[Physics/Math]/Physics

# Fragmented Bose Einstein Condensates

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# Fragmented Bose Einstein Condensates BK21? 이었나? 학회 비스무리한거 할 때 발표했던 자료. 상 받았던 걸로 기억하는데... 맞나? ## PH
• 2023-09-27 : First posting.
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## LaTeX Beamer source code 공개 [.scrollable.lang-tex] \documentclass[10pt]{beamer} \usepackage{pgf,pgfarrows,pgfnodes,pgfautomata,pgfheaps,pgfshade} \usepackage{bm} \usepackage{amsmath,amssymb} \usepackage[latin1]{inputenc} \usepackage{colortbl} \usepackage[english]{babel} \usepackage{graphicx} \usepackage{epsfig} \usepackage{psfrag} \usepackage{beamerthemesplit} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\ea}{\end{eqnarray}} \newcommand{\bdm}{\begin{displaymath}} \newcommand{\edm}{\end{displaymath}} %\newcommand{\na}{\mbox{\boldmath$\nabla$}} %\newcommand{\f}[1]{\mbox{\boldmath$#1$}} %\newcommand{\fk}[1]{\mbox{\boldmath$\scriptstyle#1$}} %\newcommand{\ord}{{\cal O}} \newcommand{\bc}{\begin{center}} \newcommand{\ec}{\end{center}} \newcommand{\fs}{\footnotesize} \title{Fragmented Bose Einstein Condensates} \author{K.-S. Lee, Bo Xiong, M.-K. Kang, \\and Uwe R. Fischer} \institute{Seoul National University, Department of Physics and Astronomy Center for Theoretical Physics, 151-747 Seoul, Korea} \begin{document} \frame{ \titlepage } \section{Introduction} \subsection{Many-Body states} \frame{ \bc {\bf Identical Particles: \alert{Indistinguishable}} \\ \medskip Fermions: \underline{totally anti-symmetric} under interchange of any pair \\ Bosons: \underline{totally symmetric} under interchange of any pair \begin{displaymath} \begin{split} \textrm{Fermions: } &\hat{P}_{mn}|\Psi\rangle = -|\Psi\rangle, \textrm{(ex) }|i\rangle |j\rangle - |j\rangle |i\rangle \\ \textrm{Bosons: } &\hat{P}_{mn}|\Psi\rangle = +|\Psi\rangle, \textrm{(ex) }|i\rangle |j\rangle + |j\rangle |i\rangle \\ \end{split} \end{displaymath} \bigskip Introducing {\bf creation} (and it\'s Hermitian conjugate {\bf annihilation}) {\bf operators}, ($\zeta=-1$ for fermions, $\zeta=+1$ for bosons) \begin{displaymath} \begin{split} \hat{a}_{\alert{i}}^{\dag} = &|\alert{i}\rangle \langle 0| + \frac{1}{\sqrt{2}} \sum_{k}{\left[ |\alert{i}\rangle |k\rangle + \zeta |k\rangle |\alert{i}\rangle \right] } \langle k| \\ &+ \frac{1}{\sqrt{3}} \sum_{k,l}{\left[ |\alert{i}\rangle |k\rangle |l\rangle + \zeta |k\rangle |\alert{i}\rangle |l\rangle + \zeta^2 |k\rangle |l\rangle |\alert{i}\rangle \right] } \langle k| \langle l| + \cdots \end{split} \end{displaymath} with completeness relation $\sum_{k}{|k\rangle\langle k|}=1$ and {\bf vacuum state} $|0\rangle$. \ec } \subsection{Second Quantization} \frame{\frametitle{Second Quantization} \bc Therefore \begin{displaymath} \boxed{ \hat{a}_{i}^{\dag} \hat{a}_{j}^{\dag} \hat{a}_{k}^{\dag} \cdots |0\rangle = \frac{1}{\sqrt{N!