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

^{1}

^{2}

^{*}

^{†}

^{‡}

## Abstract

**:**

## 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

## References

- Redfield, A. The theory of relaxation processes. In Advances in Magnetic and Optical Resonance; Academic Press: Cambridge, MA, USA, 1965; Volume 1, pp. 1–32. [Google Scholar]
- Lindblad, G. On the generators of quantum dynamical semigroups. Commun. Math. Phys.
**1976**, 48, 119–130. [Google Scholar] [CrossRef] - Van Kampen, N.G. Stochastic Processes in Physics and Chemistry; Elsevier Science B.V.: Amsterdam, The Netherlands, 1992; Volume 1. [Google Scholar]
- Van Kampen, N. A soluble model for quantum mechanical dissipation. J. Stat. Phys.
**1995**, 78, 299–310. [Google Scholar] [CrossRef] - Van Kampen, N. A new approach to noise in quantum mechanics. J. Stat. Phys.
**2004**, 115, 1057–1072. [Google Scholar] [CrossRef] - Hillery, M.; O’Connell, R.; Scully, M.; Wigner, E. Distribution functions in physics: Fundamentals. Phys. Rep.
**1984**, 106, 121. [Google Scholar] [CrossRef] - Zurek, W.H. Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys.
**2003**, 75, 715–775. [Google Scholar] [CrossRef][Green Version] - Caldeira, A.O.; Leggett, A.J. Path integral approach to quantum Brownian motion. Phys. A Stat. Mech. Appl.
**1983**, 121, 587–616. [Google Scholar] [CrossRef] - Suárez, A.; Silbey, R.; Oppenheim, I. Memory effects in the relaxation of quantum open systems. J. Chem. Phys.
**1992**, 97, 5101–5107. [Google Scholar] [CrossRef] - Romero-Rochin, V.; Oppenheim, I. Relaxation properties of two-level systems in condensed phases. Phys. A Stat. Mech. Appl.
**1989**, 155, 52–72. [Google Scholar] [CrossRef] - Caballero-Benitez, S.F.; Romero-Rochín, V.; Paredes, R. Intrinsic decoherence in an ultracold Bose gas confined in a double-well potential. J. Phys. B At. Mol. Opt. Phys.
**2010**, 43, 095301. [Google Scholar] [CrossRef] - Camacho-Guardian, A.; Paredes, R. Intrinsic decoherence and purity in a Bose quantum fluid in a triple well potential. Laser Phys.
**2014**, 24, 085501. [Google Scholar] [CrossRef][Green Version] - Engel, G.S.; Calhoun, T.R.; Read, E.L.; Ahn, T.K.; Mančal, T.; Cheng, Y.C.; Blankenship, R.E.; Fleming, G.R. Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems. Nature
**2007**, 446, 782–786. [Google Scholar] [CrossRef] [PubMed] - Collini, E.; Wong, C.Y.; Wilk, K.E.; Curmi, P.M.; Brumer, P.; Scholes, G.D. Coherently wired light-harvesting in photosynthetic marine algae at ambient temperature. Nature
**2010**, 463, 644–647. [Google Scholar] [CrossRef] [PubMed] - Marques, B.; Matoso, A.; Pimenta, W.; Gutiérrez-Esparza, A.; Santos, M.; Pádua, S. Experimental simulation of decoherence in photonics qudits. Sci. Rep.
**2015**, 5, 16049. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ballmann, S.; Härtle, R.; Coto, P.B.; Elbing, M.; Mayor, M.; Bryce, M.R.; Thoss, M.; Weber, H.B. Experimental evidence for quantum interference and vibrationally induced decoherence in single-molecule junctions. Phys. Rev. Lett.
**2012**, 109, 056801. [Google Scholar] [CrossRef] - Gu, B.; Franco, I. Quantifying early time quantum decoherence dynamics through fluctuations. J. Phys. Chem. Lett.
**2017**, 8, 4289–4294. [Google Scholar] [CrossRef] - Landau, L.; Lifshitz, E. Statistical Physics I; Pergamon Press: Oxford, UK; London, UK; Edinburgh, UK; New York, NY, USA; Paris, France; Frankfurt, Germany, 1980. [Google Scholar]
- Anderson, M.H.; Ensher, J.R.; Matthews, M.R.; Wieman, C.E.; Cornell, E.A. Observation of Bose–Einstein Condensation in a Dilute Atomic Vapor. Science
