Monday, 04 June 2018

TBA

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

Parity and time-reversal (PT) systems are open systems with balanced gain and loss. Their dynamics are described by an effective non-Hermitian Hamiltonian, and they undergo PT breaking transition when the strength of the gain-loss term exceeds a threshold. This transition is also present in systems with localized loss. I will present experimental results for multi-particle correlations and quantum information across the PT transition, obtained by using a quantum photonic chip. This quantum example will be followed by experimental results for a passive PT transition in ultracold atoms and in synthetic electrical circuits, both of which have a time-periodic (Floquet) Hamiltonian. This work is done in collaboration with Dr. Anthony Laing, Bristol, UK (quantum photonic chip), Prof. Le Luo, SYSU, China (ultracold atoms), and Prof. Roberto Leon, UNAM, Mexico (synthetic circuits). It is supported by an NSF CAREER award.

In this talk, I will discuss the dynamics and applications of certain novel mechanical and optical $\rho$$\tau$ symmetric systems. I will point out the existence of two-fold and many-fold $\rho$$\tau$ symmetric systems and explain the spontaneous symmetry breaking of each of the $\rho$$\tau$ symmetries. With a remarkable integrable model, the spontaneous symmetry breaking phenomenon will also be explained from exact solution point of view. Then, the occurence of $\rho$$\tau$ symmetry breaking phenomenon will be demonstrated in the case of coupled waveguide systems. The observed symmetry broken states have possible applications in the construction of unidirectional optical devices like optical diodes. I will show the ability to achieve unidirectional light transport in lossgain free systems and also the possibility to control blow-up responses in a nonlinear loss-gain system.

**References**:

- S. Karthiga, V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, Phys. Rev. A 93, 012102 (2016).
- S. Karthiga, V. K. Chandrasekar, M. Senthilvelan, and M. Lakshmanan, Phys. Rev. A 94, 023829 (2016).
- S. Karthiga, V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, Phys. Rev. A 95, 033829 (2017).
- H. Ramezani, T. Kottos, R. El-Ganainy and D. N. Christodoulides, Phys. Rev. A 82, 043803 (2010).
- Y. Kominis, T. Bountis, and S. Flach, Sci. Rep. 6, 33699 (2016).
- I. V. Barashenkov, G. S. Jackson, and S. Flach, Phys. Rev. A 88, 053817 (2013).

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.

T_{1} = e ^{iαqˆ0} , T_{2} = e^{−iαpˆ0} ,

where ˆq_{0} = 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 constructive feasibility of the Hermitization of the pseudo-Hermitian 1+1-dimensional Klein-Gordon equation is shown achieved via a discretization of the spatial coordinate. The model is shown fairly realistic while still representing one of the most user-friendly, mathematically correct implementations of quantum theory in its Dyson-inspired three-Hilbert-space formulation. A number of comments is added concerning, e.g., the older, model-related indefinite inner product approaches as well as some general conceptual questions concerning the so called non-Hermitian Schrödinger-picture formulation of quantum mechanics.

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.

We introduce the concept of super periodic potential (SPP) of arbitrary order n,n ∈ I+,in one dimension. General theory of wave propagation through SPP of order n is presented and the reflection and transmission coefficients are derived in their closed analytical form by transfer matrix formulation. We present scattering features of super periodic rectangular potential and super periodic delta potential as special cases of SPP. It is found that the symmetric self-similarity is the special case of super periodicity. Thus by identifying a symmetric fractal potential as special cases of SPP, one can obtain the tunnelling amplitude for a particle from such fractal potential. By using the formalism of SPP we obtain the close form expression of tunnelling amplitude of a particle for general Cantor and Smith–Volterra–Cantor potentials.

Tuesday, 05 June 2018

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.

We discuss a stochastic approach to non-equilibrium quantum spin systems based on recent insights linking quantum and classical dynamics. Exploiting a sequence of exact transformations, quantum expectation values can be recast as averages over classical stochastic processes. We illustrate this approach for the quantum Ising model by extracting the Loschmidt amplitude and the magnetization dynamics from the numerical solution of stochastic differential equations. We show that dynamical quantum phase transitions are accompanied by clear signatures in the associated classical distribution functions, including the presence of enhanced fluctuations. We demonstrate that the method is capable of handling integrable and nonintegrable problems in a unified framework, including those in higher dimensions. [1] S. De Nicola, B. Doyon and M. J .Bhaseen; arXiv:1805.05350

