic potentials. The ground state of this system is threefold fragmented. By suddenly strongly distorting the trap potential, the system performs complex many-body quantum dynamics. At long times it reveals a tendency to an oscillatory behavior around a threefold fragmented state. These oscillations are strongly suppressed and damped by quantum depletions. In spite of the richness of the observed dynamics, the three time-adaptive orbitals of MCTDHB(M=3M=3M=3) are capable of describing the many-boson quantum dynamics of the system for short and intermediate times. For longer times, however, more self-consistent time-adaptive orbitals are needed to correctly describe the nonequilibrium many-body physics. The convergence of the MCTDHB(MMM) method with the number MMM of self-consistent time-dependent orbitals used is demonstrated.