Interaction effects on $\mathcal{PT}$-symmetry breaking transition in atomic gases

TL;DR

This paper investigates how interatomic interactions influence the parity-time ($ ext{PT}$) symmetry breaking transition in ultracold atomic gases, revealing that the transition point depends on interaction parameters and system size.

Contribution

It introduces a model for interacting bosons under $ ext{PT}$-symmetric Hamiltonian and analyzes how interactions modify the symmetry breaking transition point.

Findings

Transition point $ ext{Γ}_{ m tr}$ decreases with interaction strength and particle number.

In the large interaction limit, $ ext{Γ}_{ m tr}$ scales as $| ext{δg}|^{-(N-1)}$.

Signatures of $ ext{PT}$ phases are proposed for experimental detection.

Abstract

Non-Hermitian systems having parity-time ($\mathcal {PT}$) symmetry can undergo a transition, spontaneously breaking the symmetry. Ultracold atomic gases provide an ideal platform to study interaction effects on the transition. We consider a model system of $N$ bosons of two components confined in a tight trap. Radio frequency and laser fields are coupled to the bosons such that the single particle Non-Hermitian Hamiltonian $h_{\mathcal PT}=-i \Gamma\sigma_z+J\sigma_x$, which has $\mathcal {PT}$-symmetry, can be simulated in a \emph{passive} way. We show that when interatomic interactions are tuned to maintain the symmetry, the $\mathcal {PT}$-symmetry breaking transition is affected only by the SU(2) variant part of the interactions parameterized by $\delta g$. We find that the transition point $\Gamma_{\rm tr}$ decreases as $|\delta g|$ or $N$ increases; in the large $|\delta g|$ limit, $\Gamma_{\rm tr}$ scales as $\sim|\delta g|^{-(N-1)}$. We also give signatures of the $\mathcal {PT}$-symmetric and the symmetry breaking phases for the interacting bosons in experiment.

Figures

Equations

$$H_{int}=\sum _{j\ne k}$$

$$-g_{\uparrow \downarrow }\frac{{\sigma }_z^{(j)}+1}{2}\frac{{\sigma }_z^{(k)}-1}{2}],$$

$$H=$$

$$H_{\mathcal{P}T}=$$

$$C=$$

$${\left(H_{\mathcal{P}T}\right)}_{m{m}^{\prime }}\equiv$$

$$=$$

$$+J\sqrt{(N/2)(N/2+1)-m^2+m}{\delta }_{m,m^{\prime }+1}$$

$$+J\sqrt{(N/2)(N/2+1)-m^2-m}{\delta }_{m,m^{\prime }-1}.$$

$$\frac{d\rho }{dt}=-i[{\mathcal{H}}_s,\rho ]-2\Gamma (b_{\uparrow }^{†}{b}_{\uparrow }\rho +\rho b_{\uparrow }^{†}{b}_{\uparrow })+4\Gamma b_{\uparrow }\rho b_{\uparrow }^{†},$$

$$H_{\mathcal{P}T}=\left[\begin{array}{ccc}-2i\Gamma +\delta g& \sqrt{2}J& 0\\ \sqrt{2}J& 0& \sqrt{2}J\\ 0& \sqrt{2}J& 2i\Gamma +\delta g\end{array}\right].$$

$$f(\lambda )$$

$$=-{\lambda }^3+2\delta g{\lambda }^2+[4(J^2-{\Gamma }^2)-\delta g^2]\lambda -4J^2\delta g,$$

$$\Delta (J,\Gamma ,\delta g)=$$

$$\displaystyle-\Gamma^{2}\delta g^{4}\big{]}$$

$$H_{\mathcal{P}T,L}=$$

$$V=$$

$$H_{eff}=$$

Full text

Study of non-Hermitian systems is constantly enriching our knowledge derived from Hermitian ones Bender ; Muss ; Moi ; Guo ; Kip ; Ueda ; Segev ; Khaj ; Fu . Of particular interest are a class of non-Hermitian systems having the parity-time ($\mathcal{PT}$) symmetry Bender1999 ; Moibook ; Benderbook . A representative model of the class is a two-level system whose Hamiltonian is of the form $h_{\mathcal{P}T}=-i\Gamma\sigma_{z}+J\sigma_{x}$; under the combined transformation of complex conjugate and the swap $\ket{\uparrow}\leftrightarrow\ket{\downarrow}$, $h_{\mathcal{P}T}$ is invariant Swanson . Parameter tuning across the critical point $\Gamma_{\rm tr}=J$ gives rise to the transition of the two-level system from the $\mathcal{PT}$-symmetric phase to the symmetry breaking phase where exponentially growing or decaying modes set in. $\mathcal{PT}$-symmetry breaking transition has been widely investigated in electromagnetic Guo ; Kip ; Kottos ; Peschel ; Schafer ; Yang , and mechanical systems Bender2013 . The transition is the cornerstone of exceptional properties regarding light propagation Lin2011 ; Scherer2011 ; Scherer2013 , lasing Yang2014 ; Khaj2014 ; Zhang2014 and topological energy transfer Rotter2016 ; Harris2016 .

