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Register a new userInformation retrieval and eigenstates coalescence in a non-Hermitian quantum system with anti-$mathcal{PT}$ symmetry
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Non-Hermitian systems with parity-time reversal ($mathcal{PT}$) or anti-$mathcal{PT}$ symmetry have attracted a wide range of interest owing to their unique characteristics and counterintuitive phenomena. One of the most extraordinary features is the presence of an exception point (EP), across which a phase transition with spontaneously broken $mathcal{PT}$ symmetry takes place. We implement a Floquet Hamiltonian of a single qubit with anti-$mathcal{PT}$ symmetry by periodically driving a dissipative quantum system of a single trapped ion. With stroboscopic emission and quantum state tomography, we obtain the time evolution of density matrix for an arbitrary initial state, and directly demonstrate information retrieval, eigenstates coalescence, and topological energy spectra as unique features of non-Hermitian systems.
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The recently theoretical and experimental researches related to $mathcal{PT}$-symmetric system have attracted unprecedented attention because of various novel features and potentials in extending canonical quantum mechanics. However, as the counterpart of $mathcal{PT}$-symmetry, there are only a few researches on anti-$mathcal{PT}$-symmetry. Here, we propose an algorithm for simulating the universal anti-$mathcal{PT}$-symmetric system with quantum circuit. Utilizing the protocols, an oscillation of information flow is observed for the first time in our Nuclear Magnetic Resonance quantum simulator. We will show that information will recover from the environment completely when the anti-$mathcal{PT}$-symmetry is broken, whereas no information can be retrieved in the symmetry-unbroken phase. Our work opens the gate for practical quantum simulation and experimental investigation of universal anti-$mathcal{PT}$-symmetric system in quantum computer.
The attractive inverse square potential arises in a number of physical problems such as a dipole interacting with a charged wire, the Efimov effect, the Calgero-Sutherland model, near-horizon black hole physics and the optics of Maxwell fisheye lenses. Proper formulation of the inverse-square problem requires specification of a boundary condition (regulator) at the origin representing short-range physics not included in the inverse square potential and this generically breaks the Hamiltonians continuous scale invariance in an elementary example of a quantum anomaly. The systems spectrum qualitatively changes at a critical value of the inverse-square coupling, and we here point out that the transition at this critical potential strength can be regarded as an example of a $mathcal{PT}$ symmetry breaking transition. In particular, we use point particle effective field theory (PPEFT), as developed by Burgess et al [J. High Energy Phys., 2017(4):106, 2017], to characterize the renormalization group (RG) evolution of the boundary coupling under rescalings. While many studies choose boundary conditions to ensure the system is unitary, these RG methods allow us to systematically handle the richer case of nonunitary physics describing a source or sink at the origin (such as is appropriate for the charged wire or black hole applications). From this point of view the RG flow changes character at the critical inverse-square coupling, transitioning from a sub-critical regime with evolution between two real, unitary fixed points ($mathcal{PT}$ symmetric phase) to a super-critical regime with imaginary, dissipative fixed points ($mathcal{PT}$ symmetry broken phase) that represent perfect-sink and perfect-source boundary conditions, around which the flow executes limit-cycle evolution.
Bose-Einstein condensates with balanced gain and loss in a double-well potential have been shown to exhibit PT-symmetric states. As proposed by Kreibich et al [Phys. Rev. A 87, 051601(R) (2013)], in the mean-field limit the dynamical behaviour of this system, especially that of the PT-symmetric states, can be simulated by embedding it into a Hermitian four-well system with time-dependent parameters. In this paper we go beyond the mean-field approximation and investigate many-body effects in this system, which are in lowest order described by the single-particle density matrix. The conditions for PT symmetry in the single-particle density matrix cannot be completely fulfilled by using pure initial states. Here we show that it is mathematically possible to achieve exact PT symmetry in the four-well many-body system in the sense of the dynamical behaviour of the single-particle density matrix. In contrast to previous work, for this purpose, we use mixed initial states fulfilling certain constraints and use them to calculate the dynamics.
Eigenspectra of a spinless quantum particle trapped inside a rigid, rectangular, two-dimensional (2D) box subject to diverse inner potential distributions are investigated under hermitian, as well as non-hermitian antiunitary $mathcal{PT}$ (composite parity and time-reversal) symmetric regimes. Four sectors or stripes inscribed in the rigid box comprising contiguously conjoined parallel rectangular segments with one side equaling the entire width of the box are studied. The stripes encompass piecewise constant potentials whose exact, complete energy eigenspectrum is obtained employing matrix mechanics. Various striped potential compositions, viz. real valued ones in the hermitian regime as well as complex, non-hermitian but $mathcal{PT}$ symmetric ones are considered separately and in conjunction, unraveling among typical lowest lying eigenvalues, retention and breakdown scenarios engendered by the $mathcal{PT}$ symmetry, bearing upon the strength of non-hermitian sectors. Some states exhibit a remarkable crossover of symmetry `making and `breaking: while a broken $mathcal{PT}$ gets reinstated for an energy level, higher levels may couple to continue with symmetry breaking. Further, for a charged quantum particle a $mathcal{PT}$ symmetric electric field, furnished with a striped potential backdrop, also reveals peculiar retention and breakdown $mathcal{PT}$ scenarios. Depictions of prominent probability redistributions relating to various potential distributions both under norm-conserving unitary regime for hermitian Hamiltonians and non-conserving ones post $mathcal{PT}$ breakdown are presented.
$mathcal{PT}$-symmetric quantum mechanics has been considered an important theoretical framework for understanding physical phenomena in $mathcal{PT}$-symmetric systems, with a number of $mathcal{PT}$-symmetry related applications. This line of research was made possible by the introduction of a time-independent metric operator to redefine the inner product of a Hilbert space. To treat the dynamics of generic non-Hermitian systems under equal footing, we advocate in this work the use of a time-dependent metric operator for the inner-product between time-evolving states. This treatment makes it possible to always interpret the dynamics of arbitrary (finite-dimensional) non-Hermitian systems in the framework of time-dependent $mathcal{PT}$-symmetric quantum mechanics, with unitary time evolution, real eigenvalues of an energy observable, and quantum measurement postulate all restored. Our work sheds new lights on generic non-Hermitian systems and spontaneous $mathcal{PT}$-symmetry breaking in particular. We also illustrate possible applications of our formulation with well-known examples in quantum thermodynamics.
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