Spectroscopy and spin dynamics for strongly interacting few spinor bosons in one-dimensional traps

TL;DR

This paper investigates the spin dynamics and energy level structure of a few strongly interacting spin-1 bosons in one-dimensional traps, revealing how their quantum states evolve and how symmetry and interactions influence their behavior.

Contribution

It provides a detailed analysis of the energy spectrum, level crossing, and spin-mixing dynamics of strongly interacting spinor bosons beyond the SU(3) integrable point.

Findings

Energy spectrum degeneracy is lifted away from the SU(3) point.

Spin-mixing dynamics are highly sensitive to the ratio g0/g2.

Effective spin chain models outperform Heisenberg models in state transfer accuracy.

Abstract

We consider a one-dimensional trapped gas of strongly interacting few spin-1 atoms which can be described by an effective spin chain Hamiltonian. Away from the SU(3) integrable point, where the energy spectrum is highly degenerate, the rules of ordering and crossing of the energy levels and the symmetry of the eigenstates in the regime of large but finite repulsion have been elucidated. We study the spin-mixing dynamics which is shown to be very sensitive to the ratio between the two channel interactions g0/g2 and the effective spin chain transfers the quantum states more perfectly than the Heisenberg bilinear-biquadratic spin chain.

Equations

$$H=\sum _{i=1}^{N}h\left(x_i\right)+\sum _{i<j}^N\left(c_0+c_2{\mathbf{s}}_i\cdot {\mathbf{s}}_j\right)\delta \left(x_i-x_j\right),$$

$$h\left(x\right)=-\frac{{\hslash }^2}{2M}\frac{d^2}{d{x}^2}+\frac{1}{2}M{\omega }^2{x}^2-Gx{s}^z+\Omega s^x,$$

$$\Psi \left(x_1,s_1;\cdots x_N,s_N\right)$$

$$H_{eff}=-\sum _{i=1}^{N-1}{J}_i\left(\frac{1}{g_0}{P}_0\left(i,i+1\right)+\frac{1}{g_2}{P}_2\left(i,i+1\right)\right).$$

$$P_0\left(i,i+1\right)=\frac{{\left({\mathbf{s}}_i\cdot {\mathbf{s}}_{i+1}\right)}^2-1}{3}$$

$$P_2\left(i,i+1\right)=\frac{{\left({\mathbf{s}}_i\cdot {\mathbf{s}}_{i+1}\right)}^2}{6}+\frac{{\mathbf{s}}_i\cdot {\mathbf{s}}_{i+1}}{2}+\frac{1}{3}$$

$$J_i$$

$$H=\sum _i\left({\mathbf{s}}_i\cdot {\mathbf{s}}_{i+1}+\beta ({\mathbf{s}}_i\cdot {\mathbf{s}}_{i+1}{)}^2\right).$$

$$|E_2^1,0,1⟩$$

$$|E_0^1,0,1⟩$$

$$|E_1^2,0,-1⟩$$

$$H_{eff}^{\prime }=H_{eff}-G\sum _i{D}_i{s}_i^z,$$

$$a_{j}=\left(\begin{array}[]{ccc}0&0&1\\ 0&1&0\\ 1&0&0\end{array}\right)$$

$$T|E_i,S_z⟩=|E_i,-S_z⟩.$$

$${\rho }_m(x)=\sum _i{\rho }_m^{(i)}{\rho }^{(i)}(x)$$

$${\rho }_m^{\left(i\right)}=\sum _{m_1,\cdots ,m_N}{\left|⟨m_1,\cdots ,m_N|\chi ⟩\right|}^2{\delta }_{m,m_i}$$

$${\rho }^{(i)}(x)=N!\int d{x}_1\cdots d{x}_N\delta (x-x_i)\theta (x_1,\cdots ,x_N){\left|{\varphi }_F\right|}^2.$$

$$|G_{odd}⟩=\frac{1}{\sqrt{2}}\left(|E_0,S_z=+1⟩+|E_0,S_z=-1⟩\right),$$

$$|\chi \left(t\right)⟩=e^{-\frac{i}{\hslash }{H}_{eff}t}|\chi \left(0\right)⟩=\sum _i{c}_i{e}^{-\frac{i}{\hslash }{E}_{i}t}|{\varphi }_i⟩,$$

