Multi-Tone Self-Phase-Referenced PDH

• Multi-Tone Self-Phase-Referenced PDH is a frequency stabilization method that synthesizes three coherent tones to cancel phase drifts and amplitude noise.
• It employs IQ modulation and direct digital synthesis to precisely control carrier and sideband tones for robust readout in superconducting qubit and laser spectroscopy applications.
• The approach achieves enhanced signal-to-noise ratio through active RAM suppression and intrinsic self-phase referencing, improving performance in quantum and precision measurement setups.

The Multi-Tone Self-Phase-Referenced Pound-Drever-Hall (PDH) technique is a modern generalization of the classical PDH method for frequency stabilization and dispersive readout, characterized by the synthesis and manipulation of multiple, independently controlled carrier and sideband tones. It offers enhanced immunity to phase drifts and amplitude noise, broader signal-to-noise advantages, and superior application breadth in both quantum microwave and precision optical domains. This approach circumvents the limitations of residual amplitude modulation and uncontrolled sideband production endemic to standard modulation architectures, and enables robust locking and readout in systems requiring high stability such as superconducting-qubit dispersive measurement and ultra-stable laser spectroscopy (Adisa et al., 2 Dec 2025, Kedar et al., 2023, Cialdi et al., 2021).

1. Multi-Tone Field Generation and Self-Phase Referencing

The hallmark of this technique is the synthesis of three distinct tones: the carrier at $\omega_0$, an upper sideband at $\omega_+=\omega_0+\Omega$, and a lower sideband at $\omega_-=\omega_0-\Omega$. In microwave implementations for qubit readout, tones are generated by IQ modulation using a single microwave source and arbitrary waveform generation, with independent amplitude and phase control for each component and suppression of higher-order sidebands (Adisa et al., 2 Dec 2025). In the optical domain, direct digital synthesis (DDS) modules create triplet RF frequencies, which are shifted to the optical domain using acousto-optic modulators (AOM), yielding highly controlled optical triplet fields (Kedar et al., 2023). Notably, all tones share a common reference clock, ensuring that any phase drift or slip affecting the generator or the local oscillator is identical for all tones. By exploiting the coherent self-subtraction of common-mode drift in the error signal, this configuration achieves intrinsic self-phase referencing, with the PDH discriminator remaining stationary at the true resonance regardless of slow path-length or timing changes.

2. PDH Error Signal Construction and Theory

Upon reflection from a dispersive cavity—whether hanger geometry in circuit QED or high-finesse optical cavity—the incident multi-tone field acquires amplitude and phase changes determined by the cavity's scattering coefficient $S(\omega)$ or reflection coefficient $r(\omega)$. The detected signal is the quadratic sum of the three tones: $$\mathcal{E}_{\text{det}}(t) = E_- e^{i(\omega_- t + \phi_-)} + E_0 e^{i(\omega_0 t + \phi_0)} + E_+ e^{i(\omega_+ t + \phi_+)}$$ where $E_i$, $\phi_i$ denote amplitudes and phases. The PDH error signal emerges from demodulation at the modulation frequency and is decomposed into in-phase and quadrature terms: $$\varepsilon_I = E_+ E_0 \cos(\phi_0-\phi_+) + E_- E_0 \cos(\phi_0-\phi_-)$$

$$\varepsilon_Q = E_+ E_0 \sin(\phi_0-\phi_+) - E_- E_0 \sin(\phi_0-\phi_-)$$

In optical cases with digital synthesis, phase relations are fixed ($\phi_+-\phi_0 = \pi$, $\phi_--\phi_0 = 0$), guaranteeing orthogonal beat notes and undistorted quadrature separation (Kedar et al., 2023). In transmon readout, state-dependent cavity shifts directly imprint dispersive signals on $\varepsilon_Q$’s zero-crossing (Adisa et al., 2 Dec 2025). This compact complex formalism enables robust discrimination even in the presence of large phase noise.

3. Experimental Architecture and Signal Processing

Microwave implementations route the three-tone readout through the dilution refrigerator, reflect from a hanger cavity containing the superconducting qubit, and amplify via near-quantum-limited Josephson parametric amplifiers followed by high electron mobility transistor (HEMT) amplification. Room-temperature processing splits the output into three channels, followed by simultaneous heterodyne detection against a common local oscillator; digital postprocessing reconstructs the PDH quadratures synthetically via triple-downconversion and vector operations: $$\varepsilon_I = v_+ \cdot v_0 + v_0 \cdot v_-, \qquad \varepsilon_Q = (v_+ \times v_0)_z - (v_- \times v_0)_z$$ where $v_i = [I_i, Q_i, 0]$ denotes the I/Q baseband vectors for each tone (Adisa et al., 2 Dec 2025).

