#### Prof. Dr. Ralf Schützhold

Theoretical Physics

University of Duisburg-Essen

##### Contact

- ralf.schuetzhold@uni-due.de
- +49 203 379 4720
- personal website

##### Hub

**Heisenberg limit for detecting vacuum birefringence**

Ahmadiniaz, N. and Cowan, T.E. and Sauerbrey, R. and Schramm, U. and Schlenvoigt, H.-P. and Schützhold, R.*Physical Review D*101 (2020)Quantum electrodynamics predicts the vacuum to behave as a nonlinear medium, including effects such as birefringence. However, for experimentally available field strengths, this vacuum polarizability is extremely small and thus very hard to measure. In analogy to the Heisenberg limit in quantum metrology, we study the minimum requirements for such a detection in a given strong field (the pump field). Using a laser pulse as the probe field, we find that its energy must exceed a certain threshold depending on the interaction time. However, a detection at that threshold, i.e., the Heisenberg limit, requires highly nonlinear measurement schemes - while for ordinary linear-optics schemes, the required energy (Poisson or shot noise limit) is much larger. Finally, we discuss several currently considered experimental scenarios from this point of view. © 2020 authors. Published by the American Physical Society.view abstract 10.1103/PhysRevD.101.116019 **Quantum radiation in dielectric media with dispersion and dissipation**

Lang, S. and Schützhold, R. and Unruh, W.G.*Physical Review D*102 (2020)By a generalization of the Hopfield model, we construct a microscopic Lagrangian describing a dielectric medium with dispersion and dissipation. This facilitates a well-defined and unambiguous ab initio treatment of quantum electrodynamics in such media, even in time-dependent backgrounds. As an example, we calculate the number of photons created by switching on and off dissipation in dependence on the temporal switching function. This effect may be stronger than quantum radiation produced by variations of the refractive index Δn(t) since the latter are typically very small and yield photon numbers of order (Δn)2. As another difference, we find that the partner particles of the created medium photons are not other medium photons but excitations of the environment field causing the dissipation (which is switched on and off). © 2020 American Physical Society.view abstract 10.1103/PhysRevD.102.125020 **Relaxation dynamics in a Hubbard dimer coupled to fermionic baths: Phenomenological description and its microscopic foundation**

Kleinherbers, E. and Szpak, N. and König, J. and Schützhold, R.*Physical Review B*101 (2020)We study relaxation dynamics in a strongly interacting two-site Fermi-Hubbard model that is induced by coupling each site to a local fermionic bath. To derive the proper form of the Lindblad operators that enter an effective description of the system-bath coupling in different temperature regimes, we employ a diagrammatic real-time technique for the time evolution of the reduced density matrix. In spite of a local coupling to the baths, the found Lindblad operators are nonlocal in space. We compare with the local approximation, where those nonlocal effects are neglected. Furthermore, we propose an improvement on the commonly used secular approximation (rotating-wave approximation), referred to as coherent approximation, which turns out superior in all studied parameter regimes (and equivalent otherwise). We look at the relaxation dynamics for several important observables and compare the methods for early and late times in various temperature regimes. © 2020 American Physical Society.view abstract 10.1103/PhysRevB.101.125131 **Analog of cosmological particle creation in electromagnetic waveguides**

Lang, S. and Schützhold, R.*Physical Review D*100 (2019)We consider an electromagnetic waveguide with a time-dependent propagation speed v(t) as an analog for cosmological particle creation. In contrast to most previous studies which focus on the number of particles produced, we calculate the corresponding two-point correlation function. For a small steplike variation δv(t), this correlator displays characteristic signatures of particle pair creation. As another potential advantage, this observable is of first order in the perturbation δv(t), whereas the particle number is second order in δv(t) and thus more strongly suppressed for small δv(t). © 2019 American Physical Society.view abstract 10.1103/PhysRevD.100.065003 **Boltzmann relaxation dynamics in the strongly interacting Fermi-Hubbard model**

Queisser, F. and Schützhold, R.*Physical Review A*100 (2019)Via the hierarchy of correlations, we study the Mott insulator phase of the Fermi-Hubbard model in the limit of strong interactions and derive a quantum Boltzmann equation describing its relaxation dynamics. In stark contrast to the weakly interacting case, we find that the scattering cross sections strongly depend on the momenta of the colliding quasiparticles and holes. Therefore, the relaxation towards equilibrium crucially depends on the spectrum of excitations. For example, for particle-hole excitations directly at the minimum of the (direct) Mott gap, the scattering cross sections vanish such that these excitations can have a very long lifetime. © 2019 American Physical Society.view abstract 10.1103/PhysRevA.100.053617 **Boltzmann relaxation dynamics of strongly interacting spinless fermions on a lattice**

Queisser, F. and Schreiber, S. and Kratzer, P. and Schützhold, R.*Physical Review B*100 (2019)Motivated by the recent interest in nonequilibrium phenomena in quantum many-body systems, we study strongly interacting fermions on a lattice by deriving and numerically solving quantum Boltzmann equations that describe their relaxation to thermodynamic equilibrium. The derivation is carried out by inspecting the hierarchy of correlations within the framework of the 1/Z expansion. Applying the Markov approximation, we obtain the dynamic equations for the distribution functions. Interestingly, we find that in the strong-coupling limit, collisions between particles and holes dominate over particle-particle and hole-hole collisions-in stark contrast to weakly interacting systems. As a consequence, our numerical simulations show that the relaxation timescales strongly depend on the type of excitations (particles or holes or both) that are initially present. © 2019 American Physical Society.view abstract 10.1103/PhysRevB.100.245110 **Dynamically assisted nuclear fusion**

Queisser, F. and Schützhold, R.*Physical Review C*100 (2019)We consider deuterium-tritium fusion as a generic example for general fusion reactions. For initial kinetic energies in the keV regime, the reaction rate is exponentially suppressed due to the Coulomb barrier between the nuclei, which is overcome by tunneling. Here, we study whether the tunneling probability could be enhanced by an additional electromagnetic field, such as an x-ray free electron laser (XFEL). We find that the XFEL frequencies and field strengths required for this dynamical assistance mechanism should come within reach of present-day or near-future technology. © 2019 American Physical Society.view abstract 10.1103/PhysRevC.100.041601 **Environment-induced prerelaxation in the Mott-Hubbard model**

