Amplification and Detection of Single Itinerant Microwave Photons

TL;DR

The paper tackles the challenge of detecting single itinerant microwave photons and proposes the Inelastic-Cooper-Pair-Tunneling Photon Multiplier (ICTPM), a Josephson-photonics device that multiplies one input photon into $n$ output photons under a dc bias satisfying $2eV_{\mathrm{dc}}/\hbar + \omega_a \approx n \omega_b$, enabling easier detection via heterodyne readout. It models the device with a two-mode resonator system connected to a dc-biased Josephson junction and analyzes both single-stage and cascaded two-stage implementations using a Mølmer-inspired formalism and cascaded Master equations to treat quantum pulses. Key contributions include design guidelines for achieving near-deterministic photon conversion, a detailed study of output-mode structure through the first-order coherence $G^{(1)}$, and practical readout schemes with quantum-limited amplification that yield detection probabilities around $84.5\%$ with very low dark counts in realistic parameter regimes. The results demonstrate an ICTPM-based detector that is dead-time-free and scalable, offering competitive performance for microwave quantum technologies and enabling further improvements through additional stages or higher multiplication factors.

Abstract

Single-photon detectors are an essential part of the toolbox of modern quantum optics for implementing quantum technologies and enabling tests of fundamental physics. The low energy of microwave photons, the natural signal path for superconducting quantum devices, makes their detection much harder than for visible light. Despite impressive progress in recent years and the proposal and realization of a number of different detector architectures, the reliable detection of a single itinerant microwave photon remains an open topic. Here, we investigate and simulate a detailed protocol for single-photon multiplication and subsequent amplification and detection. At its heart lies a Josephson-photonics device which uses inelastic Cooper-pair tunneling driven by a dc bias in combination with the energy of an incoming photon to create multiple photons, thus compensating for the low-energy problem. Our analysis provides clear design guidelines for utilizing such devices, which have previously been operated in an amplifier mode with a continuous wave input, for counting photons. Combining a formalism recently developed by Mølmer to describe the full quantum state of in- and outgoing photon pulses with stochastic Schrödinger equations, we can describe the full multiplication and detection protocol and calculate performance parameters, such as detection probabilities and dark count rates. With optimized parameters, a high population of a single output mode can be achieved that can then be easily distinguished from vacuum noise in heterodyne measurements of quadratures with a conventional linear amplifier. Realistic devices with two multiplication stages with multiplication of $16$ reach for an impinging Gaussian pulse of length $T$ a detection probability of $84.5\%$ with a dark count rate of $10^{-3}/T$, and promise to outperform competing schemes.

Figures (9)