Recently $\mathcal{PT}$-symmetry breaking transition was successfully demonstrated in a gas of two component noninteracting 6Li atoms in a passive way Luo ; in the experiment, the application of a radio-frequency field and a laser inducing loss in one component of the atoms leads to, apart from kinetic energy, the single particle Hamiltonian $h=-i\Gamma+h_{\mathcal{P}T}$, as the term $-i\Gamma$ gives rise to an overall decay. This approach circumvents the difficulty of realizing an atom gain in quantum simulation of $\mathcal{PT}$-symmetric non-Hermitian Hamiltonians in atomic gases Moi . On the other hand, Feshbach resonance enables unprecedented control of interactions in ultracold atomic gases Chin , and deterministic preparation is achievable for a sample of variable $N$ atoms Bloch2010 ; Greiner2010 ; Bloch2011 ; Jochim . These capabilities make ultracold atoms an ideal platform to probe interaction effects on $\mathcal{PT}$-symmetry breaking transition Konotop2016 .

In this work, we consider $N$ interacting two component bosons confined in a tight harmonic trap such that their spatial wave-function is frozen to be the ground harmonic state. The bosons are subject to the radio frequency field and the laser as in Ref. Luo . Feshbach resonance is used to tune the interaction Hamiltonian of the $N$ bosons to maintain the $\mathcal{P}T$-symmetry. We find that in this interacting system, $\mathcal{PT}$-symmetry breaking transition depends on only the SU(2) variant part of the interactions parameterized by $\delta g$. The transition point $\Gamma_{\rm tr}$ decreases as $|\delta g|$ or $N$ increases. In the large $|\delta g|$ limit, $\Gamma_{\rm tr}$ is suppressed as $\sim|\delta g|^{-(N-1)}$. Finally we show how the modification on the transition by the interactions can be detected experimentally.

Figure (1) gives a schematic of the system that we consider. Bosonic atoms with two internal states denoted by $\ket{\uparrow}$ and $\ket{\downarrow}$ are confined in a harmonic trap potential $V(\mathbf{r})=\frac{1}{2}m\omega_{0}^{2}r^{2}$, where $m$ is the atomic mass. For simplicity, we assume the confinement being so tight, i.e., $\omega_{0}$ is much larger than any other energy scales to be considered, that the spatial wave-function of the bosons is frozen to be the single particle ground state $\phi_{0}(\mathbf{r})$ of the harmonic trap. A radio-frequency field of frequency equal to the internal energy difference $E_{\uparrow}-E_{\downarrow}$ is used to flip the atoms between the two internal states ${\ket{\uparrow}}$ and $\ket{\downarrow}$ with Rabi frequency $J$. An additional laser is coupled to the atoms in state $\ket{\uparrow}$ and results in a loss rate $4\Gamma$ of the atom number in the state. We take $\hbar=1$ throughout.

In the absence of interatomic interactions, the Hamiltonian for each bosons spanned by $\ket{\uparrow}$ and $\ket{\downarrow}$ is non-Hermitian and is given by $h=-i\Gamma+h_{\mathcal{P}T}$, apart from the ground harmonic state energy Luo . The interatomic interaction Hamiltonian of the bosons is given by

[TABLE]

where $g_{\sigma\sigma^{\prime}}=(4\pi a_{\sigma\sigma^{\prime}}/m)\int d\mathbf{r}|\phi_{0}(\mathbf{r})|^{4}$ and $a_{\sigma\sigma^{\prime}}$ is the s-wave scattering length Pethick ; Bloch , and $\boldsymbol{\sigma}^{(j)}$ are the Pauli matrices for the $j$th boson. The interaction Hamiltonian is $\mathcal{PT}$-symmetric only if $g_{\uparrow\uparrow}=g_{\downarrow\downarrow}$; in atomic gases, this condition can be experimentally fulfilled by the technique of Feshbach resonance Chin . We focus on this situation $g\equiv g_{\uparrow\uparrow}=g_{\downarrow\downarrow}$ in the following discussion.