Queisser, F. and Schützhold, R.*Physical Review B*99 (2019)Via the hierarchy of correlations, we study the strongly interacting Fermi-Hubbard model in the Mott insulator state and couple it to a Markovian environment that constantly monitors the particle numbers nμ↑ and nμ↓ for each lattice site μ. As expected, the environment induces an imaginary part γ (i.e., the decay rate) of the quasiparticle frequencies ωk→ωk-iγ, and it tends to diminish the correlations between lattice sites. Surprisingly, the environment also steers the state of the system on intermediate timescales O(1/γ) to a prerelaxed state very similar to the prethermalized state after a quantum quench (i.e., suddenly switching on the hopping rate J). Full relaxation or thermalization occurs via local on-site heating and takes much longer. © 2019 American Physical Society.view abstract 10.1103/PhysRevB.99.155110 **Phonon Pair Creation by Inflating Quantum Fluctuations in an Ion Trap**

Wittemer, M. and Hakelberg, F. and Kiefer, P. and Schröder, J.-P. and Fey, C. and Schützhold, R. and Warring, U. and Schaetz, T.*Physical Review Letters*123 (2019)Quantum theory predicts intriguing dynamics during drastic changes of external conditions. We switch the trapping field of two ions sufficiently fast to tear apart quantum fluctuations, i.e., create pairs of phonons and, thereby, squeeze the ions' motional state. This process can be interpreted as an experimental analog to cosmological particle creation and is accompanied by the formation of spatial entanglement. Hence, our platform allows one to study the causal connections of squeezing, pair creation, and entanglement and might permit one to cross-fertilize between concepts in cosmology and applications of quantum information processing. © 2019 American Physical Society.view abstract 10.1103/PhysRevLett.123.180502 **WKB approach to pair creation in spacetime-dependent fields: The case of a spacetime-dependent mass**

Oertel, J. and Schützhold, R.*Physical Review D*99 (2019)Besides tunneling in static potential landscapes, for example, the Wentzel-Kramers-Brillouin (WKB) approach is a powerful nonperturbative approximation tool to study particle creation due to time-dependent background fields, such as cosmological particle production or the Sauter-Schwinger effect, i.e., electron-positron pair creation in a strong electric field. However, our understanding of particle creation processes in background fields depending on both space and time is rather incomplete. In order to venture into this direction, we propose a generalization of the WKB method to truly spacetime-dependent fields and apply it to the case of a spacetime-dependent mass. © 2019 authors. Published by the American Physical Society.view abstract 10.1103/PhysRevD.99.125014 **Analog Sauter-Schwinger effect in semiconductors for spacetime-dependent fields**

Linder, M.F. and Lorke, A. and Schützhold, R.*Physical Review B*97 (2018)The Sauter-Schwinger effect predicts the creation of electron-positron pairs out of the quantum vacuum via tunneling induced by a strong electric field. Unfortunately, as the required field strength is extremely large, this fundamental prediction of quantum field theory has not been verified experimentally yet. Here, we study under which conditions and approximations the interband tunneling in suitable semiconductors could be effectively governed by the same (Dirac) Hamiltonian, especially for electric fields which depend on space and time. This quantitative analogy would allow us to test some of the predictions (such as the dynamically assisted Sauter-Schwinger effect) in this area by means of these laboratory analogs. © 2018 American Physical Society.view abstract 10.1103/PhysRevB.97.035203 **Discrete worldline instantons**

Schneider, C. and Torgrimsson, G. and Schützhold, R.*Physical Review D*98 (2018)The semiclassical approximation of the worldline path integral is a powerful tool to study nonperturbative electron-positron pair creation in spacetime-dependent background fields. Finding solutions of the classical equations of motion, i.e., worldline instantons, is possible analytically only in special cases, and a numerical treatment is nontrivial as well. We introduce a completely general numerical approach based on an approximate evaluation of the discretized path integral that easily and robustly gives the full semiclassical pair production rate in nontrivial multidimensional fields, and apply it to some example cases. © 2018 authors. Published by the American Physical Society.view abstract 10.1103/PhysRevD.98.085009 **Interaction of a Bose-Einstein condensate with a gravitational wave**

Schützhold, R.*Physical Review D*98 (2018)Partly motivated by recent proposals for the detection of gravitational waves, we study their interaction with Bose-Einstein condensates. For homogeneous condensates at rest, the gravitational wave does not directly create phonons (to lowest order) but merely affects existing phonons or indirectly creates phonon pairs via quantum squeezing - an effect which has already been considered in the literature. For inhomogeneous condensate flows such as a vortex lattice, however, the impact of the gravitational wave can directly create phonons. This more direct interaction can be more efficient and could perhaps help bring such a detection mechanism for gravitational waves a step closer towards experimental realizability - even though there is still a long way to go. Finally, we argue that super-fluid helium might offer some advantages in this respect. © 2018 American Physical Society.view abstract 10.1103/PhysRevD.98.105019 **Ion-trap analog of particle creation in cosmology**

Fey, C. and Schaetz, T. and Schützhold, R.*Physical Review A*98 (2018)We consider the transversal modes of ions in a linear radio-frequency trap where we control the time-dependent axial confinement to show that we can excite quanta of motion via a two-mode squeezing process. This effect is analogous to phenomena predicted to occur in the early universe, in general out of reach for experimental investigation. As a substantial advantage of this proposal in comparison to previous ones we propose to exploit the radial and axial modes simultaneously to permit experimental access of these effects based on state-of-the-art technology. In addition, we propose to create and explore entanglement between the two ions. © 2018 American Physical Society.view abstract 10.1103/PhysRevA.98.033407 **Sauter-Schwinger pair creation dynamically assisted by a plane wave**

Torgrimsson, G. and Schneider, C. and Schützhold, R.*Physical Review D*97 (2018)We study electron-positron pair creation by a strong and constant electric field superimposed with a weaker transversal plane wave which is incident perpendicularly (or under some angle). Comparing the fully nonperturbative approach based on the world-line instanton method with a perturbative expansion into powers of the strength of the weaker plane wave, we find good agreement - provided that the latter is carried out to sufficiently high orders. As usual for the dynamically assisted Sauter-Schwinger effect, the additional plane wave induces an exponential enhancement of the pair-creation probability if the combined Keldysh parameter exceeds a certain threshold. © 2018 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the »https://creativecommons.org/licenses/by/4.0/» Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.view abstract 10.1103/PhysRevD.97.096004 **Avalanche of entanglement and correlations at quantum phase transitions**

