Interacting non-Hermitian ultracold atoms in a harmonic trap: Two-body exact solution and a high-order exceptional point

We study interacting ultracold atoms in a three-dimensional (3D) harmonic trap with spin-selective dissipations, which can be effectively described by non-Hermitian parity-time ($\mathcal{PT}$) symmetric Hamiltonians. By exactly solving the non-Hermitian two-body problem of spin-1/2 (spin-1) bosons in a 3D harmonic trap, we find that the system can exhibit third-order (fifth-order) exceptional points (EPs) with ultrasensitive cube-root (fifth-root) spectral response due to interaction anisotropies in spin channels. We also present the general principle for the creation of high-order EPs and their spectral sensitivities with arbitrary particle number $N$ and arbitrary spin $s$. Generally, with spin-independent interactions, the EP order of bosons can be as high as $2Ns+1$, and the spectral response around EP can be as sensitive as $\sim {\epsilon }^{1/(2ks+1)}$ under a $k$-body interaction anisotropy. Moreover, we propose to detect the ultrasensitive spectral response through the probability dynamics of certain state. These results suggest a convenient route towards more powerful sensor devices in spinor cold atomic systems.

Figure 1

Schematics of experimental setup for ultracold atoms with spin-selective dissipations. (a) Two-species (spin-1/2) system where a rf field is used to couple the two-spin states with coupling strength $\mathrm{\Omega }$. A resonant laser is applied to generate a spin-dependent dissipation ($\mathrm{\Gamma }$) on a spin-down state. (b) Three-species (spin-1/2) system where two rf fields couple the three spin states and two additional lasers with different strengths (${\mathrm{\Gamma }}_1$, ${\mathrm{\Gamma }}_2$) are used to transfer the states $|0\rangle$, $|-1\rangle$ to excited atomic states $|e_1\rangle$, $|e_2\rangle$.

Figure 2

Exact spectrum of non-Hermitian two spin-1/2 boson system in a 3D harmonic trap. (a) Energy levels as functions of coupling strength. Here we take $a_s^{1,1}=a_s^{1,-1}\equiv a_s$ and $a_s^{1,0}=a_s+\delta$ with $\delta =0.005l$, $\mathrm{\Omega }/{\omega }_T=0.3$, and $\mathrm{\Gamma }/\mathrm{\Omega }=0.5$. The location of repulsive branch in the weak-coupling regime ($0<a_s/l\ll 1$) is marked. (b) Exact solutions of the three lowest energy levels at repulsive branch as functions of $\mathrm{\Gamma }$, in comparison with the mean-field prediction (dashed lines). Here we take $a_s^{1,\pm 1}=a_s=0.1l$ and $a_s^{1,0}=a_s+\delta$ with $\delta =0.005l$. (c) The same as panel (b) except for $\delta =0$, i.e., when the interaction is spin independent. In this case, $\mathrm{\Gamma }=\mathrm{\Omega }$ is the location of third-order EP. (d) Energy splitting ($\mathrm{\Delta }E$) at EP as a function of interaction anisotropy $\epsilon$ in the $\{S=1,M=0\}$ channel. The dashed line shows the analytical fit (see text).

Figure 3

Exact spectrum of non-Hermitian two spin-1 boson system in a 3D harmonic trap. (a) Energy levels as functions of coupling strength. Here we take $a_s^{2,2}=a_s^{2,1}=a_s^{2,-1}=a_s^{2,-2}\equiv a_s$ and $a_s^{2,0}=a_s+\delta$ with $\delta =0.005l$, $\mathrm{\Omega }/\omega =0.3$, and $\mathrm{\Gamma }/\mathrm{\Omega }=0.5$. The location of the repulsive branch in the weak-coupling regime ($0<a_s/l\ll 1$) is marked. (b) Exact solutions of the five lowest energy levels at repulsive branch as functions of $\mathrm{\Gamma }$, in comparison with the mean-field prediction (dashed lines). Here we take $a_s^{2,\pm 2}=a_s^{2,\pm 1}=a_s=0.1l$ and $a_s^{2,0}=a_s+\delta$ with $\delta =0.005l$. (c) The same as panel (b) except for $\delta =0$, i.e., when the interaction is spin independent. In this case, $\mathrm{\Gamma }=\mathrm{\Omega }$ is the location of fifth-order EP. (d) Energy splitting ($\mathrm{\Delta }E$) at EP as a function of interaction anisotropy $\epsilon$ in $\{S=2,M=0\}$ channel. The dashed line shows the analytical fit (see text).

