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We describe an efficient numerical method for simulating the dynamics and steady states of collective spin systems in the presence of dephasing and decay. The method is based on the Schwinger boson representation of spin operators and uses an extension of the truncated Wigner approximation to map the exact open system dynamics onto stochastic differential equations for the corresponding phase space distribution. This approach is most effective in the limit of very large spin quantum numbers, where exact numerical simulations and other approximation methods are no longer applicable. We benchmark this numerical technique for known superradiant decay and spin-squeezing processes and illustrate its application for the simulation of non-equilibrium phase transitions in dissipative spin lattice models., 21 pages, 7 figures

The effect of PT-symmetry breaking in coupled systems with balanced gain and loss has recently attracted considerable attention and has been demonstrated in various photonic, electrical and mechanical systems in the classical regime. Here we generalize the definition of PT symmetry to finite-dimensional open quantum systems, which are described by a Markovian master equation. Specifically, we show that the invariance of this master equation under a certain symmetry transformation implies the existence of stationary states with preserved and broken parity symmetry. As the dimension of the Hilbert space grows, the transition between these two limiting phases becomes increasingly sharp and the classically expected PT-symmetry breaking transition is recovered. This quantum-to-classical correspondence allows us to establish a common theoretical framework to identify and accurately describe PT-symmetry breaking effects in a large variety of physical systems, operated both in the classical and quantum regimes., Comment: 15 pages, 3 figures

Strong coupling between light and matter is possible with a variety of organic materials. In contrast to the simpler inorganic case, organic materials often have a complicated spectrum, with vibrationally dressed electronic transitions. Strong coupling to light competes with this vibrational dressing, and if strong enough, can suppress the entanglement between electronic and vibrational degrees of freedom. By exploiting symmetries, we can perform exact numerical diagonalization to find the polaritonic states for intermediate numbers of molecules, and use these to define and validate accurate expressions for the lower polariton states and strong-coupling spectrum in the thermodynamic limit. Using this approach, we find that vibrational decoupling occurs as a sharp transition above a critical matter-light coupling strength. We also demonstrate how the polariton spectrum evolves with the number of molecules, recovering classical linear optics results only at large $N$., 14+3 pages, 8 figures. v3 updated to published version

We study the collective decay of two-level emitters coupled to a nonlinear waveguide, for example, a nanophotonic lattice or a superconducting resonator array with strong photon-photon interactions. Under these conditions a new decay channel into bound photon pairs emerges, through which spatial correlations between emitters are established by regular interference as well as interactions between the photons. We derive an effective Markovian theory to model the resulting decay dynamics of an arbitrary distribution of emitters and identify collective effects beyond the usual phenomena of super- and subradiance. Specifically, in the limit of many close-by emitters, we find that the system undergoes a super-correlated decay process where either all the emitters are in the excited state or in the ground state, but not in any of the intermediate states. The predicted effects can be probed in state-of-the-art waveguide QED experiments and provide a striking example of how the dynamics of open quantum systems can be modified by many-body effects in a non-harmonic environment., Comment: 6+13 pages 4+7 figures, more details are discussed

We study the magnetic phases of a non-equilibrium spin chain, where coherent interactions between neighboring lattice sites compete with alternating gain and loss processes. This competition between coherent and incoherent dynamics induces transitions between magnetically aligned and highly mixed phases, across which the system changes from a low- to an effective infinite-temperature state. We show that the origin of these transitions can be traced back to the dynamical effect of parity-time-reversal symmetry breaking, which has no counterpart in the theory of equilibrium phase transitions. This mechanism also results in very atypical features and we find first-order transitions without phase co-existence and mixed-order transitions which do not break the underlying $U(1)$ symmetry, even in the appropriate thermodynamic limit. Thus, despite its simplicity, the current model considerably extends the phenomenology of non-equilibrium phase transitions beyond that commonly assumed for driven-dissipative spins and related systems., 14 pages, 9 figures

Following experimental realizations of room temperature polariton lasing with organic molecules, we present a microscopic model that allows us to explore the crossover from weak to strong matter-light coupling. We consider a non-equilibrium Dicke-Holstein model, including both strong coupling to vibrational modes and strong matter-light coupling, providing the phase diagram of this model in the thermodynamic limit. We discuss the mechanism of polariton lasing, uncovering a process of self-tuning, and identify the relation and distinction between regular dye lasers and organic polariton lasers., 6 pages, 4 figures, plus supplemental material

