## Publications

40 results found

Graefe E-M, Schubert R, Christie R,
et al., 2022, Quantum-jump vs stochastic Schrödinger dynamics for Gaussian states with quadratic Hamiltonians and linear Lindbladians, *Journal of Physics A: Mathematical and Theoretical*, Vol: 55, Pages: 1-31, ISSN: 1751-8113

The dynamics of Gaussian states for open quantum systems described by Lindblad equations can be solved analytically for systems with quadratic Hamiltonians and linear Lindbladians, showing the familiar phenomena of dissipation and decoherence. It is well known that the Lindblad dynamics can be expressed as an ensemble average over stochastic pure-state dynamics, which can be interpreted as individual experimental implementations, where the form of the stochastic dynamics depends on the measurement setup. Here we consider quantum-jump and stochastic Schrödinger dynamics for initially Gaussian states. While both unravellings converge to the same Lindblad dynamics when averaged, the individual dynamics can differ qualitatively. For the stochastic Schrödinger equation, Gaussian states remain Gaussian during the evolution, with stochastic differential equations governing the evolution of the phase-space centre and a deterministic evolution of the covariance matrix. In contrast to this, individual pure-state dynamics arising from the quantum-jump evolution do not remain Gaussian in general. Applying results developed in the non-Hermitian context for Hagedorn wavepackets, we formulate a method to generate quantum-jump trajectories that is described entirely in terms of the evolution of an underlying Gaussian state. To illustrate the behaviours of the different unravellings in comparison to the Lindblad dynamics, we consider two examples in detail, which can be largely treated analytically, a harmonic oscillator subject to position measurement and a damped harmonic oscillator. In both cases, we highlight the differences as well as the similarities of the stochastic Schrödinger and the quantum-jump dynamics.

Graefe E-M, Malzard S, Melanathuru VR, 2022, Landau-Zener transitions through a pair of higher order exceptional points, *Physical Review A: Atomic, Molecular and Optical Physics*, Vol: 106, Pages: 1-11, ISSN: 1050-2947

Non-Hermitian quantum systems with explicit time dependence are of ever-increasing importance.There are only a handful of models that have been analytically studied in this context. Here, aPT-symmetric non-Hermitian N-level Landau-Zener type problem with two exceptional points ofN-th order is introduced. The system is Hermitian for asymptotically large times, far away fromthe exceptional points, and has purely imaginary eigenvalues between the exceptional points. Thefull Landau-Zener transition probabilities are derived, and found to show a characteristic binomialbehaviour. In the adiabatic limit the final populations are given by the ratios of binomial coefficients.It is demonstrated how this behaviour can be understood on the basis of adiabatic analysis, despitethe breakdown of adiabaticity that is often associated with non-Hermitian systems.

Mudute-Ndumbe S, Graefe E-M, 2020, Quantum chaos in a non-Hermitian PT-symmetric kicked top, *New Journal of Physics*, Vol: 22, ISSN: 1367-2630

A non-Hermitian PT-symmetric version of the kicked top is introduced to study the interplay of quantum chaos with balanced loss and gain. The classical dynamics arising from the quantum dynamics of the angular momentum expectation values are derived. It is demonstrated that the presence of PT-symmetry can lead to 'stable' mixed regular chaotic behaviour without sinks or sources for subcritical values of the gain–loss parameter. This is an example of what is known in classical dynamical systems as reversible dynamical systems. For large values of the kicking strength a strange attractor is observed that also persists if PT-symmetry is broken. The intensity dynamics of the classical map is investigated, and found to provide the main structure for the Husimi distributions of the subspaces of the quantum system belonging to certain ranges of the imaginary parts of the quasienergies. Classical structures are also identified in the quantum dynamics. Finally, the statistics of the eigenvalues of the quantum system are analysed and it is shown that if most of the eigenvalues are complex (which is the case already for fairly small non-Hermiticity parameters) the nearest-neighbour distances of the (unfolded) quasienergies follow a two-dimensional Posisson distribution when the classical dynamics is regular. In the chaotic regime, on the other hand they are in line with recently identified universal complex level spacing distributions for non-Hermitian systems, with transpose symmetry ÂT = Â. It is demonstrated how breaking this symmetry (by introducing an extra term in the Hamiltonian) recovers the more familiar universality class for non-Hermitian systems given by the complex Ginibre ensemble. Both universality classes display cubic level repulsion. The PT-symmetry of the system does not seem to influence the complex level spacings. Similar behaviour is also observed for the spectrum of a PT-symmetric extension of the triadic Baker map.

