Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Bose-Einstein condensation of ideal photons in a one-dimensional barrel cavity

Ze Cheng

Published Feb 22, 2016• 13 citations
Publisher

Abstract

Our experimental scheme is based on a barrel optical microresonator filled with a dye solution. The barrel mirror provides a confining potential, a chemical potential, and an effective mass for a photon, making the system formally equivalent to a one-dimensional gas of harmonically trapped, number-conserving, and massive bosons. Within the framework of quantum statistical mechanics, we propose an exact analytical solution to the problem of Bose-Einstein condensation in harmonically trapped, one-dimensional, and ideal photons. It is found that the photon number of vapor is characterized by an analytical function, which involves a $q$-digamma function in mathematics. The numerical calculation of the analytical solution gives many interesting results. In the thermodynamic limit, the analytical expressions of the critical temperature and the condensate fraction are derived. We find that the spectral radiance of a one-dimensional barrel cavity has a sharp peak at the frequency of the cavity cutoff when the photon number exceeds the critical value determined by a temperature.

DOI
10.1103/physreva.93.023829

Related papers

Statistical properties of ideal photons in a two-dimensional dye-filled spherical cap cavity

Ze 泽 Cheng 成

Abstract Within the framework of quantum statistical mechanics, we have proposed an exact analytical solution to the problem of Bose–Einstein condensation (BEC) of harmonically trapped two-dimensional (2D) ideal photons. We utilize this analytical solution to investigate the statistical properties of ideal photons in a 2D dye-filled spherical cap cavity. The results of numerical calculation of the analytical solution agree completely with the foregoing experimental results in the BEC of harmonically trapped 2D ideal photons. The analytical expressions of the critical temperature and the condensate fraction are derived in the thermodynamic limit. It is found that the 2D critical photon number is larger than the one-dimensional (1D) critical photon number by two orders of magnitude. The spectral radiance of a 2D spherical cap cavity has a sharp peak at the frequency of the cavity cutoff when the photon number exceeds the critical value determined by a temperature.

Chinese Physics B•Aug 8, 2024•0 citations

Statistical properties of ideal photons in a two-dimensional dye-filled spherical cap cavity

Ze Cheng

Abstract Within the framework of quantum statistical mechanics, we have proposed an exact analytical solution to the problem of Bose-Einstein condensation (BEC) of harmonically trapped, two-dimensional, and ideal photons. We utilize this analytical solution to investigate the statistical properties of ideal photons in a two-dimensional dye-filled spherical cap cavity. The results of numerical calculation of the analytical solution agree completely with the foregoing experimental results in the BEC of harmonically trapped, two-dimensional, and ideal photons. The analytical expressions of the critical temperature and the condensate fraction are derived in the thermodynamic limit. It is found that the two-dimensional critical photon number is larger than the one-dimensional critical photon number by two orders of magnitude. The spectral radiance of a two-dimensional spherical cap cavity has a sharp peak at the frequency of the cavity cutoff when the photon number exceeds the critical value determined by a temperature. PACS numbers: 42.50.-p, 03.75.Hh, 67.85.Hj

•Feb 10, 2023•0 citations

Exact analytical study of ideal Bose atoms in a one-dimensional harmonic trap

Ze Cheng

Within the framework of quantum statistical mechanics, we have proposed an analytical solution to the problem of Bose–Einstein condensation (BEC) in harmonically trapped, one-dimensional and ideal atoms. It is found that the number of atoms of vapor is characterized by an analytical function, which involves a q -digamma function in mathematics. The results of the numerical calculation of the analytical solution agree completely with the previous experimental results in the BEC of harmonically trapped, one-dimensional and ideal atoms. The analytical expressions of the critical temperature and the condensate fraction are derived in the thermodynamic limit.

Journal of Statistical Mechanics Theory and Experiment•Sep 16, 2015•8 citations

Exact analytical study of ideal Bose atoms in a two-dimensional isotropic harmonic trap

Ze Cheng

Abstract Within the framework of quantum statistical mechanics, we have proposed an analytical solution to the problem of Bose–Einstein condensation (BEC) of harmonically trapped, two-dimensional, and ideal atoms. It is found that the number of atoms in vapor is characterized by an analytical function, which involves a series of q -digamma functions in mathematics. The results of numerical calculation of the analytical solution agree completely with the foregoing experimental results in the BEC of harmonically trapped, two-dimensional, and ideal atoms. The analytical expressions of the critical temperature and the condensate fraction are derived in the thermodynamic limit. It is found that the two-dimensional critical temperature is smaller than the one-dimensional critical temperature by an order of magnitude or more.