}} \sum_{l,m,n,\cdots}{\epsilon_{l,m,n,\cdots}^{(i,j,k,\cdots)\pm} |l\rangle |m\rangle |n\rangle \cdots} } \end{displaymath} where \begin{displaymath} \epsilon_{l,m,n,\cdots}^{(i,j,k,\cdots)\pm} = \left\{ \begin{array}{ll} (\pm 1)^{\#} & \textrm{when l,m,n,}\cdots\textrm{ are \#-th permutation of (i,j,k,}\cdots\textrm{).} \\ 0 & \textrm{otherwise} \\ \end{array} \right. \end{displaymath} and Commutation relations becomes ($[ \hat{A}, \hat{B} ]_{\zeta} = \hat{A}\hat{B} - \zeta\hat{B}, \hat{A}$) \begin{displaymath} \boxed{ [ \hat{a}_i^\dag, \hat{a}_j^\dag ]_{\zeta} = 0, \quad [ \hat{a}_i, \hat{a}_j ]_{\zeta} = 0, \quad [ \hat{a}_i, \hat{a}_j^\dag ]_{\zeta} = \delta_{ij} } \end{displaymath} \ec } \frame{ \bc In SQ, Many things become {\bf \alert{Counting problems}}. \begin{displaymath} \boxed{ \hat{n}_i \equiv \hat{a}_i^\dag \hat{a}_i } \end{displaymath} Many-body Hamiltonian (with two-body interaction) is generally written \begin{displaymath} \boxed{ \hat{H} = \sum_{i,j}{ \hat{a}_{i}^{\dag} \langle i|\hat{H}_{trap} |j \rangle \hat{a}_{j} } + \frac{1}{2} \sum_{i,j,k,l}{ \hat{a}_{i}^{\dag} \hat{a}_{j}^{\dag} \langle i| \langle j| \hat{V}_{int} |l \rangle |k \rangle \hat{a}_{k} \hat{a}_{l} } } \end{displaymath} where $\hat{H}_{trap}$ is a single-particle Hamiltonian operator \\ and $\hat{V}_{int}$ is a two-body interaction operator. \begin{displaymath} \begin{split} &\hat{H}_{trap} |E_n \rangle = E_n |E_n \rangle \\ &\langle \vec{r}_i| \langle \vec{r}_j| \hat{V}_{int} |\vec{r}_l \rangle |\vec{r}_k \rangle = V_{int}(\vec{r}_i, \vec{r}_j) \delta^3 (\vec{r}_i-\vec{r}_l) \delta^3 (\vec{r}_j-\vec{r}_k) \end{split} \end{displaymath} \ec } \subsection{Bose-Einstein Condensation} \frame{\frametitle{Bose-Einstein Condensation} \bc Bosons, unlike fermions, are not subject to the {\bf Pauli exclusion principle}. \\ \bigskip {\bf Noninteracting Bose Gases: \\ \alert{Bose-Einstein Condensation}} (1924) \\ \smallskip \underline{Whole Particles are in \alert{the Lowest Single-Particle State} at $T=0$.} \\ \bigskip Bose-Einstein Distribution Fuction \begin{displaymath} \boxed{ n_i = \frac{g_i}{e^{(\epsilon_i-\mu)/kT}-1} } \end{displaymath} where $n_i$ is the number of particles, $g_i$ is the degeneracy, $\epsilon_i$ is the energy of $i$-th state, and $\mu$ is the chemical potential. \ec } \frame{ \bc {\bf Experimental Result for BEC} \medskip \includegraphics[width=.60\textwidth]{figures/V-distri_Rb.eps} \smallskip Velocity distribution of Rb atoms, taken by means of the expansion method %(Color) Images of the velocity distribution of rubidium atoms in the experiment by Anderson et al. (1995), taken by means of the expansion method. The left frame corresponds to a gas at a temperature just above condensation; the center frame, just after the appearance of the condensate; the right frame, after further evaporation leaves a sample of nearly pure condensate. The field of view is 200mm3270mm, and corresponds to the distance the atoms have moved in about 1/20 s. The color corresponds to the number of atoms at each velocity, with red being the fewest and white being the most. From Cornell (1996). {\fs [Anderson et al. Science {\bf 269}, 198 (1995)]} \ec } \section{Fragmented Condensates} \subsection{Definition of Fragmented Condensates} \frame{\frametitle{Fragmented Condensates} \bc Whole Particles are \alert{always} in the Lowest Single-Particle State at $T=0$? \\ \medskip {\bf No!