Longstaff B, Graefe E-M, 2020, Bloch oscillations in a Bose-Hubbard chain with single-particle losses, *Journal of Physics B: Atomic, Molecular and Optical Physics*, Vol: 53, Pages: 1-13, ISSN: 0953-4075

We theoretically investigate Bloch oscillations in a one-dimensionalBose-Hubbard chain, with single-particle losses from the odd lattice sites,described by the Lindblad equation. For a single particle the time evolution ofthe state is completely determined by a non-Hermitian effective Hamiltonian. Weanalyse the spectral properties of this Hamiltonian for an infinite lattice andlink features of the spectrum to observable dynamical effects, such asfrequency doubling in breathing modes. We further consider the case of manyparticles in the mean-field limit leading to complex nonlinear Schr\"odingerdynamics. Analytic expressions are derived for the generalised nonlinearstationary states and the nonlinear Bloch bands. The interplay of nonlinearityand particle losses leads to peculiar features in the nonlinear Bloch bands,such as the vanishing of solutions and the formation of additional exceptionalpoints. The stability of the stationary states is determined via theBogoliubov-de Gennes equation and is shown to strongly influence the mean-fielddynamics. Remarkably, even far from the mean-field limit, the stability of thenonlinear Bloch bands appears to effect the quantum dynamics. This isdemonstrated numerically for a two-particle system.

Graefe E-M, 2019, QUANTUM OPTICS PT symmetry dips into two-photon interference, *NATURE PHOTONICS*, Vol: 13, Pages: 822-823, ISSN: 1749-4885

Longstaff B, Graefe E-M, 2019, Non-adiabatic transitions through exceptional points in the band structure of a PT-symmetric lattice, *Physical Review A: Atomic, Molecular and Optical Physics*

Exceptional points, at which two or more eigenfunctions of a Hamiltoniancoalesce, occur in non-Hermitian systems and lead to surprising physicaleffects. In particular, the behaviour of a system under parameter variation candiffer significantly from the familiar Hermitian case in the presence ofexceptional points. Here we analytically derive the probability of anon-adiabatic transition in a two-level system driven through two consecutiveexceptional points at finite speed. The system is Hermitian far away from theexceptional points. In the adiabatic limit an equal redistribution between thestates coalescing in the exceptional point is observed, which can beinterpreted as a loss of information when passing through the exceptionalpoint. For finite parameter variation this gets modified. We demonstrate howthe transition through the exceptional points can be experimentally addressedin a PT-symmetric lattice using Bloch oscillations.

Graefe EM, Longstaff B, Plastow T,
et al., 2018, Lindblad dynamics of Gaussian states and their superpositions in the semiclassical limit, *Journal of Physics A: Mathematical and Theoretical*, Vol: 51, ISSN: 1751-8113

The time evolution of the Wigner function for Gaussian states generated by Lindblad quantum dynamics is investigated in the semiclassical limit. A new type of phase-space dynamics is obtained for the centre of a Gaussian Wigner function, where the Lindblad terms generally introduce a non-Hamiltonian flow. In addition to this, the Gaussian approximation yields dynamical equations for the covariances. The approximation becomes exact for linear Lindblad operators and a quadratic Hamiltonian. By viewing the Wigner function as a wave function on a coordinate space of doubled dimension, and the phase-space Lindblad equation as a Schrödinger equation with a non-Hermitian Hamiltonian, a further set of semiclassical equations are derived. These are capable of describing the interference terms in Wigner functions arising in superpositions of Gaussian states, as demonstrated for a cat state in an anharmonic oscillator subject to damping.