Journal of Statistical Mechanics Theory and Experiment•Jun 1, 2017•5 citations

Photon Bose-Einstein Condensation in a Dye-Filled Microcavity

Karindra Perrier

Bose-Einstein condensation (BEC) is the transition to a macroscopic occupation of the ground state of a system of bosonic particles. In 1995 the first experimental realization of BEC was achieved by condensing a dilute atomic gas [1,2]. It was long thought only massive particles could undergo BEC. However, in 2010 it was shown that photon BEC was achieved in an optical dye-filled microcavity [3]. In 2015, we became the second group to successfully repeat these experiments.\n\nBEC is a phase transition and thus it occurs at thermal equilibrium. To achieve a thermal photon gas the cavity is filled with a fluorescent dye solution. The dye molecules thermalize the photons by means of repeated absorption-emission cycles. We optically pump the cavity. With the pump we determine the number of photons absorbed by the dye, consequently defining a nonvanishing chemical potential. This ensures a conserved photon number. The photons are trapped by the harmonic potential induced by the concave cavity mirrors. Due to the cutoff energy determined by the cavity length, the trapped photons have a dispersion similar to that of a massive particle in a quantum harmonic oscillator. The cavity length is tuned such that the free spectral range is larger than the bandwidth of the dye. Therefore, only one longitudinal mode fits inside the cavity, making the photon gas effectively two-dimensional. We end up with a thermalized two-dimensional photon gas that has the eigenenergies of a quantum harmonic oscillator. The Bose-Einstein distribution function predicts a critical photon number. Pumping the thermal photon gas beyond this critical number will lead to photon BEC.\n\nWe measure the spectral- and spatial distributions of our two-dimensional photon gas. In both distributions we observe photon BEC. Results show the characteristics of a phase transition: the appearance of a sharp peak on top of the thermal cloud that coincides with a converging chemical potential to zero. Both the spectral- and spatial distribution are in agreement with the theoretical distribution of photons with an effective mass in a quantum harmonic oscillator. Expansion of the condensate has been measured. This could indicate nonlinear optical effects such as thermal lensing or repulsive photon-photon interactions.

•May 31, 2016•0 citations

Exact spatial density of ideal Bose atoms in a one-dimensional harmonic trap

Ze Cheng

Abstract We have proposed an exact analytical solution to the problem of Bose–Einstein condensation (BEC) of harmonically trapped, one-dimensional, and ideal atoms. It is found that the number of atoms in vapor is characterized by an analytical function, which involves a q -digamma function in mathematics. We employ the q -digamma function to calculate the spatial density n ( z;T , N ) of ideal Bose atoms in a one-dimensional harmonic trap. The first main finding in this paper is that when Bose atoms are in the normal state, the density profile exhibits Friedel oscillations. The second main finding is that when Bose atoms are in the BEC state, the density profile exhibits a sharp peak with extremely narrow width. The third main finding is that the central peak of the spatial density is a monotonically increasing function of the number of atoms N but is a monotonically decreasing function of temperature T .

