Controllable Josephson junction for photon Bose-Einstein condensates

Josephson junctions are the basis for the most sensitive magnetic flux detectors, the definition of the unit volt by the Josephson voltage standard, and superconducting digital and quantum computing. They result from the coupling of two coherent quantum states, as they occur in superconductors, superfluids, atomic Bose Einstein Condensates (BECs), and exciton-polariton condensates. In their ground state, Josephson junctions are characterized by an intrinsic phase jump. Controlling this phase jump is fundamental for applications in computing. Here, we experimentally demonstrate controllable phase relations between photon BECs resulting from particle exchange in a thermo-optically tunable potential landscape. Our experiment realizes an optical analog of a controllable 0,$\pi$-Josephson junction. By connecting several junctions, we can study a reconfigurable 4-condensate system demonstrating the potential of our approach for analog spin-glass simulation. More generally, the combination of static and dynamic nanostructuring techniques introduced in our work offers a powerful platform for the implementation of adaptive optical systems for paraxial light in and outside of thermal equilibrium.

Figure 1

Controllable Josephson junction for photon BECs. (a) The experimental setup is based on a high-finesse dye microcavity, in which optical photons propagate paraxial to the optical axis ($z$ axis) and are repeatedly absorbed and reemitted by dye molecules. The latter leads to a thermalization and condensation of the photon gas at room temperature. The potential landscape for the two-dimensional photon gas is created by a combination of two experimental techniques: A static nanostructuring of one of the mirror surfaces and an in situ variation of the index of refraction of the optical medium using a thermo-responsive polymer (pNIPAM) and a heating laser. (b) Static confinement potential. A rectangular surface structure is created on one of the cavity mirrors, which restricts the flow of light to the confined area. This area is furthermore divided into three parts by introducing two shallower barriers; see the cross-section in the lower graph. The height map of the mirror was determined by Mirau interferometry. (c) The Josephson junction in our experiment consists of two photon BECs that are created in local potential minima at the two ends of the junction. By tunneling into the central region of the junction, a particle exchange is created that leads to in-phase or anti-phase relations between the condensates (upper two graphs). The acquired phase delay is controlled by the thermo-optically induced potential (lower graph).

Figure 2

Controllable phase relations. (a) Photon density in the microresonator plane for six different heating energies as determined by a camera capturing the light transmitted by one of the cavity mirrors. All density profiles can be assigned to symmetric wave functions (0-states) or antisymmetric wave functions ($\pi$-states). The numbers in the red boxes indicate the number of observed intensity maxima between the condensates. (b) Potential landscape in the microresonator. Starting from the measured photon density $\rho ={\left|\psi (x,y)\right|}^2$, we reconstruct the wave function of the photons $\psi \left(x\right)=\epsilon \left(x\right)\sqrt{\rho \left(x\right)}$, in which $\epsilon \left(x\right)$ switches between $+1$ and $-1$ at every node in the wave function. The potential $V$ follows via $V-\text{const}=-E_{\text{kin}}=({\hslash }^2/2m)(d^2\psi /d{x}^2){\psi }^{-1}$ (points). Our method breaks down at the nodes of the wave function. These regions (shaded areas) are excluded from the polynomial fit (line). The reconstructed potentials reveal both the static potential due to the nanostructured mirror and the dynamically induced potential due to the thermo-responsive polymer. (c) Interferometric images of the coupled condensate system. The transmitted light is guided through a Mach-Zehnder interferometer to test for the coherence of the two condensates. Data for the interferometric visibility $v=(I_{\max }-I_{\min })/(I_{\max }+I_{\min })$ are given in the inset. The estimated degree of first-order coherence, which is generally found to be close to unity, takes into account the (small) population imbalances between the condensates.

Figure 3

State-resolved statistics. (a) State-resolved statistical information on the operation of the junction. For every heating energy, 50 experimental runs with identical control parameters are measured and analyzed. The data indicate that the formation of a particular junction state, as defined by the number of observed intensity maxima between the condensates (red boxes), is deterministic for a significant fraction of the investigated heating energies. (b) Probability of finding the junction in a 0- or $\pi$-state as a function of the heating energy.

