👀Quantum Optics Unit 5 – Quantized Electromagnetic Field

Key Concepts and Foundations

• Electromagnetic field described by Maxwell's equations governs the behavior of light and its interaction with matter
• Classical electromagnetic field theory treats light as continuous waves propagating through space and time
• Quantum mechanics introduces the concept of quantization, where physical quantities are restricted to discrete values
• Photons are the fundamental quantum particles of light, exhibiting both wave-like and particle-like properties
• Wave-particle duality is a central concept in quantum mechanics, exemplified by the double-slit experiment
• Heisenberg's uncertainty principle sets a fundamental limit on the precision of simultaneous measurements of certain pairs of physical quantities (position and momentum)
• Quantum states are mathematical descriptions of a quantum system, represented by state vectors in a complex Hilbert space
• Observables are physical quantities that can be measured, represented by Hermitian operators acting on the state vectors

Quantization of the Electromagnetic Field

• Quantization procedure involves promoting the classical field variables to quantum operators, satisfying specific commutation relations
• Electric and magnetic field operators are expressed in terms of creation and annihilation operators, which add or remove photons from the field
• Creation operator $\hat{a}^{\dagger}$ adds a photon to the field, while the annihilation operator $\hat{a}$ removes a photon
• Commutation relation between creation and annihilation operators: $[\hat{a}, \hat{a}^{\dagger}] = 1$
  • This commutation relation ensures the bosonic nature of photons and leads to the quantization of energy
• Hamiltonian of the quantized electromagnetic field is expressed in terms of the number operator $\hat{n} = \hat{a}^{\dagger}\hat{a}$
• Energy eigenvalues of the quantized field are discrete, given by $E_n = (n + \frac{1}{2})\hbar\omega$, where $n$ is the number of photons and $\omega$ is the angular frequency
• Quantization of the electromagnetic field provides a framework for describing the interaction of light with matter at the quantum level

Photon States and Fock Space

• Fock states, denoted as $|n\rangle$, represent the quantum states with a definite number of photons
• Fock space is the Hilbert space spanned by the Fock states, providing a convenient basis for describing the quantum states of light
• Vacuum state $|0\rangle$ is the Fock state with zero photons, representing the ground state of the electromagnetic field
• Creation operator acting on a Fock state increases the photon number by one: $\hat{a}^{\dagger}|n\rangle = \sqrt{n+1}|n+1\rangle$
• Annihilation operator acting on a Fock state decreases the photon number by one: $\hat{a}|n\rangle = \sqrt{n}|n-1\rangle$
  • The annihilation operator acting on the vacuum state gives zero: $\hat{a}|0\rangle = 0$
• Photon number operator $\hat{n}$ has Fock states as eigenstates: $\hat{n}|n\rangle = n|n\rangle$
• Fock states form a complete orthonormal basis, satisfying $\langle m|n\rangle = \delta_{mn}$, where $\delta_{mn}$ is the Kronecker delta

Field Operators and Commutation Relations

• Electric and magnetic field operators are expressed in terms of the creation and annihilation operators
  • Electric field operator: $\hat{E}(r, t) \propto \hat{a} e^{-i(\omega t - k \cdot r)} + \hat{a}^{\dagger} e^{i(\omega t - k \cdot r)}$
  • Magnetic field operator: $\hat{B}(r, t) \propto \hat{a} e^{-i(\omega t - k \cdot r)} - \hat{a}^{\dagger} e^{i(\omega t - k \cdot r)}$
• Field operators satisfy the equal-time commutation relations, which are the quantum analog of the classical Poisson brackets
  • $[\hat{E}(r, t), \hat{B}(r', t)] = i\hbar c \nabla \times \delta^{(3)}(r - r')$
  • $[\hat{E}(r, t), \hat{E}(r', t)] = [\hat{B}(r, t), \hat{B}(r', t)] = 0$
• Commutation relations between field operators and creation/annihilation operators:
  • $[\hat{E}(r, t), \hat{a}] = -i\hbar \omega \hat{a} e^{-i(\omega t - k \cdot r)}$
  • $[\hat{E}(r, t), \hat{a}^{\dagger}] = i\hbar \omega \hat{a}^{\dagger} e^{i(\omega t - k \cdot r)}$
• Field operators acting on Fock states create or annihilate photons at specific positions and times
• Commutation relations ensure the compatibility of the quantum description with the classical Maxwell's equations

