Dispersive detection of single microwave photons with quantum dots

Technical Summary: Dispersive Detection of Single Microwave Photons with Quantum Dots

Problem Statement
Detecting single itinerant microwave photons remains a significant challenge in circuit quantum electrodynamics (circuit QED) due to the low energy of microwave photons compared to their optical counterparts. While absorption-based detectors (e.g., Josephson junction switching, engineered dissipation) exist, they destroy the photon. Non-absorbing detection schemes are desirable for quantum optics applications such as correlation measurements and remote entanglement, but achieving high efficiency in the continuous-wave or single-photon regime is difficult. Previous attempts using double quantum dots (DQDs) to detect photons via photo-assisted tunneling have demonstrated proof of principle but suffered from modest detection efficiencies or challenges in resolving individual tunneling events against noise.

Methodology
The authors propose a theoretical framework for a minimal setup consisting of a resonant microwave cavity dispersively coupled to a double quantum dot (DQD) that is tunnel-coupled to a single electronic lead. The system is modeled using a quantum cascade approach to describe the time-resolved dynamics of a single-photon wavepacket impinging on the cavity.

1. Effective Model: The system is reduced to a single-electron level (the bonding state of the DQD) dispersively interacting with the cavity mode. The Hamiltonian includes the cavity frequency, the electronic level energy, and a dispersive coupling term $\lambda \hat{n}_c \hat{n}_{QD}$. The antibonding state is shown to be irrelevant in the suitable parameter regime.
2. Dynamics: The full quantum dynamics are governed by a Lindblad-like master equation.
   • Tunneling: A dissipative superoperator is derived for electron tunneling between the DQD and the lead, conditioned on the photon occupation of the cavity. This accounts for the energy shift of the electronic level induced by the photon.
   • Photon Input: The incoming photon is modeled via a "quantum cascade" approach, involving a source mode coupled to a chiral transmission line, which provides a time-resolved description of the wavepacket scattering.
3. Detection Scheme: The detection mechanism relies on monitoring the charge state of the DQD. Initially, the electronic level is below the lead's chemical potential (occupied). When a photon enters the cavity, the dispersive interaction shifts the level energy above the chemical potential, allowing the electron to tunnel out. Once the photon leaves, the level shifts back, and the dot refills. This transient change in occupation is monitored by a phenomenological charge detector.

Key Contributions and Results

• Time-Resolved Dynamics: The authors provide an exact solution for the coupled linear equations of motion for the density matrix elements, allowing for a detailed analysis of the transient behavior of the cavity and electron occupations.
• Measurement-Induced Backaction: A central finding is the observation of a significant backaction effect. The dispersive coupling suppresses the probability of the photon entering the cavity (cavity occupation) compared to the uncoupled case. This occurs because the presence of the electron in the dot detunes the cavity from the incoming photon frequency. If the tunneling rate ($\Gamma_{QD}$) is fast, the electron leaves quickly, shifting the cavity out of resonance and preventing further photon absorption. This effect is pronounced when the dispersive coupling strength $|\lambda|$ exceeds the cavity linewidth $\gamma_c$.
• Parameter Dependence:
  • Coupling Strength ($|\lambda|$): Increasing $|\lambda|$ generally reduces the maximum cavity occupation due to backaction but increases the potential signal for charge depletion.
  • Tunneling Rate ($\Gamma_{QD}$): An optimal regime exists where $\Gamma_{QD}$ is fast enough to allow electron tunneling (enabling detection) but not so fast that it completely suppresses photon entry via backaction before detection occurs. The paper identifies a non-monotonic behavior in detection efficiency as a function of $\Gamma_{QD}$.
  • Temperature ($T$): Thermal fluctuations are a critical limiting factor. For $T \gtrsim |\lambda|$, thermal noise smears out the initial occupation and the tunneling dynamics, drastically reducing the detection contrast and efficiency.
• Detection Efficiency: Using a phenomenological model for a charge detector (modeled as a two-level system triggered by the empty state of the QD), the authors estimate the detection efficiency $\eta$. They find that high efficiency is achievable in the regime where the dark count rate (thermal tunneling) is low ($T \ll |\lambda|$) and the detector is sufficiently fast ($\gamma_{det} \gg \Gamma_{QD}$).

Significance and Claims
The paper claims to demonstrate a theoretical pathway for non-absorptive, single-shot detection of microwave photons using hybrid circuit QED architectures. The authors highlight that:

1. Dispersive Interaction: The scheme leverages the dispersive coupling to induce a measurable change in electronic occupation without absorbing the photon, preserving the quantum state of the field.
2. Feasibility: The required parameters (dispersive coupling $\sim 130$ MHz, cavity linewidth $< 1$ MHz, tunneling rates in the tens of MHz) are within the reach of state-of-the-art hybrid devices.
3. Thermal Constraints: The authors modestly but critically identify the electron temperature as the primary experimental bottleneck. They argue that while achieving sub-10 mK electron temperatures is challenging, it is within the realm of contemporary cryogenic technology (e.g., via liquid helium immersion or nuclear demagnetization), provided parasitic heating is minimized.
4. Backaction as a Feature/Constraint: The work explicitly characterizes the measurement-induced backaction not just as a nuisance, but as a fundamental feature of the dispersive regime that must be balanced against detection efficiency.

The paper concludes that while the proposed setup offers a promising route for microwave photon detection, its practical realization depends heavily on minimizing thermal noise and optimizing the interplay between tunneling rates and dispersive coupling strengths.