Dorje C. Brody, Eva-Maria Graefe

A model is proposed that describes the evolution of a mixed state of a quantum system for which gain and loss of energy or amplitude are present. Properties of the model are worked out in detail. In particular, invariant subspaces of the space of density matrices corresponding to the fixed points of the dynamics are identified, and the existence of a transition between the phase in which gain and loss are balanced and the phase in which this balance is lost is illustrated in terms of the time average of observables. The model is extended to include a noise term that results from a uniform random perturbation generated by white noise. Numerical studies of example systems show the emergence of equilibrium states that suppress the phase transition.

http://arxiv.org/abs/1208.5297

Quantum Physics (quant-ph); Mathematical Physics (math-ph)

Eva-Maria Graefe, Alexei A. Mailybaev, Nimrod Moiseyev

In atomic physics, adiabatic evolution is often used to achieve a robust and efficient population transfer. Many adiabatic schemes have also been implemented in optical waveguide structures. Recently there has been increasing interests in the influence of decay and absorption, and their engineering applications. Here it is shown that contrary to what is often assumed, even a small decay can significantly influence the dynamical behaviour of a system, above and beyond a mere change of the overall norm. In particular, a small decay can lead to a breakdown of adiabatic transfer schemes, even when both the spectrum and the eigenfunctions are only sightly modified. This is demonstrated for the decaying version of a STIRAP scheme that has recently been implemented in optical waveguide structures. It is found that the transfer property of the scheme breaks down at a sharp threshold, which can be estimated by simple analytical arguments.

http://arxiv.org/abs/1207.5235

Quantum Physics (quant-ph); Mathematical Physics (math-ph); Optics (physics.optics)

Carl M. Bender, David J. Weir

Non-Hermitian PT-symmetric quantum-mechanical Hamiltonians generally exhibit a phase transition that separates two parametric regions, (i) a region of unbroken PT symmetry in which the eigenvalues are all real, and (ii) a region of broken PT symmetry in which some of the eigenvalues are complex. This transition has recently been observed experimentally in a variety of physical systems. Until now, theoretical studies of the PT phase transition have generally been limited to one-dimensional models. Here, four nontrivial coupled PT-symmetric Hamiltonians, \(H=p^2/2+x^2/2+q^2/2+y^2/2+igx^2y\), \(H=p^2/2+x^2/2+q^2/2+y^2+igx^2y\), \(H=p^2/2+x^2/2+q^2/2+y^2/2+r^2/2+z^2/2+igxyz\), and \(H=p^2/2+x^2/2+q^2/2+y^2+r^2/2+3z^2/2+igxyz\) are examined. Based on extensive numerical studies, this paper conjectures that all four models exhibit a phase transition. The transitions are found to occur at \(g\approx 0.1\), \(g\approx 0.04\), \(g\approx 0.1\), and \(g\approx 0.05\). These results suggest that the PT phase transition is a robust phenomenon not limited to systems having one degree of freedom.

http://arxiv.org/abs/1206.5100

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

Carl M. Bender, Bjorn K. Berntson, David Parker, E. Samuel

If a Hamiltonian is PT symmetric, there are two possibilities: Either the eigenvalues are entirely real, in which case the Hamiltonian is said to be in an unbroken-PT-symmetric phase, or else the eigenvalues are partly real and partly complex, in which case the Hamiltonian is said to be in a broken-PT-symmetric phase. As one varies the parameters of the Hamiltonian, one can pass through the phase transition that separates the unbroken and broken phases. This transition has recently been observed in a variety of laboratory experiments. This paper explains the phase transition in a simple and intuitive fashion and then describes an extremely elementary experiment in which the phase transition is easily observed.

http://arxiv.org/abs/1206.4972

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

Eva-Maria Graefe

The understanding of nonlinear PT-symmetric quantum systems, arising for example in the theory of Bose-Einstein condensates in PT-symmetric potentials, is widely based on numerical investigations, and little is known about generic features induced by the interplay of PT-symmetry and nonlinearity. To gain deeper insights it is important to have analytically solvable toy-models at hand. In the present paper the stationary states of a simple toy-model of a PT-symmetric system are investigated. The model can be interpreted as a simple description of a Bose-Einstein condensate in a PT-symmetric double well trap in a two-mode approximation. The eigenvalues and eigenstates of the system can be explicitly calculated in a straight forward manner; the resulting structures resemble those that have recently been found numerically for a more realistic PT-symmetric double delta potential. In addition, a continuation of the system is introduced that allows an interpretation in terms of a simple linear matrix model.

http://arxiv.org/abs/1206.4806

Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas)

Carl M. Bender, Daniel W. Hook

The PT-symmetric Hamiltonian \(H=p^2+x^2(ix)^\epsilon\) (\(\epsilon\) real) exhibits a phase transition at \(\epsilon=0\). When \(\epsilon\geq0$\) the eigenvalues are all real, positive, discrete, and grow as \(\epsilon\) increases. However, when \(\epsilon<0\) there are only a finite number of real eigenvalues. As \(\epsilon\) approaches -1 from above, the number of real eigenvalues decreases to one, and this eigenvalue becomes infinite at \(\epsilon=-1\). In this paper it is shown that these qualitative spectral behaviors are generic and that they are exhibited by the eigenvalues of the general class of Hamiltonians \(H^{(2n)}=p^{2n}+x^2(ix)^\epsilon\) (\(\epsilon\) real, n=1, 2, 3, …). The complex classical behaviors of these Hamiltonians are also examined.