$$F\left(t\right)\equiv {\left|⟨\Psi \left(t\right)|1\cdots 1,-1⟩\right|}^2,$$

Full text

I Introduction

Spinor quantum gases have attracted a lot of attentions due to their rich physics in quantum coherence, spin dynamics Kurn2013 ; Law ; Chang ; Naylor ; Zhang , long-range order Yi ; Santos , quantum magnetism Campbell and symmetry breaking Navon . The majority of researches in many-body system involved trapped spinor atoms such as 23Na and 87Rb experimentally realized in many cold atom labs Stamper ; Liu2009 ; Vengalattore ; Jacob ; Vinit ; Fukuhara . Among these, spin $s=1$ system plays a central role in the fundamental understanding of topological quantum phase transition of condensed materials and in modern technologies including, for instance, data storage Morozov , spin currents Sinova , spin vortex Yi ; Li , etc. In low-dimensional system, owing to the liberation of the spin degrees of freedom, a major focus is the understanding of quantum magnetism of higher spin, which have their origin in the underlying microscopic processes between elementary spins.

Recently the experiment in Heidelberg showed strong evidences that the spin chain of few cold atoms in one-dimensional system without an underlying lattice can be realized in vicinity of a scattering resonance Murmann2015 . On the theoretical side, the general form of the effective spin-chain model for strongly interacting atomic gases with an arbitrary spin in the one-dimensional(1D) traps has been presented Deuretzbacher2008 ; Deuretzbacher ; Yang ; Yang1 ; Yang2 ; Lindgren ; Volosniev . This provides a platform for the research of basic magnetic processes in few-body system. For the two-component system with a large but finite strong $s$-wave interaction, the transition between the ferromagnetism (FM) and anti-ferromagnetism (AFM) phase, the theorem on the level crossing between singlet ground state and the maximum spin state have been studied Cui ; Hu ; Guan ; Lieb . In addition, the effect of a weak additional $p$-wave interaction on the magnetic orders of the ground state has already been addressed Yang2015 ; Hu1 . Due to the existence of two-channel interaction, the spin-1 system exhibits much richer phenomena than the two-component system, which is expected to give rise to rich magnetic properties in the strongly interacting limit.

Based on the effective spin-chain model to strongly interacting trapped boson gases with spin-1, we study the properties of the magnetic ground state in the strongly repulsive regime and explore the ordering and crossing of the energy levels when the interaction transfers from the FM to AFM. By analyzing the symmetry of system in the presence of a spin-dependent magnetic gradient and a transverse magnetic field, we show how the ground state may be manipulated and the atoms would collide in the spin-mixing dynamics depending mainly on the ratio between of interaction strengths in the two collisional channel. Spin-1 systems, in comparison to spin-1/2 systems, offer a better security for encoding and transferring quantum information, primarily due to their larger Hilbert spaces. We further study the quantum state transfer (QST) between the two ends of the spin chain. The effective spin-1 chain provides a site-dependent spin-coupling protocol, due to background trap potential and the resultant inhomogeneous particle density. The effective spin-chain coupling protocol is expected to show more beneficial features than the Heisenberg bilinear-biquadratic (HBB) spin-chain coupling.

The rest of the paper is organized as follows. In Section II, we give the detailed spectrum and eigenstates of the effective spin-1 chain model. In Sec. III, based on the effective spin-chain model, we study the spin-changing dynamics and the efficiency and advantage of the QST with this effective spin chain. Finally we conclude in section IV.

II Spectroscopy

We consider $N$ interacting atoms of hyperfine spin $s=1$ and mass $M$ in a one-dimensional harmonic trap. The system Hamiltonian is

[TABLE]

where $\mathbf{s}_{i}=\left(s_{i}^{x},s_{i}^{y},s_{i}^{z}\right)$ is the spin-1 matrix for the $i$-th atom, $c_{0}=\left(g_{0}+2g_{2}\right)/3$ and $c_{2}=\left(g_{2}-g_{0}\right)/3$ with $g_{0}$ and $g_{2}$ the coupling constants in the scattering channels with total spin $S=0$ and $2$, respectively Ho ; Ohmi . For $F=1$ 87Rb, both the sign and the magnitude of $c_{2}$ and hence the magnetic nature of the system can be altered significantly in a relatively wide rang by means of a high resolution photoassociation spectroscopy of the atoms to some excited molecular states as shown in Chapman’s experiment Hamley . On the other hand, it has been proposed that a multichannel scattering resonance can be achieved for spinor bosons confined in one-dimension geometry with an additional spin-flipping rf field and the interaction in the two channels $S=0$ and $2$ in our system can be tuned to be large simultaneously near the resonance Cui2014 . In the single-particle Hamiltonian

