Coherence measurements of polaritons in thermal equilibrium reveal a power law for two-dimensional condensates.

We have created a spatially homogeneous polariton condensate in thermal equilibrium, up to very high condensate fraction. Under these conditions, we have measured the coherence as a function of momentum and determined the total coherent fraction of this boson system from very low density up to density well above the condensation transition. These measurements reveal a consistent power law for the coherent fraction as a function of the total density over nearly three orders of its magnitude. The same power law is seen in numerical simulations solving the two-dimensional Gross-Pitaevskii equation for the equilibrium coherence.

Critical fluctuations in a confined driven-dissipative quantum condensate.

Phase fluctuations determine the low-energy properties of quantum condensates. However, at the condensation threshold, both density and phase fluctuations are relevant. While strong emphasis has been given to the investigation of phase fluctuations, which dominate the physics of the quantum system away from the critical point, number fluctuations have been much less explored even in thermal equilibrium. In this work, we report experimental observation and theoretical description of fluctuations in a circularly confined nonequilibrium Bose-Einstein condensate of polaritons near the condensation threshold. We observe critical fluctuations, which combine the number fluctuations of a single-mode condensate state and competition between different states. The latter is analogous to mode hopping in photon lasers. Our theoretical analysis indicates that this phenomenon is of a quantum character, while classical noise of the pump is not sufficient to explain the experiments. The manifestation of a critical quantum state competition unlocks possibilities for the study of condensate formation while linking to practical realizations in photonic lasers.

Observation of quantum depletion in a non-equilibrium exciton-polariton condensate.

Superfluidity, first discovered in liquid 4He, is closely related to Bose-Einstein condensation (BEC) phenomenon. However, even at zero temperature, a fraction of the quantum liquid is excited out of the condensate into higher momentum states via interaction-induced fluctuations-the phenomenon of quantum depletion. Quantum depletion of atomic BECs in thermal equilibrium is well understood theoretically but is difficult to measure. This measurement is even more challenging in driven-dissipative exciton-polariton condensates, since their non-equilibrium nature is predicted to suppress quantum depletion. Here, we observe quantum depletion of a high-density exciton-polariton condensate by detecting the spectral branch of elementary excitations populated by this process. Analysis of this excitation branch shows that quantum depletion of exciton-polariton condensates can closely follow or strongly deviate from the equilibrium Bogoliubov theory, depending on the exciton fraction in an exciton polariton. Our results reveal beyond mean-field effects of exciton-polariton interactions and call for a deeper understanding of the relationship between equilibrium and non-equilibrium BECs.

Erratum: Bose-Einstein Condensation of Long-Lifetime Polaritons in Thermal Equilibrium [Phys. Rev. Lett. 118, 016602 (2017)].

This corrects the article DOI: 10.1103/PhysRevLett.118.016602.

Ultra-low threshold polariton condensation.

We demonstrate the condensation of microcavity polaritons with a very sharp threshold occurring at a two orders of magnitude pump intensity lower than previous demonstrations of condensation. The long cavity lifetime and trapping and pumping geometries are crucial to the realization of this low threshold. Polariton condensation, or "polariton lasing" has long been proposed as a promising source of coherent light at a lower threshold than traditional lasing, and these results indicate some considerations for optimizing designs for lower thresholds.

Bose-Einstein Condensation of Long-Lifetime Polaritons in Thermal Equilibrium.

The experimental realization of Bose-Einstein condensation (BEC) with atoms and quasiparticles has triggered wide exploration of macroscopic quantum effects. Microcavity polaritons are of particular interest because quantum phenomena such as BEC and superfluidity can be observed at elevated temperatures. However, polariton lifetimes are typically too short to permit thermal equilibration. This has led to debate about whether polariton condensation is intrinsically a nonequilibrium effect. Here we report the first unambiguous observation of BEC of optically trapped polaritons in thermal equilibrium in a high-Q microcavity, evidenced by equilibrium Bose-Einstein distributions over broad ranges of polariton densities and bath temperatures. With thermal equilibrium established, we verify that polariton condensation is a phase transition with a well-defined density-temperature phase diagram. The measured phase boundary agrees well with the predictions of basic quantum gas theory.

A new type of half-quantum circulation in a macroscopic polariton spinor ring condensate.