http://arxiv.org/abs/1205.4425

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

Carl M. Bender, Daniel W. Hook, S. P. Klevansky

The non-Hermitian PT-symmetric quantum-mechanical Hamiltonian \(H=p^2+x^2(ix)^\epsilon\) has real, positive, and discrete eigenvalues for all \(\epsilon\geq 0\). These eigenvalues are analytic continuations of the harmonic-oscillator eigenvalues \(E_n=2n+1\) (n=0, 1, 2, 3, …) at \(\epsilon=0\). However, the harmonic oscillator also has negative eigenvalues \(E_n=-2n-1\) (n=0, 1, 2, 3, …), and one may ask whether it is equally possible to continue analytically from these eigenvalues. It is shown in this paper that for appropriate PT-symmetric boundary conditions the Hamiltonian \(H=p^2+x^2(ix)^\epsilon\) also has real and {\it negative} discrete eigenvalues. The negative eigenvalues fall into classes labeled by the integer N (N=1, 2, 3, …). For the Nth class of eigenvalues, \(\epsilon\) lies in the range \((4N-6)/3<\epsilon<4N-2\). At the low and high ends of this range, the eigenvalues are all infinite. At the special intermediate value \(\epsilon=2N-2\) the eigenvalues are the negatives of those of the conventional Hermitian Hamiltonian \(H=p^2+x^{2N}\). However, when \(\epsilon\neq 2N-2\), there are infinitely many complex eigenvalues. Thus, while the positive-spectrum sector of the Hamiltonian \(H=p^2+x^2(ix)^\epsilon\) has an unbroken PT symmetry (the eigenvalues are all real), the negative-spectrum sector of \(H=p^2+x^2(ix)^\epsilon\) has a broken PT symmetry (only some of the eigenvalues are real).

http://arxiv.org/abs/1203.6590

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

Carl M. Bender, Hugh F. Jones

In a previous paper it was shown that a one-turning-point WKB approximation gives an accurate picture of the spectrum of certain non-Hermitian PT-symmetric Hamiltonians on a finite interval with Dirichlet boundary conditions. Potentials to which this analysis applies include the linear potential \(V=igx\) and the sinusoidal potential \(V=ig\sin(\alpha x)\). However, the one-turning-point analysis fails to give the full structure of the spectrum for the cubic potential \(V=igx^3\), and in particular it fails to reproduce the critical points at which two real eigenvalues merge and become a complex-conjugate pair. The present paper extends the method to cases where the WKB path goes through a *pair* of turning points. The extended method gives an extremely accurate approximation to the spectrum of \(V=igx^3\), and more generally it works for potentials of the form \(V=igx^{2N+1}\). When applied to potentials with half-integral powers of \(x\), the method again works well for one sign of the coupling, namely that for which the turning points lie on the first sheet in the lower-half plane.

http://arxiv.org/abs/1203.5702

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

Ray J. Rivers

An analysis of classical mechanics in a complex extension of phase space shows that a particle in such a space can behave in a way redolant of quantum mechanics; additional degrees of freedom permit ‘tunnelling’ without recourse to instantons and lead to time/energy uncertainty. In practice, ‘classical’ particle trajectories with additional degrees of freedom have arisen in several different formulations of quantum mechanics. In this talk we compare the extended phase space of the closed time-path formalism with that of complex classical mechanics, to suggest that $\hbar$ has a role in our understanding of the latter. However, differences in the way that trajectories are used make a deeper comparison problematical. We conclude with some thoughts on quantisation as dimensional reduction.

http://arxiv.org/abs/1202.4117

Quantum Physics (quant-ph); Classical Physics (physics.class-ph)

Eva-Maria Graefe, Roman Schubert

The complex geometry underlying the Schr\”odinger dynamics of coherent states for non-Hermitian Hamiltonians is investigated. In particular two seemingly contradictory approaches are compared: (i) a complex WKB formalism, for which the centres of coherent states naturally evolve along complex trajectories, which leads to a class of complexified coherent states; (ii) the investigation of the dynamical equations for the real expectation values of position and momentum, for which an Ehrenfest theorem has been derived in a previous paper, yielding real but non-Hamiltonian classical dynamics on phase space for the real centres of coherent states. Both approaches become exact for quadratic Hamiltonians. The apparent contradiction is resolved building on an observation by Huber, Heller and Littlejohn, that complexified coherent states are equivalent if their centres lie on a specific complex Lagrangian manifold. A rich underlying complex symplectic geometry is unravelled. In particular a natural complex structure is identified that defines a projection from complex to real phase space, mapping complexified coherent states to their real equivalents.

http://arxiv.org/abs/1111.1877

Mathematical Physics (math-ph); Quantum Physics (quant-ph)