Lecture

Coherent States: Quantum Optics

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Description

This lecture introduces coherent states in quantum optics, which are eigenstates of the annihilation operator. The lecture covers the definition of coherent states, their properties, and their relation to Fock states. It also discusses the displacement operator, the Baker-Campbell-Hausdorff formula, and the dynamics introduced by Schroedinger's equation. The lecture concludes with the time evolution of Fock states and coherent states, emphasizing their usefulness in quantum metrology.

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Physics

Quantum mechanics: Mathematical formulation of quantum mechanics

Related concepts (32)

Displacement operator

In the quantum mechanics study of optical phase space, the displacement operator for one mode is the shift operator in quantum optics, where is the amount of displacement in optical phase space, is the complex conjugate of that displacement, and and are the lowering and raising operators, respectively. The name of this operator is derived from its ability to displace a localized state in phase space by a magnitude . It may also act on the vacuum state by displacing it into a coherent state.

Fock state

In quantum mechanics, a Fock state or number state is a quantum state that is an element of a Fock space with a well-defined number of particles (or quanta). These states are named after the Soviet physicist Vladimir Fock. Fock states play an important role in the second quantization formulation of quantum mechanics. The particle representation was first treated in detail by Paul Dirac for bosons and by Pascual Jordan and Eugene Wigner for fermions.

Quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.

Creation and annihilation operators

Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually denoted ) lowers the number of particles in a given state by one. A creation operator (usually denoted ) increases the number of particles in a given state by one, and it is the adjoint of the annihilation operator.

Quantum state

In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a quantum mechanical prediction for the system represented by the state. Knowledge of the quantum state together with the quantum mechanical rules for the system's evolution in time exhausts all that can be known about a quantum system. Quantum states may be defined in different ways for different kinds of systems or problems.

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