# Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double-Well

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## Abstract

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## 1. Introduction

## 2. Hamiltonian and Methods

#### 2.1. Hamiltonian of Trapped Interacting Bosons

#### 2.2. Numerical Methods and Observables

## 3. Structure of Spectrum and Eigenstates

#### 3.1. Few-Body Excitation Spectra

#### 3.2. Eigenstate Structure and Few-Body Correlations

## 4. Dynamics in the Double-Well

#### 4.1. Static Potential: Two-Body Excited State Dynamics

#### 4.2. Time-Dependent Double-Well Potential: From Few- to Many-Body Dynamics

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Fourier Grid Hamiltonian Method

## Appendix B. Bose–Hubbard Model in the Continuum

## Appendix C. Multiconfigurational Time-Dependent Hartree Method for Indistinguishable Particles

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**Figure 1.**Single-particle eigenenergies ${E}_{n}^{1P}$ of Equation (10), (

**a**) as a function of the tunneling barrier height ${A}_{\mathrm{max}}$, and (

**b**) for ${A}_{\mathrm{max}}=30$ in the double-well potential (red). The red line in (

**a**) indicates the central barrier’s height on the energy axis. Even- (blue lines) and odd-parity (black dashed) states become nearly degenerate as ${A}_{\mathrm{max}}$ is increased. Employed parameter values for the FGH method (see Appendix A): ${x}_{\mathrm{max}}=-{x}_{\mathrm{min}}=40$, ${N}_{\mathrm{cut}}=330$, and ${N}_{\mathrm{Grid}}=2047$.

**Figure 2.**Two-particle eigenenergies ${E}_{n}^{2P}$ per particle, Equation (4), as a function of the (static) tunneling barrier height ${A}_{\mathrm{max}}$, for interaction strengths (

**a**) $\lambda =0$ and (

**b**) $\lambda =1$. Finite interactions partially or totally lift the degeneracy of the eigenenergies, depending on the considered quantum number. The red line indicates the effective potential barrier height—which is twice the barrier height for a single particle, i.e., $2{A}_{\mathrm{max}}$. Parameter values for the FGH method: ${x}_{\mathrm{max}}=-{x}_{\mathrm{min}}=40$, ${N}_{\mathrm{cut}}=330$, and ${N}_{\mathrm{Grid}}=2047$.

**Figure 3.**Two-particle eigenenergies ${E}_{n}^{2P}$ per particle, Equation (4), as a function of the inter-particle interaction strength $\lambda $, in a deep double-well with ${A}_{\mathrm{max}}=30$. Flat energies (continuous lines) correspond to the situation where the particles are almost completely localized in opposite wells and do not interact. Increasing $\lambda $ tends to induce a degeneracy between even and odd states (fermionization process). FGH parameters: ${x}_{\mathrm{max}}=-{x}_{\mathrm{min}}=40$, ${N}_{\mathrm{cut}}=330$, and ${N}_{\mathrm{Grid}}=2047$.

**Figure 4.**Three-particle eigenenergies ${E}_{n}^{3P}$ per particle, Equation (4), as a function of the inter-particle interaction strength $U\equiv \lambda {\sum}_{i}\phantom{\rule{4pt}{0ex}}{|{w}_{0i}|}^{4}$ (see Appendix B), with ${A}_{\mathrm{max}}=30$. Dashed (continuous) lines represent eigenstates with three (two) particles on the same well, and the red horizontal line indicates the Tonks-Girardeau (TG) limit for the ground state. Parameters employed for the BH method (see App. B): ${x}_{\mathrm{max}}=-{x}_{\mathrm{min}}=10$, and $L=231$.

**Figure 5.**Probability densities ${\left|{\psi}_{n}({x}_{1},{x}_{2})\right|}^{2}$ of the nth eigenstates of two non-interacting particles ($\lambda $ = 0), in configuration space $({x}_{1},{x}_{2})$, with variable barrier height from the single (${A}_{\mathrm{max}}=0$) to the double-well (${A}_{\mathrm{max}}\ne 0$) scenario, cf. Equation (2). The densities are plotted on a linear scale which interpolates between vanishing probability (dark blue) and the maximum probability density ${\left|\psi \right|}_{\mathrm{max}}^{2}$ of the given eigenstate. FGH parameters: ${x}_{\mathrm{max}}=-{x}_{\mathrm{min}}=40$, ${N}_{\mathrm{cut}}=330$, and ${N}_{\mathrm{Grid}}=2047$.

