Phase-Controlled Coherent Tunability

• Phase-controlled coherent tunability is the ability to continuously adjust quantum interference via precise phase settings in photonic circuits, enabling programmable absorption and transmission.
• It employs Mach-Zehnder interferometers and ancillary modes to emulate nonunitary dynamics, allowing deterministic quantum state routing and loss engineering.
• Enhanced phase sensitivity and metrology are achieved through coherent manipulation of single-photon and NOON states, approaching performance near the Heisenberg limit.

Phase-controlled coherent tunability refers to the ability to manipulate quantum or classical states of light and matter so that their interference properties, transmission, or response can be continuously controlled by adjusting a phase parameter. This tunability is central to quantum optics, integrated photonics, and quantum metrology, enabling applications such as programmable absorption, quantum-enhanced phase estimation, and quantum state engineering. Recent work demonstrates such tunability in programmable integrated linear photonic circuits, where ancilla-mediated loss allows emulation of non-Hermitian dynamics, and in quantum interferometry using @@@@1@@@@ constructed from deformed algebraic structures (Krishna et al., 2 Oct 2025, Abouelkhir et al., 29 Aug 2025).

1. Programmable Phase-Controlled Absorption in Integrated Photonics

Phase-controlled coherent tunability is realized in programmable integrated photonic circuits by emulating a lossy (nonunitary) beam splitter, where the absorption experienced by an input quantum state can be tuned by the internal phase and amplitude settings of Mach–Zehnder interferometers (MZIs). The effective scattering matrix takes the form: $$S = \begin{bmatrix} t & r \ r & t \end{bmatrix},$$ where $t = |t|e^{i\phi_t}$, $r = |r|e^{i\phi_r}$, and the internal phase difference $\phi_{rt} = \phi_r - \phi_t$ sets the interference conditions. The loss is programmed according to

$$|A|^2 = 1 - |t|^2 - |r|^2, \quad 2|t||r| \cos\phi_{rt} = \pm |A|^2.$$

A single-mode ancilla is used to extend this nonunitary operation to a lossless, unitary transformation in a larger Hilbert space via a Clements mesh of MZIs. By setting the phase on a designated MZI ($\theta_{\text{MZI}_2} = 2\cos^{-1}(\sqrt{2|A|^2})$), the absorption coefficient $|A|^2$ is directly programmed, providing coherent tunability between perfect transmission, perfect absorption, and all intermediate cases.

The circuit is probed with phase-encoded single-photon and NOON states: $$|\Psi_{1\text{ph}}\rangle = \frac{1}{\sqrt{2}} (e^{i\phi}|1,0\rangle - |0,1\rangle),\quad |\Psi_{\text{NOON}}\rangle = \frac{1}{\sqrt{2}} (e^{2i\phi}|2,0\rangle - |0,2\rangle).$$ Varying $\phi$ allows deterministic quantum routing between output ports and ancilla, thereby coherently controlling whether the quantum state is transmitted, absorbed, or mapped onto a specific superposition of output Fock states (Krishna et al., 2 Oct 2025).

2. Phase-Sensitive Quantum State Manipulation and Loss Engineering

The photonic circuit exploits non-Hermitian (lossy) quantum dynamics by coupling the system to a programmable ancilla mode such that photons are not truly lost but rerouted—this loss is entirely reversible. For single-photon inputs, interference controlled by the input phase $\phi$ enables routing between full transmission (antisymmetric state) and full absorption (symmetric state). With two-photon NOON states, interference fringes in the output probabilities exhibit π-periodicity: deterministic single-photon absorption or probabilistic two-photon absorption arises depending on the phase.

At the heart of this mechanism is the mapping of multiphoton quantum interference into tunable absorption: the coherent absorption of quantum light is achieved not via decoherence but by a controllable redistribution of quantum amplitudes across accessible modes, enabled by the phase-programmable circuit.

3. Nonclassical Interference, Output Modulation, and Phase Sensitivity

Phase-controlled coherent tunability in these setups reveals quantum interference effects such as anti-coalescence (anti-bunching) and photon bunching. By programming the phase offsets and loss coefficients, the circuit modulates photon number distributions across outputs and ancilla, including outcomes like $|2,0,0\rangle$, $|1,1,0\rangle$, or $|0,2,0\rangle$ (where the three modes are two signal and one ancilla).