The Dicke model describes the coupling between a quantized cavity field and a large ensemble of two-level atoms. When the number of atoms tends to infinity, this model can undergo a transition to a superradiant phase, belonging to the mean-field Ising universality class. The superradiant transition was first predicted for atoms in thermal equilibrium and was recently realized with a quantum simulator made of atoms in an optical cavity, subject to both dissipation and driving. In this Progress Report, we offer an introduction to some theoretical concepts relevant to the Dicke model, reviewing the critical properties of the superradiant phase transition, and the distinction between equilibrium and nonequilibrium conditions. In addition, we explain the fundamental difference between the superradiant phase transition and the more common lasing transition. Our report mostly focuses on the steady states of atoms in single-mode optical cavities, but we also mention some aspects of real-time dynamics, as well as other quantum simulators, including superconducting qubits, trapped ions, and using spin-orbit coupling for cold atoms. These realizations differ in regard to whether they describe equilibrium or non-equilibrium systems., Invited Progress Report; 21 pages, 6 figures

We consider the orientational alignment of dipoles due to strong matter light coupling, for a non-vanishing density of excitations. We compare various approaches to this problem in the limit of large numbers of emitters, and show that direct Monte Carlo integration, mean-field theory, and large deviation methods match exactly in this limit. All three results show that orientational alignment develops in the presence of a macroscopically occupied polariton mode, and that the dipoles asymptotically approach perfect alignment in the limit of high density or low temperature., 7 pages, 4 figures

HMC acknowledges studentship funding from EPSRC under grant no. EP/G03673X/1. PGK acknowledges support from EPSRC (EP/M010910/1). BWL acknowledges support from EPSRC (EP/K025562/1). PRE acknowledges funding from SFI (15/IACA/3402). JK acknowledges financial support from EPSRC programs “TOPNES” (EP/I031014/1) and “Hybrid-Polaritonics” (EP/M025330/1). The interaction of a quantum system with its environment causes decoherence, setting a fundamental limit on its suitability for quantum information processing. However, we show that if the system consists of coupled parts with different internal energy scales then the interaction of one part with a thermal bath need not lead to loss of coherence from the other. Remarkably, we find that the protected part can remain coherent for longer when the coupling to the bath becomes stronger or the temperature is raised. Our theory will enable the design of decoherence-resistant hybrid quantum computers. Publisher PDF

Funding: “Laidlaw Research Internship” scheme at the University of St Andrews (RIM), EPSRC (EP/M010910/1) (PGK), EPSRC programs “TOPNES” (EP/I031014/1) and “Hybrid polaritonics” (EP/M025330/1) (JK). It has previously been shown that a dye-filled microcavity can produce a Bose-Einstein condensate of photons. Thermalization of photons is possible via repeated absorption and re-emission by the dye molecules. In this paper, we theoretically explore the behavior of the polarization of light in this system. We find that in contrast to the near complete thermalization between different spatial modes of light, thermalization of polarization states is expected to generally be incomplete. We show that the polarization degree changes significantly from below to above threshold, and explain the dependence of polarization on all relevant material parameters. Postprint

We show that dephasing of individual atoms destroys the superradiance transition of the Dicke model, but that adding individual decay toward the spin down state can restore this transition. To demonstrate this, we present a method to give an exact solution for the $N$ atom problem with individual dephasing which scales polynomially with $N$. By comparing finite size scaling of our exact solution to a cumulant expansion, we confirm the destruction and restoration of the superradiance transition holds in the thermodynamic limit., 5 pages, 2 figures, updated to published version

We investigate different geometries of spin-1/2 nitrogen impurity channels for distributing entanglement between pairs of remote nitrogen vacancy centers (NVs) in diamond. To go beyond the system size limits imposed by directly solving the master equation, we implement a matrix product operator method to describe the open system dynamics. In so doing, we provide an early demonstration of how this technique can be used for simulating real systems. For a fixed NV separation there is an interplay between incoherent impurity spin decay and coherent entanglement transfer: Long transfer time, few-spin systems experience strong dephasing that can be overcome by increasing the number of spins in the channel. We examine how missing spins and disorder in the coupling strengths affect the dynamics, finding that in some regimes a spin ladder is a more effective conduit for information than a single spin chain., 5 pages, 4 figures

Finding efficient descriptions of how an environment affects a collection of discrete quantum systems would lead to new insights into many areas of modern physics. Markovian, or time-local, methods work well for individual systems, but for groups a question arises: does system-bath or inter-system coupling dominate the dissipative dynamics? The answer has profound consequences for the long-time quantum correlations within the system. We consider two bosonic modes coupled to a bath. By comparing an exact solution to different Markovian master equations, we find that a smooth crossover of the equations-of-motion between dominant inter-system and system-bath coupling exists -- but requires a non-secular master equation. We predict a singular behaviour of the dynamics, and show that the ultimate failure of non-secular equations of motion is essentially a failure of the Markov approximation. Our findings justify the use of time-local theories throughout the crossover between system-bath dominated and inter-system-coupling dominated dynamics., 10 pages, 3 figures. v2: improved introduction, added analysis of positivity, corrected typos. v3: corrected typos in eqs. 5 and 23

