Abstract
The finitesize effect plays a key role in onedimensional Bose–Einstein condensation (BEC) of photons since such condensation cannot occur in the thermodynamic limit due to the linear dispersion relation of photons. However, since a divergence difficulty arises, the previous theoretical analysis of the finitesize effect often only gives the leadingorder contribution. In this article, by using an analytical continuation method to overcome the divergence difficulty, we give an analytical treatment for the finitesize effect in BEC. We show that the deviation between experiment and theory becomes much smaller by taking into account the nexttoleading correction.
1 Introduction
The Bose–Einstein condensation (BEC) of photons was generally believed to be impossible since the number of photons is not conserved and the extremely weak interaction between photons cannot thermalize the gas. However, the situation changed in recent years. By trapping photons in a dyefilled microcavity, the BEC has been realized in twodimensional systems [1,2, 3,4]. In these experiments, the photons are trapped between two curved mirrors. The fixed longitudinal momentum gives an effective mass to the photon and a nonvanishing chemical potential to the photon gas. The repeated absorbtion and emission cycle of the dye molecules thermalizes the photon gas. Recently, a onedimensional photon condensation is also reported [5]. In the experiment, the photons are confined in a closed Erbium–Ytterbium codoped fiber with a cutoff wavelength. The existence of the cutoff wavelength gives the photons a nonvanishing chemical potential.
In the experiments of the photon condensation, the finite particle number makes the behavior of the phase transition different from the thermodynamic limit case. In particular, the finitesize effect in the onedimensional condensation is of special interest since such a condensation cannot occur in thermodynamic limit due to the linear dispersion relation of photons. The finitesize effect has a significant influence in this case and needs to be carefully analyzed.
Many studies have been devoted to the finitesize effect in BEC. However, besides the numerical calculation method [6,7], the previous approximate methods can only give the leadingorder correction to the critical temperature and the condensate fraction [8,9,10]. The main obstacle for accurately studying the finitesize effect is the divergence problem: When taking into account the contribution from the discrete energy levels of the trapped particles accurately, most terms in the expressions of thermodynamic quantities become divergent at the transition point.
To overcome the divergence difficulty, we will use an analytical continuation method [11,12] to give an analytical treatment to the problem of photon condensation. In this way, we will obtain more accurate expressions of critical temperature and condensate fraction with nexttoleading corrections. We will also give the analytical expression of the chemical potential, which is hard to obtain before. Our result shows that the chemical potential is linear in temperature at low temperature, which is quite different from the thermodynamic limit case. The comparison with the numerical solution confirms our result.
In the experiment of onedimensional photon condensation [5], the deviation of the critical particle number between experiment and theory is about 5.6%. However, according to our result, this deviation is mainly caused by the inaccurate estimate of the finitesize effect in the previous studies. If the finitesize effect is correctly taken into account, the deviation between experiment and theory will reduce to about 1.4%, i.e., the agreement is actually very well.
This article is organized as follows. In Section 2, we give an analytical treatment to the finitesize effect of the photon condensation in one dimension. In Section 3, we compare our result with the experiment. Conclusions and some discussions are presented in Section 4.
2 Critical temperature and chemical potential
Consider photons in a onedimensional closed fiber with length
where
The photon in such a system has the same energy spectrum as that of nonrelativistic particles in a onedimensional harmonic trap, and hence, these two kinds of systems should show the same transition behavior.
As we know, the BEC occurs when the number of excited particles
where
where
In the thermodynamic limit,
On the other hand, in finite systems, the energy spectrum is discrete and the first excited energy is not 0, the summation in Eq. (3) should be convergent and a finite critical temperature can be obtained. In ref. [5], the summation is approximately converted to an integral similar to Eq. (4), but the lower limit of the integral is replaced by the first excited energy
In this treatment, the interval between the ground state and the first excited state is taken into account, but the higher levels are still regarded as continuous. In fact, many previous studies of BEC in finite systems along the similar line. The finitesize effect of the BEC in onedimensional harmonic trap is also discussed in refs. [8,9,10]. Although the treatments have some difference, they all depended on similar approximations and can only give the leadingorder correction similar to Eq. (6) (may differ by a factor).
Obviously, a more rigorous treatment of Eq. (3) is to perform the summation directly. To do this, one can Taylor expand every term in the summation as follows:
where
is the global heat kernel [13,14,15]. For small
with the coefficients
Substituting the heat kernel expansion (9) into Eq. (7), we have