Vacuum State and Zero-Point Energy

• Vacuum state $|0\rangle$ is the lowest energy state of the quantized electromagnetic field, containing no photons
• Vacuum state is not a state of absolute nothingness; it has a non-zero energy called the zero-point energy
• Zero-point energy arises from the Heisenberg uncertainty principle, which prevents the field from having zero energy
• Vacuum expectation values of the field operators are zero: $\langle 0|\hat{E}(r, t)|0\rangle = \langle 0|\hat{B}(r, t)|0\rangle = 0$
• Vacuum fluctuations are the quantum fluctuations of the electromagnetic field in the vacuum state
  • These fluctuations manifest as virtual photons that constantly appear and disappear
• Casimir effect is a physical manifestation of the vacuum fluctuations, resulting in an attractive force between two uncharged conducting plates
• Lamb shift is a small difference in the energy levels of hydrogen atoms, caused by the interaction of the electron with vacuum fluctuations
• Zero-point energy and vacuum fluctuations have important implications in quantum field theory and cosmology

Coherent States and Squeezed States

• Coherent states, denoted as $|\alpha\rangle$, are quantum states that closely resemble classical electromagnetic waves
  • They are eigenstates of the annihilation operator: $\hat{a}|\alpha\rangle = \alpha|\alpha\rangle$, where $\alpha$ is a complex number
• Coherent states have a well-defined amplitude and phase, and their photon number distribution follows a Poisson distribution
• Displacement operator $\hat{D}(\alpha) = e^{\alpha \hat{a}^{\dagger} - \alpha^* \hat{a}}$ generates coherent states from the vacuum state: $|\alpha\rangle = \hat{D}(\alpha)|0\rangle$
• Coherent states are not orthogonal, but they form an overcomplete basis in the Fock space
• Squeezed states are quantum states in which the uncertainty in one quadrature (amplitude or phase) is reduced below the vacuum level, at the expense of increased uncertainty in the other quadrature
  • Squeezing operator $\hat{S}(\zeta) = e^{\frac{1}{2}(\zeta^* \hat{a}^2 - \zeta \hat{a}^{\dagger 2})}$ generates squeezed states from the vacuum state, where $\zeta$ is the squeezing parameter
• Squeezed states have applications in precision measurements, such as gravitational wave detection, where they can enhance the sensitivity by reducing the quantum noise
• Squeezed light can be generated experimentally using nonlinear optical processes, such as parametric down-conversion or four-wave mixing

Interaction of Light with Matter

• Light-matter interaction is a fundamental process in quantum optics, describing how photons interact with atoms, molecules, and solid-state systems
• Electric dipole approximation is often used to describe the interaction, assuming that the wavelength of light is much larger than the size of the atomic system
• Jaynes-Cummings model is a simplified model of light-matter interaction, considering a single two-level atom interacting with a single mode of the electromagnetic field
  • The model Hamiltonian includes the atomic energy, field energy, and the dipole interaction term: $\hat{H} = \hbar \omega_a \hat{\sigma}_z + \hbar \omega \hat{a}^{\dagger}\hat{a} + \hbar g (\hat{a}^{\dagger}\hat{\sigma}_- + \hat{a}\hat{\sigma}_+)$
• Rabi oscillations occur when an atom is driven by a resonant electromagnetic field, resulting in periodic oscillations between the ground and excited states
• Purcell effect describes the modification of the spontaneous emission rate of an atom when it is placed in a cavity or near a nanostructure
• Cavity quantum electrodynamics (cavity QED) studies the interaction of atoms with the quantized electromagnetic field in high-finesse cavities
  • Strong coupling regime is achieved when the atom-field coupling strength exceeds the decay rates of the atom and the cavity
• Optomechanics explores the interaction between light and mechanical systems, such as nanomechanical resonators or levitated nanoparticles
  • Radiation pressure force and optomechanical coupling enable the control and manipulation of mechanical motion using light

Applications and Experimental Techniques

• Quantum cryptography uses the principles of quantum mechanics to ensure secure communication
  • BB84 protocol is a widely used quantum key distribution scheme that relies on the encoding of information in the polarization states of single photons
• Quantum computing harnesses the properties of quantum systems, such as superposition and entanglement, to perform computations
  • Photonic quantum computing uses photons as qubits, exploiting their low decoherence and ease of manipulation
• Quantum metrology aims to enhance the precision of measurements by utilizing quantum resources, such as entangled states or squeezed states
  • Quantum-enhanced sensing can surpass the classical shot-noise limit and approach the Heisenberg limit
• Quantum imaging techniques, such as ghost imaging and quantum illumination, exploit the correlations between photons to obtain images with improved resolution or sensitivity
• Quantum simulation uses well-controlled quantum systems to simulate the behavior of complex quantum systems that are difficult to study directly
  • Photonic quantum simulators can emulate various quantum phenomena, such as topological phases or many-body physics
• Experimental techniques in quantum optics include:
  • Single-photon sources and detectors for generating and measuring individual photons
  • Quantum state tomography for reconstructing the quantum state of a system from a set of measurements
  • Homodyne and heterodyne detection for measuring the quadratures of the electromagnetic field
  • Quantum interference and Hong-Ou-Mandel effect for demonstrating the bosonic nature of photons and their indistinguishability