[TABLE]

with $\omega$ being the trapping frequency, each particle feels a transverse magnetic field of strength $\Omega$ and a spin-dependent magnetic gradient of strength $G$, which are extremely small perturbations and will not cause any noticeable effects in a weakly interacting system Cui . Note that both $\Omega$ and $G$ have absorbed in them the Landé factor $g_{s}$ and the Bohr magneton $\mu_{B}$. This will change a lot in the strongly interacting system as can be seen in this study.

Notably, in the absence of the external field, the eigenfunctions of the few-particle system have been exactly solved Girardeau1 ; Girardeau2 ; Deuretzbacher by means of the Bose-Fermi mapping in the Tonks-Girardeau limit. The ground state wave function of a spin-1 system with infinite interaction is described by Deuretzbacher ; Yang ; Yang1

[TABLE]

where $\theta\left(x_{1}\cdots x_{N}\right)=1$ if $x_{1}\leq\cdots\leq x_{N}$ and zero otherwise, $x_{i}$ and $s_{i}=1,0,-1$ are position and spin indices of the $i$-th particle, respectively. The wave function $\phi_{F}$ is taken as the ground state of $N$ spinless fermions, i.e. the Slater determinant made up of the lowest $N$-level of eigenstates, while the spin wave function $\left|\chi\right\rangle$ can be written as a superposition of spin Fock states $\left|m_{1}m_{2}\cdots m_{N}\right\rangle$, which means the $i$th spin is in the $m_{i}$ state, i.e. $\left|m_{i}\right\rangle=\delta_{s_{i}m_{i}}$. The permutation $P$ acts on both the spatial and spin wave functions and ensures the symmetry upon particle exchange. The model in the regime of large but finite repulsion can be mapped to an effective ferromagnetic chain of spin-1 bosons to the first order of $g_{0}^{-1},g_{2}^{-1}$ Yang

[TABLE]

Instead of representing the model in terms of the permutation operators $P_{ij}$ of neighboring spins Deuretzbacher , we here classify the states according to the collisional channels of total spin $S$ of the two sites. For spin-1 atoms we define the projection operators in the total spin $S=0$ and $S=2$ channels as

[TABLE]

and

[TABLE]

in the direct sum of the spin space $S=0\oplus S=1\oplus S=2$. The effective spin-exchange interaction

[TABLE]

depends on the overlap between the wave functions of neighboring atoms. The structure of the Hamiltonian takes the form of a HBB spin-$1$ chain Yip2003 ; Affleck

[TABLE]

The only difference is that here the coupling constants of neighbor spins are different, due to the background trap potential and the resultant inhomogeneous particle density. The effective spin Hamiltonian $H_{eff}$, constructed from variational approach and perturbation theory Deuretzbacher ; Yang ; Yang1 ; Lindgren , conserves the square of the total spin operator $\mathbf{S}=\sum_{i=1}^{N}\mathbf{s}_{i}$, its $z$ component $S_{z}=\sum_{i=1}^{N}s_{i}^{z}$, and the parity operator $\Pi=P_{1,N}P_{2,N-1}\cdots$, such that the eigenstates of $H_{eff}$ can be classified in terms of the three quantum numbers: the total spin $S$, the total magnetization $S_{z}$ and the parity $\Pi$.

It is intuitive to examine first the eigenvalues and eigenstates of the $H_{eff}$ in the regime of large but finite repulsion $g_{0},g_{2}\gg 0$. In the simplest case of two particles $N=2$, we can easily see that the eigenvalues in the channel $S=2,1,0$ are respectively $-J_{1}/g_{2}$, [math] and $-J_{1}/g_{0}$. While in the anti-ferromagnetic spin chain people pay more attention to the degenerate point of singlet and triplet $\beta=-1/3$, which corresponds to TG limit $g_{0}\rightarrow+\infty$ in our case, we focus on the $SU(3)$ integrable point $g_{0}=g_{2}$ where the quintuplet and the singlet have the same energy for the ferromagnetic spin chain. Note that the sign of $J$, hence the order of the energy levels, is inverted for these two cases, which gives different level crossing point for the ground state. In Figure 1, we show the energy level dependence on the ratio $g_{0}/g_{2}$, in which each level is $2S+1$-fold degenerate in the total spin $S$ channel. We find there exist generally plenty of level crossing in the degenerate point which can be classified into different bunches. To specify them, one needs to denote the eigenstates as $\left|E_{S}^{n},S_{z},\Pi\right\rangle$, where $n$ labels the bunch of degenerate states occurring at $g_{0}=g_{2}$. The two-particle eigenstates with zero magnetization $S_{z}=0$ can be constructed as

