Phase-Shifted Coherent States

• Phase-shifted coherent states are nonclassical quantum states engineered via deformed algebraic structures and superpositions to exhibit tunable phase properties.
• They enable precise phase manipulation and measurement through advanced techniques such as homodyne and difference-intensity detection, optimizing quantum Fisher information.
• Their tunable nonclassicality supports applications in quantum information, high-precision metrology, and quantum key distribution by surpassing standard coherent state limits.

Phase-shifted coherent states are nonclassical quantum states in which the phase degree of freedom—arising from the superposition or manipulation of optical or bosonic field states—is pivotal to their structure and applications. These states appear in contexts ranging from quantum information processing to high-precision metrology, and include generalized cat states, entangled coherent states, phase-encoded alphabets, and coherent states defined in nonstandard algebras. The following sections comprehensively survey their algebraic constructions, phase space properties, methodologies for phase manipulation and measurement, and implications for quantum technologies.

1. Algebraic Construction and Generalization

The formal construction of phase-shifted coherent states extends beyond the canonical (Glauber–Sudarshan) coherent states by involving deformed harmonic oscillator and generalized algebraic structures. Two prominent frameworks are:

• Generalized Heisenberg Algebra (GHA):
  • Defined by ladder operators $B$, $B^\dagger$ and a Hamiltonian $\mathcal{H}$. The commutation and action relations are

  $$\mathcal{H} B^\dagger = B^\dagger \varphi(\mathcal{H}), \qquad [B, B^\dagger] = \varphi(\mathcal{H}) - \mathcal{H},$$

  where $\varphi(\mathcal{H})$ is a deformation function (choice of parameters such as $q$ controls the deviation from the standard oscillator algebra). - Fock space: $\mathcal{H}|m\rangle = \epsilon_m |m\rangle$, $B^\dagger |m\rangle = N_m |m+1\rangle$ with $N_m^2 = \epsilon_{m+1} - \epsilon_0$. - Generalized coherent states: $B|\zeta\rangle = \zeta |\zeta\rangle$, expanded as

  $$|\zeta\rangle = N(|\zeta|) \sum_{m=0}^\infty \frac{\zeta^m}{N_{m-1}!} |m\rangle,$$

  with normalization and recursive generalized factorial.

• Generalized (deformed) SU(1,1) algebra:

  • Ladder operators $G_+$, $G_-$ with a Hamiltonian $\mathcal{H}$ satisfying

  $$[G_+, G_-] = (\mathcal{H} - \varphi(\mathcal{H})) (\mathcal{H} + \varphi(\mathcal{H}) - 1).$$ - Fock space representation parallels the GHA case with $\mathcal{H}|m\rangle = \epsilon_m |m\rangle$ and $G_+ |m\rangle = \mathcal{N}_m |m+1\rangle$. - The energy spectrum’s deformation (e.g., $\epsilon_m = m + \beta(m)$ with tunable $\beta$) enables control over photon-number distributions and nonclassical features.

The flexibility in constructing coherent states via analytic deformation functions or spectrum engineering is central to achieving tunable phase sensitivity and photon statistics, critical for quantum-enhanced interferometry and sensing (Abouelkhir et al., 29 Aug 2025).

2. Phase Space Structure and Properties

The phase-space characterization of phase-shifted coherent states is intricate, particularly for superpositions such as generalized Gaussian cat states. Notable features include:

• Wigner Function Structure: For a superposition $$|\Psi\rangle = a |{\cal U}, u\rangle + b |{\cal V}, v\rangle$$ of Gaussian states, the Wigner function captures both individual “Gaussian hills” and an interference term:

$$W_\Psi(x) = |a|^2 W_{|{\cal U}, u\rangle}(x) + |b|^2 W_{|{\cal V}, v\rangle}(x) + |ab|\, \mathcal{I}(x),$$

where the interference term embodies the nonclassicality.

• Interference Term Characterization:

$$\mathcal{I}(x) = 2|K| \mathrm{Re} \left\{ \exp[i(x \wedge \zeta/\hbar + \varphi)] \mathcal{G}(x; {\cal G}, \eta) \right\}$$

($\eta$ is the phase-space midpoint, $\zeta$ the displacement vector). - In standard coherent state superpositions, the phase factor is linear in $x$, resulting in linear interference fringes. - For generalized superpositions, the quadratic form in the phase yields hyperbolic (or, for mixed states, even elliptical) fringe patterns, relevant for quantifying nonclassical interference (Nicacio et al., 2010).

