Thermodynamics and State Preparation in a Two-State System of Light

TL;DR

This work addresses the thermodynamics of photons populating a two-state system coupled to a heat bath, with a freely tunable chemical potential realized in a dye-filled optical microcavity. The two eigenmodes form a double-well–like two-state system with splitting $ΔE$, and the setup enables Josephson oscillations under pulsed pumping and thermalization under quasi-CW pumping. Thermalization is mediated by dye molecules exhibiting Kennard-Stepanov scaling, driving the photon gas toward a Bose–Einstein distribution with tunable μ. Under $ΔE ≪ k_B T$, low-$N$ populations are nearly equal, while for $N ≫ N_c$ the ground state dominates due to bosonic stimulation, with the experiment reaching about 93% ground-state occupancy and $N_c^{exp} ≈ 48$. The results agree with the elementary $N$-boson, two-state model and highlight potential for photonic state preparation in quantum technologies and explorations of quantum thermodynamics, including non-unitary beam-splitter behavior at high occupations.

Abstract

The coupling of two-level quantum systems to the thermal environment is a fundamental problem, with applications ranging from qubit state preparation to spin models. However, for the elementary problem of the thermodynamics of an ensemble of bosons populating a two-level system despite its conceptual simplicity experimental realizations are scarce. Using an optical dye microcavity platform, we thermalize photons in a two-mode system with tunable chemical potential, demonstrating N bosons populating a two-level system coupled to a heat bath. Under pulsed excitation, Josephson oscillations between the two quantum states demonstrate the possibility for coherent manipulation. In contrast, under stationary conditions the thermalization of the two-mode system is observed. As the energetic splitting between eigenstates is two orders of magnitude smaller than thermal energy, at low occupations an almost equal distribution of the modes occupation is observed, as expected from Boltzmann statistics. For larger occupation, we observe efficient population of the ground state and saturation of the upper level at high filling, expected from quantum statistics. Our experiment holds promise for state preparation in quantum technologies as well as for quantum thermodynamics studies.

Figures (4)

• Figure 1: (a) Photons are confined in a dye-solution filled optical microresonator consisting of one plane and one mirror with a laterally microstructured reflective surface profile. While trapped in the microresonator, photons thermalize with the dye by repeated absorption re-emission processes. (b) Height profile of the microstructured cavity mirror. (c) The black solid line gives a cut of the corresponding expected potential for cavity photons along the axis of the double well, realizing a two-level system with ground ($\ket{\mathrm{g}}$) and excited ($\ket{\mathrm{e}}$) states, see the dashed black lines for the (calculated) position of energy levels. The blue and red lines give the spatial variation of the probability densities of the two optical eigenstates along this axis respectively. (d) (Top panel) Tunneling dynamics of photons in a double-well structure where the oscillations at the used pump pulse duration can be resolved. (Bottom panel) The connected blue circles and red squares give the time dependence of the population signals in the upper and lower sites respectively at $x=\pm 7µm$ (dashed lines, top panel).
• Figure 2: The top panel gives the measured wavelength as a function of the transverse position along the axis of the double well ($y$-axis) for three different values of the photon number $N$ in the microcavity: (a) for $N \simeq 0.1 \cdot N_\mathrm{c,exp}$, (b) for $N \simeq N_\mathrm{c,exp}$, and (c) for $N \simeq 100 \cdot N_\mathrm{c,exp}$, where $N_\mathrm{c,exp}$ denotes the characteristic photon number. The two peaks visible on the right hand side at lower photon energy (higher wavelength) correspond to the signal from photons in the (symmetric) ground state, the peaks on the left hand side to the (antisymmetric) excited state of the two-level system. The bottom panels show spectra obtained by integrating the data along the $y$-axis. While at low photon numbers (a) the population distribution between sublevels is comparable, upon increasing the photon number (b,c) the relative population in the ground state, see the right hand side peak, clearly dominates. The scale of the color code in the top panels is normalized to the total photon number in each of the respective images (a-c).
• Figure 3: (a) Variation of the observed populations in upper (red squares) and lower (blue dots) states of the two-level system with total photon number. The solid lines are theory fits, accounting for a finite instrumental spectrometer resolution. The vertical dashed line gives the position of the characteristic photon number $N_\mathrm{c,exp}$, and the green solid line gives the chemical potential (see main text). (b) Corresponding variation of the measured relative photon populations in upper (red squares) and lower (blue dots) states, along with fits. The data shows that clearly below the characteristic photon number the populations in upper and lower levels are almost equally distributed, while at high photon numbers a significantly enhanced population in the lower energetic ground state is observed. Data points are the mean values of $\sim 150$ experimental realizations per binned photon number and error bars show the statistical standard deviations.
• Figure 4: Variation of the characteristic photon number $N_\mathrm{c,exp}$ (blue dots) on the low-frequency cutoff wavelength $\lambda_\mathrm{cutoff}$ of the microcavity. The data points are fitted with a constant term (see the blue line), which corresponds to expectations from thermodynamic theory. The red solid line shows the corresponding calculated dependence of the number of dye molecular excitations $M_\mathrm{e}$, in units of the total molecule number $M$, on the cutoff wavelength. Fitted values for the measured contrast of populations are $C$ = 0.912 ± 0.001, 0.921 ± 0.001, and 0.928 ± 0.001 for $\lambda_\mathrm{cutoff}$ = 578.2nm, 584.8nm, and 589.6nm respectively.