} in general. \underline{Interactions are crucial.} \\ \bigskip Penrose and onsager (1956): Single-Particle Density Matrix \begin{displaymath} \boxed{ \rho_{ij} \equiv \langle \hat{a}_i^\dagger \hat{a}_j \rangle = \sum_{\mu, \nu}{\langle\mu |i\rangle \langle \hat{a}_{\mu}^\dagger \hat{a}_{\nu} \rangle \langle j|\nu\rangle } } \end{displaymath} How many Eigenvalues are \alert{Macroscopic} ($= \mathcal{O}(N)$) \\ \medskip $\Longrightarrow$ {\bf Single}(Coherent) or {\bf Fragmented} Condensate \\ \bigskip \smallskip Invariant way(indep. of basis) of telling {\bf How many Modes are Macroscopically Occupied} in a specific many-body state. \ec } \subsection{Two-Mode Hamiltonian} \frame{ \bc Neglecting $\mathcal{O}(1)$ fluctuations, so truncating after the first two modes, \\ \bigskip {\bf General Two-Mode Hamiltonian} \begin{displaymath} \boxed{ \begin{split} \hat{H} = &\epsilon_0 \hat{a}_0^\dagger \hat{a}_0 +\epsilon_1 \hat{a}_1^\dagger \hat{a}_1 +\frac{A_1}{2} \hat{a}_0^\dagger \hat{a}_0^\dagger \hat{a}_0 \hat{a}_0 +\frac{A_2}{2} \hat{a}_1^\dagger \hat{a}_1^\dagger \hat{a}_1 \hat{a}_1 \\ &+\frac{A_3}{2} \left( \hat a_0^\dagger \hat a^\dagger_0 \hat a_1 \hat a_1 + \textrm{h.c.} \right) +\frac{A_4}{2} \hat a_1^\dagger \hat a_1 \hat a_0^\dagger \hat a_0 \end{split} } \end{displaymath} This is one step beyond the familiar semiclassical Gross-Pitaevskii theory {\fs[A. J. Leggett, Rev. Mod. Phys. {\bf 73}, 307 (2001)]}. \\ \bigskip The two modes are determined by the multiorbital mean-field method delineated in {\fs[O. E. Alon et al., Phys. Rev. A {\bf 77}, 033613 (2008)]}. \ec } \frame{ \bc {\bf General Two-Mode Hamiltonian} \begin{displaymath} \boxed{ \begin{split} \hat{H} = &\epsilon_0 \hat{a}_0^\dagger \hat{a}_0 +\epsilon_1 \hat{a}_1^\dagger \hat{a}_1 +\frac{A_1}{2} \hat{a}_0^\dagger \hat{a}_0^\dagger \hat{a}_0 \hat{a}_0 +\frac{A_2}{2} \hat{a}_1^\dagger \hat{a}_1^\dagger \hat{a}_1 \hat{a}_1 \\ &+\frac{A_3}{2} \left( \hat a_0^\dagger \hat a^\dagger_0 \hat a_1 \hat a_1 + \textrm{h.c.} \right) +\frac{A_4}{2} \hat a_1^\dagger \hat a_1 \hat a_0^\dagger \hat a_0 \end{split} } \end{displaymath} \begin{displaymath} \begin{split} \alert{\mbox{Pair-Exchange Coefficient}} \qquad& A_3 = g\int d^3{\bm r} \Psi^2_0 ({\bm r}) \Psi^2_1({\bm r}) \\ \mbox{Ansatz for Wave Function} \qquad& |\Psi\rangle = \sum_{l=0}^N\psi_l|N-l,l\rangle \\ \mbox{Single-Particle Density Matrix} \qquad& \rho_{\mu\nu}^{(1)} = \langle \hat{a}_{\mu}^{\dag} \hat{a}_{\nu} \rangle \\ \mbox{Degree of Fragmentation} \qquad& \mathfrak{F} = 