Journal of Statistical Mechanics Theory and Experiment•May 1, 2016•4 citations
Our experimental scheme is based on a barrel optical microresonator filled with a dye solution. The barrel mirror provides a confining potential, a chemical potential, and an effective mass for a photon, making the system formally equivalent to a one-dimensional gas of harmonically trapped, number-conserving, and massive bosons. Within the framework of quantum statistical mechanics, we propose an exact analytical solution to the problem of Bose-Einstein condensation in harmonically trapped, one-dimensional, and ideal photons. It is found that the photon number of vapor is characterized by an analytical function, which involves a qqq-digamma function in mathematics. The numerical calculation of the analytical solution gives many interesting results. In the thermodynamic limit, the analytical expressions of the critical temperature and the condensate fraction are derived. We find that the spectral radiance of a one-dimensional barrel cavity has a sharp peak at the frequency of the cavity cutoff when the photon number exceeds the critical value determined by a temperature.
Abstract Within the framework of quantum statistical mechanics, we have proposed an exact analytical solution to the problem of Bose–Einstein condensation (BEC) of harmonically trapped two-dimensional (2D) ideal photons. We utilize this analytical solution to investigate the statistical properties of ideal photons in a 2D dye-filled spherical cap cavity. The results of numerical calculation of the analytical solution agree completely with the foregoing experimental results in the BEC of harmonically trapped 2D ideal photons. The analytical expressions of the critical temperature and the condensate fraction are derived in the thermodynamic limit. It is found that the 2D critical photon number is larger than the one-dimensional (1D) critical photon number by two orders of magnitude. The spectral radiance of a 2D spherical cap cavity has a sharp peak at the frequency of the cavity cutoff when the photon number exceeds the critical value determined by a temperature.
Abstract Within the framework of quantum statistical mechanics, we have proposed an exact analytical solution to the problem of Bose-Einstein condensation (BEC) of harmonically trapped, two-dimensional, and ideal photons. We utilize this analytical solution to investigate the statistical properties of ideal photons in a two-dimensional dye-filled spherical cap cavity. The results of numerical calculation of the analytical solution agree completely with the foregoing experimental results in the BEC of harmonically trapped, two-dimensional, and ideal photons. The analytical expressions of the critical temperature and the condensate fraction are derived in the thermodynamic limit. It is found that the two-dimensional critical photon number is larger than the one-dimensional critical photon number by two orders of magnitude. The spectral radiance of a two-dimensional spherical cap cavity has a sharp peak at the frequency of the cavity cutoff when the photon number exceeds the critical value determined by a temperature. PACS numbers: 42.50.-p, 03.75.Hh, 67.85.Hj
Within the framework of quantum statistical mechanics, we have proposed an analytical solution to the problem of Bose–Einstein condensation (BEC) in harmonically trapped, one-dimensional and ideal atoms. It is found that the number of atoms of vapor is characterized by an analytical function, which involves a q -digamma function in mathematics. The results of the numerical calculation of the analytical solution agree completely with the previous experimental results in the BEC of harmonically trapped, one-dimensional and ideal atoms. The analytical expressions of the critical temperature and the condensate fraction are derived in the thermodynamic limit.
Abstract Within the framework of quantum statistical mechanics, we have proposed an analytical solution to the problem of Bose–Einstein condensation (BEC) of harmonically trapped, two-dimensional, and ideal atoms. It is found that the number of atoms in vapor is characterized by an analytical function, which involves a series of q -digamma functions in mathematics. The results of numerical calculation of the analytical solution agree completely with the foregoing experimental results in the BEC of harmonically trapped, two-dimensional, and ideal atoms. The analytical expressions of the critical temperature and the condensate fraction are derived in the thermodynamic limit. It is found that the two-dimensional critical temperature is smaller than the one-dimensional critical temperature by an order of magnitude or more.
Bose-Einstein condensation (BEC) is the transition to a macroscopic occupation of the ground state of a system of bosonic particles. In 1995 the first experimental realization of BEC was achieved by condensing a dilute atomic gas [1,2]. It was long thought only massive particles could undergo BEC. However, in 2010 it was shown that photon BEC was achieved in an optical dye-filled microcavity [3]. In 2015, we became the second group to successfully repeat these experiments.\n\nBEC is a phase transition and thus it occurs at thermal equilibrium. To achieve a thermal photon gas the cavity is filled with a fluorescent dye solution. The dye molecules thermalize the photons by means of repeated absorption-emission cycles. We optically pump the cavity. With the pump we determine the number of photons absorbed by the dye, consequently defining a nonvanishing chemical potential. This ensures a conserved photon number. The photons are trapped by the harmonic potential induced by the concave cavity mirrors. Due to the cutoff energy determined by the cavity length, the trapped photons have a dispersion similar to that of a massive particle in a quantum harmonic oscillator. The cavity length is tuned such that the free spectral range is larger than the bandwidth of the dye. Therefore, only one longitudinal mode fits inside the cavity, making the photon gas effectively two-dimensional. We end up with a thermalized two-dimensional photon gas that has the eigenenergies of a quantum harmonic oscillator. The Bose-Einstein distribution function predicts a critical photon number. Pumping the thermal photon gas beyond this critical number will lead to photon BEC.\n\nWe measure the spectral- and spatial distributions of our two-dimensional photon gas. In both distributions we observe photon BEC. Results show the characteristics of a phase transition: the appearance of a sharp peak on top of the thermal cloud that coincides with a converging chemical potential to zero. Both the spectral- and spatial distribution are in agreement with the theoretical distribution of photons with an effective mass in a quantum harmonic oscillator. Expansion of the condensate has been measured. This could indicate nonlinear optical effects such as thermal lensing or repulsive photon-photon interactions.
Abstract We have proposed an exact analytical solution to the problem of Bose–Einstein condensation (BEC) of harmonically trapped, one-dimensional, and ideal atoms. It is found that the number of atoms in vapor is characterized by an analytical function, which involves a q -digamma function in mathematics. We employ the q -digamma function to calculate the spatial density n ( z;T , N ) of ideal Bose atoms in a one-dimensional harmonic trap. The first main finding in this paper is that when Bose atoms are in the normal state, the density profile exhibits Friedel oscillations. The second main finding is that when Bose atoms are in the BEC state, the density profile exhibits a sharp peak with extremely narrow width. The third main finding is that the central peak of the spatial density is a monotonically increasing function of the number of atoms N but is a monotonically decreasing function of temperature T .