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Método de Hartree multiconfiguracional para sistemas de muitas partículas

Alex Andriati

Published Mar 11, 2022• 0 citations
OpenAlex

Abstract

In the present thesis, the multiconfigurational time-dependent Hartree method is applied to study flow properties of bosons trapped in annular geometry. This physical system provides a suitable platform to investigate beyond meanfield features, since as one spatial dimensional problem, we can explore different interacting regimes, set a rather large range for parameter values without excessive concern about the computational cost and explore relevant concepts, such as superfluidity. As a very important part of the present thesis, the multiconfigurational time-dependent Hartree method for bosons (MCTDHB) is carefully described, providing all relevant derivations up to the final equations of motion, as well as a prospectus of the numerical approach. As such, this thesis also deals with topics of numerical analysis due to its method complexity. After the MCTDHB derivation and its numerical implementation are shown, the superfluid properties of a bosonic gas in annular geometry are studied with the most general approach proposed in reference [1], without any assumptions about the condensation status of the system. In addition, the MCTDHBs capability to provide correlation functions is exploited to characterize the loss of superfluid fraction due to a potential barrier. Within the same goal to study bosonic flow properties in a ring, persistent currents are also investigated, with two alternative derivations for the metastability condition as energy local minimum. Besides, the dynamical decay of the current is analyzed after quenching a potential barrier. Fluctuations of the number operator and probabilities are calculated along the time evolution to provide information about the quantum statistical distribution embedded in experimental measures, aiming to spot relevant information on image observation of flowing states.

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Multiconfigurational Hartree method for many-particle systems

Alex Andriati

In the present thesis, the multiconfigurational time-dependent Hartree method is applied to study flow properties of bosons trapped in annular geometry.This physical system provides a suitable platform to investigate beyond meanfield features, since as one spatial dimensional problem, we can explore di erent interacting regimes, set a rather large range for parameter values without excessive concern about the computational cost and explore relevant concepts, such as superfluidity.As a very important part of the present thesis, the multiconfigurational time-dependent Hartree method for bosons (MCTDHB) is carefully described, providing all relevant derivations up to the final equations of motion, as well as a prospectus of the numerical approach.As such, this thesis also deals with topics of numerical analysis due to its method complexity.After the MCTDHB derivation and its numerical implementation are shown, the superfluid properties of a bosonic gas in annular geometry are studied with the most general approach proposed in reference [1], without any assumptions about the condensation status of the system.In addition, the MCTDHB's capability to provide correlation functions is exploited to characterize the loss of superfluid fraction due to a potential barrier.Within the same goal to study bosonic flow properties in a ring, persistent currents are also investigated, with two alternative derivations for the metastability condition as energy local minimum.Besides, the dynamical decay of the current is analyzed after quenching a potential barrier.Fluctuations of the number operator and probabilities are calculated along the time evolution to provide information about the quantum statistical distribution embedded in experimental measures, aiming to spot relevant information on image observation of flowing states.

•Mar 11, 2022•0 citations

Multiconfigurational time-dependent Hartree method for bosons with internal degrees of freedom: Theory and composite fragmentation of multicomponent Bose-Einstein condensates

Axel U. J. Lode

In this paper, the multiconfigurational time-dependent Hartree for bosons (MCTDHB) method is derived for the case of $N$ identical bosons with internal degrees of freedom. The theory for bosons with internal degrees of freedom constitutes a generalization of the MCTDHB method that substantially enriches the many-body physics that can be described. We demonstrate that the numerically exact solution of the time-dependent many-body Schr\"odinger equation for interacting bosonic particles with internal degrees of freedom is now feasible. We report on the MCTDHB equations of motion for bosons with internal degrees of freedom and their implementation for a general many-body Hamiltonian with one-body and two-body terms, both of which may depend on the internal states of the considered particles and time. To demonstrate the capabilities of the theory and its software implementation integrated in the mctdh-x software, we apply MCTDHB to the emergence of fragmentation of parabolically trapped bosons with two internal states: we study the ground state of $N=100$ bosons as a function of the separation between the state-dependent minima of the two parabolic potentials. To quantify the coherence of the system, we compute its normalized first-order correlation function. We find that the coherence within each internal state of the atoms is maintained, while it is lost between the different internal states. This is a hallmark of a kind of fragmentation absent in bosons without internal structure. We term the emergent phenomenon ``composite fragmentation.''

arXiv:1602.05791•Physical Review A•Jun 1, 2016•98 citations

Multiconfigurational time-dependent Hartree method for fermions: Implementation, exactness, and few-fermion tunneling to open space