• Decoherence and Robustness: Under thermal (Markovian) noise, the interference pattern’s amplitude decays, but the underlying hyperbolic (or more generally, nondegenerate) structure of the phase remains intact, providing robustness against decoherence.

• Generalized Coherent Phase States (CPS): Projected and phase-shifted coherent states can be constructed discretely on the unit circle, offering well-controlled phase grids and non-Gaussianity regimes (Freitas et al., 2022, Drummond et al., 2016).

3. Tunable Nonclassicality and Quantum Fisher Information

Tunable nonclassical properties—such as squeezing, sub-Poissonian statistics, and tailored photon-number distributions—can be systematically engineered through algebraic deformation and state preparation:

• Photon Statistics and Squeezing: The generalized coherent states’ photon number distribution and quadrature variances are directly tunable via parameters in the deformation function ($q$, $a$, $d$, $e$, etc.), spectrum $\epsilon_m$, or polynomial structure. This enables access to states exhibiting tailored squeezing, strong sub-Poissonian/anti-bunching effects, and non-Gaussianity pertinent for enhanced quantum metrology (Abouelkhir et al., 29 Aug 2025, Freitas et al., 2022).

• Quantum Fisher Information (QFI): The QFI,

$$F = 4[\langle\partial_\phi\psi|\partial_\phi\psi\rangle - |\langle\psi|\partial_\phi\psi\rangle|^2],$$

quantifies phase estimation sensitivity. - In interferometry (e.g., with one port populated with a generalized coherent state), QFI depends on the first and higher moments of the photon-number operator. - The optimization of QFI via state engineering enables approaching or saturating the quantum Cramér–Rao bound (QCRB): $\Delta\phi_{\text{QCRB}} = 1/\sqrt{F}$.

• Detection Schemes: The practical realization of phase sensitivity exploits three main detection approaches:

  • Difference-intensity detection: Measures the difference in photon numbers between two output ports; sensitivity is given by

  $$\Delta\phi_{df} = \frac{\Delta \hat{N}_d}{|\frac{\partial}{\partial \phi}\langle\hat{N}_d\rangle|}.$$ - Single-mode intensity detection: Uses photon number at a single output port;

  $$\Delta\phi_{sing} = \frac{\Delta\hat{m}_4}{|\frac{\partial}{\partial \phi}\langle\hat{m}_4\rangle|}.$$

  Both approaches benefit from deformed states with favorable photon number fluctuation properties. - Balanced homodyne detection: Measures output field quadrature in the presence of a strong local oscillator;

  $$\Delta\phi_{hom} = \frac{\sqrt{\Delta^2 \hat{X}_{\phi_L}}}{|\frac{\partial}{\partial\phi} \langle\hat{X}_{\phi_L}\rangle|}.$$

  Homodyne detection typically achieves sensitivities closest to the QCRB, is robust to losses, and leverages quadrature squeezing.

4. Manipulation and Measurement of Phase

The manipulation and measurement of phase in phase-shifted coherent states underpin their utility in metrology and quantum information:

• Phase-encoded Alphabets: In protocols such as coherent-state quantum key distribution (QKD), discrete phase-shifted alphabets (e.g., $N=4$ states $a_k = z e^{i2\pi k/N}$) maximize key rates in the presence of loss and noise (Papanastasiou et al., 2018). With suitable parameter choice, even four-phase-encoded states approximate the performance of infinitely dense constellations.

• Elementary Quantum Gates: Arbitrarily large phase shifts between basis states $|\alpha\rangle$ and $|-\alpha\rangle$ can be deterministically induced via displacement plus photon-subtraction operations. The choice of displacement parameter $\gamma$ yields controlled rotation in phase, critical for implementing phase gates, controlled phase gates, and universal logic (Hadamard gates) in optical quantum computing with cat codes (Marek et al., 2010).

• Distillation and Purification: Phase-insensitive operations and channels can “distill” a noisy, phase-shifted coherent thermal state $\rho(\beta, \alpha)$ into a purified state close to $|\alpha\rangle$. Asymptotically optimal distillation is achieved by combining reversible concentration with non-Gaussian channels, with the error scaling inversely with the purity of coherence (derived from the right-logarithmic derivative Fisher information metric) (Yadavalli et al., 9 Sep 2024).

• Quantum Filtering and Real-Time Estimation: In scenarios involving continuous phase measurement (e.g., in interferometry), quantum filtering equations (e.g., quantum Kalman–Bucy filters) allow real-time estimation of the phase observable based on measured signal records, updating the quantum state estimate via Itō calculus and measurement-dependent innovations processes (Gough, 2016).

