Mario Castagnino, Sebastian Fortin
There are many formalisms to describe quantum decoherence. However, many of them give a non general and ad hoc definition of “pointer basis” or “moving preferred basis”, and this fact is a problem for the decoherence program. In this paper we will consider quantum systems under a general theoretical framework for decoherence and we will present a tentative definition of the moving preferred basis. These ideas are implemented in a well-known open system model. The obtained decoherence and the relaxation times are defined and compared with those of the literature for the Lee- Friedrichs model.
http://arxiv.org/abs/1304.3190
Quantum Physics (quant-ph)
Bikashkali Midya, Rajkumar Roychoudhury
We report the existence and properties of localized modes described by nonlinear Schroedinger equation with complex PT-symmetric Rosen-Morse potential well. Exact analytical expressions of the localized modes are found in both one dimensional and two-dimensional geometry with self-focusing and self-defocusing Kerr nonlinearity. Linear stability analysis reveals that these localized modes are unstable for all real values of the potential parameters although corresponding linear Schroedinger eigenvalue problem possesses unbroken PT-symmetry. This result has been verified by the direct numerical simulation of the governing equation. The transverse power flow density associated with these localized modes has also been examined.
http://arxiv.org/abs/1304.2105
Quantum Physics (quant-ph); Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)
G. P. Tsironis, N. Lazarides
A one dimensional, parity-time \({\cal PT}\)-symmetric magnetic metamaterial comprising split-ring resonators having both gain and loss is investigated. In the linear regime, the transition from the exact to the broken \({\cal PT}\)-phase is determined through the calculation of the eigenfrequency spectrum for two different configurations; the one with equidistant split-rings and the other with the split-rings forming a binary pattern (\({\cal PT}\) dimer chain). The latter system features a two-band, gapped spectrum with its shape determined by the gain/loss coefficient as well as the inter-element coupling. In the presense of nonlinearity, the \({\cal PT}\) dimer chain with balanced gain and loss supports nonlinear localized modes in the form of novel discrete breathers below the lower branch of the linear spectrum. These breathers, that can be excited from a weak applied magnetic field by frequency chirping, can be subsequently driven solely by the gain for very long times. The effect of a small imbalance between gain and loss is also considered. Fundamendal gain-driven breathers occupy both sites of a dimer, while their energy is almost equally partitioned between the two split-rings, the one with gain and the other with loss. We also introduce a model equation for the investigation of classical \({\cal PT}\) symmetry in zero dimensions, realized by a simple harmonic oscillator with matched time-dependent gain and loss that exhibits a transition from oscillatory to diverging motion. This behaviour is similar to a transition from the exact to the broken \({\cal PT}\) phase in higher-dimensional \({\cal PT}-\)symmetric systems. A stability condition relating the parameters of the problem is obtained in the case of piecewise constant gain/loss function that allows for the construction of a phase diagram with alternating stable and unstable regions.
http://arxiv.org/abs/1304.0556
Pattern Formation and Solitons (nlin.PS); Materials Science (cond-mat.mtrl-sci); Optics (physics.optics)
V.N.Rodionov
The modified Dirac equations for the massive particles with the replacement of the physical mass \(m\) with the help of the relation \(m\rightarrow m_1+ \gamma_5 m_2\) are investigated. It is shown that for a fermion theory with a \(\gamma_5\)-mass term, the limiting of the mass specter by the value \( m_{max}= {m_1}^2/2m_2\) takes place. In this case the different regions of the unbroken \(\cal PT\) symmetry may be expressed by means of the restriction of the physical mass \(m\leq m_{max}\). It should be noted that in the approach which was developed by C.Bender et al. for the \(\cal PT\)-symmetric version of the massive Thirring model with \(\gamma_5\)-mass term, the region of the unbroken \(\cal PT\)-symmetry was found in the form \(m_1\geq m_2\) \cite{ft12}. However on the basis of the mass limitation \(m\leq m_{max}\) we obtain that the domain \(m_1\geq m_2\) consists of two different parametric sectors: i) \(0\leq m_2 \leq m_1/\sqrt{2}\) -this values of mass parameters \(m_1,m_2\) correspond to the traditional particles for which in the limit \(m_{max}\rightarrow \infty\) the modified models are converting to the ordinary Dirac theory with the physical mass \(m\); ii)\(m_1/\sqrt{2}\leq m_2 \leq m_1\) – this is the case of the unusual particles for which equations of motion does not have a limit, when \(m_{max}\rightarrow \infty\). The presence of this possibility lets hope for that in Nature indeed there are some “exotic fermion fields”. As a matter of fact the formulated criterions may be used as a major test in the process of the division of considered models into ordinary and exotic fermion theories. It is tempting to think that the quanta of the exotic fermion field have a relation to the structure of the “dark matter”.
