Synchronization of micromasers

We investigate synchronization effects in quantum self-sustained oscillators theoretically using the micromaser as a model system. We use the probability distribution for the relative phase as a tool for quantifying the emergence of preferred phases when two micromasers are coupled together. Using perturbation theory, we show that the behavior of the phase distribution is strongly dependent on exactly how the oscillators are coupled. In the quantum regime where photon occupation numbers are low we find that, although synchronization effects are rather weak, they are nevertheless significantly stronger than expected from a semiclassical description of the phase dynamics. We also compare the behavior of the phase distribution with the mutual information of the two oscillators and show that they can behave in rather different ways.

Figure 1

Schematic diagram of the coupled micromaser system. Each micromaser consists of an optical cavity that interacts with a flow of excited two-level atoms. If the cavities were separated by a partially reflecting mirror, photon tunneling would lead to a (coherent) coupling between them $\varepsilon$.

Figure 2

(a) Steady-state occupation number $\langle n\rangle /N$ and (b) associated Fano factor $F$ for an uncoupled micromaser as a function of $\mathrm{\Theta }=\varphi N^{1/2}$ and $N$. The inset in (a) shows the full number distribution $P_n$ for the case where $N=20$.

Figure 3

Comparison of the relative phase probability distributions calculated numerically for coherent (solid black line) and dissipative (dashed red line) coupling with $\varepsilon =0.1, N=5$, and $\mathrm{\Theta }=\varphi N^{1/2}=2$. The inset shows the strengths of the peaks in the relative phase distributions, characterized by $S$, as a function of the coupling; the dotted lines are from the perturbation theory described in Sec. 4.

Figure 4

(a) Strength of the relative phase locking, measured by $S=2\pi \left[P{\left({\phi }_{-}\right)}_{\max }\right]-1$, as a function of $\mathrm{\Theta }=\varphi N^{1/2}$ for dissipative coupling with $N=5$ and $\varepsilon =0.01$, 0.05, and 0.1. In each case the results of full numerical calculations are shown as solid lines and the results from a semiclassical calculation using the Fokker-Planck equation are dotted lines. The inset shows the relative difference between the quantum and semiclassical calculations for the same parameters. (Note that the small peak around $\mathrm{\Theta }=5$ corresponds to the $n=1$ trapping state.) (b) Behavior of the average photon number for the same parameters compared with the uncoupled case $\varepsilon =0$ (dotted line).

Figure 5

Relative difference between the quantum (perturbation theory) and semiclassical calculations of the strength of the phase locking $S$ for different values of $N$ with $\varepsilon =0.0001$.

Figure 6

Mutual information as a function of the coupling strength for coherent (solid black line) and dissipative (dashed red line) couplings. Here $N=5$ and $\varphi N^{1/2}=2$.

Figure 7

Entanglement for the coherent (solid black line) and dissipative (dashed red line) couplings for $N=5$. (a) Logarithmic negativity for $\varepsilon =0.5$ over a range of $\mathrm{\Theta }=\varphi N^{1/2}$. The dotted lines indicate the locations of the trapping states ($n=1,2,3,4,5$). (b) Logarithmic negativity as a function of the coupling strength varied around the trapping state at $\mathrm{\Theta }=4.97$.