Coherent states are quantum states that behave most like classical waves. They're crucial in quantum optics, describing laser light and minimizing uncertainty in position and momentum simultaneously. This makes them ideal for studying the boundary between quantum and classical physics.

In the quantum harmonic oscillator, coherent states evolve in time while maintaining their shape. They follow classical-like trajectories, with expectation values of position and momentum oscillating sinusoidally. This behavior bridges quantum and classical descriptions of oscillatory systems.

Properties of Coherent States

Mathematical Representation and Key Characteristics

• Coherent states exhibit classical-like behavior and minimize the uncertainty principle in quantum mechanics
• Mathematical representation given by $|α⟩ = e^(-|α|^2/2) Σ (α^n / √n!) |n⟩$, where α represents a complex number and |n⟩ denotes number states
• Function as eigenstates of the annihilation operator â, satisfying $â|α⟩ = α|α⟩$
• Position and momentum operators' expectation values follow classical trajectories in coherent states
• Form an overcomplete basis without orthogonality to each other
• Probability distribution for finding a coherent state in a number state follows a Poisson distribution
• Generated by applying the displacement operator $D(α) = e^(α↠- αâ)$ to the ground state |0⟩

Uncertainty and Probability Properties

• Maintain minimum uncertainty in both position and momentum simultaneously
• Uncertainty remains constant over time, preserving the minimum uncertainty relationship
• Probability distribution of photon number measurements follows a Poisson distribution with mean |α|^2
• Exhibit a well-defined phase and amplitude, unlike number states with definite excitation numbers
• Possess a continuous spectrum of possible measurement outcomes, contrasting with the discrete spectrum of number states

Physical Interpretation and Visualization

• Represent quantum states closest to classical oscillatory motion in various systems (harmonic oscillators, electromagnetic fields)
• Visualized as wave packets maintaining their shape while oscillating in a harmonic potential
• Serve as displaced vacuum states, differing from thermal states (mixed states with number state distributions)
• Bridge quantum and classical descriptions of systems, facilitating studies of quantum-classical transitions
• Provide a quantum description of laser light, connecting the coherent state amplitude α to classical electric field amplitude

Coherent States in the Harmonic Oscillator

Time Evolution and Dynamics

• Time evolution in a quantum harmonic oscillator described by $|α(t)⟩ = |αe^(-iωt)⟩$, where ω represents the oscillator's angular frequency
• Position and momentum expectation values oscillate sinusoidally with time, mimicking classical behavior
• Maintain constant uncertainty in position and momentum over time, preserving minimum uncertainty
• Energy expectation value given by $⟨E⟩ = ℏω(|α|^2 + 1/2)$
• Not stationary states, evolving in time even for time-independent Hamiltonians

Comparison with Classical Behavior

• Most closely resemble classical oscillatory motion among quantum states in the harmonic oscillator
• Position and momentum expectation values follow classical trajectories
• Maintain constant shape of the wave packet while oscillating, unlike other quantum states
• Exhibit minimum uncertainty in both position and momentum simultaneously, resembling a classical point particle

Advanced Concepts and Modifications

• Squeezed coherent states can be created to further reduce uncertainty in one quadrature at the expense of increased uncertainty in the conjugate quadrature
• Schrödinger cat states formed by superpositions of coherent states with opposite phases, exhibiting quantum interference effects
• Dynamics of coherent states in the harmonic oscillator provide insights into the quantum-classical transition
• Serve as a basis for studying more complex quantum states and their behavior in harmonic potentials

Coherent States vs Other Quantum States

Comparison with Number States and Energy Eigenstates

• Coherent states possess well-defined phase and amplitude, while number states (Fock states) have definite excitation numbers
• Unlike energy eigenstates, coherent states are not stationary and evolve in time for time-independent Hamiltonians
• Coherent states exhibit a continuous spectrum of possible measurement outcomes, contrasting with the discrete spectrum of number states
• Number states have definite energy, while coherent states have a distribution of energies centered around $⟨E⟩ = ℏω(|α|^2 + 1/2)$

Uncertainty Properties and Squeezed States

• Coherent states maintain minimum uncertainty in both position and momentum simultaneously
• Squeezed states reduce uncertainty in one variable at the expense of increased uncertainty in the conjugate variable
• Thermal states are mixed states with a distribution of number states, unlike coherent states which are pure states
• Coherent states can be considered as displaced vacuum states, while thermal states represent a statistical mixture of number states

Quantum Superpositions and Interference

• Schrödinger cat states form superpositions of coherent states with opposite phases, exhibiting quantum interference
• Cat states demonstrate quantum superposition principles not present in individual coherent states
• Coherent states serve as building blocks for creating more complex quantum states through superposition and manipulation
• Interference effects in superpositions of coherent states reveal fundamental aspects of quantum mechanics (wave-particle duality)

Coherent States for Describing Light

Quantum Description of Laser Light

• Coherent states provide a quantum description of laser light, with photon number uncertainty following a Poisson distribution
• Amplitude α of a coherent state relates to the classical electric field amplitude, bridging quantum and classical electromagnetic descriptions
• Describe the output of a single-mode laser operating well above threshold
• Quadrature operators X̂ and P̂ for coherent states correspond to in-phase and out-of-phase components of the electromagnetic field

Quantum Optics Phenomena and Experiments

• Coherent states of light exhibit photon antibunching, a purely quantum effect absent in classical electromagnetic theory
• Play crucial roles in quantum optics experiments (homodyne detection, quantum state tomography)
• Displacement operator D(α) applied to the vacuum state interprets as a classical current source creating a coherent state of light
• Serve as reference states in quantum optics, allowing measurement and characterization of non-classical light states

Applications in Quantum Technologies

• Form the basis for continuous-variable quantum information processing in optical systems
• Enable quantum-enhanced metrology and sensing applications, improving precision beyond classical limits
• Facilitate the study of light-matter interactions in cavity quantum electrodynamics (cavity QED)
• Provide a framework for developing quantum communication protocols using coherent state encoding