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arXiv:1307.1667 · Physical Review A

Excitation spectra of many-body systems by linear response: General theory and applications to trapped condensates

Julian Grond, Alexej I. Streltsov, Axel U. J. Lode, Kaspar Sakmann, Lorenz S. Cederbaum, Ofir E. Alon

Published Aug 12, 2013• 33 citations
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Abstract

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.

DOI
10.1103/physreva.88.023606

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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 \ensuremathδ\ensuremath{\delta}\ensuremathδ 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.
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.
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.
Linear response of simple (i.e., condensed) Bose-Einstein condensates is known to lead to the Bogoliubov--de Gennes equations. Here, we derive linear response for fragmented Bose-Einstein condensates, i.e., for the case where the many-body wave function is not a product of one, but of several single-particle states (orbitals). This gives one access to excitation spectra and response amplitudes of systems beyond the Gross-Pitaevskii description. Our approach is based on the number-conserving variational time-dependent mean-field theory [Alon, Streltsov, and Cederbaum, Phys. Lett. A 362, 453 (2007)], which describes the time evolution of best-mean-field states. Correspondingly, we call our linear-response theory for fragmented states LR-BMF. In the derivation it follows naturally that excitations are orthogonal to the ground-state orbitals. As applications excitation spectra of Bose-Einstein condensates in double-well potentials are calculated. Both symmetric and asymmetric double wells are studied for several interaction strengths and barrier heights. The cases of condensed and twofold fragmented ground states are compared. Interestingly, even in such situations where the response frequencies of the two cases are computed to be close to each other, which is the situation for the excitations well below the barrier, striking differences in the density response in momentum space are found. For excitations with an energy of the order of the barrier height, both the energies and the density response of condensed and fragmented systems are very different. In fragmented systems there is a class of ``swapped'' excitations where an atom is transferred to the neighboring well. The mechanism of its origin is discussed. In asymmetric wells, the response of a fragmented system is purely local (i.e., finite in either one or the other well) with different frequencies for the left and right fragments. This finding is in stark contrast to that for condensed systems.
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.
In this work, we study many-body excitations of Bose-Einstein condensates trapped in periodic one-dimensional optical lattices. In particular, we investigate the impact of quantum depletion onto the structure of the low-energy spectrum and contrast the findings to the mean-field predictions of the Bogoliubov--de Gennes (BdG) equations. Accurate results for the many-body excited states are obtained by applying a linear-response theory atop the multiconfigurational time-dependent Hartree method for bosons equations of motion. We demonstrate for condensates in a triple well that even weak ground-state depletion of around 11%1 leads to visible many-body effects in the low-energy spectrum, which deviates substantially from the corresponding BdG spectrum. We further show that these effects also appear in larger systems with more lattice sites and particles, indicating the general necessity of a full many-body treatment.
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.