The PT −symmetric V = ix3 model over the real line, x ∈ R, is IR truncated and considered as Sturm-Liouville problem over a finite interval x ∈ [−L, L] ⊂ R. Combining structures hidden in the Airy function setup of the ix model with WKB techniques developed by Bender and Jones in 2012 for the derivation of the real part of the spectrum of the ix3 model, a WKB and Stokes graph analysis for the complex spectral branches of the ix3 model as well as those of more general V = −(ix) 2n+1 models over x ∈ [−L, L] ⊂ R is performed. Complementary insights into the spectra of these models are obtained by splitting the spectral branch-structure into purely real scale factors and asymptotic spectral scaling graphs. It turns out that the corresponding (structurally very simple) scaling graphs are geometrically invariant and cutoff-independent so that the IR limit L → ∞ can be formally taken. These graphs have invariantly existing PT phase transition regions. In this way, a simple heuristic picture and complementary explanation for the unboundedness of the C−operator and the lack of quasi-Hermiticity of the ix3 Hamiltonian over R is provided.

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−x_{0}), 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).

The inverse square potential is not covered in elementary textbooks on quantum mechanics because it turns to be non-trivial: particles fall to the centre which acts like a sink (or source) and hence the problem is generically non-Hermitian. Self-adjoint extensions have been discussed in the literature for many years, but are usually of an ad-hoc nature. We introduce a systematic approach based on point-particle effective field theory which allows us to embrace the non-hermitian features. We also study the inverse problem of fall to infinity which occurs in Schwinger pair production and note a connection to quantum catastrophes (caustics in quantum fields).

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.

Wednesday, 06 June 2018

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.

Nonclassical states lucidly refer to those states which have no classical analogue. They are characterized by the negativity of P-function, and quantum supremacy can only be achieved through the appropriate uses of such states. In fact, recent developments in quantum computing to quantum communication, gravitational wave detection in LIGO to the creation of quantum random number generators require dierent types of useful nonclassical states. Useful nonclassical features have recently been observed in various optomechanical systems which are described by conventional Hermitian Hamiltonians. On the other hand, various parity-time symmetric (PT S) optomechanical systems (specially, coupled optomechanical systems) have been investigated recently to reveal optomechanically-induced transparency, parity-time-symmetry-breaking chaos, parity-time-symmetric phonon laser, etc. However, no eort has yet been made to investigate the nonclassical features of parity-time-symmetric optomechanical systems. Motivated by these facts, we have considered a passive cavity (optomechanical system) coupled, via optical tunneling, to an active cavity. The passive cavity is also assumed to contain a mechanical oscillator. This system is described by a non-Hermitian PT S Hamiltonian which can undergo a transition from PT S phase to parity-time-symmetric-broken (PT SB) phase, and the transition can be controlled by the parameters of the Hamiltonian. The existence of nonclassicality in this system is observed through dierent witnesses of nonclassicality (e.g., Mandel's Q parameter, Hillery and Zubairy's criteria for steering and entanglement, Hong-Mandel's criterion for higher-order squeezing and the standard criterion for lower-order squeezing). In most cases, it's observed that the nonclassical features are more prominent in PT S phase in comparison to the PT SB phase.

Joint work with :

Anirban Pathak, Javid Naikoo and Subhashish Banerjee

We investigate the nonlinear scattering dynamics in interacting atomic Bose-Einstein condensates under non-Hermitian dissipative conditions. We show that, by carefully engineering a momentum-dependent atomic loss profile, one can achieve matter-wave amplification through four-wave mixing in a quasi-one-dimensional nearly-free-space setup—a process that is forbidden in the counterpart Hermitian systems due to energy mismatch. Additionally, we show that similar effects lead to rich nonlinear dynamics in higher dimensions. Finally, we propose a physical realization for selectively tailoring the momentum-dependent atomic dissipation. Our strategy is based on a two-step process: (i) exciting atoms to narrow Rydberg or metastable excited states, and (ii) introducing loss through recoil; all while leaving the bulk condensate intact due to protection by quantum interference.

S. Wuester (1) and R. El-Ganainy(2)

- Department of Physics, Bilkent University, Ankara 06800, Turkey Department of Physics, Indian Institute of Science Education and Research, Bhopal, Madhya Pradesh 462 023, India
- Department of Physics and Henes Center for Quantum Phenomena, Michigan Technological University, Houghton, Michigan 49931, USA

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.

Thursday, 07 June 2018

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.

We discuss a model of random matrices which are quasi-hermitian with respect to a fixed deterministic metric $B$. This ensemble is comprised of $N\times N$ matrices $H=AB$, where $A$ is a complex-hermitian matrix drawn from a $U(N)$-invariant probability distribution (the GUE ensemble). For positive-definite $B$ (corresponding to the model introduced by Joglekar and Karr several years ago in Phys. Rev. E83 (2011) 031122), the resulting spectrum is real, because $H$ is similar to a hermitian matrix. In this talk we shall discuss the average spectrum of this ensemble for indefinite-metric (analogous to the broken $PT$-symmetry phase), in which case $H$ is no-longer similar to a hermitian matrix, and therefore its spectrum becomes complex. We will present analytical and numerical results for this spectrum in the complex plane in the large-$N$ limit, and explain its behavior as the number of negative eigenvalues (a finite fraction of $N$) of the metric $B$ increases."