**Figure 6.**Probability densities ${\left|{\psi}_{n}({x}_{1},{x}_{2})\right|}^{2}$ of the nth eigenstates of two interacting particles ($\lambda $ = 1), in configuration space $({x}_{1},{x}_{2})$, with variable barrier height from the single (${A}_{\mathrm{max}}=0$) to the double-well (${A}_{\mathrm{max}}\ne 0$) scenario, cf. Equation (2). Color coding as in Figure 5. FGH parameters: ${x}_{\mathrm{max}}=-{x}_{\mathrm{min}}=40$, ${N}_{\mathrm{cut}}=330$, and ${N}_{\mathrm{Grid}}=2047$.

**Figure 7.**Three-body probability density ${\left|{\psi}_{0}({x}_{1},{x}_{2},{x}_{3})\right|}^{2}$ (

**a**,

**d**), diagonal of the reduced two-body probability density matrix ${\left|{\psi}_{0}({x}_{1},{x}_{2})\right|}^{2}$ (

**b**,

**e**), and diagonal of the reduced one-body probability density matrix ${\left|{\psi}_{0}({x}_{1})\right|}^{2}$ (

**c**,

**f**) of the ground state of three (

**a**–

**c**) non-interacting ($U=0$) and (

**d**–

**f**) interacting ($U=1$) particles in the double-well (${A}_{\mathrm{max}}=30$), cf. Equation (2). Please note that in (

**d**), $|{\psi}_{0}{|}^{2}\approx 0$ if all bosons are in the same well (${x}_{1},{x}_{2},{x}_{3}>0$ and ${x}_{1},{x}_{2},{x}_{3}<0$), due to the interactions. The red line in (

**c**) is the profile of $|{\psi}_{0}({x}_{1}){|}^{2}$ for non-interacting particles. Parameters employed for the BH method: ${x}_{\mathrm{max}}=-{x}_{\mathrm{min}}=10$, and $L=231$.

**Figure 8.**Characteristic periods ${T}_{21}$ and ${T}_{10}$ of the two-particle tunneling dynamics as displayed in Figure 9, as a function of the interaction strength $\lambda $, for a double-well potential barrier height ${A}_{\mathrm{max}}=10$, on a double-logarithmic scale. The horizontal, black, dashed line indicates the (degenerate, see main text) period of the non-interacting case $T(\lambda =0)\simeq 12\xb7{10}^{3}$.

**Figure 9.**(

**a**) Detection probabilities, Equation (8), and time-integrated probability current, Equation (9), as a function of time, for the two-particle initial state $|{\psi}_{n=0}^{\mathrm{loc}}(t=0)\rangle $, Equation (14), and a weak interaction strength $\lambda =0.005$. (

**b**) Expansion coefficients of the initial state in the interacting two-body eigenbasis, as a function of the eigenenergy ${E}_{n}^{2P}$, for interactions $\lambda =0\phantom{\rule{3.33333pt}{0ex}}(\mathrm{circles}),\phantom{\rule{3.33333pt}{0ex}}0.001\phantom{\rule{3.33333pt}{0ex}}(\mathrm{squares}),\phantom{\rule{3.33333pt}{0ex}}0.005\phantom{\rule{3.33333pt}{0ex}}(\mathrm{diamonds})$ and $1\phantom{\rule{3.33333pt}{0ex}}(\mathrm{triangles})$. The inset zooms onto the dominant expansion coefficients. FGH parameters: ${x}_{\mathrm{max}}=-{x}_{\mathrm{min}}=40$, ${N}_{\mathrm{cut}}=330$, and ${N}_{\mathrm{Grid}}=2047$.

**Figure 10.**(

**a**) Detection probabilities (8), and time-integrated probability current (9), as a function of time, with initial two-particle state $|{\Psi}_{n=3}^{\mathrm{loc}}(t=0)\rangle $, Equation (14), and interaction strength $\lambda =0.1$. The vertical, dashed black lines indicate the period $T(\lambda =0)\simeq 19.5$ of the non-interacting case. (

**b**) Expansion coefficients of the initial state in the interacting two-body eigenbasis, as a function of the eigenenergy ${E}_{n}^{2P}$, for interaction strengths $\lambda =0\phantom{\rule{3.33333pt}{0ex}}(\mathrm{circles}),\phantom{\rule{3.33333pt}{0ex}}0.1\phantom{\rule{3.33333pt}{0ex}}(\mathrm{diamonds})$, and $1\phantom{\rule{3.33333pt}{0ex}}(\mathrm{triangles})$. The inset zooms onto the dominant expansion coefficients. FGH parameters: ${x}_{\mathrm{max}}=-{x}_{\mathrm{min}}=40$, ${N}_{\mathrm{cut}}=330$, and ${N}_{\mathrm{Grid}}=2047$.

