^{1,2,a),b)}, Sven Krönke

^{1,a),c)}, Oriol Vendrell

^{2,3,d)}and Peter Schmelcher

^{1,2,e)}

### Abstract

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.

The authors would like to thank Hans-Dieter Meyer and Jan Stockhofe for fruitful discussions on MCTDH methods and symmetry conservation. Particularly, the authors would like to thank Jan Stockhofe for the DVR implementation of the ML-MCTDHB code. We also thank Johannes Schurer and Valentin Bolsinger for carefully reading the manuscript. S.K. gratefully acknowledges a scholarship of the Studienstiftung des deutschen Volkes. L.C. and P.S. gratefully acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG) in the framework of the SFB 925 “Light induced dynamics and control of correlated quantum systems.”

I. INTRODUCTION

II. THEORY

A. ML-MCTDHB for bosonic mixtures

1. Hamiltonian and Ansatz

2. Derivation and general form of the equations of motion for the expansion coefficients

3. Details and construction of the ingredients for the equations of motion

B. ML-MCTDHB for high-dimensional bosonic systems

C. ML-MCTDHB for general bosonic systems

D. Scaling

E. Symmetry conservation

III. IMPLEMENTATION

A. Implementation of the second quantization formalism

B. ML-MCTDHB program

IV. APPLICATIONS

A. Single-species tunneling

B. Mixture tunneling

V. CONCLUSIONS AND OUTLOOK

### Key Topics

- Tunneling
- 46.0
- Equations of motion
- 32.0
- Mean field theory
- 32.0
- Multilayers
- 25.0
- Wave functions
- 23.0

## Figures

The tree diagram of a three-species bosonic system. The three species are labeled as A, B, and C bosons, with primitive DOF of x A , x B , and (x C , y C , z C ), respectively. Corresponding to the ML-MCTDHB Ansatz, a three-layer tree diagram is shown, containing the top layer, the species layer, the particle layer from top to bottom. The nodes on each non-top layer correspond to different logical DOF, and the primitive DOF are given by the square nodes at the bottom. The “+” inside the species nodes denotes the species to be a bosonic species.

The tree diagram of a three-species bosonic system. The three species are labeled as A, B, and C bosons, with primitive DOF of x A , x B , and (x C , y C , z C ), respectively. Corresponding to the ML-MCTDHB Ansatz, a three-layer tree diagram is shown, containing the top layer, the species layer, the particle layer from top to bottom. The nodes on each non-top layer correspond to different logical DOF, and the primitive DOF are given by the square nodes at the bottom. The “+” inside the species nodes denotes the species to be a bosonic species.

(a) An illustration of a general particle multi-layer MCTDHB wave function expansion for spin-one bosons in three-dimensional coordinate space, with in the figures. The dashed box indicates the various possibilities for mode combination of the primitive DOF, i.e., the spatial DOF x, y, z and the spin projection on the z-axis. (b) An example of the mode combination scheme inside the dashed box of (a), where the spatial DOF y and z are combined into one logical DOF.

(a) An illustration of a general particle multi-layer MCTDHB wave function expansion for spin-one bosons in three-dimensional coordinate space, with in the figures. The dashed box indicates the various possibilities for mode combination of the primitive DOF, i.e., the spatial DOF x, y, z and the spin projection on the z-axis. (b) An example of the mode combination scheme inside the dashed box of (a), where the spatial DOF y and z are combined into one logical DOF.

A tree diagram of the combined treatment of the species and particle ML-MCTDHB for a three-species bosonic system. The three species are again labeled as A, B, and C bosons as in Figure 1 . In contrast to that example, the bosons of all species may move in three-dimensional space here.

A tree diagram of the combined treatment of the species and particle ML-MCTDHB for a three-species bosonic system. The three species are again labeled as A, B, and C bosons as in Figure 1 . In contrast to that example, the bosons of all species may move in three-dimensional space here.

The population oscillation of four bosons in the left and right well, with (a) g = 0.0, (b) g = 0.5, (c) g = 2.0, and (d) g = 4.0. Lines: ML-MCTDHB results. Crosses: MCTDH results. Figures 4(a)–4(c): Six particle SPFs; Figure 4(d): Ten particle SPFs. Particularly, Figure 4(b) shows the behavior for a shorter time interval to highlight the agreement between the two methods concerning the fast oscillation process. The difference between the two methods in Figure 4(d) is attributed to the different implementations of the contact interaction in the two methods (cf. main text).

The population oscillation of four bosons in the left and right well, with (a) g = 0.0, (b) g = 0.5, (c) g = 2.0, and (d) g = 4.0. Lines: ML-MCTDHB results. Crosses: MCTDH results. Figures 4(a)–4(c): Six particle SPFs; Figure 4(d): Ten particle SPFs. Particularly, Figure 4(b) shows the behavior for a shorter time interval to highlight the agreement between the two methods concerning the fast oscillation process. The difference between the two methods in Figure 4(d) is attributed to the different implementations of the contact interaction in the two methods (cf. main text).