1 - \frac{|\lambda_0 - \lambda_1|}{N} \\ &= 1 - \frac{2}{N} \sqrt{ |\langle \hat{a}_{0}^{\dag} \hat{a}_{1} \rangle|^2 + \left( \frac{N}{2} - N_0 \right)^2 } \\ \end{split} \end{displaymath} \ec } \subsection{Fragmented Ground States in a Single Trap} \frame{ \bc It was shown that {\bf a continous variety of ground state fragmentation} can be obtained in a single trap. \\ {\fs[P. Bader and U. R. Fischer, Phys. Rev. Lett. {\bf 103}, 060402 (2009)]} \\ \medskip \includegraphics[width=.70\textwidth]{figures/F_vs_A3.eps} \\ This type of fragmentation is, importantly, \alert{robust} to perturbations coupling on the single-particle level. \ec } \section{Recent Researches} \subsection{Robustness of Fragmentation against Dynamic Perturbations} \frame{ \bc Recently, we investigated whether {\bf the robustness} of continous fragmentation also {\bf persists} against rapid changes on the dynamical many-body instead of the single-particle level, i.e. {\bf when interaction couplings rapidly change}. \\ {\fs [Uwe R. Fischer, Kang-Soo Lee, and Bo Xiong, arXiv:1011.6444v1 (30 Nov 2010)]} \begin{displaymath} \begin{split} \hat H = \sum_{i=0,1} \epsilon_i \hat n_i +\frac{A_1}2 \hat n_0 (\hat n_0-1) %\hat a^\dagger_0\hat a^\dagger_0 \hat a_0 \hat a_0 +\frac{A_2}2 \hat n_1 (\hat n_1-1) \\ %\nn & & %\hat a^\dagger_1\hat a^\dagger_1\hat a_1\hat a_1\label{Ha}\qquad +\frac{\alert{A_3 (t)}}{2}\left(\hat a_0^\dagger\hat a^\dagger_0 \hat a_1\hat a_1+ {\rm h.c.}\right) +\frac{A_4}2 \hat n_0 \hat n_1 %- \frac{\Omega} 2 \left( \hat a_0^\dagger\hat a_1 + {\rm h.c.} \right) %\hat a_1^\dagger \hat a_1 \hat a_0^\dagger\hat a_0 \end{split} \end{displaymath} Assuming an exponential sweep of the form \begin{displaymath} A_3(t) = (A_i-A_f) \exp[-\alpha t] + A_f \end{displaymath} \ec } \frame{ \begin{displaymath} i \partial_t \psi_{l} = \frac{A_{3}(t)}2 \left[ d_{l}\psi_{l+2} + d_{l-2}\psi_{l-2}\right] +c_{l}\psi_{l} \end{displaymath} \bc are solved numerically, where the coefficients $c_l = \frac12A_1(N-l)(N-l-1)+\frac12A_2 l(l-1)+\frac12A_4 (N-l)l$ and $d_l = \sqrt{(l+2)(l+1)(N-l-1)(N-l)}$. \\ \bigskip Due to structure of the Hamiltonian, \alert{even and odd $l$ sectors decouple from each other.} And one gets a solution of the matrix equations for $\psi_l$ with a state %$\ket{\phi}=\sum_{l}\phi_{l}\ket{2l}$ for the even %and $\ket{\Phi}=\sum_l\Phi_l\ket{2l+1}$ for the odd sector for even and odd $l$ separately, \begin{displaymath} |\phi \rangle = \sum_{\textrm{even}~l}{\phi_l |l \rangle}, \qquad |\Phi \rangle = \sum_{\textrm{odd}~l}{\Phi_l |l \rangle} \end{displaymath} cf) $\mathfrak{F} = 1 - \frac{2}{N} \sqrt{ |\langle \hat{a}_{0}^{\dag} \hat{a}_{1} \rangle|^2 + \left( \frac{N}{2} - N_0 \right)^2 }$ \\ \medskip where $\langle \hat{a}_{0}^{\dag} \hat{a}_{1} \rangle = \sum_{l=0}^{N-1}{ \psi_l^* \psi_{l+1} \omega_l }$, \\ and $N_0 = \langle \hat{a}_{0}^{\dag} \hat{a}_{0} \rangle = \sum_{l=0}^{N}{ |\psi_l|^2 (N-l) } = N - N_1$ \\ \bigskip \underline{\alert{The relative phase between even and odd sectors} are important!