**Figure 11.**Time evolution of the two-body density ${\left|\psi ({x}_{1},{x}_{2};t)\right|}^{2}$, launched in the initial two-particle ground state of a harmonic trap, with interaction strength $\lambda =1$, for (

**a**–

**d**) a diabatically switched central barrier with amplitude ${A}_{\mathrm{max}}=30$ (${T}_{\mathrm{ramp}}\to 0$, with FGH parameters ${x}_{\mathrm{max}}=-{x}_{\mathrm{min}}=40$, ${N}_{\mathrm{cut}}=330$, and ${N}_{\mathrm{Grid}}=2047$), and (

**e**–

**h**) (quasi-) adiabatic switching to ${A}_{\mathrm{max}}=30$ (${T}_{\mathrm{ramp}}=30$, with MCTDH-X parameters ${x}_{\mathrm{max}}=-{x}_{\mathrm{min}}=12$, ${N}_{x}=512$, and $M=20$).

**Figure 12.**Two-body energy expectation, Equation (17), versus ramping time, after a fixed evolution time ${t}_{0}=200$, for ${A}_{\mathrm{max}}=30$ and $\lambda =1$. The inset zooms into the range ${T}_{\mathrm{ramp}}\ge 8$, where the horizontal dashed lines indicate the low-lying eigenenergies of Equation (1), computed by FGH. FGH parameters: ${x}_{\mathrm{max}}=-{x}_{\mathrm{min}}=40$, ${N}_{\mathrm{cut}}=330$, and ${N}_{\mathrm{Grid}}=2047$; MCTDH-X parameters employed for the time-propagation: ${x}_{\mathrm{max}}=-{x}_{\mathrm{min}}=12$, ${N}_{x}=512$, and $M=16$.

**Figure 13.**Von Neumann entropy (6) of the interacting two-particle state launched in the harmonic oscillator (interacting) two-particle ground state, as a function of time, for short and long ramping times, ${T}_{\mathrm{ramp}}=0.001$ (red) and ${T}_{\mathrm{ramp}}=30$ (blue), and strong ($\lambda =1$, (

**a**)) and weak ($\lambda =0.1$, (

**b**)) interaction, respectively. For small ${T}_{\mathrm{ramp}}$, the entropy increases and finally saturates, whereas it oscillates for long ramping times. MCTDH-X parameters: ${x}_{\mathrm{max}}=-{x}_{\mathrm{min}}=12$, ${N}_{x}=512$, and $M=16$.

**Figure 14.**Time evolution of the von Neumann entropy $S({T}_{\mathrm{ramp}},t)$, Equation (6), as a function of the ramping time ${T}_{\mathrm{ramp}}$, for a final barrier height ${A}_{\mathrm{max}}=30$, increasing particle number $N=2,3,10$ (from left to right), and interaction strengths $\lambda =1$ (

**a**–

**c**) and $\lambda =0.1$ (

**d**–

**f**). The red line indicates the full switching duration $t={T}_{\mathrm{ramp}}$ for the ramp to reach its maximum (Parameter values employed in the MCTDH-X calculation: ${x}_{\mathrm{max}}=-{x}_{\mathrm{min}}=12$, ${N}_{x}=512$, $M=8$).

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**MDPI and ACS Style**

Schäfer, F.; Bastarrachea-Magnani, M.A.; Lode, A.U.J.; de Parny, L.d.F.; Buchleitner, A.
Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double-Well. *Entropy* **2020**, *22*, 382.
https://doi.org/10.3390/e22040382

**AMA Style**

Schäfer F, Bastarrachea-Magnani MA, Lode AUJ, de Parny LdF, Buchleitner A.
Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double-Well. *Entropy*. 2020; 22(4):382.
https://doi.org/10.3390/e22040382

**Chicago/Turabian Style**

Schäfer, Frank, Miguel A. Bastarrachea-Magnani, Axel U. J. Lode, Laurent de Forges de Parny, and Andreas Buchleitner.
2020. "Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double-Well" *Entropy* 22, no. 4: 382.
https://doi.org/10.3390/e22040382