Temporal and spatiotemporal correlation functions for trapped Bose gases

Kohnen, M.; Nyman, Robert A.

doi:10.1103/physreva.91.033612

Cited by 3 publications

(7 citation statements)

References 33 publications

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“…We find the expected correlation function Eq. ( 3 ) for the case of the two-dimensional photon gas in a harmonic trap similar to earlier work 4 , 43 . Briefly, to find g (1) ( r ,− r ; τ ), we expand the electric field operators in eigenfunctions of the harmonic oscillator and assume that the photon gas is in thermal equilibrium at temperature T .…”

Section: Resultssupporting

confidence: 69%

First-order spatial coherence measurements in a thermalized two-dimensional photonic quantum gas

et al. 2017

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Phase transitions between different states of matter can profoundly modify the order in physical systems, with the emergence of ferromagnetic or topological order constituting important examples. Correlations allow the quantification of the degree of order and the classification of different phases. Here we report measurements of first-order spatial correlations in a harmonically trapped two-dimensional photon gas below, at and above the critical particle number for Bose–Einstein condensation, using interferometric measurements of the emission of a dye-filled optical microcavity. For the uncondensed gas, the transverse coherence decays on a length scale determined by the thermal de Broglie wavelength of the photons, which shows the expected scaling with temperature. At the onset of Bose–Einstein condensation, true long-range order emerges, and we observe quantum statistical effects as the thermal wave packets overlap. The excellent agreement with equilibrium Bose gas theory prompts microcavity photons as promising candidates for studies of critical scaling and universality in optical quantum gases.

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Section: Resultssupporting

confidence: 69%

First-order spatial coherence measurements in a thermalized two-dimensional photonic quantum gas

et al. 2017

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“…4 (middle), we compare experiments to a thermal equilibrium theory without dissipation (see Supplementary Material). The theory is based on a series expansion of the correlation function [42,43] which agrees with exact calculations [44]. There are no free parameters below threshold (solid lines), but the scaling of the horizontal axis is imprecise above threshold (shown as dashed lines), as the number of photons varies non-linearly with pump power [25].…”

mentioning

confidence: 53%

“…The theory of correlations of a non-interacting trapped Bose gas at thermal equilibrium in the absence of dissipation is well established [48,49], and can be extended from spatial to temporal correlations [50]. Our plano-spherical microcavity provides a symmetric, 2D, harmonic trapping potential for photons.…”

Section: Thermal Equilibrium Theorymentioning

confidence: 95%

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Spatiotemporal coherence of non-equilibrium multimode photon condensates

et al. 2016

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We report on the observation of quantum coherence of Bose-Einstein condensed photons in an optically pumped, dye-filled microcavity. We find that coherence is long-range in space and time above condensation threshold, but short-range below threshold, compatible with thermalequilibrium theory. Far above threshold, the condensate is no longer at thermal equilibrium and is fragmented over non-degenerate, spatially overlapping modes. A microscopic theory including cavity loss, molecular structure and relaxation shows that this multimode condensation is similar to multimode lasing induced by imperfect gain clamping. BEC means that interferometry is one of the most urgent measurements to be made with a condensate [4,5]. Where thermal equilibrium is not completely reached, coherence is the defining characteristic of non-BEC quantum condensation, e.g for semiconductor exciton-polaritons [6-9] and organic polaritons [10,11]. In nonideal Bose gases, such as ultracold atoms, interactions tend to reduce but not destroy the coherence [12][13][14].The long range coherence behaviour of two-dimensional (2D) microcavity condensates is currently an open question. The coherence of interacting, equilibrium 2D Bose gases decays with a power law at large distances. The exponent is no greater than 1/4, which is reached at the threshold for the Berezinskii-Kosterlitz-Thouless transition [15]. It is known that the equation of motion for the local phase of a non-equilibrium drivendissipative 2D condensate is in the universality class of the Kardar-Parisi-Zhang (KPZ) equation [16], and nonpower-law decays are possible. Since the long-range coherence only shows non-equilibrium behaviour for systems which are very large compared to interaction length scales (such as the healing length), it has proven difficult to observe the true long-range behaviour, mainly due to unavoidable pumping inhomogeneities [17].Photon condensates in dye-filled microcavities are weakly interacting [18][19][20][21], inhomogeneous [22, 23], dissipative Bose gases close to thermal equilibrium at room temperature [24][25][26][27][28]. It is worth noting that the physical system has some similarities to a dye laser, with the decisive difference being that lasing is necessarily a non-equilibrium effect whereas photons can also undergo BEC in thermal equilibrium. Consequently BEC implies macroscopic occupation of the ground state independently of the loss and gain properties, whereas a laser is characterised by a large occupation of exactly the mode that has the most gain [29].Unique among physical realisations of BEC, in dye-microcavity photon BEC the particles thermalise only with a bath and not directly among themselves. This implies that the establishment of phase coherence in the OPEN ACCESS RECEIVED

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“…The theory plotted is based on Ref. [27] which makes use of a decomposition of the photon fieldannihilation operator in a basis of the trap states. Taking a density operator which describes an equilibrium distribution at room temperature with energy spacings h×1.42 and 1.48 THz for the two axes, we then calculate the ex- When thermalisation through photon re-absorption is no faster than cavity loss, multiple modes condense (middle panel, λ0 = 563 nm).…”

Section: Coherencementioning

confidence: 99%

Driven-dissipative non-equilibrium Bose–Einstein condensation of less than ten photons

et al. 2018

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Coherence is a defining feature of quantum condensates. These condensates are inherently multimode phenomena and in the macroscopic limit it becomes extremely difficult to resolve populations of individual modes and the coherence between them. In this work we demonstrate non-equilibrium Bose-Einstein condensation (BEC) of photons in a sculpted dye-filled microcavity, where threshold is found for 8 ± 2 photons. With this nanocondensate we are able to measure occupancies and coherences of individual energy levels of the bosonic field. Coherence of individual modes generally increases with increasing photon number, but at the breakdown of thermal equilibrium we observe multimode-condensation phase transitions wherein coherence unexpectedly decreases with increasing population, suggesting that the photons show strong inter-mode phase or number correlations despite the absence of a direct nonlinearity. Experiments are well-matched to a detailed non-equilibrium model. We find that microlaser and Bose-Einstein statistics each describe complementary parts of our data and are limits of our model in appropriate regimes, which informs the debate on the differences between the two [1, 2].

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Temporal and spatiotemporal correlation functions for trapped Bose gases

Cited by 3 publications

References 33 publications

First-order spatial coherence measurements in a thermalized two-dimensional photonic quantum gas

First-order spatial coherence measurements in a thermalized two-dimensional photonic quantum gas

Spatiotemporal coherence of non-equilibrium multimode photon condensates

Driven-dissipative non-equilibrium Bose–Einstein condensation of less than ten photons

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