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arXiv:1902.04769 · Physical Review A

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

Lei Pan, Shu Chen, Xiaoling Cui

Published Jun 19, 2019• 24 citations
arXivHTMLPDF

Abstract

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 $\ensuremath{\sim}{\ensuremath{\epsilon}}^{1/(2ks+1)}$ 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.

DOI
10.1103/physreva.99.063616

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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 (PT\mathcal{PT}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 NNN and arbitrary spin sss. Generally, with spin-independent interactions, the EP order of bosons can be as high as 2Ns+12Ns+12Ns+1, and the spectral response around EP can be as sensitive as \ensuremath∼\ensuremathϵ1/(2ks+1)\ensuremath{\sim}{\ensuremath{\epsilon}}^{1/(2ks+1)}\ensuremath∼\ensuremathϵ1/(2ks+1) under a kkk-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.
Exceptional points (EPs) have seen substantial advances in both experiment and theory. However, in quantum systems, higher-order EPs remain of great interest and possess numerous intriguing properties yet to be fully explored. Here we describe a parity-time symmetry-protected three-level non-Hermitian system with dissipative spin-orbit-coupled fermions in which a third-order exceptional point (EP3) emerges when both the eigenvalues and eigenstates of the system collapse into one. The band structure and its spin dynamics are explored for the 173Yb^{173}\mathrm{Yb}173Yb fermion. We highlight the enhanced sensitivity to the external perturbation of the EP3 with cubic-root energy dispersion. Additionally, we investigate the second-order EP (EP2) with square-root energy dispersion in a three-level quantum system in the absence of parity symmetry, which proves that the enhanced sensitivity closely relates to the symmetries of the non-Hermitian system. Furthermore, we analyze the encircling behavior of the EP3 in terms of the slowly evolving limit and the nonadiabatic dynamics beyond the slowly evolving limit and discover some results different from those of the EP2.
High-order exceptional points (EPs) emerging in non-Hermitian systems have attracted broad interest for their significantly enhanced sensitivity to perturbations. However, quantum sensing schemes based on high-order EPs remain scarce, due to the experimental challenge of fine-tuning the system to such an extremely sensitive isolated point. Here we propose a four-channel dissipative coupling model that supports both fourth-order exceptional surfaces and second-order exceptional volumes. This non-Hermitian model can be realized in a thermal atomic system, and its complex energy spectra can be determined via electromagnetically induced transparency spectroscopy. The proposed model exhibits a characteristic fourth-order response to multiple physical quantities such as the laser detuning and the distance between optical channels, significantly surpassing the response of second-order EPs. We further reveal the sensitivity-robustness trade-off under experimental noise. Our work opens a route toward high-performance sensing leveraging higher-order EPs.
Exceptional points (EPs) are non-Hermitian spectral singularities exhibiting fractional-power responses, yet their implications for multipartite entanglement of interacting quantum many-body systems remain largely unexplored. Here we develop a general framework that links higher-order non-Hermitian degeneracies to the scaling behavior of genuine multipartite entanglement in interacting identical-qubit systems. Permutation symmetry of the identical qubits decomposes the exponentially large Hilbert space into independent irreducible-representation sectors, thereby constraining the maximal EP order of NNN qubits to N+1N+1N+1 rather than 2N2^N2N. Near an nnnth-order EP, genuine multipartite entanglement inherits the spectral response and generically exhibits a fractional-power scaling under weak perturbations. Explicit examples show that conventional two-body interactions support third- and fourth-order EPs with the corresponding entanglement responses, whereas higher-order EPs with genuine multipartite-entangled coalesced states require additional independent interaction channels, such as three-body interactions. Our results establish a fundamental connection among non-Hermitian degeneracies, multipartite entanglement, and symmetry, extending higher-order EP physics from spectral singularities to genuine many-body quantum correlations.
Abstract High-order exceptional points (EPs) emerging in non-Hermitian systems have attracted broad interest for their significantly enhanced sensitivity to perturbations. However, quantum sensing schemes based on high-order EPs remain scarce, due to the experimental challenge of fine-tuning the system to such an extremely sensitive isolated point. Here we propose a four-channel dissipative coupling model that supports both fourth-order exceptional surfaces and second-order exceptional volumes. This non-Hermitian model can be realized in a thermal atomic system, and its complex energy spectra can be determined via electromagnetically induced transparency spectroscopy. The proposed model exhibits a characteristic fourth-order response to multiple physical quantities such as the laser detuning and the distance between optical channels, significantly surpassing the response of second-order EPs. We further reveal the sensitivity-robustness trade-off under experimental noise. Our work opens a route toward high-performance sensing leveraging higher-order EPs.