Driving a quantum system at finite frequency allows one to explore its dynamics. This has become a well mastered resource for controlling the quantum state of two level systems in the context of quantum information processing. However, this can also be of fundamental interest, especially with many-body systems which display an intricate finite frequency behavior. In condensed matter, the Kondo e ffect epitomizes strong electronic correlations, but the study of its dynamics and the related scaling laws has remained elusive so far. Here, we fill this gap by studying a carbon nanotube based Kondo quantum dot driven by a microwave signal. Our findings not only confirm longstanding theoretical predictions, but also allow us to establish a simple ansatz for the scaling laws on the Kondo problem at finite frequency. More generally, our technique opens a new path for understanding the dynamics of complex quantum dot circuits in the context of quantum simulation of strongly correlated electron fluids.

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.

Quantum walks are synthetic quantum systems whose dynamics are described by time-evolution operators. It is further interesting that the quantum walk possesses novel topological phases akin to those of Floquet topological phases, which are topological insulators driven by a time-periodic field. Recently, we have theoretically discuss PT symmetry and topological properties of quantum walks with effects of gain and loss[1,2]. Furthermore, we have experimentally observed long-time surviving edge states, which are peculiar to the PT symmetric quantum walks with loss, in quantum regime[3]. In this work, we theoretically study a PT symmetric quantum walk with higher topological numbers, since the topological number in the previous work was effectively restricted to one. According to the higher topological number, degenerated edge states appear in the present setup. We also discuss how to distinguish degeneracy of edge states from the probability distribution which is an only available experimental data in usual setup.

[1] K. Mochizuki, D. Kim, and H. Obuse, Physical Revew A 93, 062116 (2016).

[2] D. Kim, K. Mochizuki, and H. Obuse, arXiv:1609.09650.

[3] L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, Nat. Phys. 13, 1117 (2017).

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.

Friday, 08 June 2018

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)."

In this talk we will discuss how Calogero-Sutherland models of $N$ identical particles on a circle are deformed away from hermiticity but retaining a $\cal PT$ symmetry, preserving the the integrability structure. The interaction potential gets completely regularized, which adds to the energy spectrum an infinite tower of previously non-normalizable states. For integral values of the coupling, extra degeneracy occurs and a nonlinear conserved charge enlarges the ring of Liouville charges.

Symmetries have characteristic effect on the energy spectrum of quantum systems. This is also the case with PT symmetric quantum mechanics: the energy eigenvalues of PT-symmetric complex potentials are either real,

or appear as complex conjugate pairs, depending on whether PT symmetry is unbroken or broken. There are, however, further symmetry concepts characterizing quantum systems, which also have influence on their

energy spectrum. In supersymmetric quantum mechanics (SUSYQM), superymmetric transformations connect energy eigenstates belonging to the same energy, but different potentials. Similarly to PT symmetry, supersymmetry can also be broken, and this also leads to consequences concerning the energy spectrum.

We discuss complex quantum mechanical potentials characterized by both PT symmetry and supersymmetry and discuss the combined effect of these symmetries on the energy spectrum. We pay special attention to situations

in which either or both of these symmetries break down. We illustrate the interplay of these symmetry concepts with exactly solvable potentials. We demonstrate that some potentials are also characterized by symmetries

associated with Lie algebras, and discuss the relation of this symmetry concept with PT symmetry and supersymmetry

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).

Atom laser is one of the great discoveries in modern science serving as an alternative theory to quantum optics. Atom laser replaces the study of many fundamental effects of quantum theory by using massive particles rather than the photons. Apart from that, it has several other applications including the high-accuracy interferometry. Squeezed atom laser is one step ahead of the ordinary atom lasers, as they provide more efficiency in producing the coherent laser beam. We construct a squeezed atom laser from the deformed Bose-Einstein condensate motivated by the genralized uncertainty principle. These modified quantum mechanical framework is under intense investigation in recent days and is a strong candidate for the quantum theory compatible with gravity. However, these models are mostly non-Hermitian and, therefore, PT-symmetry and non-Hermitian quantum mechanics play a great role in such scenarios. We shall discuss the detailed technicalities for the construction of the squeezed atom laser using the notions of Non-Hermitian quantum mechanics.

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.

