Circuit-QED Photon Monitoring

• Circuit-QED photon monitoring comprises techniques that use superconducting circuits to non-destructively measure microwave photon states via strong dispersive coupling.
• It employs quantum non-demolition counting and continuous detection methods, achieving fidelities exceeding 90% and near-unity photon detection efficiency in optimized setups.
• These protocols enable real-time tracking of photon quantum jumps, underpinning applications in quantum error correction, state engineering, and microwave quantum networking.

Circuit-QED photon monitoring encompasses a large class of measurement techniques by which the photon population or quantum state of microwave modes—whether localized in high-Q resonators, propagating through transmission lines, or distributed over multiple modes—is interrogated using superconducting quantum circuits. The field has evolved to include quantum non-demolition (QND) counting, continuous detection of itinerant photons, real-time monitoring of photon-number quantum jumps, correlation measurements, and modular observables, leveraging strong light-matter coupling, advanced dispersive engineering, and parametric control of microwave quantum hardware.

1. Fundamental Principles and Dispersive Photon Monitoring

Circuit-QED platforms exploit the dispersive regime of the Jaynes–Cummings interaction, where the qubit transition $\omega_q$ is far detuned from the cavity mode frequency $\omega_r$. The resulting effective Hamiltonian includes a photon-number-dependent shift of the qubit frequency:

$$H_{\text{disp}} = \omega_r a^\dagger a + \frac{1}{2}\omega_q \sigma_z + \chi a^\dagger a \sigma_z$$

where $\chi = g^2/\Delta$ is the dispersive shift ($g$ is the coupling, $\Delta = \omega_q - \omega_r$) (Johnson et al., 2010, Moon, 2013). Each photon shifts the qubit line, enabling photon-number-resolved spectroscopy, and conversely, photon-number fluctuations induce qubit dephasing that can be harnessed for indirect photon sensing (Atalaya et al., 2023, Yan et al., 2018).

QND photon counting is realized by mapping the photon number onto the state of a dispersively coupled qubit via selective control pulses resonant with the photon-dependent qubit transition. For instance, in single-shot mode, a photon occupation $|n\rangle$ is mapped to the excited qubit state by a conditional $\pi$-pulse, with near-unity fidelity for the $\pm65$ MHz photon-number splittings reported for realistic parameter regimes (Johnson et al., 2010). QND behavior is validated via repeated interrogations, with observed fidelities exceeding 90%.

For continuous monitoring, photon-number fluctuations $\delta n(t)$ produce a stochastic phase evolution of the qubit; measurement of the qubit’s coherence decay using dynamical decoupling sequences (e.g., CPMG, spin-locking relaxometry) allows indirect extraction of extremely small average photon numbers (down to $2\times10^{-4}$) by fitting the induced dephasing rate to theoretical models valid in the non-Gaussian noise regime (Atalaya et al., 2023, Yan et al., 2018).

2. Single-Photon Detection of Itinerant and Propagating Microwave States

Efficient detection of itinerant microwave photons—both in continuous and pulsed regimes—remains a challenge because of the low photon energies and thermal backgrounds. Multiple approaches have been proposed and implemented:

• Absorber-based “click” detectors: Josephson photomultiplier-type circuits (JPMs), modeled as metastable three-level systems, are embedded in semi-infinite transmission lines terminated by mirrors. By tuning the coupling and decay rates, perfect photodetection with near 100% quantum efficiency is achievable in the reflected-port geometry, robust to moderate variations in device parameters and immune to losses up to 10% (Peropadre et al., 2010, Schöndorf et al., 2016). For moving photons, proper impedance matching (equality of absorber-induced decay and tunneling rate) maximizes the detection probability for arbitrary input wavepackets, with optimal performance for bandwidth-matched pulses ($\kappa \sim \gamma_1$) (Schöndorf et al., 2016).
• Engineered $\Lambda$-systems for deterministic capture: By utilizing dressed-state engineering of a transmon qubit dispersively coupled to a pair of resonators (A and B), impedance matching can be achieved to deterministically trap an incident photon in resonator A while monitoring the event via continuous dispersive readout from resonator B. This architecture achieves $\eta\gtrsim 0.9$ detection efficiency with $\sim$10 MHz bandwidth (Koshino et al., 2015).
• Itinerant photon QND detection with dark states: Ensembles of inhomogeneous qubits coupled to a resonator realize dark and bright collective modes that extend the photon–detector interaction time, increasing the probability of a non-destructive detection. Detection fidelities above 96% are achieved for single-photon pulses, with signal-to-noise ratios limited primarily by quantum efficiency in the readout and residual dark-count rates (Royer et al., 2017).
• Traveling Schrödinger cat and photon-subtraction states: Josephson traveling-wave parametric amplifiers (JTWPA) are used to generate squeezed vacuum, which passes through a beamsplitter and encounters a single-photon detector realized via a dispersively readout qubit. Heralding on the detection event generates odd cat states with 95–99% fidelity, dependent on squeezing parameters and detector efficiency (Joo et al., 2016).