Figure 4

Higher-order optical circuit composed of four Josephson junctions. (a) Height map of the used cavity mirror. Optical pumping with nanosecond pulses is carried out at the four corners of the structure. (b) Photon density for three different coupling configurations as denoted by the labels 0 and $\pi$. By varying the spatial heat profile in the microcavity, the phase relations of the four condensates can be switched between in-phase and anti-phase. The top panel shows experimental results, while the bottom panel contains theoretical results derived from the numerical solution of a driven-dissipative Schrödinger equation (see Appendix pp2). The white arrows correspond to the phases of the photonic wave function at the position of the condensates and can be interpreted as the angular orientation of four XY spins. (c) Interferometric imaging of the four coupled condensates using a Mach-Zehnder interferometer. The top right condensate in one interferometer path is superimposed on the bottom left condensate in the other interferometer path. The observed interference fringes indicate coherence close to unity (see cross-section).

Figure 5

Proposed scheme for optical spin-glass simulation with photonic Josephson junctions. (a) Mapping between XY spin glasses and systems of coupled BECs. We propose to use photonic Josephson junctions to solve optimization problems, namely the ground-state problem in XY spin glasses and mathematical equivalent problems. The ground-state problem corresponds to finding the minimum of $H_{XY}=-{\sum }_{i>j}{J}_{ij}cos({\varphi }_i-{\varphi }_j)$ with optimization variables ${\varphi }_i\in [0,2\pi )$. A particular instance of such a problem is defined on a (random) 3-regular planar graph with antiferromagnetic couplings $J_{ij}=-1$. The problem is mapped onto the potential landscape in the microcavity system by introducing a triangular lattice potential, in which the lattice sites are connected by attractive step potentials. The depth of the steps is chosen to favor the formation of $\pi$-states. (b) Amplitude of the simulated light field $\left|\psi \right(x,y\left)\right|$. We have performed numerical simulations to investigate the condensation process in the given potential landscape. A two-dimensional stochastic Schrödinger equation with gain and frequency-dependent loss terms is numerically integrated using the Runge-Kutta method (see Appendix pp2). At the start of the simulation, the optical gain on the triangular lattice is set close to the condensation threshold and photons start to populate the cavity. The phases of the light field, indicated by the arrows, are adjusted to maximize the gain. (c) Total photon number and simulated energy as a function of time. The data comprise 300 numerically obtained stochastic time evolutions of the system. The relative frequency of the total photon number $N$ and the simulated energy $H_{XY}$, derived from the phases of the light field, are color coded. The photon number is normalized to the noise level $N_0$ in the stochastic Schrödinger equation (at zero gain). The coupled condensate system is capable of finding good approximative solutions of the given ground-state problem within a period as short as $20\text{ps}$.

Figure 6

Two coupled photon BECs with frequency-dependent gain-loss scheme. The latter follows the KS law for fluorescent media and is responsible for a thermalization process between the coupled condensate system and its environment.

Figure 7

Potential step model. The model assumes that two BECs exchange particles via an attractive potential step. Depending on the potential energy $V$ and gain ${\mathrm{\Gamma }}_{\uparrow }$ the system selects either 0- or $\pi$-state. We assume spatially symmetric and piecewise constant functions $V\left(x\right)$, ${\mathrm{\Gamma }}_{\uparrow }\left(x\right)$.

Figure 8

Complex energy difference $\mathrm{\Delta }E=E_0-E_{\pi }$ as a function of potential step depth $V_2$ for three parameter regimes. For positive $\text{Im}\left(\mathrm{\Delta }E\right)$, the system predominantly realizes the 0-state. For negative $\text{Im}\left(\mathrm{\Delta }E\right)$, the system chooses to be in the $\pi$-state. For all parameter sets, the gain periodically switches sign as the depth of the potential step is increased. For large gain gradients ${\mathrm{\Gamma }}_{\uparrow ,1}-{\mathrm{\Gamma }}_{\uparrow ,2}$ (left column) or high temperatures $T$ (right column), the system maximizes the spatial overlap with the high gain regions, as indicated by the quantity ${\overline{\mathrm{\Gamma }}}_{\uparrow }^0-{\overline{\mathrm{\Gamma }}}_{\uparrow }^{\pi }$ (see text). For small gradients or low temperatures (middle column), the system minimizes (the real part of) the energy. Further parameters: $m=6.5\times {10}^{-36}\text{kg}$, $a=9\mu \text{m}$, $b=7\mu \text{m}$.