Photon-BEC in an optical microcavity a research training report by Tobias Rexin Supervisor: Priv.-Doz. Dr. Axel Pelster July 11, 2011 1 Introduction 1.1 Bose-Einstein-condensation of photons in optical microcavity Bose-Einstein condensation (BEC) is the macroscopic accumulation of bosonic particles in the energetic ground state level below a critical temperature Tcrit. This phenomenom has been demonstrated in several dierent physical systems as, for instance, dilute ultracold Bose gases such as sodium(1) or exitons in solid state matter(2), but for one of the most ob- vious Bose gases, namely blackbody radiation, it is yet unobserved. Blackbody radiation is electromagnetic radiation which is in thermal equilibrium with the cavity walls. In- stead of undergoing BEC the photons disappear in the cavity walls when the temperature 3 4 T is lowered corresponding to a vanishing chemical potential. Recent experiments( )( ) with a dye-lled optical micro resonator, performed by the group of Martin Weitz at the University of Bonn, achieved thermalization of photons in a number-conserving way. The curvature of the micro resonator provides two important ingredients which are prereq- uisites for BEC: a conning potential and a non-vanishing eective photon mass. This experiment gives new opportunities for creating coherent light. In contrast to the fty-year old laser, which operates far from thermal equilibrium, the photon BEC gains coherence by an equilibrium phase transition. Before this experiment one had several verications that massive particles behave like waves for example the interference of fullerene(5). But now the quantized electromagnetic waves in the cavity are given an eective mass, so here it is the other way around waves behave like massive particles. In the next sections I report about the details of the exper- imental setup and then I investigate theoretically those `particles` of light (6). First I give a short overview of the experimental setup and some important experimental facts. In the second part I am focusing on the calculation of the eective action, then I deduce the critical particle number Ncrit for both the non-interactioning and the interacting Bose gas. The interesting feature here is that we have to deal with an eective 2-dimensional Bose gas with a given temperature. Thus we can calculate all physical observables quantum mechanical exact and do not get any divergencies from the semiclassical approach and even nite size corrections are already included. 1.2 Experimental setup Photons are trapped in a curved optical micro resonator, where the curvature of the mirrors induce an eective harmonic trapping potential (see gure 1.1c). Outside the 2 Figure 1.1: a. Schematic spectrum of cavity modes with absorption coecient α(ν) and uorescence strength f(ν) b. Dispersion relation of pho- tons in the cavity (solid line) with xed longitudinal mode (q = 7) and the free photon dispersion (dashed line) c. Schematic experi- mental setup with trapping potential imposed by the curvd mirrors. center of the mirrors the distance d becomes shorter and the allowed wave vectors j~kj = 2π grow. Thus we conclude that the energy E = cj~kj of the photons to maintain dndye ~ in this region of the cavity is higher than in the center. The two mirrors are spaced D = 1:56 µm away from each other which is exactly 3.5 optical wavelengths. Indeed , if we look at gure 1.1a one can clearly see at quantum number q = 7, which corresponds a wave length λ = 585 nm, that there is a great overlap between the absorption coecient α(ν) and the uorescence strength f(ν) of the dye. This modies spontaneous emissions such that the emission of longitudinal mode with quantum number q = 7 dominates over all other emission processes. Due to the extremely short distances of the mirrors there 14 is an eective longitudinal low frequency cut-o !cut = ckz = 2π · 5:1 · 10 Hz where 2π λ kz = and D = 7 · , because no larger wavelength ts into the microcavity and ndyeD 2 of course the speed of light in vacuum c is modied when entering the dye solution by the refraction index . Furthermore the energy of the longitudinal mode ndye = 1:33 E = ~!cut is far above thermal energy at room-temperature . kz exp [−~!cut=(kBT )] ≈ exp[−80] The photon statistics, e.g. the photon number nph of a classical blackbody radiator, is determined by T 4 due to the Stefan-Boltzmann law whereas thermal excitations in the lowest available energy level inside the cavity are suppressed by the factor / ~!cut exp[−80]. The cavity photon number nph is almost not altered by the temperature T of the surrounding dye solution. This means that the thermalization process conserves the average photon number. Thermal equilibrium can be achieved by absorption and re- emission processes in the dye solution which is acting as a heat bath for photons. As can be seen from the gure 1.a the re-emission in the longitudinal mode kz dominates over all other modes. There is the largest overlap between emission coecient and uorescence strength. This means that most of the (q = 7)-photons absorbed by the dye will be re-emitted inside the cavity whereas for all other longitudinal modes there is a disbalance between absorption and emission. The kz-mode is frozen out and the kr-modes can thermalise due to the rovibrational energy levels of the dye solution. 