ICTS

We report recent progress made on time-dependent non-Hermitian Hamiltonian quantum systems. The general framework for treating such type of systems is explained. The main novelty is that the dual nature of the Hamiltonian being simultaneous the generator of time-evolution and the energy operator no longer holds. For a simple two level system and a two dimensional system with infinite Hilbert space it is demonstrated that the spontaneously broken PT-regime in the time-independent case becomes meaningful when the coupling constants are taken to be explicitly dependent on time. We construct new types of time-dependent anti-linear operators to explain this. Comparing various equivalent solution procedures for the time-dependent Dyson equation we conclude that the Lewis-Riesenfeld method of invariants is simpler as it structures into three well-defined separate steps. We report recent progress made on time-dependent non-Hermitian Hamiltonian quantum systems. The based on:

A. Fring, T. Frith, Physical Review A 95, 010102(R) (2017)

A. Fring, T. Frith, Physics Letters A 381 (2017) 2318-2323

A. Fring, T. Frith, European Physics Journal Plus (2018) 133:57(9)

A. Fring, T. Frith, J. of Phys A: Math. and Theor. (2018) 10.1088/1751-8121/aac57b

The standard $\kappa$q-representation can be used to prove that a certain discrete subset of coherent states are complete in the Hilbert space of the square-integrable functions. This representation is based on the existence of common generalized eigenfunctions of the unitary operators. 

T₁ = e ^iαqˆ0 , T₂ = e^−iαpˆ0 ,

where ˆq₀ = ˆq ^† ₀  and ˆp0 = ˆp^† ₀ . Here, motivated by the appearance of pseudo-bosonic operators in several quantum systems, we introduce non self-adjoint position and momentum-like operators, and we use these to construct an extended kq-representation, sharing similar properties with the standard one. Then we use these properties to analyze the completeness of a set of bi-coherent states.

Hamiltonian formulation of generic manyparticle systems with spacedependent balanced loss and gain will be presented. It will be shown that the resulting Hamiltonian can always be interpreted as defining a manyparticle system on a pseudoEuclidean plane that is subjected to external inhomogeneous magnetic field having a functional form identical with spacedependent loss/gain coefficient. Two distinct classes of partially integrable Hamiltonian will be presented corresponding to systems with (i) translational symmetry or (ii) rotational invariance in a pseudoEuclidean space . A few exactly solvable systems with balanced loss and gain, including a coupled chain of nonlinear oscillators and a many particle Calogerotype model with fourbody inverse square plus twobody pairwise harmonic interactions, will be considered. Quantization of Hamiltonian system with balanced loss and gain will be discussed along with the examples of a few solvable models. The Calogerotype model with fourbody inversesquare interaction will be shown to be exactly solvable at the quantum level. The quantum bound states, its normalization in appropriate Stoke wedges and exact nbody correlation functions will be presented.

The nonlinear Schrodinger equation (NLSE) is known to be a very accurate model for wave propagation of pulses in the picosecond regime. For high-intensity shorter pulses in the femtosecond regime, one requires some higher order correction terms. The Hirota equation is a well known example of such an integrable higher order extension of the NLSE. Here we present a set of new integrable nonlocal Hirota equations that can be obtained by exploiting PT-symmetry present in the zero curvature condition. We will show how solutions can be constructed for these nonlocal systems by means of Hirota’s method and Darboux-Crum transformations. To end, we will discuss degenerate solitons for the Hirota equation.

TBA

In many studies on the non-Hermiticity including the PT symmetry, the non-Hermiticity of the Hamiltonian results from complex scaler potentials. On the other hand, there has been a stream of studies on the non-Hermiticity due to complex vector potentials. I will show that the complex vector potential causes a non-Hermitian flow in the system. The flow competes with quantum localization and gives rise to a delocalization transition even in one dimension, where the quantum localization is the strongest.