The phase sensitivity of the device is quantified by classical Fisher information

$$F^{(\text{max})}_i = \max_\phi\left\{ \frac{1}{P_i(\phi)} \left(\frac{dP_i(\phi)}{d\phi}\right)^2 \right\},$$

where $P_i(\phi)$ is the detection probability in output mode $i$ as a function of phase. Experimental results show peak Fisher information values of 0.79 for single-photon states and 3.7 for NOON states, surpassing the shot-noise limit of 2 and approaching the Heisenberg limit of 4, highlighting the enhanced tunability from multiphoton entanglement and π-periodic response.

The internal phase in the device is set by: $$\phi_{\text{MZI}_3} - \phi_{\text{MZI}_2} = \tan^{-1}\left( \frac{2|t||r|\sin(\phi_{rt})}{|t|^2 - |r|^2} \right) + \frac{\theta_{\text{MZI}_2}}{2} + \frac{\pi}{2},$$ thereby linking phase-programmable operations directly to physical device parameters.

4. Generalized Coherent States and Phase Tunability in Quantum Metrology

Phase-controlled coherent tunability also appears in the broader context of quantum metrology, as evidenced by interferometric schemes input with generalized coherent states built from deformed Heisenberg or su(1,1) algebras (Abouelkhir et al., 29 Aug 2025). These states are characterized by a deformed underlying spectrum: $\varepsilon_n = n + \beta(n),$ where parameters $a, d, e, k$ can be tuned to achieve nonclassical photon statistics and flexible photon number variance. The resulting states allow one to optimize the quantum Fisher information (QFI) for phase estimation, with QFI values controlled by the state parameters and the beam splitter configuration (e.g., balance |t| = |r| = 1/$\sqrt{2}$). The phase sensitivity, given by the quantum Cramér–Rao bound (QCRB), is: $\Delta\phi_{\text{QCRB}} = 1/\sqrt{\mathcal{F}},$ where $\mathcal{F}$ is the QFI.

Comparison of realistic detection schemes—difference intensity, single-mode intensity, and balanced homodyne detection—shows that optimal phase sensitivity approaches the QCRB for balanced interferometry and suitable state parameter regimes.

5. Practical Implications, Applications, and Outlook

Phase-controlled coherent tunability in programmable photonic circuits enables:

• Programmable quantum state engineering and state filtering, by selective routing based on input phase and circuit configuration.
• Adaptive and multiplexed quantum sensing and metrology, through dynamic control of loss and phase sensitivity.
• Emulation and simulation of non-Hermitian quantum systems, using ancilla-assisted loss mechanisms embedded in unitary circuit meshes.
• Robust, integrated quantum photonic processors capable of operating close to quantum fundamental limits of phase sensitivity.

The flexibility to coherently tune absorption/transmission profiles and sensitivity makes ancilla-assisted, phase-programmable photonic devices powerful resources for quantum information processing, quantum thermal dynamics simulation, and non-classical light measurement regimes. The incorporation of generalized coherent states further offers tunability at the quantum-state level, providing a customizable platform for precision measurements and quantum-enhanced metrology.

6. Comparative Overview: Key Features

Platform / Method	Main Tunability Mechanism	Quantum Sensitivity (Fisher Info)	Notable Effects
Ancilla-programmable photonic circuit (Krishna et al., 2 Oct 2025)	Programmable phase in MZIs; ancilla-assisted nonunitary transformation	Up to 3.7 (NOON state, Heisenberg limit = 4)	Perfect transmission/absorption, anti-coalescence, bunching
Deformed coherent state MZI (Abouelkhir et al., 29 Aug 2025)	Algebraic deformation parameters	Approaches QCRB	Tunable nonclassical statistics, optimal phase detection

A plausible implication is that future quantum photonic circuits will integrate both circuit-level phase programmability and quantum-state-level tunability, yielding devices that merge programmable nonunitary transformations with designer quantum resource states for ultimate control over quantum interference and phase sensitivity.