3 1.3 Proving BEC and experimental results Figure 1.2: a. Spectral intensity distributions (connected circles) transmitted through one cavity mirror, as measured with a spectrometer for dif- ferent pump powers b. Images of the spatial radiation distribution below criticality (upper panel) and above criticality (lower panel). The BEC of the photons inside the dye-lled cavity has been proven experimentally by investigating both the spatial and temporal coherence. One has measured spectral distri- butions (see gure 1.2a) which show a classical thermal distribution at room temperature and an increased intensity for λ = 585 nm. The latter corresponds to the frozen longitudi- nal wave vector which yields 2π . So far this is a proof for BEC in the frequency kz λ = 3:5·k domain. The experimental setup also allowsz to check the spatial domain. Hence another evidence for the achieved BEC is the in-situ spot (see gure 1.2b) captured by the camera (see gure 1.1c). Below criticality there is only a thermal 'cloud' of the cavity photons but above criticality one nds a rather sharp yellow spot sitting in the center of the cavity in the minimum of the harmonic potential, which is induced by the curvature of the cavity mirrors. 1.4 Hamiltonian for photons in an optical micro cavity We start with the relativistic energy-wave vector relation for photons where we explicitly ~ decompose jkj into its longitudinal component kz and the radial symmetric component kr: ~ p 2 2 (1.1) E = ~cjkj = ~c kz + kr ; 4 2π where kz = is the longitudinal wave vector component with the mirror distance ndyed(r;D) D depending on the xed quantum number q = 7. In general the longitudinal wave vector component depends on the distance d(r) between the two curved mirrors with curvature . From a few geometrical considerations for this biconvex cavity one gets that R = 1 m p d(r) = D − 2(R − R2 − r2) where r is now the radius to the optical axis, so that for r = 0 we once again get the mirror distance D = 1:56 µm in the center of the cavity. First we can assume kr kz which is reasonable because kr corresponds to the room tem- perature thermalised tranversal modes which are whereas / (kBT )=(~c) kz / 80(pkBT )=(~c) due to the small resonator distance. The known binomic formula for small x :( 1 + x ≈ 1 + 1=2 · x) is applied to the square-root of the energy where we have explicitly pulled out the so 2 2 yields, kz(r) x = (kr) =kz 2 ~ p 2 2 (kr) (1.2) E = ~cjkj = ~c kz + kr ≈ ~ckz(r) + ~c : 2kz(r) Furthermore we have r R in our setup, i.e. r / µm R = 1 m and the Taylor expansion in kz at kz(r = 0) gives 2π 1 1 2 3 (1.3) kz ≈ + 2 r + O(r ) : ndyed(r) D RD Inserting (1.3) into (1.2) then yields (~k )2 ( k )2 1 r 2 ~ r 2 2 (1.4) E ≈ ~ckz(r) + ~c ≈ mphc + + mph! r 2kz(r) 2mph 2 with eective photon mass ~!cut ~kz(0) −36 which is about the mph = c2 = c = 6:7 · 10 kg magnitude 1010 smaller than the usual atomic masses. Furthermore this approximation gives an eective trap frequency pc 10 which about a factor 8 ! = DR = 2π · 4:1 · 10 Hz 10 higher than the usual atomic BEC trap frequencies.2 Thus the system in an optical microcavity is equivalent to a non-relativistic 2-dimensional Bose gas. The dispersion relation is quadratic for small transversal wave numbers as is illustrated in the dispersion relation in gure 1.1 b. Note that for kr = 0 we still have an `eective rest mass` intimidly related to the frozen kz-mode, which appears in zeroth order of the kz(r)-expansion. 5 2 Theoretical description of Photon-BEC 2.1 Ultracold vs room temperature Bose gas The determination of the phase boundary between the gas and the BEC phase is of funda- mental interest. In the case of ultracold Bose gases one considers the critical temperature Tcrit(N) as a function of the particle number N, this time we have a given temperature T , namely the room temperature T = 300 K, and then calculate the critical particle number for the onset of Bose-Einstein condensation Ncrit(T ).