Nonequilibrium phase transition in a single-electron micromaser

Phase transitions occur in a wide range of physical systems and are characterized by the abrupt change of a physical observable in response to the variation of an external control parameter. Phase transitions are not restricted to equilibrium situations but can also be found in nonequilibrium settings, both for classical and quantum mechanical systems. Here, we investigate a nonequilibrium phase transition in a single-electron micromaser consisting of a microwave cavity that is driven by the electron transport in a double quantum dot. For weak electron-photon couplings, only a tiny fraction of the transferred electrons are accompanied by the emission of photons into the cavity, which essentially remains empty. However, as the coupling is increased, many photons are suddenly emitted into the cavity. Employing ideas and concepts from full counting statistics and Lee-Yang theory, we analyze this nonequilibrium phase transition based on the dynamical zeros of the factorial moment generating function of the electronic charge transport statistics, and we find that the phase transition can be predicted from short-time measurements of the higher-order factorial cumulants. These results pave the way for further investigations of critical behavior in open quantum systems.

Figure 1

Single-electron micromaser. (a) The single-electron micromaser consists of a microwave cavity (dark gray) that is coupled with amplitude $g$ to two quantum dots (light green) and with rate $\kappa$ to a thermal bath (not shown). The average occupation number of the bath is $\nu$. The micromaser is pumped by electrons entering the quantum dots from a biased source electrode (blue, $S$) and leaving through any of the two grounded drains (blue, $L$ and $R$). (b) Illustration of the single-electron pumping cycle: (I) an electron tunnels into the left dot from $S$ at the rate $\mathrm{\Gamma }$; (II) the electron interacts coherently with the cavity, emitting (absorbing) a photon as it tunnels to the right (left) dot; (III) the electron leaves the double dot system through one of the drains (here, $R$) at the rate $\gamma \gg \mathrm{\Gamma }$ before the next electron enters from the source. The inset of panel (a) shows the average output current $I_R$ in drain $R$ and the noise $S_R$ (defined in the main text) as functions of the pump parameter $\alpha =g/\gamma \sqrt{2\mathrm{\Gamma }/\kappa }$ for different values of $\mathrm{\Gamma }$, with $\nu =0.1$. Around $\alpha \simeq 1$, a phase transition from a low-activity to a high-activity phase occurs for $\mathrm{\Gamma }\gg \kappa$.

Figure 2

Simulated time traces of the right drain current for (a) $\alpha =0.2$, (b) $\alpha =1$, and (c) $\alpha =3$. (d) The corresponding (unnormalized) probability distributions $P(m,t)$ obtained from ${10}^5$ simulations for each value of $\alpha$ and $\mathrm{\Gamma }t=5\times {10}^3$. For $\alpha =3$, the Gaussian distribution (dark green, solid) from Eq. (31) is plotted for comparison. The inset shows $P(m,t)$ for $\alpha =0.2$ together with the Poissonian distribution (dark blue, solid) obtained from Eq. (27). (e) Average drain current $I_R\left(s\right)={\langle m\rangle }_F\left(s\right)/t$ as a function of the counting variable $s$ and the pump parameter $\alpha$ obtained by exact diagonalization of the Liouvillian with $N=175$ cavity states. The white dots show the critical value of $s$ for different values around $\alpha =1$ determined with the cumulant method. The dashed lines shows the phase transition line for $\mathrm{\Gamma }\to \infty$. In all panels, $\nu =0.1$ and $\mathrm{\Gamma }/\kappa =250$.

Figure 3

Zeros of the factorial moment generating function. We show the absolute value of the factorial generating function in the complex plane of $s$ for (a) $\alpha =0.1$, (b) $\alpha =0.5,$ and (c) $\alpha =1$ with $N=20$ cavity states and $\nu =0.1, \mathrm{\Gamma }/\kappa =10,$ and $t\kappa =1$. The factorial moment generating function is plotted on a logarithmic scale, so that the zeros appear as white points.

Figure 4

Factorial cumulants and determination of poles. (a) Higher-order factorial cumulants $\langle \langle m^k\rangle \rangle$ of the drain current as functions of $t$ for $\alpha =0.1, \nu =0.1,$ and $\mathrm{\Gamma }/\kappa =250$. (b) The leading pole (closest to $s=0$) of the factorial moment generating function as a function of $t$, extracted from the factorial cumulants up to order $k=10$. (c) The leading pole for different times (blue) and the analytic convergence point in the infinite-time limit (red) in the complex plane of $s$. For the numerical calculations, $N=135$ cavity states were used. For comparison, the dashed lines in all three panels indicate the analytic results obtained from the factorial moment generating function in Eq. (21).

Figure 5

Factorial cumulants and determination of Lee-Yang zeros. (a) Factorial cumulants $\langle \langle m^k\rangle \rangle$ of the drain current as functions of $t$ for $\alpha =1, \nu =0.1,$ and $\mathrm{\Gamma }/\kappa =250$. (b) The real (blue) and imaginary (green) parts of the leading zeros (closest to $s=0$) of the factorial moment generating function as a function of $t$, extracted from the cumulants up to tenth order. (c) The leading zeros for different times (blue) and the estimated convergence point (red) shown in the complex plane of $s$. For the numerical calculations, $N=135$ cavity states were used.

Figure 6

Extrapolation of the Lee-Yang zeros. (a) The real (blue) and imaginary (green) parts of the convergence points as a function of $\kappa /\mathrm{\Gamma }$ with $\alpha =1$ and $\nu =0.1$. (b, c) The real (blue) and imaginary (green) parts of the convergence points as a function of $\alpha$ with $\mathrm{\Gamma }/\kappa =250$ and $\nu =0.1$. The real part for $\mathrm{\Gamma }/\kappa =250$ is indicated in the phase diagram in Fig. 2.