Description of Photon Quantum States (such as Fock states, Glauber states)

In the realm of quantum optics, the quantization of the electromagnetic field represents a fundamental departure from classical electrodynamics. While classical theory treats light as a continuous wave of oscillating electric and magnetic fields, quantum mechanics reveals that light is composed of discrete, quantized excitations known as photons. To describe these excitations, we rely on the mathematical framework of quantum states, which provides a probabilistic description of the field's properties.

Among the various possible configurations of a light field, two types of states stand as the fundamental pillars of the field: Fock states and Glauber states (also known as coherent states). These two states represent opposite ends of the spectrum regarding the relationship between photon number and phase, embodying the dual particle-wave nature of light.
Fock states, often referred to as number states, are the most direct representation of the particle nature of light. In this regime, the focus is entirely on the discrete count of photons within a specific mode of the electromagnetic field.

Mathematical Definition and Properties

A Fock state, denoted as $|n\rangle$, is defined as the eigenstate of the photon number operator $\hat{n} = \hat{a}^\dagger \hat{a}$. When a field is in the state $|n\rangle$, the number of photons is strictly and deterministically known to be the integer $n$. Mathematically, these states are constructed by repeatedly applying the creation operator $\hat{a}^\dagger$ to the vacuum state $|0\rangle$:

$$ |n\rangle = \frac{(\hat{a}^\dagger)^n}{\sqrt{n!}} |0\rangle $$

Where:

• $|0\rangle$ represents the vacuum state, containing zero photons.
• $\hat{a}^\dagger$ is the creation operator, which adds a single photon to the mode.
• $n!$ is the factorial of the photon number.

The Uncertainty Trade-off

The defining characteristic of a Fock state is its absolute certainty in photon number ($\Delta n = 0$). However, according to the Heisenberg Uncertainty Principle, this precision comes at a significant cost. Because the photon number operator $\hat{n}$ and the phase operator $\hat{\phi}$ do not commute, a state with a perfectly defined number of photons possesses a completely undefined phase.

In a Fock state, the phase is uniformly distributed over $[0, 2\pi)$, meaning there is no stable phase reference. This makes Fock states "non-classical" in a profound way, as they lack the phase coherence required to behave like a traditional wave.

Applications

Fock states are indispensable in modern quantum technologies. The single-photon state ($|1\rangle$) is the fundamental resource for:

• Quantum Key Distribution (QKD): Ensuring secure communication through the detection of individual quanta.
• Quantum Computing: Serving as the basis for discrete-variable quantum logic gates.
• Nonlinear Optics: Driving processes where single-photon interactions are required.

Glauber States: The Coherent Wave Paradigm

If Fock states embody the "particle" aspect of light, Glauber states (coherent states) represent its "wave" aspect. They are the quantum states that most closely resemble the continuous, stable electromagnetic waves described by classical physics.

Mathematical Definition and Properties

A coherent state, denoted as $|\alpha\rangle$, is defined as the eigenstate of the annihilation operator $\hat{a}$. It satisfies the eigenvalue equation:

$$ \hat{a} |\alpha\rangle = \alpha |\alpha\rangle $$

Here, $\alpha$ is a complex number that encapsulates both the amplitude and the phase of the field:

• The average photon number is given by $\bar{n} = |\alpha|^2$.
• The phase of the field is determined by the argument (angle) of $\alpha$.

Coherent states can be viewed as a "displaced" vacuum state, generated by the displacement operator $\hat{D}(\alpha)$:

$$ |\alpha\rangle = \hat{D}(\alpha) |0\rangle = e^{\alpha \hat{a}^\dagger - \alpha^* \hat{a}} |0\rangle $$

Minimum Uncertainty and Phase Space

One of the most remarkable features of Glauber states is that they are minimum uncertainty states. In the context of the quadrature components of the field (analogous to position and momentum), the product of the uncertainties reaches the lower bound allowed by the Heisenberg Uncertainty Principle.

In phase space (often visualized in the $Q$ or $P$ planes), a coherent state appears as a symmetric, Gaussian-shaped "blob" or "spot" centered at the point $\alpha$. Unlike Fock states, which are spread out in phase, the coherent state maintains a well-defined phase and amplitude. As the intensity $|\alpha|$ increases, the relative uncertainty in the photon number becomes negligible, and the state behaves increasingly like a classical monochromatic wave.

Applications

Coherent states are the workhorses of optical science. They are used to describe:

• Laser Output: An ideal laser operating in a steady state produces a field that is nearly a perfect coherent state.
• Optical Communications: The carriers used in fiber-optic networks rely on the stable phase and amplitude properties of coherent light.
• Interferometry: High-precision measurements, such as those in LIGO, utilize the phase stability inherent in coherent-like fields.

Comparative Summary and Physical Significance

The distinction between Fock states and Glauber states highlights the fundamental tension between particle-like and wave-like descriptions in quantum mechanics.

| Feature | Fock State $|n\rangle$ | Glauber State $|\alpha\rangle$ |
| :--- | :--- | :--- |
| Primary Characteristic | Fixed photon number | Stable phase and amplitude |
| Mathematical Nature | Eigenstate of $\hat{n}$ | Eigenstate of $\hat{a}$ |
| Uncertainty Profile | $\Delta n = 0$, $\Delta \phi \to \infty$ | Minimum $\Delta x \Delta p$ (Balanced) |
| Classical Limit | Highly non-classical | Approaches classical wave as $|\alpha| \to \infty$ |
| Physical Intuition | Discrete "particles" | Continuous "waves" |

In practical experimental physics, achieving "pure" states is a significant challenge. Real-world lasers exhibit "shot noise," meaning they are technically weak coherent states rather than perfect Glauber states. Similarly, single-photon sources often have a non-zero probability of emitting multiple photons, deviating from the ideal $|1\rangle$ state.

Nevertheless, mastering these two fundamental states is the prerequisite for exploring more exotic phenomena. By manipulating these states—through techniques like squeezing or by creating superpositions (such as Schrödinger cat states) — researchers can push the boundaries of measurement precision and quantum information processing, unlocking the full potential of light in the quantum age.