Figure 4

Mean-field spectral responses of three harmonically trapped spin-1/2 bosons with anisotropic interactions in (a)–(c) the two-body and (d)–(f) three-body collision channels. Panels (a) and (b) show the real and imaginary parts of energy levels as a function of $\mathrm{\Gamma }$ under a two-body interaction anisotropy: $a_s^{1,1}=a_s^{1,-1}=a_s$ and $a_s^{1,0}=a_s+{\epsilon }_2$, where $a_s\equiv l/30$ and ${\epsilon }_2=0.005l$. Panel (c) shows the according energy shifts of four levels at $\mathrm{\Gamma }/\mathrm{\Omega }=1$ as a function of ${\epsilon }_2$. Panels (d) and (e) show the spectral responses with an isotropic two-body interaction and an additional three-body interaction (${\epsilon }_3=0.005l$) among the $\uparrow \uparrow \uparrow$ sector. Panel (f) shows the according energy shifts of four levels at $\mathrm{\Gamma }/\mathrm{\Omega }=1$ as a function of ${\epsilon }_3$. The dashed lines in panels (d) and (f) show analytical fits (see text). In all plots, we take $\mathrm{\Omega }/{\omega }_T=0.1$.

Figure 5

Mean-field spectral responses of three harmonically trapped spin-1 bosons with anisotropic interactions in the (a)–(c) two-body and (d)–(f) three-body collision channels. Panesl (a) and (b) show the real and imaginary parts of energy levels as a function of $\mathrm{\Gamma }$ under a two-body interaction anisotropy in the $S=2,M=0$ channel: $a_s^{2,\pm 2}=a_s^{2,\pm 1}=a_s$ and $a_s^{2,0}=a_s+{\epsilon }_2$, where $a_s=l/30$ and ${\epsilon }_2=0.005l$. Panel (c) shows the according energy shifts of seven levels at $\mathrm{\Gamma }=\mathrm{\Omega }$ as a function of ${\epsilon }_2$. Panels (c) and (d) show the spectral responses with an isotropic two-body interaction and an additional three-body interaction (${\epsilon }_3=0.005l$) in the $S=3$, $M=0$ channel. Panel (f) shows the according energy shifts of seven levels at $\mathrm{\Gamma }=\mathrm{\Omega }$ as a function of ${\epsilon }_3$. The dashed lines in panels (d) and (f) show analytical fits (see text). In all plots, we take $\mathrm{\Omega }/{\omega }_T=0.1$.

Figure 6

$(2ks+1)$ Hessenberg matrix produced by spin-$s$ bosons with number $N$ and $k$-body interaction anisotropy. The matrix has only zero entries below (above) the $(2ks+1)\mathrm{st}$ subdiagonals (superdiagonals) (including the main diagonal).

Figure 7

(a) Dynamical evolution of ${\rho }_{\uparrow \uparrow }\left(t\right)$ with initial state ${\rho }_{\uparrow \uparrow }\left(0\right)=1$. The oscillating curves are in the $\mathcal{PT}$-symmetric phase $\mathrm{\Gamma }/\mathrm{\Omega }<1$, and the monotonically increasing curves are in the $\mathcal{PT}$-symmetry-broken phase $\mathrm{\Gamma }/\mathrm{\Omega }>1$. The solid lines are from the Lindblad master equations (LME) and the dashed lines are from the effective Hamiltonian (EH). (b) Time evolution of ${\rho }_{\uparrow \uparrow }^{\prime }\left(t\right)$ at EP with a tiny interaction anisotropy $\epsilon /l=0.001$. (c) Time evolution of ${\tilde{\rho }}_{\uparrow \uparrow }\left(t\right)$ at EP with a tiny interaction anisotropy $\epsilon /l=0.001$. (d) Rescaled spectral distributions ${\tilde{\rho }}_{\uparrow \uparrow }\left(\omega \right)$ at EP with tiny perturbations induced by two-body anisotropy in the $S=1$, $M=0$ channel. In the inset of panel (d), the red stars show the spectral peak position ${\omega }_p$ as a function of perturbation strength $\epsilon$; the blue solid line shows the analytical fit ${\omega }_p\sim {\epsilon }^{1/3}$. Here we set $\mathrm{\Omega }/{\omega }_T=0.1$.