http://arxiv.org/abs/1303.7053
Quantum Physics (quant-ph); High Energy Physics – Phenomenology (hep-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)
Miloslav Znojil
The answer is yes. Via an elementary, exactly solvable crypto-Hermitian example it is shown that inside the interval of admissible couplings the innocent-looking point of a smooth unavoided crossing of the eigenvalues of Hamiltonian $H$ may carry a dynamically nontrivial meaning of a phase-transition boundary or “quantum horizon”. Passing this point requires a change of the physical Hilbert-space metric $\Theta$, i.e., a thorough modification of the form and of the interpretation of the operators of all observables.
http://arxiv.org/abs/1303.4876
Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Miloslav Znojil
A compact review is given, and a few new numerical results are added to the recent studies of the q-pointed one-dimensional star-shaped quantum graphs. These graphs are assumed endowed with certain specific, manifestly non-Hermitian point interactions, localized either in the inner vertex or in all of the outer vertices and carrying, in the latter case, an interesting zero-inflow interpretation.
http://arxiv.org/abs/1303.4331
Quantum Physics (quant-ph)
Panayotis G. Kevrekidis, Dmitry E. Pelinovsky, Dmitry Y.Tyugin
In the present work we examine both the linear and nonlinear properties of two related PT-symmetric systems of the discrete nonlinear Schrodinger (dNLS) type. First, we examine the parameter range for which the finite PT-dNLS chains have real eigenvalues and PT-symmetric linear eigenstates. We develop a systematic way of analyzing the nonlinear stationary states with the implicit function theorem at an analogue of the anti-continuum limit for the dNLS equation. Secondly, we consider the case when a finite PT-dNLS chain is embedded as a defect in the infinite dNLS lattice. We show that the stability intervals of the infinite PT-dNLS lattice are wider than in the case of a finite PT-dNLS chain. We also prove existence of localized stationary states (discrete solitons) in the analogue of the anti-continuum limit for the dNLS equation.
Numerical computations illustrate the existence of nonlinear stationary states, as well as the stability and saddle-center bifurcations of discrete solitons.
http://arxiv.org/abs/1303.3298
Pattern Formation and Solitons (nlin.PS); Dynamical Systems (math.DS)
W. D. Heiss, H. Cartarius, G. Wunner, J. Main
We consider the model of a PT-symmetric Bose-Einstein condensate in a delta-functions double-well potential. We demonstrate that analytic continuation of the primarily non-analytic term \(|\psi|^2 \psi\) – occurring in the underlying Gross-Pitaevskii equation – yields new branch points where three levels coalesce. We show numerically that the new branch points exhibit the behaviour of exceptional points of second and third order. A matrix model which confirms the numerical findings in analytic terms is given.
http://arxiv.org/abs/1303.0132
Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas); Chaotic Dynamics (nlin.CD)
Carl M. Bender, Mariagiovanna Gianfreda
The C operator in PT-symmetric quantum mechanics satisfies a system of three simultaneous algebraic operator equations, \(C^2=1\), \([C,PT]=0\), and \([C,H]=0\). These equations are difficult to solve exactly, so perturbative methods have been used in the past to calculate C. The usual approach has been to express the Hamiltonian as \(H=H_0+\epsilon H_1\), and to seek a solution for C in the form \(C=e^Q P\), where \(Q=Q(q,p)\) is odd in the momentum p, even in the coordinate q, and has a perturbation expansion of the form \(Q=\epsilon Q_1+\epsilon^3 Q_3+\epsilon^5 Q_5+\ldots\). [In previous work it has always been assumed that the coefficients of even powers of $\epsilon$ in this expansion would be absent because their presence would violate the condition that \(Q(p,q)\) is odd in p.] In an earlier paper it was argued that the C operator is not unique because the perturbation coefficient \(Q_1\) is nonunique. Here, the nonuniqueness of C is demonstrated at a more fundamental level: It is shown that the perturbation expansion for Q actually has the more general form \(Q=Q_0+\epsilon Q_1+\epsilon^2 Q_2+\ldots\) in which {\it all} powers and not just odd powers of \(\epsilon\) appear. For the case in which \(H_0\) is the harmonic-oscillator Hamiltonian, \(Q_0\) is calculated exactly and in closed form and it is shown explicitly to be nonunique. The results are verified by using powerful summation procedures based on analytic continuation. It is also shown how to calculate the higher coefficients in the perturbation series for Q.
http://arxiv.org/abs/1302.7047
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
Alexander I. Nesterov, Gennady P. Berman, Juan C. Beas Zepeda, Alan R. Bishop
A non-Hermitian quantum optimization algorithm is created and used to find the ground state of an antiferromagnetic Ising chain. We demonstrate analytically and numerically (for up to N=1024 spins) that our approach leads to a significant reduction of the annealing time that is proportional to \(\ln N\), which is much less than the time (proportional to \(N^2\)) required for the quantum annealing based on the corresponding Hermitian algorithm. We propose to use this approach to achieve similar speed-up for NP-complete problems by using classical computers in combination with quantum algorithms.
http://arxiv.org/abs/1302.6555
Quantum Physics (quant-ph); Mathematical Physics (math-ph)