Bose-Einstein condensates of quasiparticles such as exciton-polaritons, magnons and massive photons allow for new experimental possibilities as compared to the atomic Bose-Einstein condensates. In this Thesis we focus on a condensate of photons, first created at Bonn University in 2010. One of the recent achievements is the measurement of number fluctuations in such a condensate of photons. We present a general theory to calculate these number fluctuations in a harmonically trapped interacting Bose gas and apply this to the available experimental results on the condensate of photons, finding good quantitative agreement. Additionally, we investigate the fundamental phenomenon of phase diffusion, which is based on the fact that a Bose-Einstein condensate can be described as a symmetry-broken phase. However, the symmetry-broken phase is only well defined in the thermodynamic limit, such that in a finite system the phase of the Bose-Einstein condensate can have nontrivial dynamics. We propose a new type of interference experiment involving a condensate of photons to measure this dynamical behavior of the phase. By calculating the effects of quantum and thermal fluctuations we find different time scales on which the interference pattern vanishes. Based on these time scales, we conclude that phase diffusion is experimentally observable within the precision of current devices.
Bose-Einstein condensates of quasiparticles such as exciton-polaritons, magnons and massive photons allow for new experimental possibilities as compared to the atomic Bose-Einstein condensates. In this Thesis we focus on a condensate of photons, first created at Bonn University in 2010. One of the recent achievements is the measurement of number fluctuations in such a condensate of photons. We present a general theory to calculate these number fluctuations in a harmonically trapped interacting Bose gas and apply this to the available experimental results on the condensate of photons, finding good quantitative agreement. Additionally, we investigate the fundamental phenomenon of phase diffusion, which is based on the fact that a Bose-Einstein condensate can be described as a symmetry-broken phase. However, the symmetry-broken phase is only well defined in the thermodynamic limit, such that in a finite system the phase of the Bose-Einstein condensate can have nontrivial dynamics. We propose a new type of interference experiment involving a condensate of photons to measure this dynamical behavior of the phase. By calculating the effects of quantum and thermal fluctuations we find different time scales on which the interference pattern vanishes. Based on these time scales, we conclude that phase diffusion is experimentally observable within the precision of current devices.
In this thesis we study Bose-Einstein condensation of photons. In Chapter 1 we describe the current experimental set-up and we explain the two important features of the Bose-Einstein condensate of light: dissipation through the interaction with the external reservoir of dye molecules and driving by the external pumping. In Chapter 2 we use the Schwinger-Keldysh formalism to describe the complete time evolution of the light. We show that the finite-lifetime effects of the photons due to the interaction with the dye can be characterized by a single dimensionless parameter, that depends on the power of the external laser pumping the dye. We also determine spectral functions and collective modes of the photon gas in both the normal and the Bose-Einstein condensed phases. In Chapter 3 we demonstrate that the finite-lifetime effects on the spatial first-order correlation functions of the condensed light are negligible, but are important for the temporal first-order correlations. Furthermore, we show that in the normal state the first-order correlations of the photons are suppressed by the interaction with the dye. In Chapter 4 we develop a model to describe the effect of interactions on the number fluctuations in Bose-Einstein condensates. We find good agreement between the observed number fluctuations in the light Bose-Einstein condensate and the results of our model for different interaction strenghts. In Chapter 5 we propose an experiment to observe phase diffusion in the Bose-Einstein condensate of photons and we predict the outcome of individual measurements. In our model we incorporate quantum and thermal fluctuations as well as the finite-lifetime effects. In Chapter 6 we investigate the light in the presence of a periodic potential and we study the transition from a superfluid to a Mott-insulator. We demonstrate that in this system the true Mott-insulating state is longer present due the interactions with the dye molecules. Finally, in Chapter 7 we consider Bose-Einstein condensation of light in nano-fabricated semiconductor microcavities. We demonstrate that in this system there is also a single dimensionless damping parameter that depends on the external pumping. We propose to use the experimental freedom to investigate the superfluidity of the photons via the excitation of scissors modes. We determine the frequency and damping of these scissors modes. We also calculate the density-density correlations of the excited light fluid and we show that this contains a signature of the dynamical Casimir effect.
In this review we discuss the dynamics of the phase of trapped Bose–Einstein condensates. In particular we consider the phenomena of phase decoherence (termed also as phase collapse, or diffusion), and phase revival in systems of interacting atoms. We analyse the dependence of the collapse and revival times on the trap potential, dimensionality of the gas, atom number fluctuations, and on the coherent dynamics of the condensate. We show that in a class of experimentally relevant systems, the collapse time is relatively short, and in some cases vanishes in the limit of a large number of atoms, implying that the trapped Bose gas cannot sustain a well-defined quantum phase, and that the phase memory is lost on a relatively short time scale. Furthermore, we calculate the relative atom number fluctuations or a model of two interacting condensates, and show that the fluctuations are generically sub-Poissonian.
We study phase diffusion in a Bose--Einstein condensate of light in a dye-filled optical microcavity, i.e., the spreading of the probability distribution for the condensate phase. To observe this phenomenon, we propose an interference experiment between the condensed photons and an external laser. We determine the average interference patterns, considering quantum and thermal fluctuations as well as dissipative effects due to the dye. Moreover, we show that a representative outcome of individual measurements can be obtained from a stochastic equation for the global phase of the condensate.