[TABLE]

on which other states with $S_{z}=\pm 2,\pm 1$ can be obtained by applying spin raising or lowering operators $S^{\pm}$ repeatedly. We note that two of them belong to the first bunch $n=1$, while the second bunch $n=2$ consists of a single level. For the three-particle case with $S_{z}=0$, seven levels group into four bunches with the number of levels 2,2,2,1 in each bunch respectively; in the ground state bunch, the total spin $S=3$ state with energy $-2J_{1}/g_{2}$ competes with the $S=1$ state with energy $-J_{1}/2g_{2}\left(1/2+2\alpha+\sqrt{\left(2\alpha-1\right)^{2}+5/4}\right)$, giving rise to the level crossing point at $\alpha=1$ where $\alpha=\left(2g_{2}/g_{0}+1\right)/3$.

The spectrum of $H_{eff}$ for more particles $N=4,5$ are shown in Figure 1(c)-(d) by numerically diagonalizing the Hamiltonian (4) in the spin Fock state vector $\left|m_{1}m_{2}\cdots m_{N}\right\rangle$. It is clearly seen that the spectrum is asymmetric about the integrable point $g_{0}=g_{2}$, at which the ground state is $\left(N+1\right)\left(N+2\right)/2$ fold degenerate. The levels belonging to the same bunch have the same parity: it is always even ($\Pi=1$) for the ground state bunch, which contains nevertheless $(N+1)/2$ levels for odd $N$ and $(N+2)/2$ levels for even $N$, whereas it is always odd ($\Pi=-1$) for the first excited state bunch with $(N-1)$ levels. The total spin in the ground state bunch are $S=N,N-2,N-4,\cdots$ with step $\Delta S=2$ and for the first excited state $S=N,N-1,\cdots 1$ with step $\Delta S=1$. The two levels with highest total spin $S=N$ and lowest $S=0(1)$ for even (odd) $N$ are the only two candidates for the ground state configuration as a consequence of the ferromagnetic or anti-ferromagnetic coupling of spins. For FM coupling, $g_{0}/g_{2}>1$, away from the level crossing point, the degenerate energy levels of the bound states are eliminated for different total spin $S$. For every bunch of degenerate energy levels, the energy decreases with total spin $S$, i.e., $E_{S_{1}}^{n}<E_{S_{2}}^{n}$ when $S_{1}>S_{2}$. Therefore, the state with $S=N$ is the ground state with completely symmetric spin wave functions. For AFM coupling $g_{0}/g_{2}<1$, on the other hand, the energy increases with total spin $S$, i.e., $E_{S_{1}}^{n}<E_{S_{2}}^{n}$ when $S_{1}<S_{2}$. Therefore the states with $S=1$ (if $N$ is odd) or $S=0$ (if $N$ is even) have the lowest energy. The emergence of level crossing in the lowest bunch of the energy levels at $g_{0}=g_{2}$ clearly indicates a first-order transition between AFM and FM phases. For $N$ particles, there exist altogether $[N/2]$ independent inhomogeneous spin coupling parameters $J_{i}$, with the only result being a slight modification of the energy levels compared with the spectrum without the trapping potential, which nevertheless does not change the ordering and crossing of the levels.

The spectrum of the system is highly degenerate for the total spin $S$. We now consider the weak spin-dependent magnetic gradient introduced in the single-particle Hamiltonian (2). As schematically shown in Fig. 2, atoms of different spin components are trapped in different potential wells, with the trap center moved to the left or right by an amount $G^{\prime}=G\hbar/M\omega^{2}$ depending on the value of spin $m$. The corresponding effective spin Hamiltonian $\left(\ref{he}\right)$ in the limit of strong interaction will be modified into