Graefe EM, Korsch HJ, Rush A, 2016, Quasiclassical analysis of Bloch oscillations in non-Hermitian tight-binding lattices, *New Journal of Physics*, Vol: 18, ISSN: 1367-2630

Many features of Bloch oscillations in one-dimensional quantum lattices with a static force can be described by quasiclassical considerations for example by means of the acceleration theorem, at least for Hermitian systems. Here the quasiclassical approach is extended to non-Hermitian lattices, which are of increasing interest. The analysis is based on a generalised non-Hermitian phase space dynamics developed recently. Applications to a single-band tight-binding system demonstrate that many features of the quantum dynamics can be understood from this classical description qualitatively and even quantitatively. Two non-Hermitian and PT-symmetric examples are studied, a Hatano–Nelson lattice with real coupling constants and a system with purely imaginary couplings, both for initially localised states in space or in momentum. It is shown that the time-evolution of the norm of the wave packet and the expectation values of position and momentum can be described in a classical picture.

Graefe E-M, Rush A, Schubert R, 2016, Propagation of Gaussian beams in the presence of gain and loss, *IEEE Journal of Selected Topics in Quantum Electronics*, Vol: 22, ISSN: 1558-4542

We consider the propagation of Gaussian beams in a waveguide with gain andloss in the paraxial approximation governed by the Schr\"odinger equation. Wederive equations of motion for the beam in the semiclassical limit that arevalid when the waveguide profile is locally well approximated by quadraticfunctions. For Hermitian systems, without any loss or gain, these dynamics aregiven by Hamilton's equations for the center of the beam and its conjugatemomentum. Adding gain and/or loss to the waveguide introduces a non-Hermitiancomponent, causing the width of the Gaussian beam to play an important role inits propagation. Here we show how the width affects the motion of the beam andhow this may be used to filter Gaussian beams located at the same initialposition based on their width.

Graefe E-M, Korsch HJ, Rush A, 2016, Classical-quantum correspondence in bosonic two-mode conversion systems: Polynomial algebras and Kummer shapes, *Physical Review A*, Vol: 93, ISSN: 1094-1622

Bosonic quantum conversion systems can be modeled by many-particle single-mode Hamiltonians describing a conversion of m molecules of type A into n molecules of type B and vice versa. These Hamiltonians are analyzed in terms of generators of a polynomially deformed su(2) algebra. In the mean-field limit of large particle numbers, these systems become classical and their Hamiltonian dynamics can again be described by polynomial deformations of a Lie algebra, where quantum commutators are replaced by Poisson brackets. The Casimir operator restricts the motion to Kummer shapes, deformed Bloch spheres with cusp singularities depending on m and n. It is demonstrated that the many-particle eigenvalues can be recovered from the mean-field dynamics using a WKB-type quantization condition. The many-particle state densities can be semiclassically approximated by the time periods of periodic orbits, which show characteristic steps and singularities related to the fixed points, whose bifurcation properties are analyzed.

Graefe E-M, Mudute-Ndumbe S, Taylor M, 2015, Random matrix ensembles for PT-symmetric systems, *Journal of Physics A: Mathematical and Theoretical*, Vol: 48, ISSN: 1751-8113

Recently much effort has been made towards the introduction of non-Hermitianrandom matrix models respecting $PT$-symmetry. Here we show that there is aone-to-one correspondence between complex $PT$-symmetric matrices andsplit-complex and split-quaternionic versions of Hermitian matrices. Weintroduce two new random matrix ensembles of (a) Gaussian split-complexHermitian, and (b) Gaussian split-quaternionic Hermitian matrices, of arbitrarysizes. They are related to the split signature versions of the complex and thequaternionic numbers, respectively. We conjecture that these ensemblesrepresent universality classes for $PT$-symmetric matrices. For the case of$2\times2$ matrices we derive analytic expressions for the joint probabilitydistributions of the eigenvalues, the one-level densities and the levelspacings in the case of real eigenvalues.