Krutitsky, K.V. and Osterloh, A. and Schützhold, R.*Scientific Reports*7 (2017)We study the ground-state entanglement in the quantum Ising model with nearest neighbor ferromagnetic coupling J and find a sequential increase of entanglement depth d with growing J. This entanglement avalanche starts with two-point entanglement, as measured by the concurrence, and continues via the three-tangle and four-tangle, until finally, deep in the ferromagnetic phase for J = ∞, arriving at a pure L-partite (GHZ type) entanglement of all L spins. Comparison with the two, three, and four-point correlations reveals a similar sequence and shows strong ties to the above entanglement measures for small J. However, we also find a partial inversion of the hierarchy, where the four-point correlation exceeds the three- and two-point correlations, well before the critical point is reached. Qualitatively similar behavior is also found for the Bose-Hubbard model, suggesting that this is a general feature of a quantum phase transition. This should be taken into account in the approximations starting from a mean-field limit. © 2017 The Author(s).view abstract 10.1038/s41598-017-03402-8 **Dynamically assisted Sauter-Schwinger effect — non-perturbative versus perturbative aspects**

Torgrimsson, G. and Schneider, C. and Oertel, J. and Schützhold, R.*Journal of High Energy Physics*2017 (2017)The Sauter-Schwinger effect predicts the creation of electron-positron pairs out of the quantum vacuum by a strong and slowly varying electric field. This effect can be dynamically assisted by an additional weaker time-dependent field, which may drastically enhance the pair-creation probability. In previous studies, it has been found that the enhancement may crucially depend on the temporal shape of this weaker pulse, e.g., a Gaussian profile exp{−(ωt)2} or a Sauter pulse 1/ cosh2(ωt) behave quite differently. In order to understand this difference, we make a perturbative expansion in terms of the weaker field while treating the strong electric field non-perturbatively. For a large class of profiles including the Sauter pulse, already the sum of the zeroth-order and the first-order amplitudes of this perturbative expansion yields good agreement. For other cases, such as a Gaussian or sinusoidal profile, this is not true in general and higher orders can yield the dominant contribution — where the dominant order depends on the chosen parameters. Our findings are confirmed by numerical simulations and help us to sort previous results into a bigger picture. © 2017, The Author(s).view abstract 10.1007/JHEP06(2017)043 **Four-concurrence in the transverse XY spin-1/2 chain**

Osterloh, A. and Schützhold, R.*Physical Review A*96 (2017)We analyze the entanglement measure C4 for specific mixed states in general and for the ground state of the transverse XY spin-1/2 chain. We find that its factorizing property for pure states does not easily extend to mixed states. For cases where the density matrix is a tensor product, C4 is definitely upper bounded by the product of the corresponding concurrences. In transverse XY chains, we find that for large distances this condition goes conform with the working hypotheses of a factorizing property of density matrices in this limit. Additionally, we find that C4 together with the genuine multipartite negativity makes it impossible to decide - at the present state of knowledge - which type of entanglement prevails in the system. In particular, this is true for all entanglement measures that detect SL-invariant genuine n-partite entanglement for different n. Further measures of SL-invariant genuine multipartite entanglement have to be considered here. C4 is, however, of the same order of magnitude as the genuine multipartite negativity in Phys. Rev. B 89, 134101 (2014)PRBMDO1098-012110.1103/PhysRevB.89.134101 and shows the same functional behavior, which we read as a hint towards the Greenberger-Horne-Zeilinger (GHZ) type of entanglement. Furthermore, we observe an interesting feature in the C4 values that resembles a destructive interference with the underlying concurrence. © 2017 American Physical Society.view abstract 10.1103/PhysRevA.96.012331 **Giant magneto-photoelectric effect in suspended graphene**

Sonntag, J. and Kurzmann, A. and Geller, M. P. and Queisser, F. and Lorke, A. and Schützhold, R.*New Journal of Physics*19 (2017)We study the optical response of a suspended, monolayer graphene field-effect transistor structure in magnetic fields of up to 9 T (quantum Hall regime). With an illumination power of only 3 μW, we measure a photocurrent of up to 400 nA (without an applied bias) corresponding to a photo-responsivity of 0.13 A W-1, which we believe to be one of the highest values ever measured in single-layer graphene. We discuss possible mechanisms for generating this strong photo-response (17 electron-hole pairs per 100 incident photons). Based on our experimental findings, we believe that the most likely scenario is a ballistic two-stage process including carrier multiplication via Auger-type inelastic Coulomb scattering at the graphene edge. © 2017 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.view abstract 10.1088/1367-2630/aa739d **Multiple particle-hole pair creation in the harmonically driven Fermi-Hubbard model**

Ten Brinke, N. and Ligges, M. and Bovensiepen, U. and Schützhold, R.*Physical Review B*95 (2017)We study the Fermi-Hubbard model in the strongly correlated Mott phase under the influence of a harmonically oscillating hopping rate J(t)=J0+ΔJcos(ωt). Apart from the well-known fundamental resonance, where the frequency ω of this oscillation equals (or a little exceeds) the Mott gap, we also find higher-order resonances where multiple particle-hole pairs are created when ω is near an integer multiple of the gap. These findings should be relevant for experimental realizations such as ultracold fermionic atoms in optical lattices or pump-probe experiments using laser pulses incident on correlated electrons in solid-state materials. © 2017 American Physical Society.view abstract 10.1103/PhysRevB.95.195123 **Derivation of Hawking radiation in dispersive dielectric media**

Linder, M.F. and Schützhold, R. and Unruh, W.G.*Physical Review D - Particles, Fields, Gravitation and Cosmology*93 (2016)Motivated by recent experimental efforts, we study a black hole analog induced by the propagation of a strong laser pulse in a nonlinear dielectric medium. Based on the Hopfield model (one pair of Sellmeier coefficients), we perform an analytic and fully relativistic microscopic derivation of the analog of Hawking radiation in this setup. The Hawking temperature is determined by the analog of the surface gravity (as expected), but we also find a frequency-dependent gray-body factor (i.e., a nonthermal spectrum at infinity) due to the breaking of conformal invariance in this setup. © 2016 American Physical Society.view abstract 10.1103/PhysRevD.93.104010 **Dicke superradiance as nondestructive probe for the state of atoms in optical lattices**

ten Brinke, N. and Schützhold, R.*European Physical Journal D*70 (2016)We present a proposal for a probing scheme utilizing Dicke superradiance to obtain information about ultracold atoms in optical lattices. A probe photon is absorbed collectively by an ensemble of lattice atoms generating a Dicke state. The lattice dynamics (e.g., tunneling) affects the coherence properties of that Dicke state and thus alters the superradiant emission characteristics – which in turn provides insight into the lattice (dynamics). Comparing the Bose-Hubbard and the Fermi-Hubbard model, we find similar superradiance in the strongly interacting Mott insulator regime, but crucial differences in the weakly interacting (superfluid or metallic) phase. Furthermore, we study the possibility to detect whether a quantum phase transition between the two regimes can be considered adiabatic or a quantum quench. © EDP Sciences, Società Italiana di Fisica, Springer-Verlag 2016.view abstract 10.1140/epjd/e2016-60710-0 **Doubly assisted Sauter-Schwinger effect**

