Quantum efficiency of a single microwave photon detector based on a semiconductor double quantum dot

Motivated by recent interest in implementing circuit quantum electrodynamics with semiconducting quantum dots, we consider a double quantum dot (DQD) capacitively coupled to a superconducting resonator that is driven by the microwave field of a superconducting transmission line. We analyze the DQD current response using input-output theory and show that the resonator-coupled DQD is a sensitive microwave single photon detector. Using currently available experimental parameters of DQD-resonator coupling and dissipation, including the effects of $1/f$ charge noise and phonon noise, we determine the parameter regime for which incident photons are completely absorbed and near-unit $\gtrsim 98$% efficiency can be achieved. We show that this regime can be reached by using very high quality resonators with quality factor $Q\simeq {10}^5$.

Figure 1

Top left: A transmission line carries a continuous wave of microwave photons with flux $\dot{N}$ and frequency ${\omega }_{\mathrm{in}}$. Right dashed box: The photon detector is a single-port superconducting resonator with frequency ${\omega }_a$, capacitively coupled to a double quantum dot (DQD) with a coupling $g$. The DQD is set to zero source-drain bias, with the lead chemical potentials set in between the ground and excited state. Absorption of a resonator photon causes transitions between the DQD states $\left|g\right.〉$ and $\left|e\right.〉$, resulting in current flow. Lower left box: Energy levels (horizontal lines) and transition processes (single directed arrows) in the resonator-DQD system. A single photon received by the resonator ($\left|g\right.〉\left|n+1\right.〉$) can be emitted at the rate $\kappa$ or be absorbed by the DQD at the rate $g^2/{\mathrm{\Gamma }}_2$, exciting the metastable state $\left|e\right.〉\left|n\right.〉$ and then decaying back to the system ground state $\left|g\right.〉\left|n\right.〉$ at the inelastic decay rate ${\mathrm{\Gamma }}_1$.

Figure 2

As a function of the double-dot bias $\epsilon$, for $(t_c,g)/2\pi =(2,0.05)$ GHz and $\dot{N}=1$ MHz: (a) Jaynes-Cummings energy levels $E_{1\pm }$ of the DQD-resonator system. (b) DQD-resonator coupling $g=g_{0}sin\theta$ for $g_0/2\pi =50$ MHz, transverse relaxation rate ${\mathrm{\Gamma }}_2$ and depolarization rate ${\mathrm{\Gamma }}_1$ of the effective two-level DQD. (c) Cooperativity of the DQD-resonator system $C=g^2/{\mathrm{\Gamma }}_2\kappa$; the real and imaginary parts of the DQD susceptibility ${\chi }_{\sigma }$. (d) DQD polarization $m_z$ and photon-induced polarization $\mathrm{\Delta }{m}_z$.

Figure 3

For optimal parameters $\kappa /2\pi =76$ kHz and $t_c/2\pi =0.5$ GHz, on resonance ${\mathrm{\Delta }}_{ab}=0$: (a) Logarithmic plot of the photon-induced current $〈\mathrm{\Delta }I〉$ as a function of input flux $\dot{N}$ for ${\mathrm{\Gamma }}_l=1$ GHz. (b) Maximum efficiency as a function of the incoherent dot-lead tunnel rate ${\mathrm{\Gamma }}_l$ with $\dot{N}=1$ MHz.

Figure 4

As a function of the double-dot bias $\epsilon$, for $(t_c,g)/2\pi =(2,0.05)$ GHz and $\dot{N}=1$ MHz: (a) Magnitude $\left|r\right|$ and phase ${\varphi }_r$ of the reflection coefficient. (b) Photon number $n_a$ and power absorbed $P_a$.

Figure 5

(a) Photon detector efficiency on resonance ${\eta }_{\mathrm{res}}$ as a function of DQD-resonator coupling $g_0$ and interdot DQD tunnel coupling $t_c$, for $\kappa /2\pi =1$ MHz. (b) Efficiency ${\eta }_{\mathrm{res}}$ as a function of resonator leakage rate $\kappa$.

Figure 6

Efficiency of photon detector $\eta$ as a function of DQD bias $\epsilon$, with the tunnel couplings $t_c/2\pi =(0.5,1,1.5)$ GHz and DQD-resonator coupling $g_0=50$ MHz, for (a) nominal resonator leakage rate, $\kappa /2\pi =1$ MHz, and (b) optimal resonator leakage rate, $\kappa /2\pi =76$ kHz, Eq. (36).

Figure 7

The DQD charge relaxation rates, $T_1$ relaxation rate ${\gamma }_1$, difference in excitation and relaxation rates due to thermal phonons ${\gamma }_s$, and quasistatic dephasing rate due to charge noise ${\gamma }_{\varphi }$ as a function of the DQD bias $\epsilon$.