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Dynamics of Phase Coherence Onset in Bose Condensates of Photons by Incoherent Phonon Emission

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Abstract

Recent experiments with photons equilibrating inside a dye medium in a cavity have raised the question of whether Bose condensation can occur in a system with only incoherent interaction with phonons in a bath but without particle-particle interaction. Analytical calculations analogous to those done for a system with particle-particle interactions indicate that a system of bosons interacting only with incoherent phonons can indeed undergo Bose condensation and furthermore can exhibit spontaneous amplification of quantum coherence. We review the basic theory for these calculations.

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Notes

  1. Deviations from this model, in which the number of photons is not conserved, have been considered in [4].

  2. We ignore the fact that the system is a quasi-2D system of finite size.

  3. M. Weitz, private communication. The effective interaction comes from heating of the dye by the phonon emission from the photons, which then can shift the index of refraction and therefore the cavity photon resonance energy. Because it involves heating, this is a very slow process, with frequency in the kHz range, well below the relevant optical frequencies.

  4. Just as in superconducting pairing, a coherent particle-particle interaction induced by exchange of virtual quantum phonons is less effective at high temperatures where real scattering from thermal phonons occurs. Here, however, since the particles are bosons, condensation and enphasing can still occur.

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Acknowledgements

This work has been supported by the National Science Foundation through grants DMR-1104383 and DMR-1004406.

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Correspondence to D. W. Snoke.

Appendix: Derivation of the Evolution Equations, and Numerical Solution

Appendix: Derivation of the Evolution Equations, and Numerical Solution

In this section we briefly summarize the method of Ref. [11] and the numerical methods used to get results like those shown in Fig. 1. For a review of experimental applications of this method, see Ref. [42].

The change in the average number of particles in state \(\vec{k}\) is given by

(A.1)

where \(\hat{V}(t) = e^{iH_{0}t/\hbar}\hat{V}e^{-iH_{0}t/\hbar}\) in the standard interaction notation. The exponential terms are expanded, and only terms up to second order are kept, which is valid for weak interactions.

For the two-body interaction (1), the lowest-order term of the expansion is

(A.2)

where U D and U E are the direct and exchange interaction terms, with + for bosons and—for fermions. If |ψ i 〉 is a Fock state, this term vanishes exactly; if the two creation operators restore the same two states that the destruction operators removed, then the operator has the form \(N_{k}N_{k_{1}}\) and is equal for both terms, which cancel. More generally, if |ψ i 〉 is not a Fock state, this term depends on the value of “off-diagonal” elements of the form \(\langle \hat{\rho}^{(2)}_{k,k_{3},k_{2},k_{1}}\rangle \equiv \langle \psi_{i} |a^{\dagger}_{k} a^{\dagger}_{k_{3}}a_{k_{2}}a_{k_{1}} | \psi_{i} \rangle\).

For the second-order term in the expansion, as in the standard derivation of Fermi’s golden rule (e.g., Ref. [28], Sect. 4.7), when the states are close enough together to be taken as a continuum, one finds terms of the form

$$ \biggl(\int_{0}^t dt' e^{i\omega t'} \biggr) \biggl(\int_0^t dt'' e^{-i\omega t''} \biggr)= \biggl \vert \frac{e^{i\omega t}-1}{\omega}\biggr \vert ^2 . $$
(A.3)

which becomes δ(ω)2πt in the limit of large t; i.e., this term is also linear in t. One can therefore pick t to be a small interval dt and divide, to get a rate \(d\langle \hat{N}_{k}\rangle/dt\). For the two-body interaction (1), one obtains the second-order contribution

(A.4)

This is almost the standard quantum Boltzmann equation, except that in the standard quantum Boltzmann equation, the averages of products such as \(\langle\hat{N}_{k_{1}}\hat{N}_{k_{2}}\rangle\) etc. are factorized into products of averages \(\langle\hat{N}_{k_{1}}\rangle\langle\hat{N}_{k_{2}}\rangle\); i.e., one assumes no correlations between different \(\vec{k}\)-states, as in Ref. [21]. Ref. [11] showed that this factorization is valid in the case of no long-range correlations, i.e., before true BEC sets in. The results for phase evolution (8) and (10) also assume this type of factorization of 〈a 0〉 and 〈N k 〉, with the same justification.

In the case of a general state, we must compute the evolution of the “off-diagonal” terms which appear in (A.2). The lowest-order contribution to these factors is

$$ d\bigl\langle \hat{\rho}^{(2)}_{k,k',k'',k'''}\bigr\rangle = \frac{it}{\hbar}(E_k +E_{k'}- E_{k''}-E_{k'''}) \bigl\langle \hat{ \rho}^{(2)}_{{k,k',k'',k'''}} \bigr\rangle . $$
(A.5)

Thus, if there are nonzero \(\hat{\rho}^{(2)}\) terms, these will rotate in phase with angular frequency proportional to the degree of violation of energy conservation.

The next higher order term is

(A.6)

Off-diagonal correlations therefore do in general accumulate due to the interactions. However, the second-order contribution to the evolution of these factors, which is given in Eq. (4) in the text, suppresses the amplitude of these terms so that they stay small, except in the case of large occupation number of bosons, e.g., when BEC occurs, so that we again have a breakdown of the assumptions of the quantum Boltzmann equation right at the point of condensation.

Numerical method

As discussed above, a full evolution equation of the form (A.4) can be simplified by factorizing the averages of products into products of averages. The quantum Boltzmann equations can be reduced to a very tractable form for numerics by the additional assumption of homogeneity in k-space. This can be justified either by the assumption of ergodicity (very fast filling of all equal-energy states) or simply by the assumption that the initial state of the system is spatially homogeneous. In this case, the angles of the momentum vectors relative to each other can be integrated over analytically, leaving an integral only over the energies of the particles.

A continuous function N(E) is then defined for the occupation number of the particles, which can deviate from the equilibrium distribution by any amount. This function is represented by a single array of numbers {N(E i )} on a grid of discrete energies {E i }, and the rate of change for each energy E i is calculated using the quantum Boltzmann equation. The value of each N(E i ) is then updated for small dt according to

$$ N\bigl(E_i(t+dt)\bigr) \leftarrow N(E_i)+ \frac{dN(E_i(t))}{dt} dt. $$
(A.7)

The time step dt is picked to keep the total change of the distribution small during any given time step. With this algorithm, the full evolution of non-equilibrium systems to equilibrium can be determined very efficiently; in single-species systems the numerics can take just a few minutes on a personal computer. These simulations have been fit to data in several experiments [42].

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Snoke, D.W., Girvin, S.M. Dynamics of Phase Coherence Onset in Bose Condensates of Photons by Incoherent Phonon Emission. J Low Temp Phys 171, 1–12 (2013). https://doi.org/10.1007/s10909-012-0854-6

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