5. Applications in Quantum Technologies and Metrology

Phase-shifted coherent states and their generalizations have profound consequences in a variety of quantum technologies:

• Precision Interferometry: Deformed coherent states and engineered cat states (number-state filtered, even/odd, or entangled coherent) allow phase estimation at or near the quantum Cramér–Rao bound. For nonlinear phase estimation (e.g., in the presence of a Kerr interaction), entangled coherent states outperform traditional NOON states at the same average photon number, retaining this advantage even with optical losses (Joo et al., 2012, Meher et al., 2019).

• Quantum Key Distribution: Phase-encoded alphabets offer a robust pathway toward high-rate and secure continuous-variable QKD, performing near the ideal limit for moderate mean photon numbers, even under channel noise and loss (Papanastasiou et al., 2018).

• Quantum Information Processing: The ability to effect large, high-fidelity phase shifts in cat codes and superpositions makes these states ideal for logic operations and error correction in superconducting resonator and photonic quantum computing architectures (Marek et al., 2010, Adam et al., 2015).

• Platform-Specific Relevance:

  • In photonic systems, phase-shifted and projected coherent phase states are especially useful for circuit simplification in boson sampling, measurement-based quantum computation, and scalable quantum memories (Drummond et al., 2016).
  • In solid-state systems (e.g., Josephson quantum computers), restricted Hilbert spaces and phase manipulation via projections and deformations optimize state resource usage and enable efficient simulation (Drummond et al., 2016).
• Fundamental Investigations: Studies of the phase operator, geometric phase accumulation, and the interplay between phase and number operators inform deeper quantum mechanical questions and provide algebraic tools for both theoretical and applied quantum optics (Kibler et al., 2012, Soto-Eguibar et al., 2014, Yang et al., 2011).

6. Summary of Key Formulas and Performance Metrics

The key mathematical expressions governing phase-shifted coherent state properties and metrological bounds include:

Quantity/Operator	Formula	Relevance
Generalized Coherent State (GHA)	$$|\zeta\rangle = N(|\zeta|) \sum_{m=0}^\infty \frac{\zeta^m}{N_{m-1}!}|m\rangle$$	State construction
Wigner Function for Gaussian Cat State	$$W_\Psi(x) = |a|^2 W_{|{\cal U}, u\rangle}(x) + |b|^2 W_{|{\cal V}, v\rangle}(x) + |ab|\, \mathcal{I}(x)$$	Phase-space structure
Interference Term in Wigner Function	$$\mathcal{I}(x) = 2 |K| \mathrm{Re} \{ e^{i(x \wedge \zeta/\hbar + \varphi)} \mathcal{G}(x; {\cal G}, \eta) \}$$	Nonclassicality signature
Quantum Fisher Information (QFI)	$$F = 4[\langle\partial_\phi\psi|\partial_\phi\psi\rangle - |\langle\psi|\partial_\phi\psi\rangle|^2]$$	Phase sensitivity (Abouelkhir et al., 29 Aug 2025)
Quantum Cramér-Rao Bound (QCRB)	$\Delta\phi_{\mathrm{QCRB}} = 1/\sqrt{F}$	Ultimate phase precision
Homodyne Detection Sensitivity	$$\Delta\phi_{hom} = \sqrt{\Delta^2 \hat{X}_{\phi_L}} / |\partial_\phi \langle\hat{X}_{\phi_L}\rangle|$$	Detection scheme

Optimal phase estimation and manipulation rely on the interplay among state construction, measurement, and channel properties, with the nonclassical structure, phase robustness, and tunability afforded by algebraic deformation and superposition being central to state-of-the-art quantum metrology.

7. Outlook and Future Directions

Current research underlines that algebraically engineered phase-shifted coherent states provide a powerful and flexible resource for achieving precision at or near fundamental quantum limits. Key open areas include the experimental realization of optimal (sometimes non-Gaussian) distillation channels for state purification (Yadavalli et al., 9 Sep 2024), further scaling of phase-encoded alphabets for QKD (Papanastasiou et al., 2018), and the development of phase-space methods for more complex quantum networks—especially those restricted by realistic resource constraints, decoherence, or nonlinearities (Drummond et al., 2016, Freitas et al., 2022).

The connections between phase-insensitive operations, resource theories of asymmetry, and operationally meaningful metrics such as the right-logarithmic derivative Fisher information metric suggest new conceptual bridges between quantum information and quantum metrology. As detection and control techniques progress, phase-shifted coherent states are poised to remain a central component of both foundational studies and technological applications across quantum science.