Therefore, the total Hamiltonian of $N$ interacting bosons is given by

[TABLE]

where $\boldsymbol{S}=\sum_{j=1}^{N}\boldsymbol{\sigma}^{(j)}/2$, and $\delta g\equiv g-g_{\uparrow\downarrow}$. Compared with the noninteracting case, the $\mathcal{PT}$-symmetry breaking transition of the interacting boson system is now determined by $H_{\mathcal{P}T}$. Since there are two internal states $\ket{\uparrow}$ and $\ket{\downarrow}$ to accommodate $N$ bosons, the dimension of the Hilbert space shall be $N+1$. It is easy to assure oneself that such a space is spanned by the streched states $|N/2,m\rangle$ with $m=-N/2,-N/2+1,\dots,N/2-1,N/2$ where $S^{2}|N/2,m\rangle=N(N+2)/4|N/2,m\rangle$ and $S_{z}|N/2,m\rangle=m|N/2,m\rangle$. In this space, the matrix element of $H_{\mathcal{P}T}$ becomes

[TABLE]

The transition occurs when some eigenvalues of $H_{\mathcal{P}T}$ coalesce and turn complex afterwards. Note that when $\delta g=0$, the interactions drop out of $H_{\mathcal{P}T}$; this is the situation that the scattering lengths $a_{\sigma\sigma^{\prime}}$ become all the same and the interatomic interactions are SU(2) invariant. This dependence of the transition on the interactions is because if the interatomic interactions are SU(2) invariant, i.e., $[\mathbf{S},H_{int}]=0$, the noninteracting $\mathcal{PT}$-symmetric non-Hermitian Hamiltonian $-2i\Gamma S_{z}+2JS_{x}$ and $H_{int}$ are commutable and can be diagonalised simultaneously.

The above non-Hermitian Hamiltonian formalism is related to the Lindbald equation describing the bosons subject to pure loss in the following way. In terms of the field operator $b_{\sigma}$ ($b^{\dagger}_{\sigma}$) which annihilates (creates) a boson of internal state $\sigma$ in the ground state of the harmonic trap, the Lindbald equation for the density matrix $\rho$ of the bosons is given by Cohen

[TABLE]

where $\mathcal{H}_{s}$ is the Hermitian Hamiltonian of the bosons in the absence of the external lossy laser coupling. Given that in experiment the initial density matrix $\rho(0)$ shall be always block diagonalized in the number of bosons, i.e., $\rho_{\alpha\beta}(t=0)$ is zero unless $\alpha=\beta$, where $\rho_{\alpha\beta}(t)\equiv P_{\alpha}\rho(t)P_{\beta}$ and $P_{\alpha}$ is the $\alpha$ boson subspace projection operator, so is $\rho(t)$. If the last term $4\Gamma b_{\uparrow}\rho b^{\dagger}_{\uparrow}$ in Eq. (6) were not there, the time dependent density matrix would be given by $\rho(t)=U(t)\rho(0)U^{\dagger}(t)$ with $U(t)=e^{-i\mathcal{H}t}$ and $\mathcal{H}\equiv\mathcal{H}_{s}-i2\Gamma b^{\dagger}_{\uparrow}b_{\uparrow}$; since the projection of $\mathcal{H}$ in the $N$ boson subspace is just the non-Hermitian Hamiltonian $H$ in Eq. (2), i.e., $H=P_{N}\mathcal{H}P_{N}$, the properties of $H$ would determine the time evolution of $\rho(t)$.

To access the importance of the term $4\Gamma b_{\uparrow}\rho b^{\dagger}_{\uparrow}$ in Eq. (6) to a typical experiment starting with $N$ particles, we note that initially only $\rho_{NN}(t=0)$ is nonzero. Since the pure loss can only cause the particle number to decrease, for all the following time, $\rho_{\alpha\beta}(t)=0$ if $\alpha>N$ or $\beta>N$. Thus, from Eq. (6) one can first obtain $\rho_{NN}(t)=U(t)\rho_{NN}(0)U^{\dagger}(t)$; the term $4\Gamma b_{\uparrow}\rho b^{\dagger}_{\uparrow}$ has no effects on $\rho_{NN}(t)$ since the projection of the term involves only $\rho_{N+1,N+1}(t)$ which is identically zero. Note that $\mathcal{H}$ commutes with the total particle number. From hereon, one can show $\rho_{N-1,N-1}(t)=\int_{0}^{t}d\tau U(t-\tau)[4\Gamma b_{\uparrow}\rho_{NN}(\tau)b^{\dagger}_{\uparrow}]U^{\dagger}(t-\tau)$. Likewise, one can solve all the rest $\rho_{\alpha\alpha}(t)$ for $\alpha<N$ in a cascade; the non-diagonal parts are always zero, i.e., $\rho_{\alpha\beta}=0$ for $\alpha\neq\beta$. The above argument justifies one to study the time evolution of the purely lossy system by analyzing the non-Hermitian Hamiltonian $H$ from Eq. (2). Of course, calculations of observables should resort to the density matrix $\rho(t)$. This justification shall also apply to other similar purely lossy systems.

