Variants of the Quantum Phase Operator for the Harmonic Oscillator

Technical Summary: Variants of the Quantum Phase Operator for the Harmonic Oscillator

Problem Statement
The paper addresses the mathematical definition and analytic treatment of quantum phase operators within the framework of the Quantum Harmonic Oscillator (QHO). While the physical relevance of phase is well-established, particularly in systems where only relative phases matter, the mathematical formulation remains problematic. Specifically, the concept of phase is ill-defined in the vacuum state ($|0\rangle$), where no physically meaningful phase can be assigned. Conversely, as the photon number increases, phase determination becomes more precise. The authors aim to construct a rigorous mathematical framework for phase operators that respects the intimate connection between phase and the system's excitation level, motivated by the physically relevant case of two interacting phases.

Methodology
The authors develop a mathematical foundation based on the correspondence between Bose–Einstein statistics and the QHO. The core methodology involves:

1. Operator Construction: The paper defines time-like operators $T_0$ and $T_0^\dagger$ using the Hamiltonian $H$, the annihilation/creation operators ($a, a^\dagger$), and the inverse square root of the Hamiltonian ($H^{-1/2}$). These are defined as symmetrized products:
   $$T_0 = \frac{\sqrt{\hbar\omega}}{2} (aH^{-1/2} + H^{-1/2}a)$$
   From these, bounded, self-adjoint cosine ($C_0$) and sine ($S_0$) operators are derived.

2. Spectral and Commutator Analysis: The authors utilize the spectral mapping theorem and properties of the Hermite functions (eigenstates of $H$) to analyze the action of these operators on the number basis $|n\rangle$. They compute exact commutators, specifically $[T_0, T_0^\dagger]$ and $[C_0, S_0]$, demonstrating that these commutators are not zero but are trace-class operators.

3. Asymptotic Analysis: The behavior of the constructed operators is examined in the limit of large quantum numbers ($n \gg 1$). The authors show that $T_0$ and $T_0^\dagger$ asymptotically approximate the lowering and raising operators, respectively. Consequently, $C_0$ and $S_0$ asymptotically converge to the Susskind–Glogower operators ($T_{SG}, T_{SG}^\dagger$), which are known to behave as phase operators in the high-intensity limit.

4. Perturbation Theory: To address the ambiguity at the ground state (where phase is not unique), the authors introduce a perturbation operator $M$. They prove that the exact operators $C_0$ and $S_0$ differ from idealized phase operators (which satisfy specific shift properties on the number basis) by trace-class operators. This establishes that the constructed operators are trace-class perturbations of the Susskind–Glogower operators.

5. Two-Phase Extension: The framework is extended to a system of two identical harmonic oscillators (representing two laser fields). The authors define operators $S_{12}$ and $C_{12}$ representing the sine and cosine of the phase difference. They demonstrate that on the subspace excluding the vacuum states, the commutator $[S_{12}, C_{12}]$ vanishes up to a compact operator, and the Pythagorean identity $S_{12}^2 + C_{12}^2 \approx 2I$ holds up to a compact operator.

Key Contributions and Results

• Rigorous Definition: The paper provides a mathematically well-defined construction of quantum phase operators ($T_0, C_0, S_0$) that are bounded and self-adjoint, avoiding the unboundedness issues often associated with phase operators.
• Trace-Class Perturbations: A central result is the proof that the constructed operators are trace-class perturbations of the Susskind–Glogower operators. This implies that their spectral properties are closely related to the standard phase operators, with deviations that are mathematically "small" (trace-class).
• Ground State Ambiguity: The authors formally characterize the ambiguity of the phase at the vacuum state. They show that while the operators are not uniquely determined at $|0\rangle$, this non-uniqueness is handled via trace-class perturbations, preserving the physical consistency of the theory for excited states.
• Commutator Properties: The paper derives explicit formulas for the commutators of the sine and cosine operators. It confirms that these operators do not commute (reflecting the time-energy uncertainty principle), but the non-commutativity is a trace-class operator, vanishing in the asymptotic limit of high excitation.
• Two-Phase Consistency: For the two-phase case, the authors show that the sine and cosine of the phase difference "almost" commute and satisfy the Pythagorean identity on the subspace of non-vacuum states, aligning with classical expectations for high-intensity fields.

Significance and Claims
The authors claim that their work provides a robust mathematical foundation for quantum phase operators that bridges the gap between the problematic vacuum state and the well-behaved high-intensity limit. By demonstrating that their operators are trace-class perturbations of established models (Susskind–Glogower), they validate the physical intuition that phase becomes a well-defined observable as the number of photons increases. The paper does not propose new experimental setups but rather offers a refined spectral-theoretic treatment of existing physical concepts, specifically addressing the mathematical inconsistencies regarding the vacuum state and the non-commutativity of phase observables. The results are presented as a theoretical clarification of the operator algebra governing quantum phase in the harmonic oscillator.