3. Continuous, Time-Resolved, and Correlation Measurements

Continuous, time-resolved monitoring and correlation-function measurements are central to the characterization of quantum fields in circuit QED:

• Fluorescence-based photon-number tracking: By driving a dispersively coupled qubit with a frequency comb spanning all photon-number-dependent resonances, and monitoring the qubit fluorescence via heterodyne detection, one can infer the instantaneous photon number with single-shot resolution. High measurement rates (information gain $\Gamma_{\mathrm{meas}}^{-1} \sim 5\ \mu$s) are achievable compared to photon lifetime ($T_c > 200\ \mu$s), enabling tracking of individual photon quantum jumps in real time. Measurement backaction and induced dephasing rates $\Gamma_\phi$ can be directly compared with theoretical predictions, and trade-offs are characterized as a function of drive parameters and detection efficiency (Hutin et al., 2024).
• Quadrature (linear) detection and photon correlations: Homodyne and IQ-mixer-based quadrature measurements allow observation of first- and second-order coherence functions $g^{(1)}(\tau)$ and $g^{(2)}(\tau)$. Despite high amplifier noise, careful experimental design and subtraction protocols enable direct extraction of photon statistics, including antibunching in single-photon experiments. The approach offers temporal resolution down to tens of nanoseconds and is fully compatible with large-bandwidth quantum amplifiers (Silva et al., 2010).
• Photon-number counting statistics and measurement optimization: The full statistics of photon emission from driven resonators are calculated analytically and numerically using master-equation approaches with counting fields. Deviations from Poissonian statistics—due to Kerr squeezing and Purcell decay—are explicitly quantified. Measurement protocols are optimized by tuning integration windows and cavity decay rates to minimize the overlap of photon-number distributions for different qubit states, achieving readout fidelities $F \approx 95\%$ with state-of-the-art dispersive parameters (Nesterov et al., 2019).

4. Multi-Mode and Modular Photon Monitoring

Advances in scalable bosonic quantum computing and quantum error correction motivate multi-mode and modular photon monitoring protocols:

• Joint-photon-number-splitting via parametric coupling: By activating strong beamsplitter coupling between two cavities (“Alice” and “Bob”), a transmon ancilla statically coupled dispersively only to “Bob” can be made sensitive to the joint photon number in both modes. In the joint-splitting regime ($g_{BS}\gg\chi_2$), transitions in the ancilla depend only on the total occupation $N$, enabling implementation of mid-circuit erasure checks. Achieved erasure rates are $\approx 3\%$ per check with missed-erasure probability $9\times10^{-4}$, Pauli error $0.31\%$, and hardware-minimal connectivity (Graaf et al., 2024).
• Photon-pressure modular quadrature measurement: Parametric engineering of a photon-pressure coupling $g\,\hat{q}a^\dag a$ between two superconducting resonators enables direct, high-squeezing measurement of modular position observables (GKP stabilizers) in a single shot. The protocol yields modular variances $\Delta_q \sim 0.15$, corresponding to $15$–$18$ dB of GKP squeezing, and leverages fast ancilla Q-switching and loss-echoed measurement to minimize errors. This provides an efficient pathway for error-syndrome extraction and bosonic code stabilization (Weigand et al., 2019).
• Single-photon blockade and controllable transmission: Strongly nonlinear cavity-QED systems with large photon blockade parameters ($\chi = g^4/\Delta^3 \gg \kappa$) permit selective photon transport or transparency, depending on drive strength and bistability regime. Photon blockade is evidenced by $g^{(2)}(0) \ll 1$ and sharp single-photon transmission resonances, while increasing the drive transitions the system into a transparent response (Liu et al., 2012).

5. Experimental Realizations and Device Considerations

Key experimental considerations enabling high-fidelity photon monitoring include:

• Device parameters: Realized dispersive shifts $2\chi/2\pi$ of 20–50 MHz, qubits and cavities with $T_1, T_2 \gg 1\ \mu$s, and resonator linewidths compatible with time-resolved readout (Moon, 2013, Johnson et al., 2010).
• Cavity Q preservation during rapid measurement: Design features like notch filters and galvanic contacts enable simultaneous ultrahigh cavity-Q and fast qubit emission (Hutin et al., 2024).
• Noise mitigation: Judicious engineering of amplifier chains, cryogenic attenuation, and filtering is required to reach thermal occupations $\bar n_{\mathrm{th}}\lesssim 2\times10^{-4}$ and maximize quantum signal-to-noise (Atalaya et al., 2023, Yan et al., 2018).
• Scalability: Modular designs, beamsplitter-based entanglement buses, and joint-ancilla architectures enable hardware-efficient, scalable photon monitoring in multi-mode quantum information processors (Graaf et al., 2024).

6. Applications, Limitations, and Outlook

Circuit-QED photon monitoring techniques underpin many applications in quantum information science:

• Quantum error correction and bosonic codes: Real-time, QND photon-number and parity measurements enable error-syndrome extraction for cat codes and GKP encodings, with protocols for modular quadrature readout and mid-circuit erasure flagging (Johnson et al., 2010, Weigand et al., 2019, Graaf et al., 2024).
• Quantum state engineering: Heralded generation of nonclassical states (Schrödinger cats, Fock states) via photon subtraction or deterministic capture (Joo et al., 2016, Koshino et al., 2015).
• Microwave quantum networking and communication: Detection and characterization of propagating microwave photons support entanglement distribution and quantum state transfer (Peropadre et al., 2010, Koshino et al., 2015).
• Limits and challenges: Measurement backaction, residual dephasing from imperfect filtering, pulse-shape optimization, and finite quantum efficiency continue to set performance bounds. Improvements in materials, quantum-limited amplification, and real-time control architectures are expected to further extend sensitivity and scalability (Atalaya et al., 2023, Royer et al., 2017).

In summary, circuit-QED photon monitoring encompasses a comprehensive set of protocols and device architectures for measuring photon number, resolving propagating quantum states, and enabling advanced quantum information processing tasks at microwave frequencies. The field continues to integrate theoretical advances and experimental innovations to enhance precision, reduce backaction, and scale up the complexity of quantum measurement systems in superconducting platforms.