Full counting statistics of photons emitted by a double quantum dot

We analyze the full counting statistics of photons emitted by a double quantum dot (DQD) coupled to a high-quality microwave resonator by electric dipole interaction. We show that at the resonant condition between the energy splitting of the DQD and the photon energy in the resonator, photon statistics exhibits both a sub-Poissonian distribution and antibunching. In the ideal case, when the system decoherence stems only from photodetection, the photon noise is reduced below one-half of the noise for the Poisson distribution and is consistent with current noise. The photon distribution remains sub-Poissonian even at moderate decoherence in the DQD. We demonstrate that Josephson-junction-based photomultipliers can be used to experimentally assess the statistics of emitted photons.

Figure 1

(a) An illustration of a DQD and a resonator (a $\lambda /2$-transmission line) coupled to a photon counter (PC). (b) In the DQD, electrons are confined to the left ($L$) and right ($R$) dots by barrier gates BL, BM, and BR, which also control electron tunneling rates between the source, $S$, and the left dot, the left and right dots, and the right dot and the drain, $D$, respectively. Electrostatic energies of two quantum dots are defined by the plunger gates, PL and PR, and the PL gate is also connected to an antinode of the transmutation line. (c) Electronic eigenstates of the DQD and tunneling from the left lead to the ground/excited state with rate ${\Gamma }_{L,g/e}$, and from the ground/excited state to the right lead with rate ${\Gamma }_{g/e,R}$.

Figure 2

Dependences of (a) the electric current $I$ through the DQD, (b) the average photon number $\overline{N}$ in the resonator, (c) the electron Fano factor $F_{\mathrm{el}}$, and (d) the Fano factor $F_{\mathrm{ph}}$ for emitted photons are shown as functions of the electrostatic bias $\hslash \varepsilon$ near the resonance at ${\varepsilon }_0=-\sqrt{{\omega }_0^2-4{\mathcal{T}}^2}$. Solid (dashed) lines represent the case for ${\gamma }_r=0$ (${\gamma }_r=2\Gamma$). Other system parameters are ${\Gamma }_{l,r}=\Gamma$, ${\omega }_0=800\Gamma$, $\mathcal{T}=200\Gamma$, $g_0=5\Gamma$, $\kappa =2\Gamma$, and ${\gamma }_{\varphi }=0$. The dotted line in panel (a) refers to elastic electric current through a noninteracting DQD.

Figure 3

(a) The second-order correlation function $g_{\mathrm{ph}}^2\left(2\right)\left(\tau \right)$ for photons as a function of time $\tau$ at resonant condition $\Omega ={\omega }_0$ for different values of energy, ${\gamma }_r$, and phase, ${\gamma }_{\varphi }$, relaxation rates. (b) The photon Fano factor $F_{\mathrm{ph}}$ as a function of relaxation rate $\gamma$ in three cases ${\gamma }_r={\gamma }_{\varphi }=\gamma$ (solid line), ${\gamma }_r=\gamma$ and ${\gamma }_{\varphi }=0$ (dotted line), and ${\gamma }_{\varphi }=\gamma$ and ${\gamma }_r=0$ (dashed line). (c) $F_{\mathrm{ph}}$ as a function of photon detection rate $\kappa$ shows a flat behavior for $\kappa \gtrsim \Gamma$. Other system parameters in both panels are ${\Gamma }_{l,r}=\Gamma$, ${\omega }_0=800\Gamma$, $\mathcal{T}=200\Gamma$, $g_0=5\Gamma$, $\kappa =2\Gamma$, and ${\gamma }_{\varphi }=0$ [for (a) and (b)].

Figure 4

Probabilities $P_n$ to have $n$ emitted photons during time $t$ at resonance $\Omega ={\omega }_0$ for (a) an ideal DQD without decoherence of electronic states, ${\gamma }_r={\gamma }_{\varphi }=0$ and $t=75/\Gamma$; (b) a DQD with inelastic relaxation, ${\gamma }_r=2\Gamma$, ${\gamma }_{\varphi }=0$, and $t=145/\Gamma$. Other system parameters in both panels are ${\Gamma }_{l,r}=\Gamma$, ${\omega }_0=800\Gamma$, $\mathcal{T}=200\Gamma$, $g_0=5\Gamma$, and $\kappa =2\Gamma$. A thin curve in both panels represents the corresponding Poisson distribution $P_n^{\left(P\right)}=e^{-\overline{n}}{\overline{n}}^n/n!$ with $\overline{n}={\sum }_{n}n{P}_n$ equal to the average number of emitted photons.

Figure 5

Dependence of photon noise Fano factor on (a) photon detection rate $\kappa$ and (b) the bare coupling constant $g_0$. In panel (a), the bare coupling constant is fixed, $g_0=5\Gamma$, and in panel (b) the photon detection rate is fixed, $\kappa =10\Gamma$. Solid (dashed) lines represent results for numerical (analytical) calculations with the method described in Sec. 2c (Sec. 4). Other system parameters are ${\Gamma }_{l,r}=\Gamma$, ${\omega }_0=800\Gamma$, and $\mathcal{T}=200\Gamma$. Both plots indicate that the analytical results agree with the numerical results in the limit $\kappa \gtrsim g_0,\Gamma$.

Figure 6

Comparison of the Fano factor measured by an ideal photon detector and JPM with the same set of parameters of a DQD. Other additional parameters for JPM model are $g_{\mathrm{JPM}}=5\Gamma$, ${\omega }_{\mathrm{JPM}}={\omega }_0$, ${\kappa }_0=0.6\Gamma$, ${\gamma }_{\mathrm{cap}}=1.5\Gamma$, and ${\gamma }_{\mathrm{t}}=1.5\Gamma$. When we introduce the energy relaxation rate ${\gamma }_d=1.5\Gamma$ for the JPM, the resultant Fano factor (green lines) increase toward unity.

Figure 7

Probabilities $P_n$ to have $n$ JJ recordings during time $t$ at resonance $\Omega ={\omega }_0$ for (a) an ideal DQD without decoherence of electronic states, ${\gamma }_r={\gamma }_{\varphi }=0$ and $t=132/\Gamma$; (b) a DQD with inelastic relaxation, ${\gamma }_r=2\Gamma$, ${\gamma }_{\varphi }=0$, and $t=293.3/\Gamma$. Other system parameters in both panels are ${\Gamma }_{l,r}=\Gamma$, ${\omega }_0=800\Gamma$, $\mathcal{T}=200\Gamma$, $g_0=5\Gamma$, and $\kappa =2\Gamma$. A thin curve in both panels represents the corresponding Poisson distribution $P_n^{\left(P\right)}=e^{-\overline{n}}{\overline{n}}^n/n!$ with $\overline{n}={\sum }_{n}n{P}_n$ equal to the average number of emitted photons.