Space fractional quantum mechanics (SFQM) was developed by replacing Brownian paths by Levy flight paths and is very useful in describing variety of processes like anomalous diffusion, turbulence, chaotic dynamics etc. In this talk I intend to present few aspects of SFQM. In the first problem we calculate the tunneling time for single and double barrier in a closed form and hence show the absence of Hartman effect in SFQM. In the second problem, non-Hermitian extension of SFQM is considered to study the various scattering properties of certain systems. In particular we explore spectral singularity (SS) and Coherent Perfect Absorption (CPA) in the domain of non-Hermitian SFQM governed by fractional Schrodinger equation which is characterized by Levy index α (1<α≤2). We observe that non-Hermitian SFQM systems have more flexibility for SS and CPA and display some new features of scattering. For the delta potential V(x)=−iρδ(x−x₀), ρ>0, the SS energy, Ess, is blue or red shifted with decreasing α depending the strength of the potential. For complex rectangular barrier in non-Hermitian SQM, it is known that the reflection and transmission amplitudes are oscillatory near the spectral singular point. It is found that these oscillations eventually develop SS in non-Hermitian SFQM. The similar features is also reported for the case of CPA phenomena from complex rectangular barrier in non-Hermitian SFQM. These observations suggest a deeper relation between scattering features of non-Hermitian SQM and non-Hermitian SFQM.

In this talk I will discuss several symmetries in quantum physics that have recently led to the observations of intriguing optical phenomena and the realizations of novel photonic functionalities. Different from the standard quantum mechanics, these symmetries implies non-Hermiticity that is difficult to realize in high-energy physics or condensed matter systems in a controlled fashion. However, thanks to absorption and radiation loss as well as light amplification, photonics provides an ideal platform to explore the ramification of these symmetries, including parity-time (PT) symmetry, non-Hermitian particle-hole symmetry, and complex mirror symmetry. PT symmetric photonics is one of the fastest growing fields in the past five years. It requires a judiciously balanced refractive index satisfying , i.e., with a symmetric real index modulation and an antisymmetric imaginary index modulation. I will talk about its spontaneous symmetry breaking [1], the coexistence of laser and anti-laser [2], generalized conservation laws for wave propagation [3], and anisotropic transmission resonances [4]. Particle-hole symmetry imposes a strong restriction on the underlying system in the Hermitian case, which exists, for example, in superconductors and protects Majorana zero modes. In photonics however, I will show that particle-hole symmetry is ubiquitous in gain and loss modulated systems with two sublattices [5,6], such as coupled waveguides and a network of optical cavities. As a consequence, there exist a large set of symmetry-protect zero modes, which can be utilized for building a unique single-mode, fixed-frequency, and spatially tunable laser, potentially useful for spatial encoding of telecommunication signals. Finally, I will discuss the most straightforward generalization of parity or mirror symmetry in non-Hermitian systems, which I term complex mirror symmetry.

[1] Ge, L. & Stone, A. D. Parity-Time Symmetry Breaking beyond One Dimension: The Role of Degeneracy. Phys. Rev. X 4, (2014).

[2] Chong, Y. D., Ge, L., Cao, H. & Stone, A. D. Coherent Perfect Absorbers: Time-Reversed Lasers. Phys. Rev. Lett. 105, 53901 (2010).

[3] Ge, L., Makris, K. G., Christodoulides, D. N. & Feng, L. Scattering in PT− and RT-symmetric multimode waveguides: Generalized conservation laws and spontaneous symmetry breaking beyond one dimension. Phys. Rev. A 92, (2015).

[4] Ge, L., Chong, Y. D. & Stone, A. D. Conservation relations and anisotropic transmission resonances in one-dimensional PT-symmetric photonic heterostructures. Phys. Rev. A 85, (2012).

[5] Ge, L. Symmetry-protected zero-mode laser with a tunable spatial profile. Phys. Rev. A 95, (2017).

