Parity-time ($\mathcal{PT}$) symmetric systems are classical, gain-loss systems whose dynamics are governed by non-Hermitian Hamiltonians with exceptional-point (EP) degeneracies. The eigenvalues of a $\mathcal{PT}$-symmetric Hamiltonian change from real to complex conjugates at a critical value of gain-loss strength that is called the $\mathcal{PT}$ breaking threshold. Here, we obtain the $\mathcal{PT}$-threshold for a one-dimensional, finite Kitaev chain -- a prototype for a p-wave superconductor -- in the presence of a single pair of gain and loss potentials as a function of the superconducting order parameter, on-site potential, and the distance between the gain and loss sites. In addition to a robust, non-local threshold, we find a rich phase diagram for the threshold that can be qualitatively understood in terms of the band-structure of the Hermitian Kitaev mo del. In particular, for an even chain with zero on-site potential, we find a re-entrant $\mathcal{PT}$-symmetric phase bounded by second-order EP contours. Our numerical results are supplemented by analytical calculations for small system sizes.
Non-Hermitian systems with $\mathcal{PT}$ symmetry can possess purely real eigenvalue spectra. In this work two one-dimensional systems with two different topological phases, the topological nontrivial Phase (TNP) and the topological trivial phase (TTP) combined with $\mathcal{PT}$-symmetric non-Hermitian potentials are investigated. The models of choice are the Su-Schrieffer-Heeger (SSH) model and the Kitaev chain. The interplay of a spontaneous $\mathcal{PT}$-symmetry breaking due to gain and loss with the topological phase is different for the two models. The SSH model undergoes a $\mathcal…
Traversability dynamics of minimal Sachdev-Ye-Kitaev Wormhole-inspired teleportation protocol with a parity-time ($\mathcal{PT}$)-symmetric non-Hermitian deformation
Indian Institute of Technology (Banaras Hindu University)（印度技术学院（贝拿勒斯印度教大学））
The Solovay-Kitaev theorem states that universal quantum gate sets can be exchanged with low overhead. More specifically, any gate on a fixed number of qudits can be simulated with error $ε$ using merely $\mathrm{polylog}(1/ε)$ gates from any finite universal quantum gate set $\mathcal{G}$. One drawback to the theorem is that it requires the gate set $\mathcal{G}$ to be closed under inversion. Here we show that this restriction can be traded for the assumption that $\mathcal{G}$ contains an irreducible representation of any finite group $G$. This extends recent work of Sardharwalla et al. [arX…
Exchange fluctuation theorems (XFTs) describe a fundamental symmetry relation for particle and energy exchange between several systems. Here we study the XFTs of a Kitaev chain connected to two reservoirs in the same temperature but different bias. By varying the parameters in the Kitaev chain model, we calculate analytically the full counting statistics of the transport current and formulate the corresponding XFTs for multiple current components. We also demonstrate the XFTs with numerical results. We find that due to the presence of the U(1) symmetry breaking terms in the Hamiltonian of the …
We systematically study a Kitaev chain with imbalanced pair creation and annihilation, which is introduced by non-Hermitian pairing terms. Exact phase diagram shows that the topological phase is still robust under the influence of the conditional imbalance. The gapped phases are characterized by a topological invariant, the extended Zak phase, which is defined by the biorthonormal inner product. Such phases are destroyed at the points where the coalescence of groundstates occur, associating with the time-reversal symmetry breaking. We find that the Majorana edge modes also exist for the open c…
We investigate a non-Hermitian extension of Kitaev chain by considering imaginary $p$-wave pairing amplitudes. The exact solution shows that the phase diagram consists two phases with real and complex \ Bogoliubov-de-gens spectra, associated with $\mathcal{PT}$-symmetry breaking, which is separated by a hyperbolic exceptional line. The exceptional points (EPs) correspond to a specific Cooper pair state $( 1+c_{k}^{\dagger }c_{-k}^{\dagger }) \left\vert 0\right\rangle $\ with movable $k$ when the parameters vary along the exceptional line. The non-Hermiticity around EP supports resonant generat…
A single unit cell contains all the information about the bulk system, including the topological feature. The topological invariant can be extracted from a finite system, which consists of several unit cells under certain environment, such as a non-Hermitian external field. We investigate a non- Hermitian finite-size Kitaev chain with PT-symmetric chemical potentials. Exact solution at the symmetric point shows that Majorana edge modes can emerge as the coalescing states at exceptional points and PT symmetry breaking states. The coalescing zero mode is the finite-size projection of the convent…
