Lasing at half the Josephson frequency with exponentially long coherence times

We describe a superconducting device capable of producing laser light in the visible range at half the Josephson generation frequency, with the optical phase of the light locked to the superconducting phase difference. An earlier proposed device, the so-called “half-Josephson laser” [Phys. Rev. Lett. 107, 073901 (2011)], cannot provide long coherence times, because of spontaneous switchings between the emitter states. To circumvent this we consider $N\gg 1$ emitters driving an optical resonator mode. We derive a general model that captures essential physics of such devices while not depending on specific microscopic details. We find the conditions under which the coherence times are exponentially long, thus surpassing the fundamental limitation on the coherence times of common lasers. For this we study the noise in the device. In particular, we are interested in the rate of large fluctuations of the light field in the limit where the typical fluctuations are small. The large fluctuations are responsible for switching of the laser between stable states of radiation and therefore determine the coherence time.

Figure 1

Lasing in the device. The top plot shows the number of photons, $n$, for the stationary solutions of Eq. (12), as a function of the rescaled detuning $\chi$, with ${\Omega }^{\prime \prime }<0$. Thick solid (dashed) lines represent stable (unstable) solutions, labeled by $n_{\pm }$. Three parameter regimes (I, II, and III) are found based on the number of stationary solutions. The vertical dotted lines represent boundaries between these regimes. Below this plot is a panel that illustrates the existing stationary solutions in the $x$-$y$ plane for each regime. Solutions with $n>0$ occur twofold with opposite sign. Solid (empty) circles represent stable (unstable) stationary states. The bottom panel shows the dependence of the regimes on the parameters $\left|A\right|$ and $\chi$. The diagonal lines are the lasing thresholds ${\chi }_{\pm }$ as discussed in the main text. The line between regimes II and III is the one where $n_{\pm }$ becomes real valued. The plot of the upper figure is taken along the dashed line. When crossing the boundary along arrow(s) (i) and (ii) [(iii)], the system undergoes a second- [first-] order phase transition.

Figure 2

Noise and the noise spectrum in the HJL with many emitters. On the left-hand side is an illustration of fluctuations in the optical field, $b=x+iy$, where we neglect for clarity the detuning noise on the $z$ axis. The value of $b$ evolves along the lines. The dashed line, indicated by $A$, represents a large fluctuation, where $b$, starting in the vicinity of a stable lasing point, crosses the unstable point to the other stable lasing point. This process shall occur on exponentially long time scales, ${\tau }_{\mathrm{large}}$ (see main text). The solid erratic lines, indicated with $B$, represent small fluctuations in the vicinity of the stable points. These fluctuations typically occur on a time scale ${\tau }_{\mathrm{small}}$ (see main text). The two kinds of fluctuation contribute independently to the noise spectrum, of which an illustration is shown on the right-hand side. The sharp peak, indicated with $A$, corresponds to the large fluctuations. It is a Lorentzian-shaped peak with exponentially small width. The small fluctuations provide a broadband background, indicated by $B$. Details on the peaks can be found in the main text.

Figure 3

A stream plot corresponding to Eq. (9) for variables $x$ (horizontal) and $y$ (vertical), defined according to $b=x+iy$. The gray streamlines represent a “force field”. When a “particle” is placed in the force field it will evolve along these lines to one of the attractors, indicated by $A$ and $A$', depending on the initial condition. When placed at the thick red (dashed) or blue (solid) line, the particle will not evolve to one of the attractors but to the saddle point, indicated by $S$. These lines form the separatrix of the system that separates the domains of attraction of the attractors.

Figure 4

The value of the action of the optimal path for several parameter regimes. Fixed parameters are denoted in the graphs. Parameters were chosen such that we are far away from the lasing threshold where the use of optimal paths is justified. The main text contains a comparison with Eqs. (36, 37, 38, 39, 40) to estimate the values $F_{1–5}$. (a) The action as a function of $n\langle {\tilde{\chi }}^2\rangle$, for three values of $\gamma$. The axes are logarithmically scaled. For small values of $n\langle {\tilde{\chi }}^2\rangle$ the action is linear in photon number and independent of $\gamma$. This is the regime where intrinsic photon number fluctuations dominate. For large $n\langle {\tilde{\chi }}^2\rangle$ the detuning fluctuations dominate and the action saturates to a value depending on $\gamma$. (b) The action as a function of $\gamma$ for two values of $n\langle {\tilde{\chi }}^2\rangle$, with logarithmically scaled axes. For $\gamma \ll 1$ the $\langle {\tilde{\chi }}^2\rangle S$ is constant, while for $\gamma \gg A$ it is proportional to $\gamma$. In the region between we see a minimum at $\gamma \simeq A$. The curve for $n\langle {\tilde{\chi }}^2\rangle =25$ has points in common with the three curves of (a). (c) The action as a function of the detuning $\chi$, for three values of $n\langle {\tilde{\chi }}^2\rangle$.

Figure 5

Optimal paths corresponding to points in Fig. 4 for various values of $n\langle {\tilde{\chi }}^2\rangle$ and $\gamma$ as indicated above the columns and beside the rows, respectively. The other parameters are fixed at $A=20$ and $\chi =0$. In the bottom two rows each box shows an optimal path from two perspectives. Initial and final point are indicated by $I$ and $F$. Note that the scales of the axes are typically different in different boxes. The left column corresponds to the regime where detuning fluctuations dominate, the right column to the regime where intrinsic fluctuations dominate and the center column to the crossover.