[6] Qi, B., Zhang, L. & Ge. L. Defect States Emerging from a Non-Hermitian Flatband of Photonic Zero Modes. Phys. Rev. Lett. 120, 093901 (2018).

A class of nonlocal nonlinear Schrodinger equation (NLSE) with a self induced parity- time symmetric potential is considered. The integrability, exact solvability and symmetry properties of such systems are investigated. In particular, a two component nonlocal vector NLSE with a self induced parity-time symmetric potential is shown to possess a Lax pair and an infinite number of conserved quantities and hence integrable. The inverse scattering method is employed to find exact soliton solutions for such systems. Further, a class of nonlocal NLSE is considered in an external potential with a space-time modulated coefficient of the nonlinear interaction term as well as confining and/or loss-gain terms. Exact soliton solutions are obtained for the inhomogeneous and/or non-autonomous nonlocal NLSE by using similarity transformation, and the method is illustrated with a few examples. It is found that only those transformations are allowed for which the transformed spatial coordinate is odd under the parity transformation of the original one. Finally, it is shown that a (d + 1)-dimensional generalization of nonlocal NLSE without the external potential possesses all the symmetries of the (d + 1)-dimensional Schrdinger group. The conserved Noether charges associated with the time translation, dilatation, and special conformal transformation are shown to be real-valued although they are not Hermitian.

Rapid, high-fidelity, quantum non-demolition measurement is a basic requirement for practical quantum information processors. In the first tutorial in this sequence we will begin with an overview of the so-called circuit-QED architecture for superconducting qubits. We will introduce and review the basic features of the family of parametrically driven, quantum limited microwave amplifiers which are now widely used in readout of superconducting qubits. 

We consider the linear damped wave equation on possibly unbounded domains with the damping being allowed to become unbounded at infinity and singular and we analyze newly arising phenomena. We show the generation of a contraction semigroup, the relation of spectra of the equation generator and the associated quadratic operator function, having the form of a Schroedinger operator with complex potential, the convergence of non-real eigenvalues in asymptotic regime of an infinite damping on a subdomain and investigate both non-real eigenvalues and the negative essential spectrum. The presence of the latter turns out to be an robust effect that cannot be easily canceled by adding a positive potential and so the exponential estimate of the semigroup is often lost. Moreover, we investigate spectral instablibity (pseudospectrum) of the semigroup generator. The analytic results will be illustrated by several examples in various dimensions. 

The treatment for time-dependent non-Hermitian Hamiltonians with time-independent metric operators has been studied [1]. The generalization to time-dependent metric operators which is introduced in [2] have advanced the ground for treating time dependent systems and opened new venues for further studies. Nevertheless, the existence of invariants (constants of the motion or first integral) introduced by Lewis- Riesenfeld [3] is an important factor in the study of time-dependent systems. We revisite the quasi hermiticity relation though the invariant operator associated to non-Hermitian Hamiltonian with a time-dependent metric. The pseudo-Hermitian invariant operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. Interesting physical applications are suggested and discussed.

In superconducting quantum circuits, qubits are indirectly read out by interrogating the cavities to which they are dispersively coupled with coherent microwave pulses. The key limitation on fidelity is the accuracy with which we can faithfully capture the quantum information in these few-photon pulses. In this tutorial, we will consider the case of superconducting readout with a phase-sensitive parametric amplifier. We will first discuss how 'weak' measurements, in which the readout pulse is deliberately too weak to project the qubit, reveal quantum features of both the qubit/cavity system and the amplification process. We will also discuss how this process can be extended to remotely entangle qubits via joint, cascaded readout followed by a phase-sensitive amplifier. 

