Stroboscopic detection of itinerant microwave photons

Abstract

We present a novel scheme to detect itinerant microwave radiation at the single photon level. Using existing Josephson-photonics devices, where two microwave cavities are coupled by a dc-voltage biased superconducting junction, we theoretically show how to implement a stroboscopically repeated, near-projective measurement of a photon impinging on one of the cavities. Optimizing rate, duration, and strength of the measurement by flux control of the junction and developing a threshold protocol to detect the photon from a homodyne measurement of the radiation output of the other cavity, we achieve highly efficient detection with low dark counts. By cascading the detector with a preamplifier, where a similar two-cavity Josephson-photonics device acts as a photon multiplier, we can further improve the device to reach a detection efficiency of $88.5 \%$ with a dark count rate of $\sim10^{-4} γ_a$, set by the resonance width $γ_a$ of the absorbing cavity. These results for a multiplication factor of two suggest that near-unity efficiencies may be reached for higher multiplication factors.

Figures (12)

• Figure 1: (a) Effective description of a microwave cavity driven by an input field. The scenario is theoretically modeled by a cascaded Master equation involving an auxiliary cavity $a_u$ that emits the quantum pulse $u(t)$ via a tunable decay coupling $g_u(t)$ (blue arrow). We assume a Gaussian mode $u(t) \sim \mathrm{e}^{-(\sigma_\omega t)^2/2}$ with spectral width $\sigma_\omega=\gamma_a$ in a single-photon Fock state $\ket{\psi}_u=\ket{1}_u$. The input field is partly reflected and partly absorbed and re-emitted by the microwave cavity. (b) Occupations of auxiliary cavity $a_u$ emitting the photon, cavity $a$ which absorbs and subsequently emits the photon, and the integrated output $n_\mathrm{out}=\int_0^t \langle{\hat{L}_a^\dagger \hat{L}_a}\rangle(t') \dd t'$. The plateau in $n_\mathrm{out}$ at $t\gamma_a \approx 4$ indicates destructive interference of contributions from direct reflection and from photon absorption and re-emission.
• Figure 2: Stroboscopic projective measurements (performed with frequency $\gamma_\mathrm{m}=0.2 \gamma_a$) of a microwave cavity $a$ that is subject to an impinging resonant Gaussian single-photon pulse ($\sigma_\omega/\gamma_a=0.1$). Each measurement projects the cavity's state either to $\ket{1}_a$ (i.e. the photon is detected) or to $\ket{0}_a$, where the photon remains undetected (c.f. schematic setup in the inset). The solid line shows a trajectory where the photon was never detected, while the dashed trajectory detects the photon in the measurement at $t_{\mathrm{m},j} =47/\gamma_a$, where the (total) state is projected to $\ket{\psi}=\ket{01}_{u,a}$. Because of the subsequent exponential decay, the cavity occupation will be (with very high probability) projected to vacuum in the following measurement. The probability to find the photon at measurement $j$, $p_j(1)=p_j(O_j=1,\,O_k=0 \,\forall\, k<j)$ is indicated by purple markers, while the total photon detection probability is $\eta=67\%$. The y-axis shows both the occupation and the probability (same scale). [Parameters: $\gamma_\mathrm{m}=0.2\gamma_a$, $\sigma_\omega=0.1\gamma_a$, $\Delta_a=0$.]
• Figure 3: Detection probability $\eta$ depending on the measurement rate $\gamma_\mathrm{m}$ for different pulse shapes. The curves were averaged over several measurement positions relative to the pulse. Shaded regions indicate the range of detection probabilities depending on the relative measurement position. A maximum detection probability of $\eta=81 \%$ can be achieved for a measurement rate $\gamma_m=0.4\gamma_a$ for spectrally narrow and resonant Gaussian input pulses (solid light blue, $\sigma_\omega=\gamma_a/10$), while it peaks at $\eta=78 \%$ for resonant exponential input pulses, $u(t)= \sqrt{\gamma_u} \, \theta(t) \, \mathrm{e}^{-\gamma_ut/2}$ (where $\gamma_u=\gamma_a/10$, dashed). For spectrally broader Gaussian input pulses ($\sigma_\omega=\gamma_a$, dark blue line and region), and for a time-reversed exponential pulse $u(t)= \sqrt{\gamma_u} \, \theta(-t) \, \mathrm{e}^{\gamma_ut/2}$ ($\gamma_u=\gamma_a$, red line and region), the maximum detection probability is significantly smaller. Due to the stricter time confinement of those pulses, the position of the measurements relative to the pulses' arrival time strongly influences the total detection probability, especially for low measurement frequencies. A preamplification of a resonant single-photon Gaussian input pulse ($\sigma_\omega=\gamma_a/20$) with multiplication factor $n=2$ (c.f. Sec. \ref{['sec:MultiCasc']}) enables near-deterministic detection, $\eta=95 \%$.
• Figure 4: Schematic electrical circuit of a Josepshon-photonics device consisting of two microwave cavities connected in series with a dc-biased Josephson junction. By tuning the voltage to the single-photon resonance of cavity $b$, $\omega_\mathrm{dc}=\omega_b$, each Cooper pair (CP) tunneling through the junction effectively creates one photon in $b$. The linear driving of cavity $b$ is renormalized by virtual excitations [c.f. the Bessel functions in Eq. \ref{['eq:RWA_H']}], and thereby depends on the state of cavity $a$. An impinging quantum pulse modifies the state of cavity $a$, which effectively modulates the driving and thus the quantum state of cavity $b$.
• Figure 5: Measuring a closed cavity $a$, $\gamma_a=0$, in a Josephson-photonics setup. The coupling $E_J$ is switched on between $t\gamma_b=20$ and $t \gamma_b=40$. The top panel shows the occupation of cavity $a$, while the bottom panel shows the quadrature $\langle{\hat{x}_b}\rangle$ of cavity $b$. If cavity $a$ is empty, cavity $b$ is driven to a (near-)coherent state $\ket{\beta=1.6}$ and returns to vacuum when $E_J$ is switched off (dashed black lines). For $\alpha_0=1$ and when $\ket{\psi}_{0,a}=\ket{1}$, the effective driving strength of $b$ vanishes, such that $b$ stays in vacuum irrespective of the coupling (dashed magenta lines). When cavity $a$ starts in a coherent superposition $\ket{\psi}_{0,a}=(\ket{0}+\ket{1})/\sqrt{2}$, the measurement projects its state to either $\ket{0}_a$ (solid orange lines) or $\ket{1}_a$ (solid blue lines) with a probability of $1/2$ each. [Parameters: $\gamma_b=1$, $\beta_0=0.3$, $\Delta_a=\Delta_b=0$, $E_J^* \beta_0/(\hbar \gamma_b)=1.6$.]