**1995**, 269, 198–201. [Google Scholar] [CrossRef][Green Version] - Davis, K.B.; Mewes, M.O.; Andrews, M.R.; van Druten, N.J.; Durfee, D.S.; Kurn, D.M.; Ketterle, W. Bose–Einstein Condensation in a Gas of Sodium Atoms. Phys. Rev. Lett.
**1995**, 75, 3969–3973. [Google Scholar] [CrossRef][Green Version] - Regal, C.A.; Greiner, M.; Jin, D.S. Observation of Resonance Condensation of Fermionic Atom Pairs. Phys. Rev. Lett.
**2004**, 92, 040403. [Google Scholar] [CrossRef][Green Version] - Vinit, A.; Raman, C. Precise measurements on a quantum phase transition in antiferromagnetic spinor Bose–Einstein condensates. Phys. Rev. A
**2017**, 95, 011603. [Google Scholar] [CrossRef][Green Version] - Gomez, P.; Mazzinghi, C.; Martin, F.; Coop, S.; Palacios, S.; Mitchell, M.W. Interferometric measurement of interhyperfine scattering lengths in
^{87}Rb. Phys. Rev. A**2019**, 100, 032704. [Google Scholar] [CrossRef][Green Version] - Gomez, P.; Martin, F.; Mazzinghi, C.; Benedicto Orenes, D.; Palacios, S.; Mitchell, M.W. Bose–Einstein Condensate Comagnetometer. Phys. Rev. Lett.
**2020**, 124, 170401. [Google Scholar] [CrossRef] [PubMed] - Ketterle, W.; Durfee, D.S.; Stamper-Kurn, D.M. Making, probing and understanding Bose–Einstein condensates. In Proceedings of the International School of Physics “Enrico Fermi”; Inguscio, S.M., Stringari, C.W., Eds.; IOS Press EBooks: Amsterdam, The Netherlands, 1999; Volume 140, pp. 67–176. [Google Scholar]
- Deutsch, J.M. Quantum statistical mechanics in a closed system. Phys. Rev. A
**1991**, 43, 2046–2049. [Google Scholar] [CrossRef] [PubMed] - Srednicki, M. Chaos and quantum thermalization. Phys. Rev. E
**1994**, 50, 888–901. [Google Scholar] [CrossRef] [PubMed][Green Version] - Rigol, M.; Dunjko, V.; Olshanii, M. Thermalization and its mechanism for generic isolated quantum systems. Nature
**2008**, 452, 854–858. [Google Scholar] [CrossRef] [PubMed][Green Version] - Reimann, P. Canonical thermalization. New J. Phys.
**2010**, 12, 055027. [Google Scholar] [CrossRef][Green Version] - Kaufman, A.M.; Tai, M.E.; Lukin, A.; Rispoli, M.; Schittko, R.; Preiss, P.M.; Greiner, M. Quantum thermalization through entanglement in an isolated many-body system. Science
**2016**, 353, 794–800. [Google Scholar] [CrossRef][Green Version] - Choi, J.y.; Hild, S.; Zeiher, J.; Schauß, P.; Rubio-Abadal, A.; Yefsah, T.; Khemani, V.; Huse, D.A.; Bloch, I.; Gross, C. Exploring the many-body localization transition in two dimensions. Science
**2016**, 352, 1547–1552. [Google Scholar] [CrossRef][Green Version] - Ho, T.L. Spinor Bose condensates in optical traps. Phys. Rev. Lett.
**1998**, 81, 742. [Google Scholar] [CrossRef][Green Version] - Ohmi, T.; Machida, K. Bose–Einstein Condensation with Internal Degrees of Freedom in Alkali Atom Gases. J. Phys. Soc. Jpn.
**1998**, 67, 1822–1825. [Google Scholar] [CrossRef][Green Version] - Damski, B.; Zurek, W.H. Quantum phase transition in space in a ferromagnetic spin-1 Bose–Einstein condensate. New J. Phys.
**2009**, 11, 063014. [Google Scholar] [CrossRef] - Kajtoch, D.; Witkowska, E. Spin squeezing in dipolar spinor condensates. Phys. Rev. A
**2016**, 93, 023627. [Google Scholar] [CrossRef][Green Version] - Imamoḡlu, A.; Lewenstein, M.; You, L. Inhibition of Coherence in Trapped Bose–Einstein Condensates. Phys. Rev. Lett.