[TABLE]

where $D_{i}=N!\int x_{i}\left|\phi_{F}\right|^{2}\theta\left(x_{1}\cdots x_{N}\right)\prod\nolimits_{j=1}^{N}dx_{j}$ represents the average position of the $i$th atom. The spin-dependent magnetic gradient destroys the total spin conservation and parity conservation, implying that $H_{eff}^{\prime}$ no longer commutes wtih $S^{2}$ and $\Pi$. However we can find that the Hamiltonian $H_{eff}^{\prime}$ commutes with an operator $T=\Pi\prod_{j=1}^{N}a_{j}$ ChenLi , where

[TABLE]

serves to flip the spin of $j$-th atom. It is straightforward to show that $\left\{T,S_{z}\right\}=0$ and $\left[T,S_{z}\right]=2TS_{z}$. As a result, $T$ applying to an energy eigenstate $\left|E_{i},S_{z}\right\rangle$ changes the state to a degenerate eigenstate $\left|E_{i},-S_{z}\right\rangle$, i.e.,

[TABLE]

One can also infer from Fig. 3 that, adding the magnetic gradient lifts partially the degeneracy of spectrum. The states with total magnetizations $S_{z}$ and $-S_{z}$ remain degenerate. The ground state of the system mixes all total spin states to achieve a maximum reduction in energy and thus occurs the spin component separation (bottom of Fig. 2). The first-order transition at $g_{0}=g_{2}$ disappears and the level with total magnetization $S_{z}=0$ proves to be the ground state (Fig. 3(a)) for even particle numbers. On the other hand, for odd particle numbers, the state with total magnetization $S_{z}=\pm 1$ has the lowest energy (Fig. 3(b)).

The density distribution of $m$-th spin component is defined as

[TABLE]

with the probability that the magnetization of the $i$th spin equals $m$,

[TABLE]

and the probability to find the $i$th atom with any spin at position $x$,

[TABLE]

We show the density distribution for a gradient $G=2\hbar^{2}\omega^{2}/g_{2}$ in the bottom of Fig. 2. To be more precise, we focus on the $SU(3)$ integrable point $g_{0}=g_{2}$ which guarantees the conservation of atoms in each spin component due to the spin-independent interaction Jiang . Surprisingly, we find that the spin-[math] component always disappears for even particle numbers in the subspace of the total magnetization $S_{z}=0$ (see Fig. 2(b)), while for odd atom numbers the density of spin-0 component remains unity (Fig. 2(a) and 2(c)). This can be understood by noting that the already fermionized atoms would fill the evenly spaced levels from bottom one by one, and it is more energetically favorable to put the additional atoms in the left and right traps which are lowered by the gradient by an amount $(G\hbar)^{2}/2M\omega^{2}$.

For an applied transverse magnetic field the effective Hamiltonian $H_{eff}^{\prime}$, with an additional term $\Omega S_{x}$ included, no longer commutes with $S_{z}$ and the degeneracy of the system is completely eliminated. However, $H_{eff}^{\prime}$ still commutes with the operator $T$. For even atom numbers, the density distributions in the ground state (see for example, Fig. 2(b)) are hardly modified after the introduction of a very small $\Omega$, which can be regarded as a perturbation to the $S_{z}=0$ ground state. For odd atom numbers, the ground state can be constructed as the superposition of two degenerate ground states with $S_{z}=\pm 1$, whose densities are shown in Fig. 4(a) and 4(d), respectively, i.e.

[TABLE]

with $T\left|G_{odd}\right\rangle=\left|G_{odd}\right\rangle$. The transverse field plays the role of coupling the two degenerate states such that the ground state is lowered by an amount $\hbar\Omega$. The density profiles of spin $+1$ and spin $-1$ are now symmetric to each other, due to the conservation of $T$ implying an combined operation of space inversion and spin flipping, as can be seen in Fig. 4(b). The population in the component spin [math] increases noticeably for a strong enough $\Omega$ (Fig. 4(c)). The off-diagonal feature of the $S_{x}$ matrix would inevitably mix excited states such as that with $S_{z}=0$, the spin [math] density of which is significant. An alternative way to introduce the spin [math] component is to bring the system away from the integrable point. We show this in Fig. 4(e) and 4(f) for the case of $g_{0}/g_{2}=2$, which shows an obvious enhancement of spin [math] density.

