# Intrinsic Decoherence and Recurrences in a Large Ferromagnetic F = 1 Spinor Bose–Einstein Condensate

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## Abstract

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## 1. Introduction

## 2. An F = 1 SBEC within the SMA Approximation—Full Quantum Diagonalization

## 3. Time Evolution of the One-Body Density Matrix in Coherent States

## 4. Decoherence and Recurrences in the Strong and Weak Interacting Regimes

#### 4.1. Strong Interaction Regime $q/N\left|\eta \right|\ll 1$

#### 4.2. Weak Interaction Regime $N\left|\eta \right|/q\ll 1$

## 5. Discussion and Final Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Magnetization $\overrightarrow{f}$ as a function of time t. Comparison of a full 3D Gross–Pitaevskii (dotted line) versus Single Mode Approximation (SMA)-Gross–Pitaevskii (GP) calculations (continuous line), for a ${}^{87}\mathrm{Rb}$$F=1$ ferromagnetic Spinor Bose–Einstein Condensate (SBEC). Blue and red lines are the x- and y-components, with black the z-component. We use ${}^{87}$Rb constants and experimentally accessible fields, $\tilde{p}=-0.7h$ MHz G${}^{-1}$, $\tilde{q}=72h$ Hz G${}^{-2}$, ${c}_{0}=50.2$ Å, ${c}_{2}=50.9$ Å with a field ${B}_{z}=84$ mG and for $N=6.8\times {10}^{4}$ atoms.

**Figure 2.**Hamiltonian structure for N = 6 particles where the Hilbert space size is 28, each blue square representing a block of magnetization M. The intensity of the color blue and the size of the blocks depend on the value of M.

**Figure 3.**Energy spectrum ${E}_{n}$ and its degeneracy $lnW$ (right inset), for $N={10}^{3}$ particles and (

**a**) $p=1$, $q=0$, and $\eta =0$; (

**b**) $p=0$, $q=0$, and $\eta =-1$; and (

**c**) $p=1$, $q=100$, and $\eta =-1$. In the left inset, we show the detail of the energy spectrum.

**Figure 4.**Time evolution of the expectation value $\langle {\widehat{f}}_{x}\rangle $ illustrating the sequence of decoherences and recurrences; (

**a**) weak interaction $N\left|\eta \right|/q\ll 1$ ($q=1$, $\eta =-{10}^{-4}$, $\theta =3\pi /10$); (

**b**) strong interaction $q/N\left|\eta \right|\ll 1$ ($q=1$, $\eta =-{10}^{4}$, $\theta =3\pi /10$); and (

**c**) crossover $q/N\left|\eta \right|\sim 1$ ($q=1$, $\eta =-{10}^{-2}$, $\theta =\pi /2$). In all cases, $N=100$.

**Figure 5.**Real part of elements (

**a**) ${\rho}_{+1,0}$, (

**b**) ${\rho}_{+1,-1}$, and (

**c**) ${\rho}_{-1,0}$ of the density matrix as a function of time, for $N=700$, $q=1$, $p=1$, and $\eta $ = −30,000, for a time period longer than $(0,2\pi \hslash N/q)$. In continuous (red) lines, we show the overlap of the recurrences predicted by Equation (22). In Panel (

**d**), we show an example of the agreement of the oscillations predicted by Equation (26), red dots, with the full quantum calculation, continuous blue line, for the real part of ${\rho}_{+1,0}$; $N=1000$, $q=1$, $p=1$, and $\eta $ = 30,000.

**Figure 6.**Evolution in time of the expectation value $\langle {\widehat{f}}_{x}\rangle $ as a function of time $\tau $ and the non-linear Zeeman strength q, for $p=1$ and $\eta $ = −30,000. Note that if $q\ll 1$, the recurrences appear more separated.

**Figure 7.**Real part of different elements of the density matrix as a function of time. (

**a**) Comparison of the full quantum calculation (blue solid line) with heuristic fit (red dots), Equation (27), of ${\rho}_{+1,-1}$. Recurrences of the real parts of (

**b**) ${\rho}_{+1,0}$, (

**c**) ${\rho}_{+1,-1}$ and (

**d**) ${\rho}_{0,0}$, within the time period $(0,2\pi \hslash /\eta )$; the full quantum calculation (blue solid line) and overlap of heuristic fit (red line), Equation (27). The (dimensionless) parameters are $N=300$, $\eta =-{10}^{-5}$; $q=7$, $\theta =7\pi /30$, $p=0$, and $\phi =0$.

**Figure 8.**Coefficient ${g}_{jk}\left(\theta \right)$ as a function of $\theta $, in Equation (29). (

**a**) ${g}_{12}\left(\theta \right)$. (

**b**) ${g}_{23}\left(\theta \right)$. (

**c**) ${g}_{13}\left(\theta \right)$. The dots are values numerically calculated, and the continuous line is a spline interpolation. The parameters are $N=300$, $\eta ={10}^{-5}$; $q=7$, $p=0$, and $\phi =0$.

**Figure 9.**Amplitude coefficient (

**a**) ${A}_{11}\left(\theta \right)$ and (

**b**) ${A}_{22}\left(\theta \right)$ as a function of $\theta $, in Equation (28). Note that both are very small. (

**c**) Coefficient ${g}_{11}\left(\theta \right)={g}_{22}\left(\theta \right)={g}_{33}\left(\theta \right)$as a function of $\theta $ in Equation (29). The dots are values numerically calculated, and the continuous line is a spline interpolation. The parameters are $N=300$, $\eta ={10}^{-5}$; $q=7$, $p=0$, and $\phi =0$.

**Figure 10.**(

**a**) Real part of element ${\rho}_{+1,0}$ of the density matrix as a function of time. (

**b**) Real part of element ${\rho}_{-1,0}$ of the density matrix as a function of time. (

**c**) Expectation value of ${\widehat{f}}_{x}$ as a function of time, full quantum calculation; (

**d**) expectation value of ${\widehat{f}}_{x}$ as a function of time, heuristic fit, Equation (27). The beating pattern in $\langle}{\widehat{f}}_{x}\left(t\right){\displaystyle \rangle$ is due to the very small difference of the high frequency oscillations of between ${\rho}_{+1,0}$ and ${\rho}_{-1,0}$; see Equation (27). The parameters are $N=300$, $\eta =-{10}^{-5}$; $q=7$, $\theta =7\pi /30$, $p=0$, and $\phi =0$.

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**MDPI and ACS Style**

Sandoval-Santana, J.C.; Zamora-Zamora, R.; Paredes, R.; Romero-Rochín, V.
Intrinsic Decoherence and Recurrences in a Large Ferromagnetic *F* = 1 Spinor Bose–Einstein Condensate. *Symmetry* **2021**, *13*, 67.
https://doi.org/10.3390/sym13010067

**AMA Style**

Sandoval-Santana JC, Zamora-Zamora R, Paredes R, Romero-Rochín V.
Intrinsic Decoherence and Recurrences in a Large Ferromagnetic *F* = 1 Spinor Bose–Einstein Condensate. *Symmetry*. 2021; 13(1):67.
https://doi.org/10.3390/sym13010067

**Chicago/Turabian Style**

Sandoval-Santana, Juan Carlos, Roberto Zamora-Zamora, Rosario Paredes, and Victor Romero-Rochín.
2021. "Intrinsic Decoherence and Recurrences in a Large Ferromagnetic *F* = 1 Spinor Bose–Einstein Condensate" *Symmetry* 13, no. 1: 67.
https://doi.org/10.3390/sym13010067