Torgrimsson, G. and Oertel, J. and Schützhold, R.*Physical Review D - Particles, Fields, Gravitation and Cosmology*94 (2016)We study electron-positron pair creation by a strong and slowly varying electric field, assisted by a weaker and more rapidly changing field (e.g., in the keV regime) plus an additional high-energy (say MeV) photon. It turns out that this combination can yield a pair creation probability which is exponentially larger than in the case where one (or more) of the three ingredients is missing. Apart from a deeper understanding of these nonperturbative phenomena, this double enhancement may pave the way for an experimental verification of this fundamental prediction. © 2016 American Physical Society.view abstract 10.1103/PhysRevD.94.065035 **Duality in spin systems via the SU(4) algebra**

Schaller, G. and Schützhold, R.*Journal of Statistical Mechanics-theory and Experiment*(2016)We provide several examples and an intuitive diagrammatic representation demonstrating the use of two-qubit unitary transformations for mapping coupled spin Hamiltonians to simpler ones and vice versa. The corresponding dualities may be exploited to identify phase transition points or to aid the diagonalization of such Hamiltonians. For example, our method shows that a suitable one-parameter family of coupled Hamiltonians whose ground states transform from an initially factorizing state to a final cluster state on a lattice of arbitrary dimension is dual to a family of trivial decoupled Hamiltonians containing local on-site terms only. As a consequence, the minimum energy gap (which determines the adiabatic run-time) does not scale with system size, which facilitates an efficient and simple adiabatic preparation of e.g. the two-dimensional cluster state used for measurement-based quantum computation.view abstract 10.1088/1742-5468/2016/05/053113 **Dynamically assisted Sauter-Schwinger effect in inhomogeneous electric fields**

Schneider, C. and Schützhold, R.*Journal of High Energy Physics*2016 (2016)Abstract: Via the world-line instanton method, we study electron-positron pair creation by a strong (but sub-critical) electric field of the profile E/ cosh2(kx) superimposed by a weaker pulse E′/ cosh2(ωt). If the temporal Keldysh parameter γω = mω/(qE) exceeds a threshold value γω crit which depends on the spatial Keldysh parameter γk = mk/(qE), we find a drastic enhancement of the pair creation probability — reporting on what we believe to be the first analytic non-perturbative result for the interplay between temporal and spatial field dependences E(t, x) in the Sauter-Schwinger effect. Finally, we speculate whether an analogous effect (drastic enhancement of tunneling probability) could occur in other scenarios such as stimulated nuclear decay, for example. © 2016, The Author(s).view abstract 10.1007/JHEP02(2016)164 **Large-coordination-number expansion of a lattice Bose gas at finite temperature**

Navez, P. and Queisser, F. and Schützhold, R.*Physical Review A - Atomic, Molecular, and Optical Physics*94 (2016)The expansion of the partition function for large coordination number Z is a long-standing method and has formerly been used to describe the Ising model at finite temperatures. We extend this approach and study the interacting Bose gas at finite temperatures. An analytical expression for the free energy is derived which is valid for weakly interacting and strongly interacting bosons. The transition line which separates the superfluid phase from Mott insulating or normal gas phase is shown for fillings (n)=1 and (n)=2. For unit filling, our findings agree qualitatively with quantum Monte Carlo results. Contrary to the well-known mean-field result, the shift of the critical temperature in the weakly interacting regime is apparent. © 2016 American Physical Society.view abstract 10.1103/PhysRevA.94.023629 **Prefactor in the dynamically assisted Sauter-Schwinger effect**

Schneider, C. and Schützhold, R.*Physical Review D - Particles, Fields, Gravitation and Cosmology*94 (2016)The probability of creating an electron-positron pair out of the quantum vacuum by a strong electric field can be enhanced tremendously via an additional weaker time-dependent field. This dynamically assisted Sauter-Schwinger effect has already been studied in several works. It has been found that the enhancement mechanism depends on the shape of the weaker field. For example, a Sauter pulse 1/cosh(ωt)2 and a Gaussian profile exp(-ω2t2) exhibit significant, qualitative differences. However, so far most of the analytical studies were focused on the exponent entering the pair-creation probability. Here, we study the subleading prefactor in front of the exponential using the worldline instanton method. We find that the main features of the dynamically assisted Sauter-Schwinger effect, including the dependence on the shape of the weaker field, are basically unaffected by the prefactor. To test the validity of the instanton approximation, we compare the number of produced pairs to a numerical integration of the full Riccati equation. © 2016 American Physical Society.view abstract 10.1103/PhysRevD.94.085015 **Dicke superradiance as a nondestructive probe for quantum quenches in optical lattices**

Ten Brinke, N. and Schützhold, R.*Physical Review A - Atomic, Molecular, and Optical Physics*92 (2015)We study Dicke superradiance as collective and coherent absorption and (time-delayed) emission of photons from an ensemble of ultracold atoms in an optical lattice. Since this process depends on the coherence properties of the atoms (e.g., superfluidity), it can be used as a probe for their quantum state. In analogy to pump-probe spectroscopy in solid-state physics, this detection method facilitates the investigation of nonequilibrium phenomena and is less invasive than time-of-flight experiments or direct (projective) measurements of the atom number (or parity) per lattice site, which both destroy properties of the quantum state such as phase coherence. © 2015 American Physical Society.view abstract 10.1103/PhysRevA.92.013617 **Inverse approach to solutions of the Dirac equation for space-time dependent fields**