} \\ \bigskip \bigskip \ec } \frame{ \bc \includegraphics[width=.90\textwidth]{figures/Fig_alpha_final.eps} \\ (a) Second-order transition, ${\cal F}_{\rm gs} = 0.6$, $\Delta {\mathcal F} = 0$ \\ % For \alert{a second-order transition (a)} $N=100, A_1 = A_4 =1, A_2 = 0.5,$ \\ % $A_3(t=0)=-0.2, A_3(\infty)=0.4$, ground state final value of ${\cal F}_{\rm gs} = 0.6$, \\ \medskip (b) (Weakly) First order transition, ${\cal F}_{\rm gs} = 0.71$, $\Delta {\mathcal F} = 1/3$ \\ % and for \alert{(weakly) first order in (b)} they are $N=200, A_2=0.6$, \\ % ${\cal F}_{\rm gs} = 0.71$, $\Delta {\mathcal F} =1/3$ with all others identical. \\ \bigskip The inset shows ${\cal F} (t)$ %upon crossing the transition point at $A_3=0$ and for $\alpha=0.01$; %(a), $\alpha=0.006$ (b); the blue arrow indicates where $A_3=0$. \ec } \subsection{Emergence of pair-coherent phase} \frame{ \bc \includegraphics[width=.90\textwidth]{figures/Phase_jumpEDITED.eps} \\ Left: The final degree of fragmentation for $\alpha=0.1$ as a function of the jump at $A_3=0$%; $N=200$ and the variation of $\Delta F$ %is achieved by varying $A_2$ from $\frac12$ to 1. \\ Right: The average phase difference at late times versus $\Delta {\mathcal F}$. %The inset shows that in the second-order %case $\Delta F=0$, the average $\theta$ approaches $\pi/2$ at late times. $${\cal F} = 1- \frac 2N \sqrt{\left[|ab| N \sin\theta \left(1-\frac{\sigma^2 + 2{\mathfrak S}^2}{N^2}\right)\right]^2\!+{\mathfrak S}^2}$$ \ec } \subsection{Sweeps entirely on the Positive Side} \frame{ \bc For sweeps {\bf entirely on the positive exchange-coupling side}, \\ \alert{No suppression} \\ %is obtained even for large sweep rates, and the dynamical degree of fragmentation remains close to its ground-state value. \\ \bigskip \includegraphics[width=.70\textwidth]{figures/pidividedby4.eps} \\ \alert{Emergence of pair-coherent phase} is due to the {\bf crossing} a singular point of vanishing pair-exchange coupling. \ec } \subsection{Conclusion} \frame{\frametitle{Conclusion} \begin{itemize} \item {\bf For a second-order} dynamical quantum phase transition, the creation of single-trap {\bf fragmentation} from a coherent state {\bf is possible only for sufficiently slow sweeps} of the pair-exchange coupling. \item {\bf For moderatedly rapid variations} of the pair-exchange coupling from negative to positive values, {\bf \alert{a new pair-coherent phase}} is created. \item We have shown {\bf single-particle density matrix is not sufficient to describe (dynamical) many-body physics.