Graefe E-M, Graney M, Rush A, 2015, Semiclassical quantization for a bosonic atom-molecule conversion system, *Physical review A: General physics*, Vol: 92, ISSN: 0556-2791

We consider a simple quantum model of atom-molecule conversion where bosonicatoms can combine into diatomic molecules and vice versa. The many-particlesystem can be expressed in terms of the generators a deformed $SU(2)$ algebra,and the mean-field dynamics takes place on a deformed version of the Blochsphere, a teardrop shaped surface with a cusp singularity. We analyse themean-field and many-particle correspondence, which shows typical features ofquantum-classical correspondence. We demonstrate that semiclassical methods canbe employed to recover full many-particle features from the mean-fielddescription in cold atom systems with atom-molecule conversion, and derive ananalytic expression for the many-particle density of states in the limit oflarge particle numbers.

Graefe E-M, Korsch HJ, Rush A,
et al., 2014, Classical and quantum dynamics in the (non-Hermitian) Swanson oscillator, *Journal of Physics A-Mathematical and General*, ISSN: 1361-6447

The non-Hermitian quadratic oscillator studied by Swanson is one of thepopular $PT$-symmetric model systems. Here a full classical description of itsdynamics is derived using recently developed metriplectic flow equations, whichcombine the classical symplectic flow for Hermitian systems with a dissipativemetric flow for the anti-Hermitian part. Closed form expressions for the metricand phase-space trajectories are presented which are found to be periodic intime. Since the Hamiltonian is only quadratic the classical dynamics exactlydescribes the quantum dynamics of Gaussian wave packets. It is shown that theclassical metric and trajectories as well as the quantum wave functions candiverge in finite time even though the $PT$-symmetry is unbroken, i.e., theeigenvalues are purely real.

Graefe E-M, Korsch HJ, Strzys MP, 2014, Bose-Hubbard dimers, Viviani's windows and pendulum dynamics, *JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL*, Vol: 47, ISSN: 1751-8113

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- Citations: 11

Graefe E-M, Liverani C, 2013, Mean-field approximation for a Bose-Hubbard dimer with complex interaction strength, *JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL*, Vol: 46, ISSN: 1751-8113

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- Citations: 4

Graefe E-M, Mailybaev AA, Moiseyev N, 2013, Breakdown of adiabatic transfer of light in waveguides in the presence of absorption, *PHYSICAL REVIEW A*, Vol: 88, ISSN: 1050-2947

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- Citations: 41

Brody DC, Graefe E-M, 2013, Information Geometry of Complex Hamiltonians and Exceptional Points, *ENTROPY*, Vol: 15, Pages: 3361-3378, ISSN: 1099-4300

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- Citations: 20

Graefe EM, 2013, Viewpoint: Quantum Chaos on Display, *Physics 6, 9*, Vol: 6

Spin-orbit-coupled cold atomic gases are proposed as a model system for observing the signatures of quantum chaos.