The search for a geometric extension of quantum mechanics (QM) is usually motivated by the desire to formulate a consistent physical theory that would reduce to quantum mechanics and general relativity in different limits. There have been various attempts to generalize QM during the past 70 or so years, but it would be fair to say that no major progress could be made. The route of the difficulty of this problem lies in the stringent and inflexible nature of the axioms of QM and the lack of experimental guidance towards their possible alternatives. In this talk I will propose a geometric extension of QM that arises as a natural reaction to a no-go theorem pertaining the conflict between the unitarity of the Schrodinger time-evolution and the observability of the Hamiltonian operator for systems with a dynamical state space. In the proposed theory the role of the Hilbert space and the Hamiltonian operator are respectively played by a complex Hermitian vector bundle E endowed with a metric-compatible connection and a global section of a real vector bundle determined by E. The axioms of QM are not replaced by others but elevated to the level of the relevant bundles. The standard description of a quantum system is recovered when one restricts to a local coordinate patch of E. The talk will include a rather extensive introductory part where the necessary results related to pseudo-Hermitian operators and vector bundles will be reviewed. It will then focus on the conceptual aspects of the subject and their consequences. Among these is a geometrical notion of “energy observable” that applies also for systems with a time-dependent Hamiltonian or a time-dependent state space.

In this talk, I will present recent theory work from my group investigating driven-dissipative quantum phenomena that can be connected to effective non-Hermitian Hamiltonians. The first part will discuss whether sensing techniques based on exceptional points in non-Hermitian coupled mode systems are advantageous in the quantum regime. The second part will discuss a surprising connection between topological physics in driven bosonic systems and effective non-Hermitian physics. 

We study the statistical properties of the eigenvalues of non-Hermitian operators assoicated with the dissipative complex systems. By considering the Gaussian ensembles of such operators, a hierarchical relation between the correlators is obtained. Further the eigenvalues are found to behave like particles moving on a complex plane under 2-body and 3-body interactions and there seems to underlie a deep connection and universality in the spectral behaviour of different complex systems.

We consider QCD in a new quadractic gauge which highlights certain characteristic of the theory in the non-Perturbative sector. By considering natural hermiticity properties of the ghost fields we cast this model as Non-Hermtian but symmetric under combined parity and time reversal transformation. We explicitly study the PT phase transition in this model. This is very first such study in the non abelian gauge theory. The ghost fields condensate as a direct consequence of spontaneous breaking of PT symmetry. This leads to realise the transition from deconfined phase to confined phase as a PT phase transition in this system. The hidden C-symmetry in this system is perceived as inner automorphism in this theory. Explicit representation is constructed for the C-symmetry.

Rapid, high-fidelity, quantum non-demolition measurement is a basic requirement for practical quantum information processors. In the first tutorial in this sequence we will begin with an overview of the so-called circuit-QED architecture for superconducting qubits. We will introduce and review the basic features of the family of parametrically driven, quantum limited microwave amplifiers which are now widely used in readout of superconducting qubits. 

"Unitary evolution of a non-relativistic quantum system ${\cal S}$ is most often described in Schroedinger picture (SP). The Hilbert space ${\cal H}^{(physical)}$ is chosen in advance (frequently, ${\cal H}^{(physical)}=L^{2}(\mathbb{R}^3)$). The generic operators of observables $q_{(SP)}(t)$ are required, due to Stone theorem, self-adjoint in ${\cal H}^{(physical)}$. The evolution of wave functions is generated by a dedicated observable $g_{(SP)}(t)$ called Hamiltonian.

According to the ``non-Hermitian Schroedinger picture'' (NSP) idea initiated by Dyson and made widely popular by Bender et al, a double change of the representation space ${\cal H}^{(physical)} \to {\cal H}_{(auxiliary)}^{(unphysical)} \to {\cal H}_{(amended)}^{(physical)}$ can be motivated by a simplifying upgrade $g_{(SP)}\to G_{(NSP)}$ of the generator in Schroedinger equation.