A similar treatment can also apply to the grand potential and other thermodynamic quantities, and these quantities are also expressed as the series of the Bose–Einstein integrals. The higherorder correction terms can describe the influence of the boundary, the potential, or the topology, depending on the details of specific systems. This heat kernel expansion approach has been applied to various problems in statistical physics [13,16]. However, a serious difficulty arises when considering the problem of BEC phase transition. Due to the asymptotic form of the Bose–Einstein integral Eq. (5), every term in equation (11) is divergent at
First, substituting the leading term of the asymptotic expansion of each Bose–Einstein integral (5) into Eq. (11) gives
The summation in the second term can be represented by the heat kernel if the gamma function is replaced by the integral form
where we have introduced a small parameter
The summation in the last term differs from the heat kernel expansion (9) only by one term corresponding to
In the last step, the definition of heat kernel (8) has been employed to perform the integral, i.e.,
For
where
Now taking the limit
where the Euler constant
From Eq. (18), the critical particle number for a given temperature
The critical temperature for a fixed particle number
where
According to the asymptotic expansion of the Lambert function
We retain the second term in the denominator since for a relative small particle number, e.g.,
The condensate fraction is straightforward from Eq. (18),
This result is not very accurate, especially near the transition point. The reason is that the chemical potential is taken as 0 below the transition point in the aforementioned calculation, which is of course an approximation. As the temperature tends to the transition point, the deviation of the chemical potential from 0 becomes larger and larger. To describe the phase transition more accurately, we need to find the expression of the chemical potential.
The chemical potential
Here, the number of excited particles
where
An interesting feature of this result is that at low temperature
which is linearly related to the temperature. This is different from the thermodynamic limit result
which is exponentially small and leads to an unreasonable large particle number in the ground state at low temperature. In Figure 1, we compare the chemical potential in Eq. (26) with the thermodynamic limit result and the exact numerical solution, and it confirms the aforementioned lowtemperature behavior.
3 Comparison with the experiment
In Eq. (19), we present the critical particle number of BEC in the onedimensional photon system. Compared with the previous result [5] given in Eq. (6), the leadingorder term is the same, but our result also gives a new nexttoleading correction term. This nexttoleading correction leads to a relative deviation as follows:
It indicates that the previous treatment in which the excited states are regarded as continuous gives a lower prediction of the order of
The following are the relevant experimental parameters [5]. The length of the fiber
Our prediction of
In the experiment [5], the measured quantity is the pump power, which is proportional to the photon number, and the measurement result is
4 Conclusion and discussion
In this article, we give a more systematic and accurate discussion on the finitesize effect in onedimensional BEC of photons. By using an analytical continuous method based on the heat kernel expansion, we overcome the divergence difficulty and obtain the nexttoleading order finitesize corrections on thermodynamic quantities. In the experiment of onedimensional photon BEC [5], the measurement value of the critical particle number is about 5.6% higher than the previous theoretical prediction. However, our result shows that the most part of the deviation arise from the inaccurate analysis of the finitesize effect. When taking into account of the nexttoleading correction, the deviation between experiment and theory reduces to about 1.4%. Moreover, the chemical potential at the low temperature given by our approach is also consistent with the exact solution, while the thermodynamiclimit result is physically unreasonable since it may lead to a too large groundstate particle number.
The magnitude of the finitesize effect is closely related to the spatial dimension of the system. In fact, the most important factor determining the statistical properties is the density of states, and the density of states is strongly affected by the spatial dimension. In some specific systems, the density of states has different behavior in different energy scales, which may significantly affect the critical temperature of BEC [17]. When considering the finitesize effect in a twodimensional harmonic trap, the leading term of the finitesize correction to the critical temperature is of the order of
The method used in this article is based on the heat kernel expansion. We know that the heat kernel expansion is a shortwavelength (highenergy) expansion. In principle, it is only applicable to the hightemperature and lowdensity case. When applying the heat kernel expansion to the problem of phase transition, the divergence problem arises indeed. In this article, however, we show that with the help of the analytical continuation method, the application range of heat kernel expansion can be extended to below the transition point, and the thermodynamic quantities can also be obtained analytically.

Funding information: The authors state no funding involved.

Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

Conflict of interest: Authors state no conflict of interest.

Data availability statement: The data that support the finding of this study are available from the corresponding author upon request.
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