**1997**, 78, 2511–2514. [Google Scholar] [CrossRef] - Plimak, L.; Weiß, C.; Walser, R.; Schleich, W.P. Quantum dynamics of atomic coherence in a spin-1 condensate: Mean-field versus many-body simulation. Opt. Commun.
**2006**, 264, 311–320. [Google Scholar] [CrossRef][Green Version] - Zamora-Zamora, R.; Romero-Rochín, V. Skyrmions with arbitrary topological charges in spinor Bose–Einstein condensates. J. Phys. B At. Mol. Opt. Phys.
**2018**, 51, 045301. [Google Scholar] [CrossRef][Green Version] - Zamora-Zamora, R.; Domínguez-Castro, G.A.; Trallero-Giner, C.; Paredes, R.; Romero-Rochín, V. Validity of Gross–Pitaevskii solutions of harmonically confined BEC gases in reduced dimensions. J. Phys. Commun.
**2019**, 3, 085003. [Google Scholar] [CrossRef] - Xue, M.; Yin, S.; You, L. Universal driven critical dynamics across a quantum phase transition in ferromagnetic spinor atomic Bose–Einstein condensates. Phys. Rev. A
**2018**, 98, 013619. [Google Scholar] [CrossRef][Green Version] - Law, C.; Pu, H.; Bigelow, N. Quantum spins mixing in spinor Bose–Einstein condensates. Phys. Rev. Lett.
**1998**, 81, 5257. [Google Scholar] [CrossRef][Green Version] - Pu, H.; Law, C.; Raghavan, S.; Eberly, J.; Bigelow, N. Spin-mixing dynamics of a spinor Bose–Einstein condensate. Phys. Rev. A
**1999**, 60, 1463. [Google Scholar] [CrossRef] - De Sarlo, L.; Shao, L.; Corre, V.; Zibold, T.; Jacob, D.; Dalibard, J.; Gerbier, F. Spin fragmentation of Bose–Einstein condensates with antiferromagnetic interactions. New J. Phys.
**2013**, 15, 113039. [Google Scholar] [CrossRef][Green Version] - Koashi, M.; Ueda, M. Exact eigenstates and magnetic response of spin-1 and spin-2 Bose–Einstein condensates. Phys. Rev. Lett.
**2000**, 84, 1066. [Google Scholar] [CrossRef][Green Version] - Santos, L.; Pfau, T. Spin-3 chromium bose-einstein condensates. Phys. Rev. Lett.
**2006**, 96, 190404. [Google Scholar] [CrossRef] [PubMed][Green Version] - Raghavan, S.; Pu, H.; Law, C.; Bigelow, N. Properties of spinor Bose condensates. J. Low Temp. Phys.
**2000**, 119, 437–460. [Google Scholar] [CrossRef] - Bookjans, E.M.; Vinit, A.; Raman, C. Quantum phase transition in an antiferromagnetic spinor Bose–Einstein condensate. Phys. Rev. Lett.
**2011**, 107, 195306. [Google Scholar] [CrossRef] [PubMed] - Dağ, C.B.; Wang, S.T.; Duan, L.M. Classification of quench-dynamical behaviors in spinor condensates. Phys. Rev. A
**2018**, 97, 023603. [Google Scholar] [CrossRef][Green Version] - Anquez, M.; Robbins, B.; Bharath, H.; Boguslawski, M.; Hoang, T.; Chapman, M. Quantum Kibble-Zurek mechanism in a spin-1 Bose–Einstein condensate. Phys. Rev. Lett.
**2016**, 116, 155301. [Google Scholar] [CrossRef] [PubMed][Green Version] - Gell-Mann, M. Symmetries of Baryons and Mesons. Phys. Rev.
**1962**, 125, 1067–1084. [Google Scholar] [CrossRef][Green Version] - Cohen-Tannoudji, C.; Diu, B.; Laloe, F. Quantum Mechanics; John Wiley and Sons: New York, NY, USA, 1977; Volume I. [Google Scholar]
- Sandoval-Santana, J.C.; Ibarra-Sierra, V.G.; Cardoso, J.L.; Kunold, A.; Roman-Taboada, P.; Naumis, G. Method for Finding the Exact Effective Hamiltonian of Time-Driven Quantum Systems. Ann. Phys.
**2019**, 531, 1900035. [Google Scholar] [CrossRef]

**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 $\phi =0$.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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