• Figure 1: Inelastic-Cooper-pair-Tunneling Photon Multiplier. (a) The system consists of a dc-biased Josephson junction (with tunable Josephson energy $E_J$), connected in series with two microwave modes, that are realized as LC-resonators and capacitively coupled to transmission lines (TLs). A single-photon pulse impinging on cavity $\hat{a}$ from the input transmission line will be absorbed by the ICTPM, which then leaks $n$ photons into the output transmission line. (b) A Cooper pair can inelastically tunnel through the junction if the microwave modes absorb its energy of $2eV_\mathrm{dc}$. By setting the dc-voltage on the resonance $2eV_\mathrm{dc}/\hbar + \omega_a= n \omega _b$, the Cooper pairs drive the transition where one cavity-$\hat{a}$ excitation is coherently transferred to $n \in \mathbb{N}$ cavity $\hat{b}$-excitations. Time-dependent input-, output- and reflected pulses moving along transmission lines which are absorbed or emitted from the ICTPM can be numerically modeled by Mølmer's approach Christiansen2023Kiilerich2019Kiilerich2020Khanahmadi2023Lund2023. Therein, the transmission lines are mimicked by auxiliary cavities which emit (absorb) modes $u(t)$ ($v(t)$) for appropriately chosen time-dependent loss rates $g_u(t)$ ($g_v(t)$).
• Figure 2: Simulation of the amplification of an incoming single-photon pulse. (a) The impinging pulse in the frequency domain, $u(\omega)$, is Gaussian and smaller than the width of the resonance curve $n_\mathrm{res}$ of cavity $\hat{a}$ (occupation of a single cavity biased at $2eV_\mathrm{dc}=\hbar\omega$), so that it can be absorbed. (b) Pulse $u(t)$ in time domain and an output mode $v_0(t)$ of the right transmission [compare Fig. \ref{['fig:fig_model']}(a, c)]. They are modeled by auxiliary cavities with time-dependent loss rates $g_u(t)$ and $g_v(t)$ (not shown) and frequency $\omega_a$ and $\omega_b$. (c) Occupation numbers gained from the simulation show how the input pulse loses its photon to the input cavity $\hat{a}$. The multiplication process driven by the Josephson junction transfers excitations from cavity $\hat{a}$ to cavity $\hat{b}$, which subsequently leaks photons into the output transmission line. Part of this output is in mode $v_0(t)$ and thus absorbed by the auxiliary cavity modeling that mode. (d) Integrating the rate of lost photons (from $\hat{a}$: 0 lost photons (red), from $\hat{b}$: $n=9$ lost photons) yields a perfect multiplication to $n$ photons when choosing the optimal Josephson driving strength $E_J$. The explicitly modeled auxiliary cavity $v_0$ from the right transmission line retains $\langle \hat{n}\rangle = 7.1$ photons [cf. blue line in (c) approaching that constant], but loses $1.91$ photons, which are contained in other, not explicitly simulated modes. (e) The mode $v_0$ is found in a mixture of Fock states as shown by its Husimi-Q function. [Parameters: $n=9$, $\gamma_b =\sigma_\omega = \gamma_a/10$, $E_J^* = E_{J, \mathrm{opt}}^*$].
• Figure 3: Two-time first order correlation functions of the output field, $G^{(1)} = \langle \hat{b}^{\dagger }_\mathrm{out}(t_2) \hat{b}_\mathrm{out}(t_1)\rangle$ with eigenmodes and their occupations $\langle \hat{n}_k\rangle = \langle \hat{b}^{\dagger }_k \hat{b}_k\rangle$. In all cases, we simulate a single input mode centered around $\omega_a$ in a Gaussian envelope with $\sigma_\omega = \gamma_a/10$ in the quantum state $\lvert 1\rangle_\mathrm{in}$ impinging on and being resonantly absorbed by cavity $\hat{a}$, and $E_J^* = E_{J, \mathrm{opt}}^*$, for optimal conversion to $n$ output photons on resonance. (a) Linear conversion, $n=1$ and $\gamma_b = 10 \gamma_a$, results in one output mode with occupation $\langle \hat{b}^{\dagger }_0 \hat{b}_0\rangle = 1$, cf. (d). (b) For nonlinear multiplication $n=9$ and $\gamma_b = \gamma_a$, the duration of the outgoing pulse (the extent of $G^{(1)}$ along the diagonal) is much longer than the inverse instantaneous bandwidth of the amplifier (off-diagonal extent). $G^{(1)}$ becomes elliptical and possesses multiple eigenmodes with similar occupations. (c) For $\gamma_a = 10 \gamma_b$, the pulse duration of the output pulse is comparable to the inverse bandwidth of the instantaneous spectrum, yielding a triangular $G^{(1)}$ diagonalizable to very few dominant eigenmodes, see (d). Ticks mark different time scales that help to characterize $G^{(1)}$.
• Figure 4: (a) Weak measurement of the output from cavity $\hat{b}$ by heterodyne detection. A single trajectory of the numerical simulation yields $\langle \hat{b}\rangle = 0$, without incoming photon, and $\langle \hat{b}\rangle \neq 0$ during the photon multiplication process of the ICTPM when a single photon (Gaussian input mode in Fock state $\lvert 1\rangle_\mathrm{in}$, orange curve) impinges on cavity $\hat{a}$. Experimentally observable, however, is only the measured signal $J_\beta$ dominated by noise, where information about photon arrivals seems lost. Integrating the measured signal with a mode envelope $v_0$ results in a single complex number, which samples the Husimi-Q function of $v_0$ at the specific integration time. (b) The probability density $f(r)$ for the radial component, $|\beta_k| =r$, clearly discriminates cases without an incoming photon (blue histogram and dashed theory curve) from the highly occupied output mode of the ICTPM after a photon arrival (red histogram and dashed theory curve). Defining a photon detection event, as $|\beta_k|>r$, for any chosen threshold $r$, the integrals $p(|\beta_k|\geq r) =\int_{r}^\infty d|\beta_k| \, f(|\beta_k| \,| 1_\mathrm{in})$ (solid red line), $1-p(|\beta_k|\geq r)$, and $\int_{r}^\infty d|\beta_k| \,f( |\beta_k| \, | 0_\mathrm{in})$ (solid blue line) are the probability for correct, false negative, and false positive detection. [Parameters: $N_\mathrm{traj} = 10^4$ trajectories simulated, $\sigma_\omega = \gamma_b = \gamma_a / 10$, $n=9$, $E_J^* = E_{J, \mathrm{opt}}^*$].
• Figure 5: (a) Convolution $\beta_\mathrm{conv}$ of two recorded trajectories with a single mode $v_0$, for a simulation with an input photon in a Gaussian mode $u(t)$. The convolution induces correlations within a correlation time $\tau_\mathrm{c}$. Measuring stroboscopically once within $\tau_\mathrm{c}$ (circles) avoids induced two-time correlations and reduces the effective number of measurement points. A photon detection event at time $t$ is defined when $|\beta_\mathrm{conv}|>R_0$. (b) The resulting probability distributions, where averaging the instances of stroboscopic measurement over an interval of length $\tau_\mathrm{c}$ for an unbiased comparison to experiment results in a slight deterioration compared to Fig. \ref{['fig:fig_1stage_scheme1_1']}(b). (c) The resulting performance curve showing detection probability versus the dark-count rate for convolution with the dominant eigenmode $v_0$ and an exponential mode $v_\mathrm{exp}(t) \propto \exp[- \gamma_bt/2]$. The comparison with an exponential mode with comparable timescales demonstrates that a small mismatch between detection mode and eigenmode (or similarly between assumed and actual input mode) only slightly degrades the performance. [Parameters see Fig. \ref{['fig:fig_1stage_scheme1_1']}].