Phase transitions to quantum condensed phases--such as Bose-Einstein condensation (BEC), superfluidity, and superconductivity--have long fascinated scientists, as they bring pure quantum effects to a macroscopic scale. BEC has, for example, famously been demonstrated in dilute atom gas of rubidium atoms at temperatures below 200 nanokelvin. Much effort has been devoted to finding a solid-state system in which BEC can take place. Promising candidate systems are semiconductor microcavities, in which photons are confined and strongly coupled to electronic excitations, leading to the creation of exciton polaritons. These bosonic quasi-particles are 10(9) times lighter than rubidium atoms, thus theoretically permitting BEC to occur at standard cryogenic temperatures. Here we detail a comprehensive set of experiments giving compelling evidence for BEC of polaritons. Above a critical density, we observe massive occupation of the ground state developing from a polariton gas at thermal equilibrium at 19 K, an increase of temporal coherence, and the build-up of long-range spatial coherence and linear polarization, all of which indicate the spontaneous onset of a macroscopic quantum phase.

Injection and decay of particles in an inhomogeneous quantum condensate can significantly change its behaviour. We model trapped, pumped, decaying condensates by a complex Gross-Pitaevskii equation and analyse the density and currents in the steady state. With homogeneous pumping, rotationally symmetric solutions are unstable. Stability may be restored by a finite pumping spot. However if the pumping spot is larger than the Thomas-Fermi cloud radius, then rotationally symmetric solutions are replaced by solutions with spontaneous arrays of vortices. These vortex arrays arise without any rotation of the trap, spontaneously breaking rotational symmetry.PACS numbers: 03.75. Kk,47.37.+q,71.36.+c,71.35.Lk While much of the possible physics of quantum condensates has been examined in experiments on atomic gases, superfluid Helium and superconductors, there has recently been much interest in examples of condensates of quasiparticle excitations, such as excitons [1,2] (bound electron-hole pairs), exciton-polaritons [3,4,5] (superpositions of quantum well excitons and microcavity photons), and magnons (spin-wave excitations) both in magnetic insulating crystals [6,7] [33] and in superfluid 3 He [8,9,10]. One particular difference shown by these systems is that the quasiparticles have finite lifetimes, and as a result, they can be made to form condensates out of equilibrium, which are best understood as a steady state balance between pumping and decay, rather than true thermal equilibrium. The effects of pumping and decay in these condensates have been the subject of several recent works [5,11,12,13,14,15,16,17,18,19,20] which have shown that even when collisions can rapidly thermalise the energy distribution of a system, there may yet be noticeable effects associated with the energy scale introduced by the pumping and decay.The Gross-Pitaevskii equation (GPE) has been applied to successfully describe many features of equilibrium condensates when far in the condensed regime, including density profiles, the dynamics of vortices, hydrodynamic modes -see e.g. [21] and Refs. therein. Using a meanfield description of the condensate, e.g. [18,19,20], one can recover a complex Gross-Pitaevskii equation (cGPE), including terms representing gain, loss and an external trapping potential. This letter studies the interplay between pumping and decay and the external trapping potential in the context of the cGPE in order to illustrate some of the differences between equilibrium and nonequilibrium condensates. In the absence of trapping, this is the celebrated complex Ginzburg-Landau equation that describes a vast variety of phenomena [22] from nonlinear waves to second-order phase transitions, from superconductivity to liquid crystals and cosmic strings and binary fluids [23]. What is of interest in this letter is how pumping and decay, described in the cGPE modify behaviour compared to the regular GPE as is widely applied to spatially inhomogeneous equilibrium quantum condensates [21]. Spatial inhomogeneity, due to either engineere...