We report the observation of coherent circulation in a macroscopic Bose-Einstein condensate of polaritons in a ring geometry. Because they are spinor condensates, half-quanta are allowed in where there is a phase rotation of π in connection with a polarization vector rotation of π around a closed path. This half-quantum behavior is clearly seen in the experimental observations of the polarization rotation around the ring. In our ring geometry, the half-quantum state that we see is one in which the handedness of the spin flips from one side of the ring to the other side in addition to the rotation of the linear polarization component; such a state is allowed in a ring geometry but will not occur in a simply connected geometry. This state is lower in energy than a half-quantum state with no change of the spin direction and corresponds to a superposition of two different elementary half-quantum states. The direction of circulation of the flow around the ring fluctuates randomly between clockwise and counterclockwise from one shot to the next; this fluctuation corresponds to spontaneous breaking of time-reversal symmetry in the system. This type of macroscopic polariton ring condensate allows for the possibility of direct control of the circulation to excite higher quantized states and the creation of Josephson junction tunneling barriers.

Enhanced coherence between condensates formed resonantly at different times.

We demonstrate coherence between exciton-polariton condensates created resonantly at different times. The coherence persists much longer than the individual particle dephasing time, and this persistence increases as the particle density increases. The observed coherence time exceeds that of the injecting laser pulse by more than an order of magnitude. We show that this significant coherence enhancement relies critically on the many-body particle interactions, as verified by its dependence on particle density, interaction strength, and bath temperature, whereas the mass of the particles plays no role in the condensation of resonantly injected polaritons. Furthermore, we observe a large nonlinear phase shift resulting from intra-condensate interaction energy. Our results provide a new approach for probing ultrafast dynamics of resonantly-created condensates and open new directions in the study of coherence in matter.

Dynamic Stark effect in strongly coupled microcavity exciton polaritons.

We present experimental observations of a nonresonant dynamic Stark shift in strongly coupled microcavity quantum well exciton polaritons--a system which provides a rich variety of solid-state collective phenomena. The Stark effect is demonstrated in a GaAs/AlGaAs system at 10 K by femtosecond pump-probe measurements, with the blueshift approaching the meV scale for a pump fluence of 2 mJ cm(-2) and 50 meV red detuning, in good agreement with theory. The energy level structure of the strongly coupled polariton Rabi doublet remains unaffected by the blueshift. The demonstrated effect should allow generation of ultrafast density-independent potentials and imprinting well-defined phase profiles on polariton condensates, providing a powerful tool for manipulation of these condensates, similar to dipole potentials in cold-atom systems.

Bose-Einstein condensation and superfluidity of trapped polaritons in graphene and quantum wells embedded in a microcavity.

The theory for spontaneous coherence of short-lived quasiparticles in two-dimensional excitonic systems is reviewed, in particular, quantum wells (QWs) and graphene layers (GLs) embedded in microcavities. Experiments with polaritons in an optical microcavity have already shown evidence of Bose-Einstein condensation (BEC) in the lowest quantum state in a harmonic trap. The theory of BEC and superfluidity of the microcavity excitonic polaritons in a harmonic potential trap is presented. Along the way, we determine a general method for defining the superfluid fraction in a two-dimensional trap, within the angular momentum representation. We discuss BEC of magnetoexcitonic polaritons (magnetopolaritons) in a QW and GL embedded in an optical microcavity in high magnetic field. It is shown that Rabi splitting in graphene is tunable by the external magnetic field B, while in a QW the Rabi splitting does not depend on the magnetic field in the strong B limit.

Simulating evolution by gene duplication of protein features that require multiple amino acid residues.

Gene duplication is thought to be a major source of evolutionary innovation because it allows one copy of a gene to mutate and explore genetic space while the other copy continues to fulfill the original function. Models of the process often implicitly assume that a single mutation to the duplicated gene can confer a new selectable property. Yet some protein features, such as disulfide bonds or ligand binding sites, require the participation of two or more amino acid residues, which could require several mutations. Here we model the evolution of such protein features by what we consider to be the conceptually simplest route-point mutation in duplicated genes. We show that for very large population sizes N, where at steady state in the absence of selection the population would be expected to contain one or more duplicated alleles coding for the feature, the time to fixation in the population hovers near the inverse of the point mutation rate, and varies sluggishly with the lambda(th) root of 1/N, where lambda is the number of nucleotide positions that must be mutated to produce the feature. At smaller population sizes, the time to fixation varies linearly with 1/N and exceeds the inverse of the point mutation rate. We conclude that, in general, to be fixed in 10(8) generations, the production of novel protein features that require the participation of two or more amino acid residues simply by multiple point mutations in duplicated genes would entail population sizes of no less than 10(9).