Oertel, J. and Schützhold, R.*Physical Review D - Particles, Fields, Gravitation and Cosmology*92 (2015)Exact solutions of the Dirac equation in external electromagnetic background fields are very helpful for understanding nonperturbative phenomena in quantum electrodynamics. However, for the limited set of known solutions, the field often depends on one coordinate only, which could be the time t, a spatial coordinate such as x or r, or a light-cone coordinate such as ct-x. By swapping the roles of known and unknown quantities in the Dirac equation, we are able to generate families of solutions of the Dirac equation in the presence of genuinely space-time dependent electromagnetic fields in 1+1 and 2+1 dimensions. © 2015 American Physical Society.view abstract 10.1103/PhysRevD.92.025055 **Monogamy of entanglement and improved mean-field ansatz for spin lattices**

Osterloh, A. and Schützhold, R.*Physical Review B - Condensed Matter and Materials Physics*91 (2015)We consider rather general spin-1/2 lattices with large coordination numbers Z. Based on the monogamy of entanglement and other properties of the concurrence C, we derive rigorous bounds for the entanglement between neighboring spins, such as C≤1/Z, which show that C decreases for large Z. In addition, we demonstrate that the concurrence C measures the deviation from mean-field behavior and can only vanish if the mean-field ansatz yields an exact ground state of the Hamiltonian. Motivated by these findings, we propose an improved mean-field ansatz by adding entanglement. © 2015 American Physical Society.view abstract 10.1103/PhysRevB.91.125114 **Partner particles for moving mirror radiation and black hole evaporation**

Hotta, M. and Schützhold, R. and Unruh, W.G.*Physical Review D - Particles, Fields, Gravitation and Cosmology*91 (2015)The partner mode with respect to a vacuum state for a given mode (like that corresponding to one of the thermal particles emitted by a black hole) is defined and calculated. The partner modes are explicitly calculated for a number of cases, in particular for the modes corresponding to a particle detector being excited by turn-on/turn-off transients, or with the thermal particles emitted by the accelerated mirror model for black hole evaporation. One of the key results is that the partner mode in general is just a vacuum fluctuation, and one can have the partner mode be located in a region where the state cannot be distinguished from the vacuum state by any series of local measurements, including the energy density. For example, "information" (the correlations with the thermal emissions) need not be associated with any energy transport. The idea that black holes emit huge amounts of energy in their last stages because of all the information which must be emitted under the assumption of black hole unitarity is found to not necessarily be the case. © 2015 American Physical Society.view abstract 10.1103/PhysRevD.91.124060 **Pulse shape dependence in the dynamically assisted Sauter-Schwinger effect**

Linder, M.F. and Schneider, C. and Sicking, J. and Szpak, N. and Schützhold, R.*Physical Review D - Particles, Fields, Gravitation and Cosmology*92 (2015)While the Sauter-Schwinger effect describes nonperturbative electron-positron pair creation from vacuum by a strong and slowly varying electric field Estrong via tunneling, the dynamically assisted Sauter-Schwinger effect corresponds to a strong (exponential) enhancement of the pair-creation probability by an additional weak and fast electric or electromagnetic pulse Eweak. Using the WKB and worldline instanton method, we find that this enhancement mechanism strongly depends on the shape of the fast pulse. For the Sauter profile 1/cosh2(ωt) considered previously, the threshold frequency ωcrit (where the enhancement mechanism sets in) is basically independent of the magnitude Eweak of the weak pulse - whereas for a Gaussian pulse exp(-ω2t2), an oscillating profile cos(ωt) or a standing wave cos(ωt)cos(kx), the value of ωcrit does depend (logarithmically) on Eweak/Estrong. © 2015 American Physical Society.view abstract 10.1103/PhysRevD.92.085009 **Equilibration and prethermalization in the Bose-Hubbard and Fermi-Hubbard models**

Queisser, F. and Krutitsky, K.V. and Navez, P. and Schützhold, R.*Physical Review A - Atomic, Molecular, and Optical Physics*89 (2014)We study the Bose-Hubbard and Fermi-Hubbard models in the (formal) limit of large coordination numbers Z≫1. Via an expansion into powers of 1/Z, we establish a hierarchy of correlations which facilitates an approximate analytical derivation of the time evolution of the reduced density matrices for one and two sites, etc. With this method, we study the quantum dynamics (starting in the ground state) after a quantum quench, i.e., after suddenly switching the tunneling rate J from zero to a finite value, which is still in the Mott regime. We find that the reduced density matrices approach a (quasi)equilibrium state after some time. For one lattice site, this state can be described by a thermal state (within the accuracy of our approximation). However, the (quasi)equilibrium state of the reduced density matrices for two sites including the correlations can not be described by a thermal state. Thus, real thermalization (if it occurs) should take a much longer time. This behavior has already been observed in other scenarios and is sometimes called "prethermalization". Finally, we compare our results to numerical simulations for finite lattices in one and two dimensions and find qualitative agreement. © 2014 American Physical Society.view abstract 10.1103/PhysRevA.89.033616 **Quasi-particle approach for lattice Hamiltonians with large coordination numbers**

Navez, P. and Queisser, F. and Schützhold, R.*Journal of Physics A: Mathematical and Theoretical*(2014)In many condensed-matter systems, it is very useful to introduce a quasi-particle approach, which is based on some sort of linearization around a suitable background state. In order to be a systematic and controlled approximation, this linearization should be justified by an expansion into the powers of some small control parameter. Here, we present a method for general lattice Hamiltonians with large coordination numbers Z 蠑 1, which is based on an expansion into the powers of 1/Z. In order to demonstrate the generality of our method, we apply it to various spin systems, as well as the Bose and Fermi-Hubbard model. © 2014 IOP Publishing Ltd.view abstract 10.1088/1751-8113/47/22/225004 **Cosmological particle creation in the lab?**

Schützhold, R. and Unruh, W.G.*Lecture Notes in Physics*870 (2013)We give an overview of some fundamental quantum vacuum effects in curved space times that may be studied in earth based laboratories. In particular we review the concept of cosmological particle creation related to a contraction or expansion of the Universe. © Springer International Publishing Switzerland 2013.view abstract 10.1007/978-3-319-00266-8-3 **Feasibility study of a nuclear exciton laser**

