Chemical potential for light by parametric coupling

Usually photons are not conserved in their interaction with matter. Consequently, for the thermodynamics of photons, while we have a concept of temperature for energy conservation, there is no equivalent chemical potential for particle number conservation. However, the notion of a chemical potential is crucial in understanding a wide variety of single- and many-body effects, from transport in conductors and semiconductors to phase transitions in electronic and atomic systems. Here we show how a direct modification of the system-bath coupling via parametric oscillation creates an effective chemical potential for photons even in the thermodynamic limit. In particular, we show that the photonic system equilibrates to the temperature of the bath, with a tunable chemical potential that is set by the frequency of the parametric coupler. Specific implementations, using circuit-QED or optomechanics, are feasible using current technologies, and we show a detailed example demonstrating the emergence of Mott insulator–superfluid transition in a lattice of nonlinear oscillators. Our approach paves the way for quantum simulation, quantum sources, and even electronlike circuits with light.

Figure 1

Thermal bath with modes ${\widehat{b}}_j$ and response functions with a cutoff ${\nu }_c$ can be parametrically coupled to a higher frequency (optical) system with modes ${\widehat{a}}_j$ near the frequency ${\nu }_p$. Additional loss via the high frequency bath can lead transport from the parametric bath through the system to the high frequency bath.

Figure 2

Coupled array of nonlinear microwave cavities provides a potential quantum simulator, where individual elements' parametric coupling to a bath provides chemical potential. The inset shows the bath coupler implementation suggested in the text using circuit QED. Specifically, a transmission line is coupled to the mode ${\mathrm{\Psi }}_X$ of the coupler. The system is connected to the mode ${\mathrm{\Psi }}_S$. The mode ${\mathrm{\Psi }}_Z=\lambda {\mathrm{\Phi }}_{0}cos{\omega }_{p}t$ is driven harmonically at frequency ${\omega }_p$ and provides the up- and down-conversion necessary for particle and hole exchange with the bath.

Figure 3

Energy eigenstates plotted as a function of energy and total photon number $N$ for a numerical solution of a four site Bose-Hubbard model (shown in the upper inset) with $(J,\mu ,\gamma ,\kappa )=(0.1,0.4,0.01,0.003)U$ and $\beta =1/10U$. The opacity of the blue dots represent the probability, in steady state, of being in the associated energy eigenstate. The lower inset shows the region near the ground state in the rotating frame; holelike excitations (lower $N$) are preferentially filled due to optical loss processes $\kappa$ only reducing particle number. The relatively high temperature leads to some thermal filling of the first particle excited state.

Figure 4

Numerical results for Mandel $Q$ (left) and coherence $\langle a\rangle \equiv \sqrt{|\langle a_i^{†}{a}_j\rangle |}$ (right) using the four-site (top) and a six-site (bottom) Bose-Hubbard model with periodic boundary conditions. We assume an Ohmic parametric bath and a flat high frequency (loss) bath. The Mandel $Q\approx -1$ regions (dark blue, left plot) are the Mott insulator states; at the same time, the finite coherence between sites on the right indicates the emergence of superfluid order (right plot) outside the Mott lobes. Finite size effects prevent observation of sharp transitions. Here ${\gamma }_0=0.07U$ and $\kappa ={\gamma }_0/30$. Overlaid are the critical values for finite occupation of particles and holes (solid black lines), with the dotted line for a higher value of $\kappa ={\gamma }_0/3$. The asymmetry of particles and holes arises due to preferential hole creation from optical loss.

Figure 5

Time-dependent simulation of a two level system with natural frequency ${\omega }_0$ interacting normally (red) and parametrically (green, yellow, blue) with a high-frequency filtered bath. For an initial excited state $\langle {\sigma }_z\rangle =1$, coupling to the bath leads to exponential decay when the oscillating frequency of the bath is set to zero, as shown in red. However, for an initial ground state $\langle {\sigma }_z\rangle =-1$, turning on the parametric coupling to the bath such that it oscillates at frequencies $\mu =\{0.9$ (blue), 1.0 (yellow), 1.1 (green)$\}{\omega }_0$ leads to inversion of the spin when $\mu >{\omega }_0$, as predicted by our more general theoretical model.