Light is a fascinating phenomenon that bridges classical and quantum physics. In classical optics, light behaves as electromagnetic waves, while quantum optics reveals its particle-like nature as photons. This duality leads to unique properties and behaviors that challenge our everyday understanding of light.

Quantum light exhibits phenomena like entanglement, non-classical correlations, and discrete energy levels. These properties form the foundation for cutting-edge technologies in quantum communication, computing, and sensing, pushing the boundaries of what's possible with light manipulation and information processing.

Classical vs Quantum Light

Properties of Classical and Quantum Light

• Classical light is described by Maxwell's equations as electromagnetic waves, while quantum light is described by photons, which are quantized particles of light
• Classical light exhibits properties such as intensity, wavelength, and polarization, while quantum light exhibits properties such as photon number, photon energy, and photon polarization
• Classical light can have any energy value, while quantum light has discrete energy levels determined by the frequency of the photons ($E = hν$)
• Classical light can be described by continuous wave functions, while quantum light is described by discrete, probabilistic wave functions

Unique Phenomena in Quantum Light

• Classical light obeys the superposition principle, while quantum light exhibits unique phenomena such as entanglement and non-classical correlations
• Entanglement occurs when two or more quantum systems (photons) are correlated in such a way that the state of one system cannot be described independently of the others, even when the systems are separated by large distances
• Non-classical correlations in quantum light, such as photon bunching and antibunching, cannot be explained by classical electromagnetic theory and require a quantum description
• Quantum light can exhibit sub-Poissonian statistics, where the fluctuations in photon number are smaller than those predicted by classical statistics (Poissonian)
• Squeezed states of light, where the uncertainty in one quadrature (phase or amplitude) is reduced below the standard quantum limit at the expense of increased uncertainty in the other quadrature, are a unique feature of quantum light

Photons in Quantum Optics

Properties of Photons

• Photons are the fundamental particles of light in quantum optics, carrying energy, momentum, and angular momentum
• The energy of a photon is given by $E = hν$, where $h$ is Planck's constant and $ν$ is the frequency of the light
• Photons exhibit wave-particle duality, displaying both wave-like and particle-like properties depending on the measurement context
• Photons have a spin of 1 and are classified as bosons, which means they follow Bose-Einstein statistics and can occupy the same quantum state

Photon Interactions with Matter

• Photons can be created and annihilated through processes such as emission and absorption by atoms or molecules
• The interaction between photons and matter is the foundation of many quantum optical phenomena, such as spontaneous and stimulated emission, absorption, and scattering
• Spontaneous emission occurs when an atom or molecule in an excited state emits a photon and transitions to a lower energy state without external stimulation
• Stimulated emission occurs when an incoming photon interacts with an atom or molecule in an excited state, causing it to emit an additional photon with the same frequency, phase, and direction as the incoming photon (the basis for lasers)
• Absorption occurs when a photon is absorbed by an atom or molecule, causing it to transition to a higher energy state
• Scattering processes, such as Rayleigh, Raman, and Compton scattering, involve the interaction between photons and matter, leading to changes in the photon's energy, frequency, or direction

Photon Entanglement

• Photons can be entangled, exhibiting non-classical correlations that are essential for quantum communication and computation
• Entangled photons can be generated through processes such as spontaneous parametric down-conversion (SPDC) or four-wave mixing (FWM)
• Entangled photon pairs exhibit correlations in properties such as polarization, energy, and momentum, which cannot be explained by classical theories
• Bell states, such as $|\Phi^+⟩ = \frac{1}{\sqrt{2}}(|HH⟩ + |VV⟩)$, are maximally entangled two-photon states that form the basis for many quantum communication protocols (quantum key distribution)
• Entanglement swapping allows the creation of entanglement between two photons that have never interacted directly, by performing a joint measurement on two other entangled photons