Brody DC, Graefe E-M, 2012, Mixed-State Evolution in the Presence of Gain and Loss, *PHYSICAL REVIEW LETTERS*, Vol: 109, ISSN: 0031-9007

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- Citations: 87

Graefe E-M, 2012, Stationary states of a PT symmetric two-mode Bose-Einstein condensate, *JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL*, Vol: 45, ISSN: 1751-8113

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- Citations: 82

Graefe E-M, Schubert R, 2012, Complexified coherent states and quantum evolution with non-Hermitian Hamiltonians, *JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL*, Vol: 45, ISSN: 1751-8113

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- Citations: 21

Demange G, Graefe E-M, 2012, Signatures of three coalescing eigenfunctions, *JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL*, Vol: 45, ISSN: 1751-8113

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- Citations: 137

Brody DC, Graefe E-M, 2011, Six-dimensional space-time from quaternionic quantum mechanics, *PHYSICAL REVIEW D*, Vol: 84, ISSN: 1550-7998

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- Citations: 11

Graefe E-M, Jones HF, 2011, PT-symmetric sinusoidal optical lattices at the symmetry-breaking threshold, *PHYSICAL REVIEW A*, Vol: 84, ISSN: 1050-2947

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- Citations: 102

Graefe E-M, Schubert R, 2011, Wave-packet evolution in non-Hermitian quantum systems, *PHYSICAL REVIEW A*, Vol: 83, ISSN: 1050-2947

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- Citations: 42

Brody DC, Graefe E, 2011, Coquaternionic quantum dynamics for two-level systems, *Acta Polytechnica: journal of advanced engineering*, Vol: 51, Pages: 14-20

The dynamical aspects of a spin-1/2 particle in Hermitian coquaternionicquantum theory is investigated. It is shown that the time evolution exhibitsthree different characteristics, depending on the values of the parameters ofthe Hamiltonian. When energy eigenvalues are real, the evolution is eitherisomorphic to that of a complex Hermitian theory on a spherical state space, orelse it remains unitary along an open orbit on a hyperbolic state space. Whenenergy eigenvalues form a complex conjugate pair, the orbit of the timeevolution closes again even though the state space is hyperbolic.

Brody DC, Graefe E-M, 2011, On complexified mechanics and coquaternions, *JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL*, Vol: 44, ISSN: 1751-8113

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- Citations: 36

Graefe E-M, Hoening M, Korsch HJ, 2010, Classical limit of non-Hermitian quantum dynamics-a generalized canonical structure, *JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL*, Vol: 43, ISSN: 1751-8113

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- Citations: 48

Graefe E, Korsch HJ, Niederle AE, 2010, Quantum Classical Correspondence for a non-Hermitian Bose-Hubbard Dimer, *Phys. Rev. A*, Vol: 82

We investigate the many-particle and mean-field correspondence for anon-Hermitian N-particle Bose-Hubbard dimer where a complex onsite energydescribes an effective decay from one of the modes. Recently a generalizedmean-field approximation for this non-Hermitian many-particle system yieldingan alternative complex nonlinear Schr\"odinger equation was introduced. Here wegive details of this mean-field approximation and show that the resultingdynamics can be expressed in a generalized canonical form that includes ametric gradient flow. The interplay of nonlinearity and non-Hermiticityintroduces a qualitatively new behavior to the mean-field dynamics: Thepresence of the non-Hermiticity promotes the self-trapping transition, whiledamping the self-trapping oscillations, and the nonlinearity introduces astrong sensitivity to the initial conditions in the decay of the normalization.Here we present a complete characterization of the mean-field dynamics and thefixed point structure. We also investigate the full many-particle dynamics,which shows a rich variety of breakdown and revival as well as tunnelingphenomena on top of the mean-field structure.

Brody DC, Graefe EM, 2010, Coherent states and rational surfaces, *J PHYS A-MATH THEOR*, Vol: 43

The state spaces of generalised coherent states associated with special unitary groups are shown to form rational curves and surfaces in the space of pure states. These curves and surfaces are generated by the various Veronese embeddings of the underlying state space into higher dimensional state spaces. This construction is applied to the parameterisation of generalised coherent states, which is useful for practical calculations, and provides an elementary combinatorial approach to the geometry of the coherent state space. The results are extended to Hilbert spaces with indefinite inner products, leading to the introduction of a new kind of generalised coherent states.

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