The natural populations and one-body densities of the system of four bosons in the double well at different time instants, with (a) and (c) for g = 2.0, as well as (b) and (d) for g = 4.0.

The natural populations and one-body densities of the system of four bosons in the double well at different time instants, with (a) and (c) for g = 2.0, as well as (b) and (d) for g = 4.0.

The population oscillation of ten bosons in the double well, with (a) g = 0.5, (b) g = 1.0. Enhanced tunneling is obtained in (a), where the amplitude of the population is around 2. A slow evolution to the quasistationary state is obtained in (b).

The population oscillation of ten bosons in the double well, with (a) g = 0.5, (b) g = 1.0. Enhanced tunneling is obtained in (a), where the amplitude of the population is around 2. A slow evolution to the quasistationary state is obtained in (b).

The natural population and one-body densities of ten bosons in the double well at different time, with (a) and (c) g = 2.0, as well as (b) and (d) g = 4.0.

The natural population and one-body densities of ten bosons in the double well at different time, with (a) and (c) g = 2.0, as well as (b) and (d) g = 4.0.

2 A and 2 B bosons initially loaded in the left well of a double well trap. Blue (red) line: Probability for finding an A (B) boson in the left well versus time. Line: ML-MCTDHB results. Crosses via ML-MCTDH. Parameters: g A = 0.3, g B = 0.5 g A , and g AB = 0.1 g A . M = 4 and m = 3 SPFs for both calculations. The dashed vertical lines: the first three Rabi-tunneling periods of a single particle.

2 A and 2 B bosons initially loaded in the left well of a double well trap. Blue (red) line: Probability for finding an A (B) boson in the left well versus time. Line: ML-MCTDHB results. Crosses via ML-MCTDH. Parameters: g A = 0.3, g B = 0.5 g A , and g AB = 0.1 g A . M = 4 and m = 3 SPFs for both calculations. The dashed vertical lines: the first three Rabi-tunneling periods of a single particle.

Evolution of several joint probabilities for the same setup as in Figure 8 : The blue (red) solid line: probability of finding both A (B) bosons in the same well; The green solid line: The probability of detecting an A and a B boson in the same well. Solid lines: ML-MCTDHB. Crosses: ML-MCTDH. Dashed vertical lines: The first three Rabi-tunneling periods.

Evolution of several joint probabilities for the same setup as in Figure 8 : The blue (red) solid line: probability of finding both A (B) bosons in the same well; The green solid line: The probability of detecting an A and a B boson in the same well. Solid lines: ML-MCTDHB. Crosses: ML-MCTDH. Dashed vertical lines: The first three Rabi-tunneling periods.

The natural populations, i.e., eigenvalues of the reduced density matrix corresponding to the whole species A (or equivalently B) are plotted versus time for the same tunneling scenario as in Figure 8 . The solid lines refer to the ML-MCTDHB and the crosses to the corresponding ML-MCTDH calculation.

The natural populations, i.e., eigenvalues of the reduced density matrix corresponding to the whole species A (or equivalently B) are plotted versus time for the same tunneling scenario as in Figure 8 . The solid lines refer to the ML-MCTDHB and the crosses to the corresponding ML-MCTDH calculation.

The dynamics of the natural populations of the reduced density matrix corresponding to an A and a B boson are shown in (a) and (b), respectively. All parameters are chosen as in Figure 8 . The solid lines: ML-MCTDHB; the crosses: ML-MCTDH calculation.

The dynamics of the natural populations of the reduced density matrix corresponding to an A and a B boson are shown in (a) and (b), respectively. All parameters are chosen as in Figure 8 . The solid lines: ML-MCTDHB; the crosses: ML-MCTDH calculation.

## Tables

The scaling of the methods MCTDH, ML-MCTDH, MCTDH-BB, and ML-MCTDHB is compared for the case of S = 2 species, m = 4 particle SPFs, and n = 250 spatial grid points. The number of bosons per species, N, and (for the multi-layer methods) the number of species SPFs, M, are varied. Each table entry contains the number of coefficients needed for the wave function expansion of the respective method and its ratio with respect to the number of ML-MCTDHB coefficients.

The scaling of the methods MCTDH, ML-MCTDH, MCTDH-BB, and ML-MCTDHB is compared for the case of S = 2 species, m = 4 particle SPFs, and n = 250 spatial grid points. The number of bosons per species, N, and (for the multi-layer methods) the number of species SPFs, M, are varied. Each table entry contains the number of coefficients needed for the wave function expansion of the respective method and its ratio with respect to the number of ML-MCTDHB coefficients.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content