III Dynamics

III.1 Spin-changing Dynamics

Realizing spin-chain Hamiltonian with trapped cold atoms offers important applications in the study of microscopic magnetic phenomena. Here we investigate the spin-changing dynamics of this system, which is different from the population dynamics of the weakly interacting system governed by the Gorss-Pitaevskii equation Chang ; Kronjager . To do this, strongly interacting atoms are initially prepared in the ground state $\left|\chi\left(0\right)\right\rangle$ with a weak spin-dependent magnetic gradient $G=2\hbar^{2}\omega^{2}/g_{2}$. The total magnetization $S_{z}$ is still the conserved quantity, so the system will evolve within one of the $S_{z}$ subspace. Then the gradient $G$ is abruptly switched off and the evolution of the system is governed by the effective spin chain Hamiltonian $H_{eff}$ in (4). The initial state is realized by obtaining the ground state of Hamiltonian $H_{eff}^{\prime}$ with non-vanishing $G$. Starting from the initial state $\left|\chi\left(0\right)\right\rangle$, the time evolution of the wave function is governed by

[TABLE]

where $c_{i}=\left\langle\phi_{i}\right.\left|\chi\left(0\right)\right\rangle$ is the overlap of the initial state and the $i$-th eigenstate of the system $\phi_{i}$ with eigenenergy $E_{i}$. We introduce the spin population $P_{m}\left(t\right)=\sum_{i}\rho_{m}^{\left(i\right)}\left(t\right)$ with $\rho_{m}^{\left(i\right)}\left(t\right)$ defined in (13) with the replacement $\chi(0)\rightarrow\chi(t)$, which measures the population of $m$-th component in the system. For spin-$1$ system, two atoms in the states $-1$ and $+1$ have a chance to coherently and reversibly scatter into final states containing two atoms in the state [math], which leads to the population transferring from $P_{1}\left(t\right)+P_{-1}\left(t\right)$ to $2P_{0}\left(t\right)$, or vice versa, subject to the conservation of the total population $\sum_{m}P_{m}\left(t\right)=N$. The system satisfies $P_{1}\left(t\right)-P_{-1}\left(t\right)=S_{z}$ at any time. In this section we only consider the dynamics in the subspace of total magnetization $S_{z}=0$ in which case one must have $P_{+1}=P_{-1}$. We illustrate the spin population dynamics in Figure 5 for both $P_{\pm 1}$ and $P_{0}$. In the case of $N=3$, the initial spin populations for both interaction parameters $g_{0}/g_{2}=0.5$ and $2$ are very close to the case of equally distributed among the three components in the integrable point $g_{0}=g_{2}$. Starting from such an initial population, Rabi-like oscillations of spin populations between the components [math] and $\pm 1$ are observed and depicted in Fig. 5(b), which is in sharp contrast to the respectively conservation of atoms in each spin component for $g_{0}=g_{2}$. In the entire range of interaction of interest, we managed to extract the amplitude and the frequency of the oscillation (Fig. 5(a)), and it turns out that the oscillation amplitudes of populations are determined by the weight coefficients of the basis vectors and the oscillation frequencies of populations are determined by the energy differences, among which three energy levels $E_{1}^{1},E_{3}^{1},E_{3}^{3}$ play dominate roles in the dynamics of the spin-changing collisions. At the integrable point we find either the frequency or the amplitude of the partial wave would vanish, which ensures the populations in each components remain constants, $P_{0}(t)=P_{\pm 1}(t)=1$ for $N=3$ and $P_{0}(t)=0$, $P_{\pm}(t)=2$ for $N=4$. The intrinsic origin of this exotic phenomenon lies in that this point is highly degenerate. Away from this point, the oscillation frequency of the primary amplitude increases significantly on both sides of $g_{0}=g_{2}$, however, a lower frequency will slow down the oscillation for $g_{0}/g_{2}=2$. More energy levels are involved in the dynamics of $N=4$ atoms, the initial spin population of which is close to the case of equally distributed on the $\pm 1$ components at $g_{0}=g_{2}$. The characteristic dynamics here may be used to detect the quantum phases of the spin-1 chain model, and moreover, may reveal the interesting spin population transfer across the phase boundary by the oscillation frequency.

III.2 Quantum state transfer

Spin chains have important applications in quantum simulation and computation. The spin chains have been proposed intensively as quantum channels to study state transfer in small quantum networks Bose2003 ; Bose2007 ; Burgarth . Perfect quantum-state transfer is very important to accomplish prospective quantum information processing through a chain of nearest-neighbor coupled spins. The interaction energy of each qutrit-qutrit pair in the translation-invariant HBB spin-$1$ chain is the same, which can be described by the Hamiltonian (8) with $\beta=(2g_{2}/g_{0}+1)/3$, but in our effective spin chain this interaction energy (7) is site dependent. Here we study the superiority of the inhomogeneous effective spin-1 chain as a quantum channel.