Optical architectures utilize a self-contained electronics card with multi-channel DDS for tone generation, AOM driving for spectral shift, and dual photodiode detection for residual amplitude modulation (RAM) monitoring. Orthogonal amplitude and phase I/Q control loops null RAM passively and actively by feeding demodulated signals to per-channel DDS controls, achieving sub-ppm amplitude stability (Kedar et al., 2023). Feedback to the cavity or laser actuator proceeds via standard high-bandwidth servos.

4. Error Suppression, Phase Stability, and RAM Mitigation

Common-mode phase drift is effectively canceled, leading to robust phase stability benchmarks: the microwave PDH phase differential remains stable to $0.44^\circ$ rms over two hours, representing more than $200\times$ improvement compared to conventional heterodyne carrier phase drift (Adisa et al., 2 Dec 2025). In single-shot readout, state discrimination persists even under arbitrary free-running phase slip. Residual amplitude modulation, which saturates at $10^{-5}$ to $10^{-6}$ in EOM-based PDH, is suppressed to $10^{-6}$ (free) and $<3\times10^{-7}$ (active) in FM triplet AOM schemes (Kedar et al., 2023).

The amplitude and phase balancing loops in the optical triplet PDH null both in-phase and quadrature RAM, enabling shot-noise-limited SNR at high modulation index, and dynamic bandwidth over 100 kHz. In OPO phase stabilization, the single modulation signal supports both cavity and phase locking without additional demodulation stages, compensating for offsets induced by pump injection (Cialdi et al., 2021).

5. Backaction, Signal Gain, and Quantum Nondemolition Properties

Off-resonant sideband tones can, in principle, induce measurement-induced state transitions (MIST) in transmons or unwanted excitations in other systems. PDH implementations quantify and mitigate this backaction via conditional probability analysis; for detunings $\Delta_p \geq 20$ MHz ($\sim 30\kappa$ cavity linewidths), no induced transitions are observed up to $+28$ dBc sideband power, enabling sidebands to exceed the carrier amplitude by tens of dB without extra decoherence and preserving quantum nondemolition (QND) character (Adisa et al., 2 Dec 2025). A plausible implication is the feasibility of readout signal enhancement far beyond traditional carrier-probe approaches.

In microwave analog PDH with a pre-amplification square-law detector, mixing provides inherent heterodyne gain ($\varepsilon \propto E_0 E_\pm$) before the amplifier chain, yielding up to $14$ dB of additional readout signal—predicted to enable faster or frequency-multiplexed single-shot readout when implemented with wide-band cryogenic detectors (Adisa et al., 2 Dec 2025). In optical realization, shot-noise-limited SNR is achieved in the absence of higher-order sidebands, with 22% improvement over EOM PDH at high modulation index (Kedar et al., 2023).

6. Applications and Extension to Quantum and Precision Measurement Domains

Multi-tone self-phase-referenced PDH has been demonstrated in superconducting-qubit dispersive readout, achieving single-shot fidelity, long-term stability, and high immunity to phase drift—critical for scaling quantum computing architectures by parallel, multiplexed readouts (Adisa et al., 2 Dec 2025). In optical frequency metrology, the technique is employed for laser stabilization with ultra-low RAM, amplitude and phase control loops, and self-contained feedback architecture, supporting frequency references for optical clocks, gravitational-wave interferometry, sub-Doppler FM spectroscopy, and searches for fundamental physics (Kedar et al., 2023). In OPOs, simultaneous stabilization of the cavity resonance and seed-pump relative phase enables controlled generation of squeezed states and precise manipulation of quantum states (Cialdi et al., 2021).

The generalizable electronic synthesis and flexible architecture permit easily reconfigurable modulation schemes, multiplexed cavity locks, and arbitrary waveform generation, supporting applications ranging from precision timekeeping to quantum sensing.

7. Limitations, Calibration, and Practical Considerations

Certain experimental constraints require attention. For OPO locks, the monotonicity region of the phase-dependent error signal restricts the phase lock operating points: exact locking at singularities ($\phi_p = \pm \pi/2$) is infeasible due to vanishing sensitivity (Cialdi et al., 2021). Long-term drift is limited by actuator travel and environmental shifts, with lock durations of hours achievable. Scaling to higher-finesse cavities necessitates tuning modulation frequency or index to maintain sideband off-resonance. Electronics for splitting and processing signals can be consolidated via synthesized architectures and commercial modules, but calibration—particularly for zero-crossing and offset subtraction—remains critical for robust, drift-free operation.

A plausible implication is that integration of direct digital synthesis and self-contained control electronics will further drive the adoption of multi-tone PDH systems for both quantum information processing and precision metrology, leveraging their stability and configurability.