$\mathcal{PT}$-symmetry-breaking enhanced cavity optomechanical magnetometer is proposed, which is achieved by monitoring the change of intensity of a nonlinear four-wave mixing (FWM) process in a gain-cavity-assisted cavity optomechanical system (COMS). Compared with the traditional single loss COMS, the FWM intensity can be enhanced by two orders of magnitude when the gain-cavity-assisted COMS operates at the $\mathcal{PT}$-symmetry-breaking phase. Meanwhile, the sensitivity of magnetic field sensing can be increased from $10^{-9}$T to $10^{-11}$T. This originally comes from the fact that th…
In multiparticle quantum interference, bosons show rather generally the tendency to bunch together, while fermions can not. This behavior, which is rooted in the different statistics of the particles, results in a higher coincidence rate $P$ for fermions than for bosons, i.e. $P^{(bos)}<P^{(ferm)}$. However, in lossy systems such a general rule can be violated because bosons can avoid lossy regions. Here it is shown that, in a rather general optical system showing passive parity-time ($\mathcal{PT}$) symmetry, at the $\mathcal{PT}$ symmetry breaking phase transition point the coincidence proba…
Non-Hermitian systems, with symmetric or antisymmetric Hamiltonians under the parity-time ($\mathcal{PT}$) operations, can have entirely real eigenvalues. This fact has led to surprising discoveries such as loss-induced lasing and topological energy transfer. A merit of anti-$\mathcal{PT}$ systems is free of gain, but in recent efforts on making anti-$\mathcal{PT}$ devices, nonlinearity is still required. Here, counterintuitively, we show how to achieve anti-$\mathcal{PT}$ symmetry and its spontaneous breaking in a linear device by spinning a lossy resonator. Compared with a Hermitian spinning…
PT-symmetry breaking with divergent potentials: lattice and continuum cases
研究了在连续极限下，长程、非厄米、$\mathcal{PT}$对称势保持有限或发散时，格点模型中的宇称和时间反演（$\mathcal{PT}$）对称性破缺.通过对开放有限格点的脆弱$\mathcal{PT}$阈值的标度分析，我们证明了连续损失-增益势$V_α(x)\propto i |x|^α\mathrm{sign}(x)$对$α>-2$有正的$\mathcal{PT}$ -破缺阈值，对$α\leq -2$有零阈值.当$α<0$局域态具有复杂的（共轭）能量的连续带中出现在较高的损失增益强度。我们研究了耦合波导中$\mathcal{PT}$对称性破缺的特征，发现局域态的出现大大缩短了$\mathcal{PT}$对称性破缺区域的时间尺度.
We analyze phase transitions induced by forbidding charges and fluxes in $\mathcal{D}(S_3)$, the simplest non-Abelian model among quantum doubles, a class of 2D spin lattice topological models introduced by Kitaev. Contrary to a topological quantum field theory, the lattice degrees of freedom allow to forbid charges and fluxes independently, resulting in a non-trivial effect on dyons. Forbidding charges and fluxes leads to only a subset of the original anyons remaining, forming a new theory. We interpret the processes the theory undergoes in terms of condensation, spontaneous symmetry breaking…
Open, non-equilibrium systems with balanced gain and loss, known as parity-time ($\mathcal{PT}$)-symmetric systems, exhibit properties that are absent in closed, isolated systems. A key property is the $\mathcal{PT}$-symmetry breaking transition, which occurs when the gain-loss strength, a measure of the openness of the system, exceeds the intrinsic energy-scale of the system. We analyze the fate of this transition in disordered lattices with non-Hermitian gain and loss potentials $\pm iγ$ at reflection-symmetric sites. Contrary to the popular belief, we show that the $\mathcal{PT}$-symmetric …
Spontaneous symmetry breaking has revolutionized the understanding in numerous fields of modern physics. Here, we theoretically demonstrate the spontaneous time-reversal symmetry breaking in a cavity quantum electrodynamics system in which an atomic ensemble interacts coherently with a single resonant cavity mode. The interacting system can be effectively described by two coupled oscillators with positive and negative mass when the two-level atoms are prepared in their excited states. The occurrence of symmetry breaking is controlled by the atomic detuning and the coupling to the cavity mode, …
A fundamental axiom of quantum mechanics requires the Hamiltonians to be Hermitian which guarantees real eigen-energies and probability conservation. However, a class of non-Hermitian Hamiltonians with Parity-Time ($\mathcal{PT}$) symmetry can still display entirely real spectra. The Hermiticity requirement may be replaced by $\mathcal{PT}$ symmetry to develop an alternative formulation of quantum mechanics. A series of experiments have been carried out with classical systems including optics, electronics, microwaves, mechanics and acoustics. However, there are few experiments to investigate $…
Aubry-Andre Harper (AAH) lattice models, characterized by reflection-asymmetric, sinusoidally varying nearest-neighbor tunneling profile, are well-known for their topological properties. We consider the fate of such models in the presence of balanced gain and loss potentials $\pm iγ$ located at reflection-symmetric sites. We predict that these models have a finite $\mathcal{PT}$ breaking threshold only for {\it specific locations} of the gain-loss potential, and uncover a hidden symmetry that is instrumental to the finite threshold strength. We also show that the topological edge-states remain…