Ten Brinke, N. and Schützhold, R. and Habs, D.*Physical Review A - Atomic, Molecular, and Optical Physics*87 (2013)Nuclear excitons known from Mössbauer spectroscopy describe coherent excitations of a large number of nuclei - analogous to Dicke states (or Dicke super-radiance) in quantum optics. In this paper, we study the possibility of constructing a laser based on these coherent excitations. In contrast to the free-electron laser (in its usual design), such a device would be based on stimulated emission and thus might offer certain advantages, e.g., regarding energy-momentum accuracy. Unfortunately, inserting realistic parameters, the window of operability is probably not open (yet) to present-day technology; but our design should be feasible in the UV regime, for example. © 2013 American Physical Society.view abstract 10.1103/PhysRevA.87.053814 **Hawking radiation with dispersion versus breakdown of the WKB approximation**

Schützhold, R. and Unruh, W.G.*Physical Review D - Particles, Fields, Gravitation and Cosmology*88 (2013)Inspired by the condensed matter analogues of black holes (a.k.a. dumb holes), we study Hawking radiation in the presence of a modified dispersion relation which becomes superluminal at large wave numbers. In the usual stationary coordinates (t,x), one can describe the asymptotic evolution of the wave packets in WKB, but this WKB approximation breaks down in the vicinity of the horizon, thereby allowing for a mixing between initial and final creation and annihilation operators. Thus, one might be tempted to identify this point where WKB breaks down with the moment of particle creation. However, using different coordinates (τ,U), we find that one can evolve the waves so that WKB in these coordinates is valid throughout this transition region, which contradicts the above identification of the breakdown of WKB as the cause of the radiation. Instead, our analysis suggests that the tearing apart of the waves into two different asymptotic regions (inside and outside the horizon) is the major ingredient of Hawking radiation. © 2013 American Physical Society.view abstract 10.1103/PhysRevD.88.124009 **Strong magnetophotoelectric effect in folded graphene**

Queisser, F. and Schützhold, R.*Physical Review Letters*111 (2013)We study electronic transport in graphene under the influence of a transversal magnetic field B(r)=B(x)ez with the asymptotics B(x→±∞)=±B0, which could be realized via a folded graphene sheet in a constant magnetic field, for example. By solving the effective Dirac equation, we find robust modes with a finite energy gap which propagate along the fold - where particles and holes move in opposite directions. Exciting these particle-hole pairs with incident (optical or infrared) photons would then generate a nearly perfect charge separation and thus a strong magnetophotoelectric or magnetothermoelectric effect - even at room temperature. © 2013 American Physical Society.view abstract 10.1103/PhysRevLett.111.046601 **Universal frozen spectra after time-dependent symmetry restoring phase transitions**

Queisser, F. and Navez, P. and Schützhold, R.*Journal of Physics Condensed Matter*25 (2013)For a general O(N) model, we study the time-dependent phase transition from a state with broken symmetry to the symmetric phase . During this non-equilibrium process, the primordial quantum (or thermal) fluctuations of the initial Goldstone modes are frozen and result in a deviation from the final ground (or thermal) state. For very slow transitions, we find that these fluctuations display a universal scaling behaviour. Their spectra are universal functions of a single parameter, which combines the initial frequency of the Goldstone modes and the sweep rate. As a result, the final two-point function is not exponentially suppressed at large distances Δr = r - r′ (as it would be in the ground state) but decays polynomially in 1/|Δr|. Finally, we exemplify this universal behaviour for the transition from the super-fluid phase to the Mott state in the Bose-Hubbard model. © 2013 IOP Publishing Ltd.view abstract 10.1088/0953-8984/25/40/404215 **Whispering gallery like modes along pinned vortices**

Marecki, P. and Schützhold, R.*JETP Letters*96 (2013)We study sound propagation in stationary and locally irrotational vortex flows where the circulation is wound around a long (rotating) cylinder, using Unruh's formalism of acoustic space-times. Apart from the usual scattering solutions, we find anomalous modes which are bound to the vicinity of the cylinder and propagate along its axis-similar to whispering gallery modes. These modes exist for subsonic and supersonic flow velocities. In the supersonic case (corresponding to an effective ergoregion in the acoustic space-time), they can even have zero frequency ω = 0 and thus the associated quasiparticles with E = h{stroke}ω = 0 are easy to excite from an energetic point of view. Hence they should be relevant for the question of stability or instability of this setup. © 2012 Pleiades Publishing, Ltd.view abstract 10.1134/S0021364012220079 **Focus on classical and quantum analogues for gravitational phenomena and related effects**

Leonhardt, U. and Maia, C. and Schützhold, R.*New Journal of Physics*14 (2012)Hawking's prediction that black holes are not black but radiate has been one of the intellectually most influential results of theoretical physics, but Hawking's theory has not so far been testable. Recent developments in analogue models of gravity might change that. This focus issue assembles a series of papers that report on steps towards this goal and related physical effects in a variety of physical systems. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.view abstract 10.1088/1367-2630/14/10/105032 **Hawking radiation from "phase horizons" in laser filaments?**

Unruh, W.G. and Schützhold, R.*Physical Review D - Particles, Fields, Gravitation and Cosmology*86 (2012)Belgiorno et al. have reported on experiments aiming at the detection of (the analogue of) Hawking radiation using laser filaments [F. Belgiorno et al., Phys. Rev. Lett. 105, 203901 (2010)]. They sent intense focused Bessel pulses into a nonlinear dielectric medium in order to change its refractive index via the Kerr effect and saw creation of photons orthogonal to the direction of travel of the pulses. Since the refractive index change in the pulse generated a "phase horizon" (where the phase velocity of these photons equals the pulse speed), they concluded that they observed the analogue of Hawking radiation. We study this scenario in a model with a phase horizon and a phase velocity very similar to that of their experiment and find that the effective metric does not quite correspond to a black hole. The photons created in this model are not due to the analogue of black hole evaporation but have more similarities to cosmological particle creation. Nevertheless, even this effect cannot explain the observations-unless the pulse has significant small scale structure in both the longitudinal and transverse dimensions. © 2012 American Physical Society.view abstract 10.1103/PhysRevD.86.064006 **Momentum dependence in the dynamically assisted Sauter-Schwinger effect**

Fey, C. and Schützhold, R.*Physical Review D - Particles, Fields, Gravitation and Cosmology*85 (2012)Recently it was found that the superposition of a strong and slow electric field with a weaker and faster pulse can significantly enhance the probability for nonperturbative electron-positron pair creation out of the vacuum-the dynamically assisted Sauter-Schwinger effect. Via the WKB method, we estimate the momentum dependence of the pair creation probability and compare it to existing numerical results. Besides the theoretical interest, a better understanding of this pair creation mechanism should be helpful for the planned experiments aiming at its detection. © 2012 American Physical Society.view abstract 10.1103/PhysRevD.85.025004 **Optical lattice quantum simulator for quantum electrodynamics in strong external fields: Spontaneous pair creation and the Sauter-Schwinger effect**