A strategy is proposed to excite particles from a Fermi sea in a noise-free fashion by electromagnetic pulses with realistic parameters. We show that by using quantized pulses of simple form one can suppress the particle-hole pairs which are created by a generic excitation. The resulting many-body states are characterized by one or several particles excited above the Fermi surface accompanied by no disturbance below it. These excitations carry charge which is integer for noninteracting electron gas and fractional for Luttinger liquid. The operator algebra describing these excitations is derived, and a method of their detection which relies on noise measurement is proposed. Controlling single electrons is one of the main avenues of research in nanoelectronics. Once advanced far enough, it will bring about a range of quantum-coherent singleparticle sources with full control over the orbital and spin degrees of freedom. Currently efforts are mostly focused on employing localized electron states, trapped on metal islands [1] or quantum dots [2] and shuttled between the dots or islands by electric pulses [1,2] or acoustic fields [3]. It is of interest, however, to extend the concept of singleparticle sources to the situation when electrons propagate freely as part of a degenerate Fermi system. If proved feasible, it would allow one to harness particle dynamics, characterized by high Fermi velocity, v F 10 8 cm=s, to transmit quantum states in a solid and, at low temperature, to use Fermi-Dirac statistics for generating many-particle entangled states [4].In this Letter we propose a scheme which allows the creation of ''clean'' electric current pulses, free of particlehole excitations. We consider a 1d electron gas, serving as a prototype for carbon nanotube, quantum wire, and point contact systems, in which current is driven by voltage pulses with a typical frequency small compared to the Fermi energy. In this quasistationary regime the electric response is described as It g 0 Vt with g 0 e 2 =h the Landauer conductance. A current pulse, which carries total charge q g 0 R Vtdt, is a collective many-body state involving a number of fermions excited to a higher energy [5]. Microscopically, such a current pulse is described by a number of particle-hole excitations, with energies of the order @=, where is the duration of the pulse. As discussed in Refs. [5][6][7] and below, these excitations can be probed by noise measurement [8].Here we show that, quite strikingly, by engineering the pulse profile one can inhibit the particle-hole excitations. We analyze the particle-hole content of current pulses in a single-channel conductor, and pose and solve the problem of minimizing the number of such excitations. The condition required for the excitation number to be small is area quantization, R Vdt nh=e, where n is an integer. We show that such pulses, carrying integer charge q ne, are accompanied by fewer excitations than noninteger pulses. Also, we show how to optimize the Vt profile, by designing pulses which are totally free...

We study spontaneous quantum coherence in an out of an equilibrium system, coupled to multiple baths describing pumping and decay. For a range of parameters describing coupling to, and occupation of the baths, a stable steady-state condensed solution exists. The presence of pumping and decay significantly modifies the spectra of phase fluctuations, leading to correlation functions that differ both from an isolated condensate and from a laser.

In order to model realistic quantum devices it is necessary to simulate quantum systems strongly coupled to their environment. To date, most understanding of open quantum systems is restricted either to weak system–bath couplings or to special cases where specific numerical techniques become effective. Here we present a general and yet exact numerical approach that efficiently describes the time evolution of a quantum system coupled to a non-Markovian harmonic environment. Our method relies on expressing the system state and its propagator as a matrix product state and operator, respectively, and using a singular value decomposition to compress the description of the state as time evolves. We demonstrate the power and flexibility of our approach by numerically identifying the localisation transition of the Ohmic spin-boson model, and considering a model with widely separated environmental timescales arising for a pair of spins embedded in a common environment.

Motivated by experiments observing self-organization of cold atoms in optical cavities we investigate the collective dynamics of the associated nonequilibrium Dicke model. The model displays a rich semiclassical phase diagram of long time attractors including distinct superradiant fixed points, bistable and multistable coexistence phases and regimes of persistent oscillations. We explore the intrinsic timescales for reaching these asymptotic states and discuss the implications for finite duration experiments. On the basis of a semiclassical analysis of the effective Dicke model we find that sweep measurements over 200ms may be required in order to access the asymptotic regime. We briefly comment on the corrections that may arise due to quantum fluctuations and states outside of the effective two-level Dicke model description.

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. This progress report offers 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, it explains the fundamental difference between the superradiant phase transition and the more common lasing transition. This report mostly focuses on the steady states of atoms in single‐mode optical cavities, but it also mentions 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 nonequilibrium systems.

In this work we introduce boundary time crystals. Here continuous time-translation symmetry breaking occurs only in a macroscopic fraction of a many-body quantum system. After introducing their definition and properties, we analyze in detail a solvable model where an accurate scaling analysis can be performed. The existence of the boundary time crystals is intimately connected to the emergence of a time-periodic steady state in the thermodynamic limit of a many-body open quantum system. We also discuss connections to quantum synchronization.