Comparison with Experiments

To examine how experiments A and B fit into the resonance scenario described above, in Fig.

3.5we present the results of our computations for experiment A withM = 4modes included, while for experiment B we tookM = 6modes into account. For the former, the agreement with the experiment is very good regarding both the frequency and the amplitude of the Josephson oscillations. The fraction of fluctuations remains below 10%, indicating that this experimental setup is away from the resonance discussed above. Note that there is no fitting of parameters involved.

For experiment B, we assume a small initial condensate occupation of the modesm = 3,4, as listed in Tab. 3.1, because of the small excitation energy of these modes with regard to the larger interaction parameters. Here, the agreement with experiment is quantitatively acceptable only for short times. At the point where the agreement starts to decrease, our results show a fast and efficient excitation of fluctuations, which indicates that this experimental setup is in the resonant regime.

Importantly, we find that the efficient creation of fluctuations for the parameters of experiment B is robust, independent of the small condensate occupation of the modes withm= 3,4as well as the precise value ofN. The reason for the reduced quantitative agreement with experiment can indeed be understood from the behavior of the fluctuation fraction. In the lower right panel of Fig.3.5, the departure of the theoretical results from the experimental data is significant for times when the non-condensate fractionδN(t)/N is large, which indicates that the HFB approximation employed in this Sec. is not sufficient and higher-order corrections should be included to account for the inelastic collisions of quasiparticles. Beyond-HFB diagrammatics will be of importance in Chp.5.

Second-Order Correlation Functions of Photon Bose-Einstein Condensates

Physical systems in thermal equilibrium are well understood. Natural systems, and in particular organic matter, however, are seldomly found in conditions of thermal equilibrium. Rather, there is energy goingintothe system via certain processes and energy goingout of the system via others.

This energy flow drives the system away from equilibrium. Earth is the most prominent example of such a non-equilibrium system: energy enters it in the form of visible light from the sun and is later on radiated away in the infrared. As a matter of fact, photon condensates are closely analogous:

light of a shorter wavelength excites the organic dye molecules and is converted into photons of longer wavelength before it leaves the cavity.

For systems in thermal equilibrium, the machinery of equilibrium statistical mechanics allows one to calculate straightforwardly, for instance, the spectral intensity distribution of a photon gas.

As we have seen in paragraph1.1.1, this results in a Bose-Einstein distribution. In non-equilibrium scenarios where Boltzmann’s considerations no longer apply (cf. the introduction to Sec.1.1), however, an equally general formalism does not exist. As we remarked in paragraph2.1.3, one consequence of this is that in non-equilibrium quantum field theory, the statistical function is no longer fixed to the Bose distribution function but becomes a degree of freedom on the same footing as the spectral function. To bridge the gap between well-understood equilibrium and still largely unexplored non-equilibrium phenomena, it seems reasonable to start by investigating a system as it is gradually driven away from equilibrium. In this respect, the photon BEC holds an ideal position because it can be operated both very close and further away from equilibrium. Though it is in fact never in a perfect thermal equilibrium state, it reliably undergoes Bose-Einstein condensation, which remains true even as the system is driven more strongly. As we will see in the following, the earliest signs of the non-equilibrium character of the driven-dissipative photon gas do not manifest in the spectral intensity distribution but in the time dependence of the intensity fluctuations. The energy flow through the system makes it non-Hermitian and induces a novel driven-dissipative transition in the second-order coherence of the photon condensate characterized by the appearance of an exceptional point. If only the static intensity spectrum was considered, this transition would go unnoticed, as would the fact that the system is not truly in equilibrium.

Chp.4is organized as follows. After introducing the non-equilibrium model of the photon condensate in Sec.4.1by deriving it from a microscopic Hamiltonian via a polaron transformation and the Born-Markov approximation, we describe the methods required to study second-order correlation functions in Sec.4.2and also present some illustrative solutions. Finally, in Sec.4.3 we apply the developed methods to understand the experimental data. Notice that parts of the work presented in this Chp. are taken from Ref. [36], of which the author of this thesis is a co-author.

4.1 Non-Equilibrium Model of the Photon BEC

As mentioned above, the photon BEC isnota closed system: the cavity mirrors are not perfect, for which reason the photons do not stay inside the cavity indefinitely, which in turn necessitates the introduction of an external optical pumping to supply excitations to the system. Furthermore, the quantum efficiency of the dye molecules only amounts to about 95%, which means that a fraction of the absorbed photons will not be re-emitted. For all of these reasons, the photon BEC should be modeled as anopen system. The cavity loss, the optical pumping, and the electronic loss due to imperfect quantum efficiency can be described within the previously encountered framework of themaster equation, as was done in Ref. [16] for the first time. A more detailed discussion of the same model may be found in [17].

The Franck-Condon principle was discussed in the introduction (cf. 1.3.1) as the physical foundation of the interaction between the photons and dye molecules. We understood in broad terms how the molecular vibrations alter the electronic transition frequency: insufficient or excess energy of an incoming photon may be compensated by destruction or creation of a corresponding amount of phonons. A two-level system with a similarly “fluctuating” transition frequency is described in Ref. [15] by means of the modified Jaynes-Cummings Hamiltonian

H_X =ω₀a^†a+g

a^†σ⁻+aσ⁺ + ∆

2σ^z+1

2X(t)σ^z, (4.1) whereX(t)is a fluctuating quantity. The model in Ref. [16] is now a physically motivated alteration of the Hamiltonian (4.1). Since the fluctuations of the dye-molecule transition are generated by the phonon coupling, a promising idea is to replaceX(t)by some function of the phonon operators.

The question then is of course: which function? The physics of the Franck-Condon principle provides the answer. As we learned above, the equilibrium positions of the constituent atoms in the dye molecules depend on the electronic state. When an electronic transition takes place, the atoms are suddenly in a non-equilibrium state and begin to vibrate. If one idealizes these molecular vibrations with the help of a single phonon mode per molecule with operatorsb_mandb^†_m, this means that the rest position of the phononic oscillator should be coupled to the electronic transition, which suggests the following Hamiltonian [16,143] for a collection ofM molecules and multiple cavity modes:

H=X

k

ω_ka^†_ka_k+

M

X

m=1

"

∆

2σ^z_m+ Ωb^†_mb_m+ Ω√ S σ^z_m

b_m+b^†_m

+gX

k

a_kσ_m⁺+a^†_kσ⁻_m

# . (4.2)

The cavity-photon modes with transverse dispersionω_kare represented by the bosonic operators a_kanda^†_k. The vibrational states of dye moleculemhave an oscillator frequencyΩ. The electronic two-level system of moleculemis described by the Pauli matrixσ^z_mand raising/lowering operators σ^±_m, the electronic transition frequency being∆. The Franck-Condon coupling is parametrized by the so-calledHuang-Rhys parameterS, where the phonon position operators arexˆ_m ∼(b_m+b^†_m).

Sinceσ^z_m = σ_m⁺σ_m⁻ −σ_m⁻σ⁺_m, the effect of this term will be a displacement of the phononic oscillator with the sign depending on whether the molecule is in the electronic ground or excited state. The effect of having several vibrational modes is discussed in [17].

For the rest of this Sec., we will follow [15–17,19] in developing the Hamiltonian (4.2) into a description of the open-system dynamics of the photon BEC. The first step is to make apolaron transformationto deal with the potentially strong Franck-Condon coupling parameterized byS.

After that, the second step is to derive a master equation by treating the phonons as a bath to which the system is weakly coupled via the optical transition parameterg.