} \end{itemize} } \section{On-going Researches} \subsection{The most General two-mode Hamiltonian} \frame{ \bc The most General Hamiltonian (in two-mode approximation) is written by \bdm \begin{split} \hat{H} = &\epsilon_0 \hat{a}_{0}^{\dag} \hat{a}_{0} + \epsilon_1 \hat{a}_{1}^{\dag} \hat{a}_{1} - \frac{\Omega}{2} \hat{a}_{0}^{\dag} \hat{a}_{1} - \frac{\Omega^*}{2} \hat{a}_{1}^{\dag} \hat{a}_{0} \\ &+ \frac{A_1}{2} \hat{a}_{0}^{\dag} \hat{a}_{0}^{\dag} \hat{a}_{0} \hat{a}_{0} + \frac{A_2}{2} \hat{a}_{1}^{\dag} \hat{a}_{1}^{\dag} \hat{a}_{1} \hat{a}_{1} + \frac{A_3}{2} \hat{a}_{0}^{\dag} \hat{a}_{0}^{\dag} \hat{a}_{1} \hat{a}_{1} +\frac{A_3^*}{2}\hat{a}_{1}^{\dag} \hat{a}_{1}^{\dag} \hat{a}_{0} \hat{a}_{0} + \frac{A_4}{2} \hat{a}_{0}^{\dag} \hat{a}_{1}^{\dag} \hat{a}_{0} \hat{a}_{1} \\ &+ \frac{A_5}{2} \hat{a}_{0}^{\dag} \hat{a}_{0}^{\dag} \hat{a}_{0} \hat{a}_{1} +\frac{A_5^*}{2}\hat{a}_{1}^{\dag} \hat{a}_{0}^{\dag} \hat{a}_{0} \hat{a}_{0} + \frac{A_6}{2} \hat{a}_{0}^{\dag} \hat{a}_{1}^{\dag} \hat{a}_{1} \hat{a}_{1} +\frac{A_6^*}{2}\hat{a}_{1}^{\dag} \hat{a}_{1}^{\dag} \hat{a}_{1} \hat{a}_{0} \\ \end{split} \edm with $A_i$\'s defined to be \begin{displaymath} \begin{split} A_1 &\equiv V_{0000}, \qquad A_2 \equiv V_{1111},\qquad A_3 \equiv V_{0011}, \quad A_3^* \equiv V_{1100} \\ A_4 &\equiv V_{0101} + V_{1010} + V_{0110} + V_{1001} \\ A_5 &\equiv V_{0001} + V_{0010} , \qquad A_5^* \equiv V_{1000} + V_{0100} \\ A_6 &\equiv V_{0111} + V_{1011} , \qquad A_6^* \equiv V_{1110} + V_{1101} \\ \end{split} \end{displaymath} where \bdm V_{ijkl} \equiv \langle i,j|\hat{V}_{int} |l,k \rangle = \int{ d^3\vec{r}_1 \int{ d^3\vec{r}_2 ~~ \psi_{i}^{*}(\vec{r}_1) \psi_{j}^{*}(\vec{r}_2) V_{int}(\vec{r}_1, \vec{r}_2) \psi_{k}(\vec{r}_2) \psi_{l}(\vec{r}_1) } } \edm \ec } \subsection{Works on-going} \frame{ \bdm \begin{split} \hat{H} = &\epsilon_0 \hat{a}_{0}^{\dag} \hat{a}_{0} + \epsilon_1 \hat{a}_{1}^{\dag} \hat{a}_{1} - \frac{\Omega}{2} \hat{a}_{0}^{\dag} \hat{a}_{1} - \frac{\Omega^*}{2} \hat{a}_{1}^{\dag} \hat{a}_{0} \\ &+ \frac{A_1}{2} \hat{a}_{0}^{\dag} \hat{a}_{0}^{\dag} \hat{a}_{0} \hat{a}_{0} + \frac{A_2}{2} \hat{a}_{1}^{\dag} \hat{a}_{1}^{\dag} \hat{a}_{1} \hat{a}_{1} + \frac{A_3}{2} \hat{a}_{0}^{\dag} \hat{a}_{0}^{\dag} \hat{a}_{1} \hat{a}_{1} +\frac{A_3^*}{2}\hat{a}_{1}^{\dag} \hat{a}_{1}^{\dag} \hat{a}_{0} \hat{a}_{0} + \frac{A_4}{2} \hat{a}_{0}^{\dag} \hat{a}_{1}^{\dag} \hat{a}_{0} \hat{a}_{1} \\ &+ \frac{A_5}{2} \hat{a}_{0}^{\dag} \hat{a}_{0}^{\dag} \hat{a}_{0} \hat{a}_{1} +\frac{A_5^*}{2}\hat{a}_{1}^{\dag} \hat{a}_{0}^{\dag} \hat{a}_{0} \hat{a}_{0} + \frac{A_6}{2} \hat{a}_{0}^{\dag} \hat{a}_{1}^{\dag} \hat{a}_{1} \hat{a}_{1} +\frac{A_6^*}{2}\hat{a}_{1}^{\dag} \hat{a}_{1}^{\dag} \hat{a}_{1} \hat{a}_{0} \\ \end{split} \edm \begin{itemize} \item Effect of \alert{$A_5$ and $A_6$} on Fragmentation? \item Effect of \alert{imaginary $A_3$ and imaginary $\Omega$}? \item \alert{Stability} of continuous fragmentation including $\Omega$ (or/and $A_5, A_6$). \item How to get \alert{the first two-modes}? \item Seeing Problems with \alert{Different Basis}. (Change of Basis) \end{itemize} } \end{document} / ## RRA
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