December 2012
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Month December 2012

Non-Hermitian spin chains with inhomogeneous coupling

Andrei G. Bytsko

An open \(U_q(sl_2)\)-invariant spin chain of spin S and length N with inhomogeneous coupling is investigated as an example of a non-Hermitian (quasi-Hermitian) model. For several particular cases of such a chain, the ranges of the deformation parameter gamma are determined for which the spectrum of the model is real. For a certain range of gamma, a universal metric operator is constructed and thus the quasi-Hermiticity of the model is established. The constructed metric operator is non-dynamical, its structure is determined only by the symmetry of the model. The results apply, in particular, to all known homogeneous \(U_q(sl_2)\)-invariant integrable spin chains with nearest-neighbour interaction. In addition, the most general form of a metric operator for a quasi-Hermitian operator in finite dimensional space is discussed.

Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)

Entanglement of Multipartite Fermionic Coherent States for Pseudo Hermitian Hamiltonians

G. Najarbashi, M. A. Fasihi, M. Nakahara, F. Mirmasoudi, S. Mirzaei

In this paper the entanglement of multi-qubit fermionic pseudo Hermitian coherent states (FPHCS) described by anticommutative Grassmann numbers is studied. The pseudo-Hermitian versions of the well known maximally entangled pure states such as Bell and GHZ, W and biseparable states are introduced through integrating over the tensor products of FPHCSs with suitable choice of Grassmannian weight functions. Meanwhile as an illustration, the method is applied to tensor product of 2 and 3 qubit pseudo Hermitian systems. Then the measures of concurrence and average entropy are applied to quantify the entanglement of the pseudo two and three qubit states respectively.

Quantum Physics (quant-ph)

Symmetry breaking in dipolar matter-wave solitons in dual-core couplers

Yongyao Li, Jingfeng Liu, Wei Pang, Boris A. Malomed

We study effects of the spontaneous symmetry-breaking (SSB) in solitons built of the dipolar Bose-Einstein condensate (BEC), trapped in a dual-core system with the dipole-dipole interactions (DDIs) and hopping between the cores. Two realizations of such a matter-wave coupler are introduced, weakly- and strongly-coupled. The former one in based on two parallel pipe-shaped traps, while the latter one is represented by a single pipe sliced by an external field into parallel layers. The dipoles are oriented along axes of the pipes. In these systems, the dual-core solitons feature the SSB of the supercritical type and subcritical types, respectively. Stability regions are identified for symmetric and asymmetric solitons, and, in addition, for non-bifurcating antisymmetric ones, as well as for symmetric flat states, which may also be stable in the strongly-coupled system, due to competition between the attractive and repulsive intra- and inter-core DDIs. Effects of the contact interactions are considered too. Collisions between moving asymmetric solitons in the weakly-symmetric system feature elastic rebound, merger into a single breather, and passage accompanied by excitation of intrinsic vibrations of the solitons, for small, intermediate, and large collision velocities, respectively. A PT-symmetric version of the weakly-coupled system is briefly considered too, which may be relevant for matter-wave lasers. Stability boundaries for PT-symmetric and antisymmetric solitons are identified.

Quantum Gases (cond-mat.quant-gas); Pattern Formation and Solitons (nlin.PS)

PT-symmetry with a system of three-level atoms

Chao Hang, Guoxiang Huang, Vladimir V. Konotop

We show that a vapor of multilevel atoms driven by far-off resonant laser beams, with possibility of interference of two Raman resonances, is highly efficient for creating parity-time (PT) symmetric profiles of the probe-field refractive index, whose real part is symmetric and imaginary part is anti-symmetric in space. The spatial modulation of the susceptibility is achieved by proper combination of standing-wave strong control fields and of Stark shifts induced by a far-off-resonance laser field. As particular examples we explore a mixture of isotopes of Rubidium atoms and design a PT-symmetric lattice and a parabolic refractive index with a linear imaginary part.

