Time-dependent Darboux (supersymmetric) transformations for non-Hermitian quantum systems

We propose time-dependent Darboux (supersymmetric) transformations that provide a scheme for the calculation of explicitly time-dependent solvable non-Hermitian partner Hamiltonians. Together with two Hermitian Hamilitonians the latter form a quadruple of Hamiltonians that are related by two time-dependent Dyson equations and two intertwining relations in form of a commutative diagram. Our construction is extended to the entire hierarchy of Hamiltonians obtained from time-dependent Darboux-Crum transformations. 
 As an alternative approach we also discuss the intertwining relations for Lewis-Riesenfeld invariants for Hermitian as well as non-Hermitian Hamiltonians that reduce the time-dependent equations to auxiliary eigenvalue equations. The working of our proposals is discussed for a hierarchy of explicitly time-dependent rational, hyperbolic, Airy function and nonlocal potentials

Exact analytical solutions for time-dependent Hermitian Hamiltonian systems from static unobservable non-Hermitian Hamiltonians

We propose a procedure to obtain exact analytical solutions to the time-dependent Schrödinger equations involving explicit time-dependent Hermitian Hamiltonians from solutions to time-independent non-Hermitian Hamiltonian systems and the time-dependent Dyson relation, together with the time-dependent quasi-Hermiticity relation. We illustrate the working of this method for a simple Hermitian Rabi-type model by relating it to a non-Hermitian time-independent system corresponding to the one-site lattice Yang-Lee model

Eternal life of entropy in non-Hermitian quantum systems

We find a different effect for the behavior of von Neumann entropy. For this we derive the framework for describing von Neumann entropy in non-Hermitian quantum systems and then apply it to a simple interacting PT-symmetric bosonic system. We show that our model is well defined even in the PT-broken regime with the introduction of a time-dependent metric and that it displays three distinct behaviors relating to the PT symmetry of the original time-independent Hamiltonian. When the symmetry is unbroken, the entropy undergoes rapid decay to zero (so-called “sudden death”) with a subsequent revival. At the exceptional point it decays asymptotically to zero and when the symmetry is spontaneously broken it decays asymptotically to a finite constant value (“eternal life”)

Quasi-exactly solvable quantum systems with explicitly time-dependent Hamiltonians

For a large class of time-dependent non-Hermitian Hamiltonians expressed in terms linear and bilinear combinations of the generators for an Euclidean Lie-algebra respecting different types of PT-symmetries, we find explicit solutions to the time-dependent Dyson equation. A specific Hermitian model with explicit time-dependence is analyzed further and shown to be quasi-exactly solvable. Technically we constructed the Lewis–Riesenfeld invariants making use of the metric picture, which is an equivalent alternative to the Schrödinger, Heisenberg and interaction picture containing the time-dependence in the metric operator that relates the time-dependent Hermitian Hamiltonian to a static non-Hermitian Hamiltonian

Mending the broken PT-regime via an explicit time-dependent Dyson map

We demonstrate that non-Hermitian Hamiltonian systems with spontaneously broken PT-symmetry and partially complex eigenvalue spectrum can be made meaningful in a quantum mechanical sense when introducing some explicit time-dependence into their parameters. Exploiting the fact that explicitly time-dependent non-Hermitian Hamiltonians are unobservable and not identical to the energy operators in such a scenario, we show that their corresponding non-Hermitian energy operators develop a different type of PT-symmetry from the Hamiltonians that ensures the reality of their energy spectra. For this purpose we analytically solve the fully time-dependent Dyson equation with all quantities involved being explicitly time-dependent giving rise to a time-dependent metric. The key auxiliary equation to be solved for the two level atomic system considered here is the nonlinear Ermakov–Pinney equation with time-dependent coefficients

Time-dependence in non-Hermitian quantum systems

In this thesis we present a coherent and consistent framework for explicit time-dependence in non-Hermitian quantum mechanics. The area of non-Hermitian quantum mechanics has been growing rapidly over the past twenty years [2]. This has been driven by the fact that PT -symmetric non-Hermitian systems exhibit real energy eigenvalues and unitary time evolution [1, 3, 4]. Historically, the introduction of time into the world of non-Hermitian quantum mechanics has been a conceptually difficult problem to address [5, 6], as it requires the Hamiltonian to become unobservable. However, we solve this issue with the introduction of a new observable energy operator [7]. We explain why its instigation is a necessary and natural progression in this setting. For the first time, the introduction of time has allowed us to make sense of the parameter regime in which the PT -symmetry is spontaneously broken. Ordinarily, in the time-independent setting, the energy eigenvalues become complex and the wave functions are asymptotically unbounded. However, we demonstrate that in the time-independent setting this broken symmetry can be mended and analysis on the spontaneously broken PT regime is indeed possible. We provide many examples of this mending on a wide range of different systems, beginning with a 2 x 2 matrix model [8] and extending to higher dimensional matrix models [9] and coupled harmonic oscillator systems with infinite Hilbert space [10, 11]. Furthermore, we use the framework to perform analysis on time-dependent quasi-exactly solvable models [12]. The ability to make sense of the spontaneously broken PT regime has revealed a vast array of new and exotic effects. We present the "eternal life" of entropy [13] in this thesis. Ordinarily, for entangled quantum systems coupled to the environments, the entropy decays rapidly to zero. However, in the spontaneously broken regime, we find the entropy decays asymptotically to a non-zero value. Finally, we create an elegant framework for Darboux and Darboux/Crum transformations for time-dependent non-Hermitian Hamiltonians [14]. This combines the area of non-Hermitian quantum mechanics with non linear differential equations and solitons

Spatial interaction and security: a review and case study of the Syrian refugee crisis

Sir Alan Wilson's ideas have been highly influential in the modelling of phenomena including migration, transport and economics. Latterly, research has explored the application of similar ideas to new problems at larger scales. Many of these studies relate to global challenges with significant policy implications. Here, we present an example in the form of original empirical work concerning forced migration associated with the current Syrian refugee crisis. We employ a spatial interaction framework to examine the flows of migrants fleeing Syria and the characteristics which influence their choice of destination country. In line with the broader literature, we find that shorter distances, economic prosperity, and cultural similarity (e.g. shared language) attract forced migrants, as does the probability of being granted asylum; a finding with potential implications for policy. Contrary to expectation, we find little influence for levels of security in potential host nations (e.g. absence of terrorism)

Neural signatures of strategic types in a two-person bargaining game

The management and manipulation of our own social image in the minds of others requires difficult and poorly understood computations. One computation useful in social image management is strategic deception: our ability and willingness to manipulate other people's beliefs about ourselves for gain. We used an interpersonal bargaining game to probe the capacity of players to manage their partner's beliefs about them. This probe parsed the group of subjects into three behavioral types according to their revealed level of strategic deception; these types were also distinguished by neural data measured during the game. The most deceptive subjects emitted behavioral signals that mimicked a more benign behavioral type, and their brains showed differential activation in right dorsolateral prefrontal cortex and left Brodmann area 10 at the time of this deception. In addition, strategic types showed a significant correlation between activation in the right temporoparietal junction and expected payoff that was absent in the other groups. The neurobehavioral types identified by the game raise the possibility of identifying quantitative biomarkers for the capacity to manipulate and maintain a social image in another person's mind