We study theoretically the effects of pump-spot size and location on photon condensates. By exploring the inhomogeneous molecular excitation fraction, we make clear the relation between spatial equilibration, gain clamping and thermalization in a photon condensate. This provides a simple understanding of several recent experimental results. We find that as thermalization breaks down, gain clamping is imperfect, leading to "transverse spatial hole burning" and multimode condensation. This opens the possibility of engineering the gain profile to control the condensate structure., Comment: Further extended, including new figures. Now 10 figures

We examine in detail the mechanisms behind thermalization and Bose-Einstein condensation of a gas of photons in a dye-filled microcavity. We derive a microscopic quantum model, based on that of a standard laser, and show how this model can reproduce the behavior of recent experiments. Using the rate equation approximation of this model, we show how a thermal distribution of photons arises. We go on to describe how the non-equilibrium effects in our model can cause thermalization to break down as one moves away from the experimental parameter values. In particular, we examine the effects of changing cavity length, and of altering the vibrational spectrum of the dye molecules. We are able to identify two measures which quantify whether the system is in thermal equilibrium. Using these we plot "phase diagrams" distinguishing BEC and standard lasing regimes. Going beyond the rate equation approximation, our quantum model allows us to investigate both the second order coherence, $g^{(2)}$, and the linewidth of the emission from the cavity. We show how the linewidth collapses as the system transitions to a Bose condensed state, and compare the results to the Schawlow--Townes linewidth., 17 pages, 12 figures

Starting from a microscopic model, we investigate the optical spectra of molecules in strongly-coupled organic microcavities examining how they might self-consistently adapt their coupling to light. We consider both rotational and vibrational degrees of freedom, focusing on features which can be seen in the peak in the center of the spectrum at the bare excitonic frequency. In both cases we find that the matter-light coupling can lead to a self-consistent change of the molecular states, with consequent temperature-dependent signatures in the absorption spectrum. However, for typical parameters, these effects are much too weak to explain recent measurements. We show that another mechanism which naturally arises from our model of vibrationally dressed polaritons has the right magnitude and temperature dependence to be at the origin of the observed data., Comment: 14 pages, 6 figure

We analyze the response of a nanomechanical resonator to an external drive when it is also coupled to a single-electron transistor (SET). The interaction between the SET electrons and the mechanical resonator depends on the amplitude of the mechanical motion leading to a strongly non-linear response to the drive which is similar to that of a Duffing oscillator. We show that the average dynamics of the resonator is well-described by a simple effective model which incorporates damping and frequency renormalization terms which are amplitude dependent. We also find that for a certain range of parameters the system displays interesting bistable dynamics in which noise arising from charge fluctuations causes the resonator to switch slowly between different dynamical states., 11 pages, 9 figures

We develop a nonequilibrium model of condensation and lasing of photons in a dye filled microcavity. We examine in detail the nature of the thermalization process induced by absorption and emission of photons by the dye molecules, and investigate when the photons are able to reach a thermal equilibrium Bose-Einstein distribution. At low temperatures, or large cavity losses, the absorption and emission rates are too small to allow the photons to reach thermal equilibrium and the behavior becomes more like that of a conventional laser., 5 pages, 3 figures

We present a formalism for calculating the non-symmetrized quantum current noise within the Born-Markov approximation for the master equation. The formalism is particularly well suited to obtaining the current noise for quantum transport in mesoscopic devices such as a superconducting single electron transistor (SSET). As an example of the method, we obtain explicit results for the double Josephson-quasiparticle (DJQP) resonance in a SSET. Our calculations reveal the asymmetries that develop in the current noise as well as clarifying the behavior at high frequencies. Our findings are consistent with recent measurements of the asymmetry in the current noise spectrum., 5 pages, 3 figures

We analyze the charge dynamics of a superconducting single-electron transistor (SSET) in the regime where charge transport occurs via Cooper-pair resonances. Using an approximate description of the system Hamiltonian, in terms of a series of resonant doublets, we derive a Born-Markov master equation describing the dynamics of the SSET. The average current displays sharp peaks at the Cooper-pair resonances and we find that the charge noise spectrum has a characteristic structure which consists of a series of asymmetric triplets of peaks. The strongest feature in the charge noise spectrum is the triplet of peaks centered at zero frequency which has a peak spacing equal to the level separation within the doublets and is similar to the triplet in the spectrum of a driven, damped, two-level system. We also explore the back-action that the SSET charge noise would have on an oscillator coupled to the island charge, measurement of which provides a way of probing the charge noise spectrum., 14 pages, 7 figures