Wave-Particle Duality of Light

Experimental Evidence

• Wave-particle duality is the concept that light exhibits both wave-like and particle-like properties, depending on the measurement context
• The double-slit experiment demonstrates the wave-like nature of light, showing interference patterns when light passes through two slits
• In the double-slit experiment, the interference pattern is observed even when photons are sent through the slits one at a time, indicating that each photon interferes with itself
• The photoelectric effect demonstrates the particle-like nature of light, showing that light is absorbed and emitted in discrete packets called photons
• In the photoelectric effect, the kinetic energy of the emitted electrons depends on the frequency of the incident light, not its intensity, supporting the photon hypothesis

Implications and Applications

• The de Broglie wavelength ($λ = h/p$) relates the wavelength of a particle to its momentum, highlighting the wave-particle duality of matter as well
• The wave-particle duality of light has implications for the interpretation of quantum mechanics, leading to concepts such as the Copenhagen interpretation and the role of measurement in determining the state of a quantum system
• The Copenhagen interpretation asserts that the wave function represents the probability distribution of the particle's position and that the act of measurement collapses the wave function, forcing the particle into a definite state
• The wave-particle duality also has practical implications for the design and operation of quantum optical devices, such as single-photon sources and detectors
• Single-photon sources, such as quantum dots or nitrogen-vacancy centers in diamond, exploit the particle-like nature of photons to generate individual photons on demand
• Single-photon detectors, such as avalanche photodiodes or superconducting nanowire detectors, rely on the particle-like nature of photons to detect individual photons with high efficiency and low noise

Coherence and Interference in Quantum Optics

Coherence Properties

• Coherence refers to the ability of light to exhibit a fixed phase relationship between different points in space and time
• Temporal coherence is related to the monochromaticity of the light source, with longer coherence times corresponding to more monochromatic light
• The coherence time ($τ_c$) is inversely proportional to the spectral bandwidth ($Δν$) of the light source: $τ_c ≈ 1/Δν$
• Spatial coherence is related to the size of the light source, with smaller sources producing more spatially coherent light
• The spatial coherence area ($A_c$) is inversely proportional to the solid angle ($Ω$) subtended by the source: $A_c ≈ λ^2/Ω$

Quantum Interference Phenomena

• Interference occurs when two or more coherent light waves overlap, resulting in a pattern of constructive and destructive interference
• Quantum interference can be observed in experiments such as the Hong-Ou-Mandel effect, where two indistinguishable photons entering a beam splitter always exit together, demonstrating the bosonic nature of photons
• In the Hong-Ou-Mandel effect, when two identical photons simultaneously enter a 50:50 beam splitter, they always exit the beam splitter together, either both in one output port or both in the other, due to destructive interference of the probability amplitudes for the photons to exit in different output ports
• Quantum interference is the foundation for many quantum information processing tasks, such as linear optical quantum computing (LOQC) and boson sampling
• In LOQC, quantum gates are implemented using linear optical elements (beam splitters and phase shifters) and single-photon sources, relying on quantum interference to perform the desired transformations on the photonic qubits
• Boson sampling is a computational task that exploits the quantum interference of multiple indistinguishable photons in a linear optical network to solve problems that are believed to be intractable for classical computers

Applications of Coherence and Interference

• Coherence and interference are crucial for applications such as quantum cryptography, quantum metrology, and quantum imaging, where the unique properties of quantum light are exploited to achieve enhanced performance or security
• In quantum key distribution (QKD), the coherence and interference properties of single photons are used to establish a secure encryption key between two parties, with the security guaranteed by the laws of quantum mechanics (BB84 protocol)
• Quantum metrology exploits the sensitivity of quantum interference to small changes in phase or other parameters to achieve ultra-precise measurements, surpassing the limits of classical metrology (LIGO gravitational wave detector)
• Quantum imaging techniques, such as ghost imaging and quantum illumination, rely on the spatial correlations and interference of entangled photon pairs to image objects with improved resolution, sensitivity, or noise reduction compared to classical imaging techniques