Szpak, N. and Schützhold, R.*New Journal of Physics*14 (2012)The spontaneous creation of electron-positron pairs out of the vacuum due to a strong electric field is a spectacular manifestation of the relativistic energy-momentum relation for the Dirac fermions. This fundamental prediction of quantum electrodynamics has not yet been confirmed experimentally, as the generation of a sufficiently strong electric field extending over a large enough space-time volume still presents a challenge. Surprisingly, distant areas of physics may help us to circumvent this difficulty. In condensed matter and solid state physics (areas commonly considered as low-energy physics), one usually deals with quasi-particles instead of real electrons and positrons. Since their mass gap can often be freely tuned, it is much easier to create these light quasi-particles by an analogue of the Sauter-Schwinger effect. This motivates our proposal for a quantum simulator in which excitations of ultra-cold atoms moving in a bichromatic optical lattice represent particles and antiparticles (holes) satisfying a discretized version of the Dirac equation together with fermionic anti-commutation relations. Using the language of second quantization, we are able to construct an analogue of the spontaneous pair creation which can be realized in an (almost) table-top experiment. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.view abstract 10.1088/1367-2630/14/3/035001 **Sauter-Schwinger-like tunneling in tilted Bose-Hubbard lattices in the Mott phase**

Queisser, F. and Navez, P. and Schützhold, R.*Physical Review A - Atomic, Molecular, and Optical Physics*85 (2012)We study the Mott phase of the Bose-Hubbard model on a tilted lattice. On the (Gutzwiller) mean-field level, the tilt has no effect, but quantum fluctuations entail particle-hole pair creation via tunneling. For small potential gradients (long-wavelength limit), we derive a quantitative analogy to the Sauter-Schwinger effect, i.e., electron-positron pair creation out of the vacuum by an electric field. For large tilts, we obtain resonant tunneling reminiscent of Bloch oscillations. © 2012 American Physical Society.view abstract 10.1103/PhysRevA.85.033625 **Comment on "hawking radiation from ultrashort laser pulse filaments"**

Schützhold, R. and Unruh, W.G.*Physical Review Letters*107 (2011)A Comment on the Letter by F. Belgiorno et al., Phys. Rev. Lett.PRLTAO0031-9007 105, 203901 (2010)10.1103/PhysRevLett.105.203901. The authors of the Letter offer a Reply. © 2011 American Physical Society.view abstract 10.1103/PhysRevLett.107.149401 **Entangling photons via the double quantum Zeno effect**

Ten Brinke, N. and Osterloh, A. and Schützhold, R.*Physical Review A - Atomic, Molecular, and Optical Physics*84 (2011)We propose a scheme for entangling two photons via the quantum Zeno effect, which describes the inhibition of quantum evolution by frequent measurements and is based on the difference between summing amplitudes and probabilities. For a given error probability P error, our scheme requires that the one-photon loss rate ξ 1γ and the two-photon absorption rate ξ 2γ in some medium satisfy ξ 1γ/ ξ 2γ=2Perror2/π2, which is significantly improved compared to previous approaches. Again based on the quantum Zeno effect, as well as coherent excitations, we present a possibility to fulfill this requirement in an otherwise linear optics setup. © 2011 American Physical Society.view abstract 10.1103/PhysRevA.84.022317 **Erratum: Hawking radiation on an ion ring in the quantum regime (New Journal of Physics (2011) 13 (045008))**

Horstmann, B. and Schützhold, R. and Reznik, B. and Fagnocchi, S. and Cirac, J.I.*New Journal of Physics*13 (2011)view abstract 10.1088/1367-2630/13/6/069601 **Hawking radiation on an ion ring in the quantum regime**

Horstmann, B. and Schützhold, R. and Reznik, B. and Fagnocchi, S. and Cirac, J.I.*New Journal of Physics*13 (2011)This paper discusses a recent proposal for the simulation of acoustic black holes with ions (Horstmann et al 2010 Phys. Rev. Lett. 104 250403). Ions are rotating on a ring with an inhomogeneous but stationary velocity profile. Phonons cannot leave a region in which the ion velocity exceeds the group velocity of the phonons, because light cannot escape from a black hole. The system is described by a discrete field theory with a nonlinear dispersion relation. Hawking radiation is emitted by this acoustic black hole, generating entanglement between the inside and the outside of the black hole. We study schemes for detecting the Hawking effect in this setup. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.view abstract 10.1088/1367-2630/13/4/045008 **Quantum simulator for the Schwinger effect with atoms in bichromatic optical lattices**

Szpak, N. and Schützhold, R.*Physical Review A - Atomic, Molecular, and Optical Physics*84 (2011)Ultracold atoms in specifically designed optical lattices can be used to mimic the many-particle Hamiltonian (whose effective parameters can be tuned in a wide range) describing electrons and positrons in an external electric field. This analogy facilitates the experimental simulation of (so far unobserved) fundamental quantum phenomena such as the Schwinger effect, i.e., spontaneous electron-positron pair creation out of the vacuum by a strong electric field. Such an experiment would also test nonperturbative aspects of these lattice systems. © 2011 American Physical Society.view abstract 10.1103/PhysRevA.84.050101 **Decoherence in a dynamical quantum phase transition**

Mostame, S. and Schaller, G. and Schützhold, R.*Physical Review A - Atomic, Molecular, and Optical Physics*81 (2010)Motivated by the similarity between adiabatic quantum algorithms and quantum phase transitions, we study the impact of decoherence on the sweep through a second-order quantum phase transition for the prototypical example of the Ising chain in a transverse field and compare it to the adiabatic version of Grover's search algorithm, which displays a first-order quantum phase transition. For site-independent and site-dependent coupling strengths as well as different operator couplings, the results show (in contrast to first-order transitions) that the impact of decoherence caused by a weak coupling to a rather general environment increases with system size (i.e., number of spins or qubits). This might limit the scalability of the corresponding adiabatic quantum algorithm. © 2010 The American Physical Society.view abstract 10.1103/PhysRevA.81.032305 **Emergence of coherence in the Mott-insulator-superfluid quench of the Bose-Hubbard model**