As one of the essential features in non-Hermitian systems coupled with environment, the exceptional point has attracted much attention in many physical fields. The phenomena that eigenvalues and eigenvectors of the system simultaneously coalesce at the exceptional point are also one of the important properties to distinguish from Hermitian systems. In non-Hermitian systems with parity-time reversal symmetry, the eigenvalues can be continuously adjusted in parameter space from all real spectra to pairs of complex-conjugate values by crossing the phase transition from the parity-time reversal symmetry preserving phase to the broken phase. The phase transition point is called an exceptional point of the system, which occurs in company with the spontaneous symmetry broken and many novel physical phenomena, such as sensitivity-enhanced measurement and loss induced transparency or lasing. Here, we focus on a two-qubit quantum system with parity-time reversal symmetry and construct an experimental scheme, prove and verify the features at its third-order exceptional point, including high-order energy response induced by perturbation and the coalescence of eigenvectors. We first theoretically study a two-qubit non-Hermitian system with parity-time reversal symmetry, calculate the properties of eigenvalues and eigenvectors, and prove the existence of a third-order exceptional point. Then, in order to study the energy response of the system induced by perturbation, we introduce an Ising-type interaction as perturbation and quantitatively demonstrate the response of eigenvalues. In logarithmic coordinates, three of the eigenvalues are indeed in the cubic root relationship with perturbation strength, while the fourth one is a linear function. Moreover, we study the eigenvectors around exceptional point and show the coalescence phenomenon as the perturbation strength becomes smaller. The characterization of the response of eigenvalues at high-order exceptional points is a quite diffcult task as it is in gneneral diffcult to directly measure eigenenergies in a quantum system composed of a few qubits. In practice, the time evolution of occupation on a particular state is used to indirectly fit the eigenvalues. In order to make the fitting of experimental data more reliable, we want to determine an accurate enough expressions for the eigenvalues and eigenstates. To this aim, we employ a perturbation treatment and show good agreement with the numerical results of states occupation obtained by direct evolution. Moreover, we find that after the system evolves for a long enough time, it will end up to one of the eigenstates, which gives us a way to demonstrate eigenvector coalescence by measuring the density matrix via tomography and parity-time reversal transformation. To show our scheme is experimentally applicable, we propose an implementation using trapped 171Yb+ ions. We can map the parity-time reversal symmetric Hamiltonian to a purely dissipative two-ion system:use microwave to achieve spin state inversion, shine a 370 nm laser to realize dissipation of spin-up state, and apply Raman operation for Mølmer-Sørensen gates to implement Ising interaction. By adjusting the corresponding microwave and laser intensity, the spin coupling strength, the dissipation rate and the perturbation strength can be well controlled. We can record the probability distribution of the four product states of the two ions and measure the density matrix by detecting the fluorescence of each ion on different Pauli basis.
Usually, when investigating exceptional points (EPs) of an open Markovian bosonic system, one deals with spectral degeneracies of a non-Hermitian Hamiltonian (NHH), which can correctly describe the system dynamics only in the semiclassical regime. A recently proposed quantum Liouvillian framework [Minganti et al., Phys. Rev. A 100, 062131 (2019)] enables the complete determination of the dynamical properties of such systems and their EPs (referred to as Liouvillian EPs, or LEPs) in the quantum regime by taking into account the effects of quantum jumps, which are ignored in the NHH formalism. Moreover, the symmetry and eigenfrequency spectrum of the NHH become a part of much larger Liouvillian eigenspace. As such, the EPs of an NHH form a subspace of the LEPs. Here we show that once an NHH of a dissipative linear bosonic system exhibits an EP of a certain finite order nnn, it immediately implies that the corresponding LEP can become of any higher order $m\ensuremath{\ge}n$ defined in the infinite Hilbert space. These higher-order LEPs can be identified by the coherence and spectral functions at the steady state. The coherence functions can offer a convenient tool to probe extreme system sensitivity to external perturbations in the vicinity of higher-order LEPs. As an example, we study a linear bosonic system of a bimodal cavity with incoherent mode coupling to reveal its higher-order LEPs; particularly, of second and third order via first- and second-order coherence functions, respectively. Accordingly, these LEPs can be additionally revealed by squared and cubic Lorentzian spectral lineshapes in the power and intensity-fluctuation spectra. Moreover, we demonstrate that these EPs can also be associated with spontaneous parity-time (PT\mathcal{PT}PT) and anti-PT\mathcal{PT}PT-symmetry breaking in the system studied. These symmetries can be switched in the output fields (the so-called supermodes) of an additional linear coupler with a properly chosen coupling strength. Thus, we show that the initial loss-loss dynamics for the supermodes can be equivalent to the balanced gain-loss evolution.