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.
http://arxiv.org/abs/1212.5034
Optics (physics.optics); Mesoscale and Nanoscale Physics (cond-mat.mes-hall)
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.
http://arxiv.org/abs/1212.4584
Quantum Physics (quant-ph)
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.
http://arxiv.org/abs/1212.4615
Mathematical Physics (math-ph); Quantum Physics (quant-ph)
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.
http://arxiv.org/abs/1212.3944
Mathematical Physics (math-ph)
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.
http://arxiv.org/abs/1212.3077
Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas)
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.
http://arxiv.org/abs/1212.2122
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)
B. Bagchi, A. Ghose Choudhury, Partha Guha
We explore the Jacobi Last Multiplier as a means for deriving the Lagrangian of a fourth-order differential equation. In particular we consider the classical problem of the Pais-Uhlenbeck oscillator and write down the accompanying Hamiltonian. We then compare such an expression with an alternative derivation of the Hamiltonian that makes use of the Ostrogradski’s method and show that a mapping from the one to the other is achievable by variable transformations. Assuming canonical quantization procedure to be valid we go for the operator version of the Hamiltonian that represents a pair of uncoupled oscillators. This motivates us to propose a generalized Pais-Uhlenbeck Hamiltonian in terms of the usual harmonic oscillator creation and annihilation operators by including terms quadratic and linear in them. Such a Hamiltonian turns out to be essentially non-Hermitian but has an equivalent Hermitian representation which is reducible to a typically position-dependent reduced mass form.
http://arxiv.org/abs/1212.2092
Mathematical Physics (math-ph); Quantum Physics (quant-ph)
Jia-wen Deng, Uwe Guenther, Qing-hai Wang
Three ways of constructing a non-Hermitian matrix with possible all real eigenvalues are discussed. They are PT symmetry, pseudo-Hermiticity, and generalized PT symmetry. Parameter counting is provided for each class. All three classes of matrices have more real parameters than a Hermitian matrix with the same dimension. The generalized PT-symmetric matrices are most general among the three. All self-adjoint matrices process a generalized PT symmetry. For a given matrix, it can be both PT-symmetric and P’-pseudo-Hermitian with respect to some P’ operators. The relation between corresponding P and P’ operators is established. The Jordan block structures of each class are discussed. Explicit examples in 2×2 are shown.
http://arxiv.org/abs/1212.1861
Quantum Physics (quant-ph); Mathematical Physics (math-ph)
K. Li, D. A. Zezyulin, V. V. Konotop, P. G. Kevrekidis
In this work, we propose a PT-symmetric coupler whose arms are birefringent waveguides as a realistic physical model which leads to a so-called quadrimer i.e., a four complex field setting. We seek stationary solutions of the resulting linear and nonlinear model, identifying its linear point of PT symmetry breaking and examining the corresponding nonlinear solutions that persist up to this point, as well as, so-called, ghost states that bifurcate from them. We obtain the relevant symmetry breaking bifurcations and numerically follow the associated dynamics which give rise to growth/decay even within the PT-symmetric phase. Our obtained stationary nonlinear solutions are found to terminate in saddle-center bifurcations which are analogous to the linear PT-phase transition.
http://arxiv.org/abs/1212.1676
Quantum Physics (quant-ph)
Keiichi Nagao, Holger Bech Nielsen
We briefly review the correspondence principle proposed in our previous paper, which claims that if we regard a matrix element defined in terms of the future state at time \(T_B\) and the past state at time \(T_A\) as an expectation value in the complex action theory whose path runs over not only past but also future, the expectation value at the present time \(t\) of a future-included theory for large \(T_B-t\) and large \(t- T_A\) corresponds to that of a future-not-included theory with a proper inner product for large \(t- T_A\). This correspondence principle suggests that the future-included theory is not excluded phenomenologically.
http://arxiv.org/abs/1211.7269
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th)