The advent of controlled experimental accessibility of Bose-Einstein condensates, as realized with e.g. cold atomic gases, exciton-polaritons, and more recently photons in a dye-filled optical microcavity, has paved the way for new studies and tests of a plethora of fundamental concepts in quantum physics. We here describe recent experiments studying a transition between laser-like dynamics and Bose-Einstein condensation of photons in the dye microcavity system. Further, measurements of the second-order coherence of the photon condensate are presented. In the condensed state we observe photon number fluctuations of order of the total particle number, as understood from effective particle exchange with the photo-excitable dye molecules. The observed intensity fluctuation properties give evidence for Bose-Einstein condensation occurring in the grand-canonical statistical ensemble regime.
This thesis investigates thermalisation, correlations and entanglement in Bose-Einstein condensates. Bose-Einstein condensates are ultra-cold collections of identical bosonic atoms which accumulate in a single quantum state, forming a mesoscopic quantum object. They are clean and controllable quantum many-body systems that permit an unprecedented degree of experimental flexibility compared to other physical systems. Further, a tractable microscopic theory exists which allows a direct and powerful comparison between theory and experiment, propelling the field of quantum atom optics forward at an incredible pace. Here we explore some of the fundamental frontiers of the field, examining how non-classical correlations and entanglement can be created and measured, as well as how non-classical effects can lead to the rapid heating of atom clouds. We first investigate correlations between two weakly coupled condensates, a system analogous to a superconducting Josephson junction. The ground state of this system contains non-classical number correlation arising from the repulsion between the atoms. Such states are of interest because they may lead to more precise measurement devices such as atomic gyroscopes. Unfortunately thermal fluctuations can destroy these correlations, and great care is needed to experimentally observe non-classical effects. We show that adiabatic evolution can drive the isolated quantum system out of thermal equilibrium and decrease thermal noise, in agreement with a recent experiment [Esteve et al. Nature 455, 1216 (2008)]. This technique may be valuable for observing and using quantum correlated states in the future. Next, we analyse the rapid heating that occurs when a condensate is placed in a moving periodic potential. The dynamical instability responsible for the heating was the subject of much uncertainty, which we suggest was due to the inability of the mean-field approximation to account for important spontaneous scattering processes. We show that a model including non-classical spontaneous scattering can describe dynamical instabilities correctly in each of the regimes where they have been observed, and in particular we compare our simulations to an experiment performed at the University of Otago deep inside the spontaneous scattering regime. Finally, we proposed a method to create and detect entangled atomic wave-packets. Entangled atoms are interesting from a fundamental perspective, and may prove useful in future quantum information and precision measurement technologies. Entanglement is generated by interactions, such as atomic collisions in Bose-Einstein condensates. We analyse the type of entanglement generated via atomic collisions and introduce an abstract scheme for detecting entanglement and demonstrating the Einstein-Podolsky-Rosen paradox with ultra-cold atoms. We further this result by proposing an experiment where entangled wave-packets are created and detected. The entanglement is generated by the pairwise scattering that causes the instabilities in moving periodic potentials mentioned above. By careful arrangement, the instability process can be controlled to to produce two well-defined atomic wave-packets. The presence of entanglement can be proven by applying a series of laser pulses to interfere the wave-packets and then measuring the output populations. Realising this experiment is feasible with current technology.
We review recent experiments on the Bose-Einstein condensation of photons in a dye-filled optical microresonator. The most well-known example of a photon gas, photons in blackbody radiation, does not show Bose-Einstein condensation. Instead of massively populating the cavity ground mode, photons vanish in the cavity walls when they are cooled down. The situation is different in an ultrashort optical cavity imprinting a low-frequency cutoff on the photon energy spectrum that is well above the thermal energy. The latter allows for a thermalization process in which both temperature and photon number can be tuned independently of each other or, correspondingly, for a nonvanishing photon chemical potential. We here describe experiments demonstrating the fluorescence-induced thermalization and Bose-Einstein condensation of a two-dimensional photon gas in the dye microcavity. Moreover, recent measurements on the photon statistics of the condensate, showing Bose-Einstein condensation in the grand-canonical ensemble limit, will be reviewed. Introduction Quantum statistical effects become relevant when a gas of particles is cooled, or its density is increased, to the point where the associated de Broglie wavepackets spatially overlap. For particles with integer spin (bosons), the phenomenon of Bose-Einstein condensation (BEC) then leads to macroscopic occupation of a single quantum state at finite temperatures [1]. Bose-Einstein condensation in the gaseous case was first achieved in 1995 by laser and subsequent evaporative cooling of a dilute cloud of alkali atoms [2, 3, 4], as detailed in Chapter 3 of this volume. The condensate atoms can be described by a macroscopic singleparticle wavefunction, similar to the case of liquid helium [1]. Bose-Einstein condensation has also been observed for exciton-polaritons, which are hybrid states of matter and light [5, 6, 7] (see Chapter 4), magnons [8] (see Chapter 25), and other physical systems. Other than material particles, photons usually do not show Bose-Einstein condensation [9]. In blackbody radiation, the most common Bose gas, photons at low temperature disappear, instead of condensing to a macroscopically occupied ground-state mode. In this system, photons have a vanishing chemical potential, meaning that the number of photons is determined by the available thermal energy and cannot be tuned independently from temperature.