In 2008 we opposed certain widespread no-go beliefs and we showed that the three-Hilbert-space NSP formalism admits a straightforward generalization. As a non-Hermitian extension of the Dirac's interaction picture (abbreviated as NIP) it offered an innovative formulation of quantum theory characterized, first of all, by the emergence of the non-Hermitian Coriolis forces $\Sigma_{(NIP)}(t)$. This causes that the non-Hermitian and non-stationary generator $G_{(NIP)}(t)$ in Schroedinger equation ceases to be observable.

In the talk we shall outline the key features of the NIP formalism and we shall review some of its domains of applicability. Marginally we will notice that the limiting-case emergence of the non-Hermitian Heisenberg picture remains fully analogous to the conventional textbook single-Hilbert-space scenario.

References:
MZ, Phys. Rev. D 78 (2008) 085003 (or arXiv: 0809.2874v1).
MZ, SIGMA 5 (2009) 001 (or arXiv: 0901.0700).
MZ, Annals of Physics 385 (2017) pp. 162-179 (or arXiv:1702.08493v2)."

Advances in control techniques for vibrational quantum states in molecules present new challenges for modelling such systems, which could be amenable to quantum simulation methods. Here, by exploiting a natural mapping between vibrations in molecules and photons in waveguides, we demonstrate a reprogrammable photonic chip as a versatile simulation platform for a range of quantum dynamic behaviour in different molecules. We begin by simulating the time evolution of vibrational excitations in the harmonic approximation for several four-atom molecules, including H2CS, SO3, HNCO, HFHF, N4 and P4. We then simulate coherent and dephased energy transport in the simplest model of the peptide bond in proteins—N-methylacetamide—and simulate thermal relaxation and the effect of anharmonicities in H2O. Finally, we use multi-photon statistics with a feedback control algorithm to iteratively identify quantum states that increase a particular dissociation pathway of NH3. These methods point to powerful new simulation tools for molecular quantum dynamics and the field of femtochemistry.

Reference:
C. Sparrow, E. Martin-Lopez, N. Maraviglia, A. Neville, C. Harrold, J. Carolan, Y.N. Joglekar, T. Hashimoto, N. Matsuda, J.L. O’Brien, D.P. Tew,
and A. Laing, Nature 557, 660 (2018).

We propose a methodical approach to enhancing non-exponential decay in quantum and optical systems by exploiting recent progress surrounding another subtle effect: the bound states in continuum [1], which have been observed in optical lattice array experiments just in the last decade [2, 3]. Specifically, by populating an initial state orthogonal to that of the bound state in continuum, it is possible to engineer system parameters for which both the usual exponential decay process and fractional decay (associated with the BIC or any other bound states present in the system) are suppressed in favor of non-exponential dynamics. We demonstrate our method using a model based on one of the previously mentioned optical lattice array experiments. We further show the energy gap between the continuum threshold and a nearby (virtual) bound state determines the timescale that characterizes the inverse power law decay [4]. In the case that the bound state in continuum eigenvalue appears directly in the middle of the scattering continuum, there appear Rabi-like oscillations that experience a π/2 phase shift when surpassing this timescale.

[1] C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, Nat. Rev. Mater. 1, 16048 (2016).
[2] Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, Phys. Rev. Lett. 107, 183901 (2011).
[3] S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, A. Tünnermann, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, Phys. Rev. Lett. 111, 240403 (2013).
[4] S. Garmon, T. Petrosky, L. Simine, and D. Segal, Fortschr. Phys. 61, 261 (2013).

A class of exactly solvable real as well as PT symmetric extended Calogero-Wolfes type three body potentials are obtained whose bound states are written in terms of newly discovered exceptional orthogonal polynomials (EOPs). These potentials are isospectral to the well known usual Calogero-Wolfes type potentials. Some of the many body systems are also extended by introducing new interaction terms and obtained their exact solutions in terms of exceptional Laguerre polynomials.

