Coherence length is a concept used in physics, particularly in the study of waves and quantum mechanics, to describe the spatial or temporal extent over which a wave maintains a specific phase relationship or coherence. It is an important parameter in understanding the behaviour of waves and their interference patterns. The specific definition and context of coherence length can vary depending on the field of physics in which it is applied:

Optics: In optics, coherence length refers to the distance over which a light wave maintains a constant phase relationship. Light from conventional sources like a light bulb or the sun has a very short coherence length, meaning that the light waves quickly lose their phase relationship, resulting in incoherent light. In contrast, lasers can produce highly coherent light with long coherence lengths, allowing for applications such as holography and interferometry.
Quantum Mechanics: In quantum mechanics, coherence length is related to the behaviour of quantum systems, such as electrons in a superconducting material or photons in a quantum state. It characterizes the distance or time scale over which quantum superposition states remain coherent before decoherence occurs. In quantum computing, for example, maintaining a long coherence length for qubits is crucial for performing complex quantum operations.
Superconductivity: Coherence length is a parameter in the study of superconductors. It describes the size scale over which the superconducting order parameter, which characterizes the superconducting state, varies significantly. A longer coherence length indicates a more extended region of superconductivity.
Acoustics: In acoustics, coherence length can refer to the spatial extent over which sound waves remain in phase. This concept is important in applications like sonar and ultrasonography.

The coherence length is inversely related to the spectral width of the wave or the energy spread of the quantum system. In practical terms, longer coherence lengths are desirable in various applications because they enable the formation of well-defined interference patterns and the preservation of quantum states. Scientists and engineers often work to extend coherence lengths through careful control of experimental conditions and materials.

London equation is a local equation because it relates the current density at a point (r) to the vector potential at the same point. But coherence length 𝜉 is the range over which we should average A to obtain J.

The plane wave equation is 𝜓(x) = e^ikx ....1

and modulated wave is

where q = modulation in the wave vector.

Modulation in wave vector, also known as wave vector modulation or spatial modulation, is a concept used in various branches of physics and engineering to describe the variation of a wave vector in space or time. The wave vector (often denoted as "k") represents the direction and magnitude of the wave's propagation in a particular medium. Modulating the wave vector means changing the direction and/or magnitude of the wave vector as the wave travels.

Here are a few contexts where modulation in wave vector is relevant:

Photonic Crystals: In photonics, specifically in the study of photonic crystals, wave vector modulation refers to the deliberate alteration of the wave vector of light as it passes through a periodic structure, such as a crystal lattice. This modulation can lead to the creation of bandgaps in the electromagnetic spectrum, where certain frequencies or wavelengths of light are forbidden from propagating through the crystal. This phenomenon is crucial for designing photonic devices like optical filters and waveguides.
Acousto-Optic Devices: In acousto-optic devices, such as acousto-optic modulators, the wave vector of light is modulated by an acoustic wave traveling through a medium, often a crystal or an optical fiber. This modulation allows for the manipulation of the intensity, frequency, or direction of light. Applications include laser beam steering and frequency shifting.
Grating Spectroscopy: Diffraction gratings are optical components that can modulate the wave vector of incident light by dispersing it into its constituent wavelengths. The angle at which the light is diffracted depends on the wavelength, resulting in spectral separation. This principle is widely used in spectroscopy for analyzing the composition of light sources.
Metamaterials: Metamaterials are engineered materials with properties not found in nature. They are designed to manipulate electromagnetic waves in novel ways, often involving wave vector modulation. Metamaterials can be used to create devices like invisibility cloaks or negative refractive index materials by controlling the wave vector of light in unconventional ways.
Signal Processing: In the context of signal processing, modulation in wave vector can refer to the manipulation of the phase or direction of signals, such as radio waves or electromagnetic waves used in communication systems. Techniques like phased-array antennas use wave vector modulation to steer the direction of transmitted or received signals.

In summary, modulation in wave vector involves altering the properties of waves, such as their direction, phase, or wavelength, as they propagate through a medium or interact with structured materials. This concept plays a crucial role in various fields, enabling the development of advanced optical and electromagnetic devices and systems.

So KE associated with modulated wave is

So, extra kinetic energy needed for modulation is ℏ²kq/ 2m when it exceed the energy gap Eg, the superconducting state will be destroyed. The critical value (qc) of the modulation in wave vector.

ℏ²/ 2m kq/c = Eg

q/c = 2mEg / ℏ²k

The intrinsic coherence length 𝜉o is given by

𝜉o = 1/qc = ℏ²k / 2mEg = ℏv_F/ 2 Eg

But actual boundary condition calculation shows that

𝜉o = 2 /π ℏv_F/ Eg

This is a simplified expression for the coherence length in a system, and it's typically associated with phenomena like superconductivity. The coherence length represents the maximum distance over which a pair of electrons in a superconducting material can maintain their correlated quantum states. It plays a crucial role in understanding the behaviour of superconductors.

To discuss this expression in the context of coherence length, you would need additional information or context regarding the specific system or physical phenomenon you are interested in. Coherence length is typically associated with wave properties, such as the spatial extent over which a wave maintains a specific phase relationship.

The expression ℏ²kq/2m may be relevant to describing the energy or momentum of a quantum particle, but it doesn't directly relate to the coherence length without further information about the specific wave or particle system you are considering. To connect this expression with coherence length, you would likely need to derive or analyze a relevant wave equation or dispersion relation that includes these parameters and understand how they relate to the coherence properties of the wave or particle under consideration.

This note is a part of the Physics Repository.

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