Transferring a known or unknown quantum state with spin-1 from one place to another has been studied in Ref. Ghosh ; Wiesniak . In the original proposal, the quantum-state transfer protocol involves initializing the spin chain of $N$ sites with the first spin in an arbitrary state $\left|\psi\right\rangle=\xi_{-1}\left|-1\right\rangle+\xi_{0}\left|0\right\rangle+\xi_{1}\left|1\right\rangle$ ($\sum_{m}\left|\xi_{m}\right|^{2}=1$) and decoupled from the rest of the chain. At $t=0$, the first and second spins abruptly couple and let the system freely evolve in the spin chain. At time $t$, the quality of the transfer of $\left|\psi\right\rangle$ to the last spin of the chain is evaluated by the fidelity of attaining $\left|\psi\right\rangle$ at site $N$. Ideal transfer would imply that at time $t^{\ast}$ the last spin of the chain is in state $\left|\psi\right\rangle$. We consider a simple case, at time $t=0$ its state was $\left|\Psi\left(0\right)\right\rangle=\left|-1,1\cdots 1\right\rangle$. Our aim is to maximize the probability of retrieving state $\left|1\cdots 1,-1\right\rangle$ at time $t^{\ast}$. We define the fidelity of state transfer as

[TABLE]

which relies only on the expansion coefficients of the eigenstates $\left|E_{S}^{n}\right\rangle$ expanded in the basis vectors $\left|1\cdots 1,-1\right\rangle$ and $\left|-1,1\cdots 1\right\rangle$. Let $F\equiv F\left(t^{\ast}\right)$ be the maximum value that achieves in the intermediate time. We plot the maximum of fidelity $F$ of the state transfer in Fig. 6 as a function of $g_{0}/g_{2}$. For a system of $N=4$ with the trap frequency $\omega=40kHz$ and the interaction strength $g_{2}=20$ in units of $\sqrt{\hbar^{3}\omega/M}$, the occurrence time of the maximal fidelity $t^{\ast}$ ranges from $10^{-2}$s to $1.5$s, which appears, however, randomly for varying $g_{0}/g_{2}$. We compare the fidelity $F$ of the HBB and effective spin chains of length $N=4$ to $8$ in Fig. 6, which reflects that the effective spin chain transfers the state more faithfully than the HBB spin chain, especially for longer spin chain. With the increase of particle number $N$, the overall trend is that $F$ decreases. At the integrable point $g_{0}/g_{2}=1$, $F$ reaches a maximum value in the both spin chain models, while the effective spin chain model always provides more efficient way for quantum state transfer in the entire interaction regime.

IV Conclusions

We have shown that a three-component system of strongly interacting bosonic atoms in a 1D harmonic trap can be represented effectively as a spin chain described by the bilinear-biquadratic spin-1 model Hamiltonian. For few atoms in the trap we have determined the energy spectrum of the ground states and obtained the rules of the ordering and crossing of energy levels near the first-order quantum phase transition, i.e. the $SU(3)$ integrable point, $g_{0}=g_{2}$. The energy levels of the eigenstates are collected into different bunches which can be labelled by the total spin and the parity. Away from the degenerate point, the ground state is either with highest total spin $S=N$ for FM coupling between atoms, or with lowest spin $S=1$ (for odd $N$) or $S=0$ (for even $N$) for AFM coupling $g_{0}/g_{2}<1$. We further introduce a magnetic gradient to remove the degeneracy on $S$, motivated by the experimental studies of coherent multi-flavour spin dynamics in a fermionic quantum gas Krauser , and subsequently study the quench dynamics of the the ground states of spin component separation when the initial magnetic gradient is removed quickly. Our results reveal the spin-change dynamics of the system governed by the ratio of interactions between the two channels. Through the study of the dynamics of the quantum state transfer, we show that the inhomogeneous qutrit-qutrit interaction of the engineered effective spin chain is more efficient in state transferring.

Acknowledgements.

Y. Zhang would like thank Han Pu for helpful discussion. This work is supported by NSF of China under Grant Nos. 11474189, 11674201, 11234008, Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT)(No. IRT13076). S. C. is supported by NSF of China under Grant Nos. 11425419, 11374354 and 11174360.