We start with analyzing the non-Hermitian Hamiltonian $H$ of two interacting bosons. For $N=2$, the Hamiltonian $H_{\mathcal{P}T}$ in the basis $\{\ket{\uparrow}\ket{\uparrow},(\ket{\uparrow}\ket{\downarrow}+\ket{\downarrow}\ket{\uparrow})/\sqrt{2},\ket{\downarrow}\ket{\downarrow}\}$ has the explicit form

[TABLE]

The corresponding characteristic polynomial is

[TABLE]

whose zeros are the eigenvalues of $H_{\mathcal{P}T}$. Since all the coefficients of the cubic polynomial $f(\lambda)$ are real, one of the three zeros of $f(\lambda)$ is real definite. When $\Gamma$ is increased from zero, the rest two zeros of $f(\lambda)$, which are also real in the first place, coalesce at the the $\mathcal{PT}$-symmetry transition point and become complex afterwards. This coalescence occurs when the discriminant of $f(\lambda)$

[TABLE]

is zero. When $\delta g=0$, $\Delta(J,\Gamma_{\rm tr},0)=0$ retrieves the known transition point $\Gamma_{\rm tr}/J=1$. For nonzero $\delta g$, Fig. (2) shows that $\Gamma_{\rm tr}/J$ is suppressed more and more as $|\delta g/J|$ increases. By Eq. (9), it is manifest from that $\Gamma_{\rm tr}/J$ is even in $\delta g/J$, and one finds $(\Gamma_{\rm tr}/J)^{2}\approx 1-3|\delta g/J|^{2/3}/2^{4/3}$ for $|\delta g/J|\ll 1$, and $\Gamma_{\rm tr}/J\approx|\delta g/J|^{-1}$ for $|\delta g/J|\gg 1$.

The suppression of $\Gamma_{\rm tr}/J$ in the limit $|\delta g/J|\gg 1$ is readily understood by inspecting the $\mathcal{PT}$-symmetric Hamiltonian, Eq. (7). In such a limit, we recast $H_{\mathcal{P}T}=H_{\mathcal{P}T,L}+V$ with

[TABLE]

To the the leading order, $H_{\mathcal{P}T,L}$ yields right away that the two states $\ket{\uparrow}\ket{\uparrow}$ and $\ket{\downarrow}\ket{\downarrow}$ are degenerate and share the same eigenvalue $\delta g$, and the eigenvalue of the third state $(\ket{\uparrow}\ket{\downarrow}+\ket{\downarrow}\ket{\uparrow})/\sqrt{2}$ is zero, well separated apart from $\delta g$. Since the $\mathcal{PT}$-symmetry transition is expected to happen at the point where eigenvalues of $H_{\mathcal{P}T}$ coalesce, to find the effects of $\Gamma$ and $J$ on the two degenerate eigenvalues originally equal to $\delta g$, we use $V$ to carry out a perturbation calculation to derive the effective Hamiltonian in the subspace spanned by the states $\ket{\uparrow}\ket{\uparrow}$ and $\ket{\downarrow}\ket{\downarrow}$; we find that to second order of $V$ the effective Hamiltonian is given by Cohen

[TABLE]

This effective Hamiltonian yields $\Gamma_{\rm tr}/J\approx|J/\delta g|^{-1}$, the same as from requiring $\Delta(J,\Gamma_{\rm tr},\delta g)=0$.

For $N>2$, we numerically diagonalise $H_{\mathcal{P}T}$ and find that the $\mathcal{PT}$ transition is always due to the coalescence of a pair of eigenvalues of $H_{\mathcal{P}T}$. Figure (2) shows that the critical value $\Gamma_{\rm tr}/J$ is also symmetric in $\delta g/J$, which is because, under the transformation $\delta g\to-\delta g$ and $\mathbf{S}\to-\mathbf{S}$, we have $H_{\mathcal{P}T}\to-H_{\mathcal{P}T}$. We find that for fixed $N$, $\Gamma_{\rm tr}/J$ decreases as $|\delta g/J|$ increases, while for fixed $\delta g/J$, as $N$ increases, $\Gamma_{\rm tr}/J$ is more and more suppressed.