Navez, P. and Schützhold, R.*Physical Review A - Atomic, Molecular, and Optical Physics*82 (2010)We study the quench from the Mott-insulator to the superfluid phase in the Bose-Hubbard model and investigate the spatial-temporal growth of phase coherence, that is, phase locking between initially uncorrelated sites. To this end, we establish a hierarchy of correlations via a controlled expansion into inverse powers of the coordination number 1/Z. It turns out that the off-diagonal long-range order spreads with a constant propagation speed, forming local condensate patches, whereas the phase correlator follows a diffusionlike growth rate. © 2010 The American Physical Society.view abstract 10.1103/PhysRevA.82.063603 **O(N) symmetry-breaking quantum quench: Topological defects versus quasiparticles**

Uhlmann, M. and Schützhold, R. and Fischer, U.R.*Physical Review D - Particles, Fields, Gravitation and Cosmology*81 (2010)We present an analytical derivation of the winding number counting topological defects created by an O(N) symmetry-breaking quantum quench in N spatial dimensions. Our approach is universal in the sense that we do not employ any approximations apart from the large-N limit. The final result is nonperturbative in N, i.e., it cannot be obtained by an expansion in 1/N, and we obtain far less topological defects than quasiparticle excitations, in sharp distinction to previous, low-dimensional investigations. © 2010 The American Physical Society.view abstract 10.1103/PhysRevD.81.025017 **On the detection of footprints from strong electron acceleration in high-intensity laser fields, including the Unruh effect**

Thirolf, P.G. and Habs, D. and Homma, K. and Hörlein, R. and Karsch, S. and Krausz, F. and Maia, C. and Osterhoff, J. and Popp, A. and Schmid, K. and Schreiber, J. and Schützhold, R. and Tajima, T. and Veisz, L. and Wulz, J. and Yamazaki, T.*AIP Conference Proceedings*1228 (2010)The ultra-high fields of high-power short-pulse lasers are expected to contribute to understanding fundamental properties of the quantum vacuum and quantum theory in very strong fields. For example, the neutral QED vacuum breaks down at the Schwinger field strength of 1.3.1018 V/m, where a virtual e+e- pair gains its rest mass energy over a Compton wavelength and materializes as a real pair. At such an ultra-high field strength, an electron experiences an acceleration of aS=2.10 28g and hence fundamental phenomena such as the long predicted Unruh effect start to play a role. The Unruh effect implies that the accelerated electron experiences the vacuum as a thermal bath with the Unruh temperature. In its accelerated frame the electron scatters photons off the thermal bath, corresponding to the emission of an entangled pair of photons in the laboratory frame. In upcoming experiments with intense accelerating fields, we will encounter a set of opportunities to experimentally study the radiation from electrons under extreme fields. Even before the Unruh radiation detection, we should run into the copious Larmor radiation. The detection of Larmor radiation and its characterization themselves have never been experimentally carried out to the best of our knowledge, and thus this amounts to a first serious study of physics at extreme acceleration. For example, we can study radiation damping effects like the Landau-Lifshitz radiation. Furthermore, the experiment should be able to confirm or disprove whether the Larmor and Landau-Lifshitz radiation components may be enhanced by a collective (N2) radiation, if a tightly clumped cluster of electrons is accelerated. The technique of laser driven dense electron sheet formation by irradiating a thin DLC foil target should provide such a coherent electron cluster with a very high density. If and when such mildly relativistic electron sheets are realized, a counterpropagating second laser can interact with them coherently. Under these conditions enhanced Larmor and Unruh radiation signals may be observed. Detection of the Unruh photons (together with its competing radiation components) is envisaged via Compton polarimetry in a novel highly granular 2D-segmented position-sensitive germanium detector. © 2010 American Institute of Physics.view abstract 10.1063/1.3426087 **Quantum correlations across the black hole horizon**

Schützhold, R. and Unruh, W.G.*Physical Review D - Particles, Fields, Gravitation and Cosmology*81 (2010)Inspired by the condensed-matter analogues of black holes, we study the quantum correlations across the event horizon reflecting the entanglement between the outgoing particles of the Hawking radiation and their in-falling partners. For a perfectly covariant theory, the total correlation is conserved in time and piles up arbitrary close to the horizon in the past, where it merges into the singularity of the vacuum two-point function at the light cone. After modifying the dispersion relation (i.e., breaking Lorentz invariance) for large k, on the other hand, the light cone is smeared out and the entanglement is not conserved but actually created in a given rate per unit time. © 2010 The American Physical Society.view abstract 10.1103/PhysRevD.81.124033 **Quantum Zeno suppression of three-body losses in Bose-Einstein condensates**

Schützhold, R. and Gnanapragasam, G.*Physical Review A - Atomic, Molecular, and Optical Physics*82 (2010)We study the possibility of suppressing three-body losses in atomic Bose-Einstein condensates via the quantum Zeno effect, which means the delay of quantum evolution by frequent measurements. It turns out that this requires very fast measurements with the rate being determined by the spatial structure of the three-body form factor, i.e., the point interaction approximation δ3(r-r') is not adequate. Since the molecular binding energy Eb provides a natural limit for the measurement rate, this suppression mechanism can only work if the form factor possesses certain special properties. © 2010 The American Physical Society.view abstract 10.1103/PhysRevA.82.022120 **System size scaling of topological defect creation in a second-order dynamical quantum phase transition**

Uhlmann, M. and Schützhold, R. and Fischer, U.R.*New Journal of Physics*12 (2010)We investigate the system size scaling of the net defect number created by a rapid quench in a second-order quantum phase transition from an O(N) symmetric state to a phase of broken symmetry. Using a controlled mean-field expansion for large N, we find that the net defect number variance in convex volumina scales as the surface area of the sample for shortrange correlations. This behaviour follows generally from spatial and internal symmetries. Conversely, if spatial isotropy is broken, e.g. by a lattice, and in addition long-range periodic correlations develop in the broken-symmetry phase, we get the rather counterintuitive result that the scaling strongly depends on the dimension being even or odd: for even dimensions, the net defect number variance scales as the surface area squared, with a prefactor oscillating with the system size, while for odd dimensions, it essentially vanishes. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.view abstract 10.1088/1367-2630/12/9/095020

#### graphene

#### optical properties

#### phase transformations

#### quantum dynamics

#### quantum information technology

#### quantum optics