Optics (physics.optics)

Optical Resonator Analog of a Two-Dimensional Topological Insulator

G. Q. Liang, Y. D. Chong

A lattice of optical ring resonators can exhibit a topological insulator phase, with the direction of rotation in the resonators playing the role of spin. This occurs when the inter-resonator coupling is sufficiently large, and the synthetic magnetic vector potential set up by the couplers is zero. Using the transfer matrix method, we derive the band structure, phase diagram, and the projected band diagram showing the existence of spin-polarized edge states. When PT (parity/time-reversal) symmetric gain and loss are introduced, the system functions as an optical diode which does not require optical nonlinearities.
Optics (physics.optics); Mesoscale and Nanoscale Physics (cond-mat.mes-hall)

Resources needed for non-unitary quantum operations

Raam Uzdin

Non-unitary operations generated by an effective non-Hermitian Hamiltonian can be used to create quantum state manipulations which are impossible in Hermitian quantum mechanics. These operations include state preparation (or cooling) and non-orthogonal state discrimination. In this work we put a lower bound on the resources needed for the construction of some given non-unitary evolution. Passive systems are studied in detail and a general feature of such a system is derived. After interpreting our results using the singular value decomposition, several examples are studied analytically. In particular, we put a lower bound on the resources needed for non-Hermitian state preparation and non-orthogonal state discrimination.
Quantum Physics (quant-ph)

Time evolution and adiabatic approximation in PT-symmetric quantum mechanics

Zhihua Guo, Huaixin Cao

In this paper, we discuss time evolution and adiabatic approximation in PT-symmetric quantum mechanics. we give the time evolving equation for a class of PT-symmetric Hamiltonians and some conditions of the adiabatic approximation for the class of PT-symmetric Hamiltonians.
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

CPT-Frames for PT-symmetric Hamiltonians

Huai-Xin Cao, Zhi-Hua Guo, Zheng-Li Chen

PT-symmetric quantum mechanics is an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose and complex conjugate) is replaced by the physically transparent condition of space-time reflection symmetry (PT-symmetry). A Hamiltonian H is said to be PT-symmetric if it commutes with the operator PT. The key point of PT-symmetric quantum theory is to build a new positive definite inner product on the given Hilbert space so that the given Hamiltonian is Hermitian with respect to the new inner product.  The aim of this note is to give further mathematical discussions on this theory. Especially, concepts of PT-frames, CPT-frames on a Hilbert space and for a Hamiltonian are proposed, their existence and constructions are discussed.
Mathematical Physics (math-ph)

A third-order exceptional point e ect on the dynamics of a single particle in a time-dependent harmonic trap

Raam Uzdin, Emanuele Dalla Torre, Ronnie Kosloff, Nimrod Moiseyev

The time evolution of a single particle in a harmonic trap with time dependent frequency omega(t) is well studied. Nevertheless here we show that, when the harmonic trap is opened (or closed) as function of time while keeping the adiabatic parameter mu = [d omega(t)/dt]/omega(t)^2 fixed, a sharp transition from an oscillatory to a monotonic exponential dynamics occurs at mu = 2. At this transition point the time evolution has a third-order exceptional point (EP) at all instants. This situation, where an EP of a time-dependent Hermitian Hamiltonian is obtained at any given time, is very different from other known cases. Our finding is relevant to the dynamics of a single ion in a magnetic, optical, or rf trap, and of diluted gases of ultracold atoms in optical traps.
Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas)

CPT-conserved effective mass Hamiltonians through first and higher order charge operator C in a supersymmetric framework

B. Bagchi, A. Banerjee, A. Ganguly

This paper examines the features of a generalized position-dependent mass Hamiltonian in a supersymmetric framework in which the constraints of pseudo-Hermiticity and CPT are naturally embedded. Different representations of the charge operator are considered that lead to new mass-deformed superpotentials which are inherently PT-symmetric. The qualitative spectral behavior of the Hamiltonian is studied and several interesting consequences are noted.
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)