Monday, 11 June 2018

One prerequisite for the construction of large-scale quantum machines is rapid, high fidelity qubit measurement. This challenge is now regularly met in superconducting qubits read out with parametrically driven microwave amplifiers. However, their near-quantum limited performance comes at the cost of virtually every other performance metric, including dynamic range, bandwidth, and directionality. I will discuss recent efforts to combat these shortcomings by both eliminating unwanted Hamiltonian terms as well as combining multiple, simultaneous parametric drives.

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.

Photons have recently emerged as a promising building block to realize synthetic materials. When confined in semiconductor microcavities, they acquire an effective mass and behave as massive particles. In the limit of 0D confinement, they can even act as artificial atoms. In our experiments at C2N, we arrange such artificial atoms into semiconductor lattices, thus forming crystals of light. Such manipulation of light not only presents potential for applications in photonics, but is also of great promise for fundamental studies, as I will demonstrate using the examples of a photonic benzene molecule and also photonic graphene recently realized recently in our lab. In the strong coupling regime with quantum well excitons, photons can finally be made to interact through formation of part-light part-matter particles called cavity polaritons. In the limit of strong enough interactions as compared to the polariton linewidth (polariton blockade), it has been predicted that polaritons may be used to implement strongly interacting systems of light. To date, this limit has nevertheless not been reached experimentally. In the second part of the talk, I will describe recent experiments performed in the group of Atac Imamoglu (ETH Zürich) attempting at increasing polariton- polariton interactions in order to reach the quantum regime.

Tuesday, 12 June 2018

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.

Using diagrammatic Keldysh non-equilibrium Green’s function and Quantum Master Equation approach, I will argue that recent quantum dot c-QED setups can be potential quantum devices. I will present both electronic and photonic properties of these systems. I will show that these systems can act as rectifiers, transistors [1,2], microwave amplifiers [3] and masers [4].

[1] J. Lu, R. Wang, J. Ren, M. Kulkarni, JH Jiang (in preparation, 2018)

[2] JH. Jiang, M. Kulkarni, D. Segal, Y. Imry, Phys. Rev. B 92, 045309 (2015),

[3] B. Agarwalla, M. Kulkarni, S. Mukamel, D. Segal, Phys. Rev. B 94, 121305, (Rapid Communications) [2016]

[4] B. Agarwalla, M. Kulkarni, D. Segal (in preparation, 2018)

In recent work, the so-called quasi-Zeno dynamics of a system has been investigated in the context of the quantum first passage problem. This dynamics considers the time evolution of a system that is subjected to a sequence of selective projective measurements made at small but finite intervals of time. This means that one has a sequence of steps, with each step consisting of a unitary transformation followed by a projection. It has been shown that this non-unitary dynamics can be effectively described by two different non-Hermitian Hamiltonians. I will discuss this connection, and an application to the problem of detection of a free quantum particle moving on a one-dimensional lattice, with a detector placed at the origin.

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.

I will consider the most general non-interacting (i.e, quadratic Hamiltonian) open quantum set-up. I will show that up to leading order in system-bath coupling, the evolution equation for the correlation matrix has the form of a Lyapunov equation, which is a well-known equation in control theory. This can be derived via the Redfield Quantum Master Equation (RQME), which is the Quantum Master Equation rigorously obtained from a system+bath approach via Born-Markov approximation. The RQME is known to violate the complete positivity requirement of Lindblad's theorem. By recasting this requirement in terms of the Lyapunov equation, I will show that the RQME may do so completely consistently within the Born-Markov approximation. The RQME actually has a weaker positivity requirement to be consistent with the Born-Markov approximation.

Wednesday, 13 June 2018

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.

One dimensional non-Hermitian PT symmetric lattice models have been extensively researched previously, but experimental realizations of such systems in coupled waveguide arrays is difficult. The challenge in probing the region of unbroken PT symmetric phases arises from the fact that the PT transition threshold is proportional to the effective coupling between the gain and loss sites, which is determined by the distance between them. This PT threshold can be increased by reducing this separation, but there is a physical limitation to it. We investigate the fate of this threshold in the presence of parallel, strongly coupled, Hermitian (neutral) chains, and find that it is increased by a factor proportional to the number of neutral chains. We present numerical results and analytical arguments for this enhancement. We then consider the effects of adding neutral sites to PT symmetric dimer and trimer configurations and show that the threshold is more than doubled, or tripled by their presence. Our results provide a surprising way to engineer the PT threshold in experimentally accessible samples.

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.

We study semi-classical asymptotics of eigenvalues and eigenfunctions for Hermitian and Non-Hermitian Schroedinger Operators on geometric graphs. These Laplacians are described in terms of extension theory of symmetric operators which leads to boundary conditions at the vertices. These conditions together with jumps of the potential at vertices lead to the appearance of eigenfunctions localized near vertices, edges and general subgraphs. We compute asymptotics of eigenvalues in terms of quantization rules and describe structure of asymptotical eigenfunctions.