Elke Faßhauer, Axel U. J. Lode

We report on an implementation of the multiconfigurational time-dependent Hartree method (MCTDH) for spin-polarized fermions (MCTDHF). Our approach is based on a mapping for operators in Fock space that allows a compact and efficient application of the Hamiltonian and solution of the MCTDHF equations of motion. Our implementation extends, builds on, and exploits the recursive implementation of MCTDH for bosons (r-mctdhb) package. Together with r-mctdhb, the present implementation of MCTDHF forms the mctdh-x package. We benchmark the accuracy of the algorithm with the harmonic interaction model and a time-dependent generalization thereof. These models consider parabolically trapped particles that interact through a harmonic interaction potential. We demonstrate that MCTDHF is capable of solving the time-dependent many-fermion Schr\"odinger equation to an arbitrary degree of precision and can hence yield numerically exact results even in the case of Hamiltonians with time-dependent one-body and two-body potentials. We study the problem of two initially parabolically confined and charged fermions tunneling through a barrier to open space. We demonstrate the validity of a model proposed previously for the many-body tunneling to open space of bosonic particles with contact interactions [Proc. Natl. Acad. Sci. USA 109, 13521 (2012)]. The many-fermion tunneling can be built up from sequentially happening single-fermion tunneling processes. The characteristic momenta of these processes are determined by the chemical potentials of trapped subsystems of smaller particle numbers: The escaped fermions convert the different chemical potentials into kinetic energy. Using the two-body correlation function, we present a detailed picture of the sequentiality of the process and are able to tell tunneling from over-the-barrier escape.

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Multiconfigurational time-dependent Hartree method for bosons: Many-body dynamics of bosonic systems

Ofir E. Alon, Alexej I. Streltsov, L. S. Cederbaum

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 $N$ 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 $\text{MCTDHB}(M)$, where $M$ 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 $M$ of orbitals is demonstrated in a specific example of four interacting bosons in a double well.

arXiv:cond-mat/0703237•Physical Review A•Mar 14, 2008•355 citations

Accurate multi-boson long-time dynamics in triple-well periodic traps

Alexej I. Streltsov, Kaspar Sakmann, Ofir E. Alon +1 more

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 $N=12$, 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($M=3$) 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($M$) method with the number $M$ of self-consistent time-dependent orbitals used is demonstrated.

arXiv:0910.5916•Physical Review A•Apr 6, 2011•50 citations

The multi-layer multi-configuration time-dependent Hartree method for bosons: Theory, implementation, and applications

Lushuai Cao, Sven Krönke, Oriol Vendrell +1 more

We develop the multi-layer multi-configuration time-dependent Hartree method for bosons (ML-MCTDHB), a variational numerically exact ab initio method for studying the quantum dynamics and stationary properties of general bosonic systems. ML-MCTDHB takes advantage of the permutation symmetry of identical bosons, which allows for investigations of the quantum dynamics from few to many-body systems. Moreover, the multi-layer feature enables ML-MCTDHB to describe mixed bosonic systems consisting of arbitrary many species. Multi-dimensional as well as mixed-dimensional systems can be accurately and efficiently simulated via the multi-layer expansion scheme. We provide a detailed account of the underlying theory and the corresponding implementation. We also demonstrate the superior performance by applying the method to the tunneling dynamics of bosonic ensembles in a one-dimensional double well potential, where a single-species bosonic ensemble of various correlation strengths and a weakly interacting two-species bosonic ensemble are considered.