A hallmark of topological band theory in periodic media is that bulk properties are not affected by boundary conditions. Remarkably, in certain non-Hermitian lattices the bulk properties are largely affected by boundaries, leading to such major effects as the non-Hermitian skin effect and violation of the bulk-boundary correspondence. Here we unveil that non-unitary discrete-time quantum walks of photons in systems involving gain and loss show rather generally non-Bloch parity-time ($\mathcal{PT}$) symmetry breaking phase transitions, and suggest a bulk probing method to detect such boundary-d…
We study the scattering properties of $N$ identical one-dimensional localized $\mathcal{PT}$-symmetric potentials, connected in series as well as in parallel. We derive a general transfer matrix formalism for parallel coupled quantum scatterers, and apply that theory to demonstrate that the spectral singularities and $\mathcal{PT}$-symmetric transitions of single scattering cells may be observed in coupled systems, at the same or distinct values of the critical parameters, depending on the connection modes under which the scattering objects are coupled. We analyse the influences of the connect…
Universidade Federal do Rio Grande do Norte（北里奥格兰德联邦大学）；Stockholm University（斯德哥尔摩大学）
We simulate the dissipative dynamics of a mesoscopic system of long-range interacting particles which can be mapped into non-Hermitian spin models with a $\mathcal{PT}$ symmetry. We find rich $\mathcal{PT}$-phase diagrams with $\mathcal{PT}$-symmetric and $\mathcal{PT}$-broken phases. The dynamical regimes can be further enriched by modulating tunable parameters of the system. We outline how the $\mathcal{PT}$ symmetries of such systems may be probed by studying their dynamics. We note that systems of Rydberg atoms and systems of Rydberg ions with strong dipolar interactions are particularly w…
Open physical systems with balanced loss and gain, described by non-Hermitian parity-time ($\mathcal{PT}$) reflection symmetric Hamiltonians, exhibit a transition which could engenders modes that exponentially decay or grow with time and thus spontaneously breaks the $\mathcal{PT}$-symmetry. Such $\mathcal{PT}$-symmetry breaking transitions have attracted many interests because of their extraordinary behaviors and functionalities absent in closed systems. Here we report on the observation of $\mathcal{PT}$-symmetry breaking transitions by engineering time-periodic dissipation and coupling, whi…
We investigate Majorana modes in a quantum spin chain with bond-dependent exchange interactions by studying its dynamics. Specifically, we consider two-time correlations for the Kitaev-Heisenberg (KH) Hamiltonian close to the so-called Kitaev critical point. Here, the model coincides with a phase boundary of two uncoupled instances of Kitaev's model for p-wave superconductors, together supporting a degenerate ground state characterized by multiple Majorana modes. In this regime, the real-time dynamics of local spins reveal a set of strong zero modes, corresponding to a set of protruding freque…
In quantum mechanics, the spaces of momentum and its conjugate, the position, are related via Fourier transforms and thus the properties are interwoven with their structure. In particular, for lattice systems possessing an underlying discrete position space, the momentum becomes finite. Moreover, if the lattice length is finite, $L<\infty$, the momentum space also becomes both finite and discrete breaking altogether the continuity of the dispersion relation. This aspect is relevant in new systems such as the topological materials. We address this point paving the path for new ways to the obser…
A unit-preserving and completely positive linear map, or a channel, $Λ\colon \mathcal{A} \to \mathcal{A}_{\mathrm{in}}$ between $C^\ast$-algebras $\mathcal{A}$ and $\mathcal{A}_{\mathrm{in}}$ is called entanglement-breaking (EB) if $ω\circ( Λ\otimes \mathrm{id}_{\mathcal{B}} ) $ is a separable state for any $C^\ast$-algebra $\mathcal{B}$ and any state $ω$ on the injective $C^\ast$-tensor product $\mathcal{A}_{\mathrm{in}} \otimes \mathcal{B} .$ In this paper, we establish the equivalence of the following conditions for a channel $Λ$ with a quantum input space and with a general outcome $C^\ast…
Emergent $\mathcal{PT}$-symmetry breaking of collective modes with topological critical phenomena
National University of Singapore（新加坡国立大学）；University of Science and Technology of China（中国科学技术大学）；CAS Center For Excellence in Quantum Information and Quantum Physics（中国科学院量子信息与量子物理学前沿卓越创新中心）
We present a complete symmetry classification of the Sachdev-Ye-Kitaev (SYK) model with $\mathcal{N}=0$, $1$ and $2$ supersymmetry (SUSY) on the basis of the Altland-Zirnbauer scheme in random matrix theory (RMT). For $\mathcal{N}=0$ and $1$ we consider generic $q$-body interactions in the Hamiltonian and find RMT classes that were not present in earlier classifications of the same model with $q=4$. We numerically establish quantitative agreement between the distributions of the smallest energy levels in the $\mathcal{N}=1$ SYK model and RMT. Furthermore, we delineate the distinctive structure…