We demonstrate the merging of levels at the exceptional point, separating the PT symmetry breaking and non-breaking phases. The emergence of bound state in the continuum as zero width resonances are also derived. The connection of a conserved correlation in exceptional point stopping light is explicated. We also show the occurrence of spontaneous breaking of SUSY in the model under consideration and a hysteresis phenomenon near the exceptional point.

Bose-Einstein condensates with balanced gain and loss in a double-well potential have been shown to exhibit PT-symmetric states. As proposed by Kreibich et al [Phys. Rev. A 87, 051601(R) (2013)], in the mean-field limit the dynamical behavior of this system, in particular that of the PT-symmetric states, can be simulated by embedding it into a fourwell system with time-dependent parameters. In this talk I shall go beyond the mean-field approximation and investigate many-body effects in the system, which can to lowest order be described by the single-particle density matrix. I show that it is mathematically possible to achieve exact PT symmetry in the four-well many-body system in terms of the dynamical behavior of the single-particle density matrix. In contrast to previous work, for this purpose, I start from impure initial states and use them to calculate the dynamics.
Work done in collaboration with Tina Mathea, Dennis Dast, Daniel Dizdarevic, Holger Cartarius, and Jörg Main.

We study spectral properties for Hermitian and Non-Hermitian Laplacians on 2D polyhedra. These Laplacians are described in terms of extension theory of symmetric operators which leads to boundary conditions at the vertices. We describe spaces of harmonic functions in terms of natural complex structures on polyhedra. We also study certain spectral properties (like trace formulas) and singularities of solutions for wave equations. In particular, we describe wave fronts and time asymptotics of the number of scatterings.

In this talk, I will report and discuss our ongoing investigation on a parity-time symmetric coupled oscillator configuration with nonlinear dissipation. We have found that nonlinear dissipation plays the role of a control parameter that can constrain the exponential growth of energy in the broken PT symmetry regime. We have also found that for zero coupling between the two oscillators, the gain oscillator does not exhibit characteristics of exponential growth of energy. But instead it exhibits oscillatory characteristics. On the other hand, on analyzing the Lyapunov spectra, it has been found that for certain choice of parameters, the phase space dynamics depicts chaotic trajectory. Furthermore, we have analytically shown how our mathematical model could be converted to that of the so-called parity-time symmetric dimer.

Apart from above-mentioned topic, I will briefly talk about some recent interesting results related to our study on the chaotic dynamics and optical power saturation in a parity-time (PT) symmetric double ring resonator.

We present a comprehensive numerical analysis of steering dynamics of bright solitons in PT - symmetric non-Kerr couplers by including the third order dispersion effect. We first identify the steering threshold condition for the PT -symmetric non-Kerr couplers of two different types and subsequently compare the ramifications to those of the conventional competing nonlinear directional couplers. We also extend our study on the role of PT -symmetric effect to various lengths of directional couplers, and pertaining intriguing steering characteristics are explored. The coupling dynamics through the spatiotem- poral evolution of optical solitons in such PT - symmetric couplers unravel a fact that combinations of all these perturbative effects are fairly controlled by the gain/loss parameter exclusively to a π/2 coupler among available total coupling lengths. 

Recent advances in ultra-fast measurement in cold atoms, as well as pump-probe spectroscopy of K3C60 films, have opened the possibility of rapidly quenching systems of interacting fermions to, and across, a finite temperature superfluid transition. However determining that a transient state has approached a second-order critical point is difficult, as standard equilibrium techniques are inapplicable. We show that the approach to the superfluid critical point in a transient state may be detected via time-resolved transport measurements, such as the optical conductivity. We leverage the fact that quenching to the vicinity of the critical point produces a highly time dependent density of superfluid fluctuations, which affect the conductivity in two ways. Firstly by inelastic scattering between the fermions and the fluctuations, and secondly by direct conduction through the fluctuations. The competition between these two effects leads to non-monotonic behavior in the time-resolved optical conductivity, providing a signature of the critical transient state.