arXiv:1305.3862•The Journal of Chemical Physics•Oct 1, 2013•139 citations
In the present thesis, the multiconfigurational time-dependent Hartree method is applied to study flow properties of bosons trapped in annular geometry. This physical system provides a suitable platform to investigate beyond meanfield features, since as one spatial dimensional problem, we can explore different interacting regimes, set a rather large range for parameter values without excessive concern about the computational cost and explore relevant concepts, such as superfluidity. As a very important part of the present thesis, the multiconfigurational time-dependent Hartree method for bosons (MCTDHB) is carefully described, providing all relevant derivations up to the final equations of motion, as well as a prospectus of the numerical approach. As such, this thesis also deals with topics of numerical analysis due to its method complexity. After the MCTDHB derivation and its numerical implementation are shown, the superfluid properties of a bosonic gas in annular geometry are studied with the most general approach proposed in reference [1], without any assumptions about the condensation status of the system. In addition, the MCTDHBs capability to provide correlation functions is exploited to characterize the loss of superfluid fraction due to a potential barrier. Within the same goal to study bosonic flow properties in a ring, persistent currents are also investigated, with two alternative derivations for the metastability condition as energy local minimum. Besides, the dynamical decay of the current is analyzed after quenching a potential barrier. Fluctuations of the number operator and probabilities are calculated along the time evolution to provide information about the quantum statistical distribution embedded in experimental measures, aiming to spot relevant information on image observation of flowing states.
In the present thesis, the multiconfigurational time-dependent Hartree method is applied to study flow properties of bosons trapped in annular geometry.This physical system provides a suitable platform to investigate beyond meanfield features, since as one spatial dimensional problem, we can explore di erent interacting regimes, set a rather large range for parameter values without excessive concern about the computational cost and explore relevant concepts, such as superfluidity.As a very important part of the present thesis, the multiconfigurational time-dependent Hartree method for bosons (MCTDHB) is carefully described, providing all relevant derivations up to the final equations of motion, as well as a prospectus of the numerical approach.As such, this thesis also deals with topics of numerical analysis due to its method complexity.After the MCTDHB derivation and its numerical implementation are shown, the superfluid properties of a bosonic gas in annular geometry are studied with the most general approach proposed in reference [1], without any assumptions about the condensation status of the system.In addition, the MCTDHB's capability to provide correlation functions is exploited to characterize the loss of superfluid fraction due to a potential barrier.Within the same goal to study bosonic flow properties in a ring, persistent currents are also investigated, with two alternative derivations for the metastability condition as energy local minimum.Besides, the dynamical decay of the current is analyzed after quenching a potential barrier.Fluctuations of the number operator and probabilities are calculated along the time evolution to provide information about the quantum statistical distribution embedded in experimental measures, aiming to spot relevant information on image observation of flowing states.
In this paper, the multiconfigurational time-dependent Hartree for bosons (MCTDHB) method is derived for the case of NNN identical bosons with internal degrees of freedom. The theory for bosons with internal degrees of freedom constitutes a generalization of the MCTDHB method that substantially enriches the many-body physics that can be described. We demonstrate that the numerically exact solution of the time-dependent many-body Schr\"odinger equation for interacting bosonic particles with internal degrees of freedom is now feasible. We report on the MCTDHB equations of motion for bosons with internal degrees of freedom and their implementation for a general many-body Hamiltonian with one-body and two-body terms, both of which may depend on the internal states of the considered particles and time. To demonstrate the capabilities of the theory and its software implementation integrated in the mctdh-x software, we apply MCTDHB to the emergence of fragmentation of parabolically trapped bosons with two internal states: we study the ground state of N=100N=100N=100 bosons as a function of the separation between the state-dependent minima of the two parabolic potentials. To quantify the coherence of the system, we compute its normalized first-order correlation function. We find that the coherence within each internal state of the atoms is maintained, while it is lost between the different internal states. This is a hallmark of a kind of fragmentation absent in bosons without internal structure. We term the emergent phenomenon ``composite fragmentation.''
We report on an implementation of the multiconfigurational time-dependent Hartree method (MCTDH) for spin-polarized fermions (MCTDHF). Our approach is based on a mapping for operators in Fock space that allows a compact and efficient application of the Hamiltonian and solution of the MCTDHF equations of motion. Our implementation extends, builds on, and exploits the recursive implementation of MCTDH for bosons (r-mctdhb) package. Together with r-mctdhb, the present implementation of MCTDHF forms the mctdh-x package. We benchmark the accuracy of the algorithm with the harmonic interaction model and a time-dependent generalization thereof. These models consider parabolically trapped particles that interact through a harmonic interaction potential. We demonstrate that MCTDHF is capable of solving the time-dependent many-fermion Schr\"odinger equation to an arbitrary degree of precision and can hence yield numerically exact results even in the case of Hamiltonians with time-dependent one-body and two-body potentials. We study the problem of two initially parabolically confined and charged fermions tunneling through a barrier to open space. We demonstrate the validity of a model proposed previously for the many-body tunneling to open space of bosonic particles with contact interactions [Proc. Natl. Acad. Sci. USA 109, 13521 (2012)]. The many-fermion tunneling can be built up from sequentially happening single-fermion tunneling processes. The characteristic momenta of these processes are determined by the chemical potentials of trapped subsystems of smaller particle numbers: The escaped fermions convert the different chemical potentials into kinetic energy. Using the two-body correlation function, we present a detailed picture of the sequentiality of the process and are able to tell tunneling from over-the-barrier escape.
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.
We develop the multi-layer multi-configuration time-dependent Hartree method for bosons (ML-MCTDHB), a variational numerically exact ab initio method for studying the quantum dynamics and stationary properties of general bosonic systems. ML-MCTDHB takes advantage of the permutation symmetry of identical bosons, which allows for investigations of the quantum dynamics from few to many-body systems. Moreover, the multi-layer feature enables ML-MCTDHB to describe mixed bosonic systems consisting of arbitrary many species. Multi-dimensional as well as mixed-dimensional systems can be accurately and efficiently simulated via the multi-layer expansion scheme. We provide a detailed account of the underlying theory and the corresponding implementation. We also demonstrate the superior performance by applying the method to the tunneling dynamics of bosonic ensembles in a one-dimensional double well potential, where a single-species bosonic ensemble of various correlation strengths and a weakly interacting two-species bosonic ensemble are considered.