Figure (3) shows that in the large $|\delta g/J|$ limit, $\Gamma_{\rm tr}/J\sim|\delta g/J|^{-(N-1)}$. This asymptotic behavior can be understood by an analysis similar to the one given above for $N=2$. For arbitrary $N$, in the large $|\delta g|$ limit, we separate $H_{\mathcal{P}T}$ from Eq. (3) as $H_{\mathcal{P}T}=H_{\mathcal{P}T,L}+V$ with $H_{\mathcal{P}T,L}=\delta g\left(S_{z}\right)^{2}$ and $V=-2i\Gamma S_{z}+2JS_{x}$. In such a limit, the leading order Hamiltonian $H_{\mathcal{P}T,L}$ gives rise to a pair of degenerate eigvenvalues in each subspaces spanned by $|N/2,m\rangle$ and $|N/2,-m\rangle$; the eigenvalues $m^{2}\delta g$ are all well separated from each other. To determine the transition, we use $V$ to derive the effective Hamiltonian $H_{eff}$ in the each two dimensional subspace. It is easy to convince oneself that to the lowest order of $J$, the diagonal elements are $\langle N/2,\pm m|H_{eff}|N/2,\pm m\rangle=m^{2}\delta g\mp 2im\Gamma$, and the off-diagonal elements are generated at order of $V^{2|m|}$, resulting in $\langle N/2,\pm m|H_{eff}|N/2,\mp m\rangle\sim J^{2|m|}/\delta g^{2|m|-1}$. Thus, by diagonalizing the $2\times 2$ matrix of $H_{eff}$, we find the transition point $\Gamma_{\rm tr}/J\sim|\delta g/J|^{-(2|m|-1)}$ for each two dimensional subspaces. Given that the maximum value of $|m|$ equals $N/2$, overall, the $N$-body system enters into the symmetry breaking phase first at $\Gamma_{\rm tr}/J\sim|\delta g/J|^{-(N-1)}$.

The relation between the non-Hermitian Hamiltonian formalism and the Lindbald equation for our system given above indicates that the signatures of the $\mathcal{PT}$-symmetric and symmetry breaking phases governed by $H_{\mathcal{P}T}$ in Eq. (3) can be detected experimentally in the following way. Let one prepare the experiment initally with $N$ bosons Bloch2010 ; Greiner2010 ; Bloch2011 such that $\rho_{NN}(t)=e^{-2N\Gamma t}e^{-iH_{\mathcal{P}T}t}\rho_{NN}(0)e^{iH_{\mathcal{P}T}^{\dagger}t}$; the quantity $e^{2N\Gamma t}\rho_{NN}(t)$ shall have qualitatively different time dependent behaviors in the symmetric and symmetry breaking phases. For example, by the high accuracy atom number detection achieved experimentally Jochim , one can measure the rescaled probability of finding $N$ bosons ${\rm P}(N,t)=e^{2N\Gamma t}\tilde{\rm P}(N,t)$ with $\tilde{\rm P}(N,t)\equiv{\rm Tr}\rho_{NN}(t)$. In contrast, the total number of atoms was measured to distinguish the two phases for the noninteracting 6Li atoms Luo . In our interacting case, the total number of atoms ceases to be a good observable for the purpose since the observable depends on not only $\rho_{NN}(t)$ but also $\rho_{\alpha\alpha}(t)$ for $\alpha<N$ whose dynamics is not determined by a single Hamiltonian $H$ for $\alpha$ bosons. Figure (4) plots the rescaled ${\rm P}(2,t)$ in an experiment starting with $N=2$ bosons and $\delta g/J=1$ for various values of $\Gamma/J$. Note that for $N=2$ and $\delta g/J=1$, $\Gamma_{\rm tr}/J\approx 0.538$. Figure (4) shows that ${\rm P}(2,t)$ is bounded in the $\mathcal{PT}$-symmetric phase, and grows exponentially in the symmetry breaking one. Due to the interactions, the point $\Gamma/J=3/4$ is already in the symmetry breaking phase while its value is still smaller than the critical value $\Gamma_{\rm tr}/J=1$ for the noninteracting case.

Acknowledgements. We thank Jiaming Li for discussions. This work is supported by NSFC Grants No. 11474179, No. 11722438, and No. 91736103.