Fock state

Fock state. In quantum mechanics and quantum field theory, a Fock state is a specific quantum state that contains a definite, well-defined number of particles. It is a fundamental concept in the formalism of second quantization, providing the basis for describing systems with variable particle numbers, such as in quantum optics and condensed matter physics. These states are the eigenstates of the particle number operator and are named after the Soviet physicist Vladimir Fock.

Definition and mathematical representation

In the formalism of second quantization, a Fock state is defined within a Fock space, which is constructed as the direct sum of tensor products of single-particle Hilbert spaces. For a system of identical bosons, the state is symmetric under particle exchange and is represented using creation operators \(a^\dagger\) acting on the vacuum state \(|0\rangle\). For example, a state with \(n\) particles in a single mode is written as \(|n\rangle = \frac{(a^\dagger)^n}{\sqrt{n!}} |0\rangle\). For fermions, which obey the Pauli exclusion principle, the construction uses anticommutation relations and Slater determinants to ensure antisymmetry. The entire framework is central to quantum electrodynamics and the description of photons in cavity quantum electrodynamics.

Properties

A key property is that Fock states are eigenstates of the number operator \(\hat{N} = a^\dagger a\), satisfying \(\hat{N} |n\rangle = n |n\rangle\), where \(n\) is a non-negative integer. They form a complete, orthonormal basis for the Fock space, with \(\langle m | n \rangle = \delta_{mn}\). Their phase is completely undefined, as they are invariant under U(1) transformations, leading to a zero expectation value for the field quadratures \(\langle \hat{X} \rangle = \langle \hat{P} \rangle = 0\). However, they exhibit particle number fluctuations of zero, \(\Delta n = 0\), while the quadrature variances are unequal, yielding a noise distribution that does not satisfy the minimum Heisenberg uncertainty principle for the electric field, a signature of non-classical light.

Generation and detection

Experimentally generating single-photon Fock states, or \(|1\rangle\) states, is a cornerstone of quantum information science. Common methods include the spontaneous emission from a single quantum emitter, such as an atom, quantum dot, or nitrogen-vacancy center, triggered by a pulsed excitation laser. Another prominent technique is spontaneous parametric down-conversion in nonlinear crystals like beta barium borate, which can produce correlated photon pairs. Detection relies heavily on single-photon avalanche photodiodes and transition edge sensors. In circuit quantum electrodynamics, microwave photon Fock states can be engineered and probed using superconducting qubits coupled to coplanar waveguide resonators, as demonstrated in laboratories like Yale University and the University of California, Santa Barbara.

Applications

Fock states are indispensable in technologies that require precise photon-number resolution. They serve as the ideal carriers for quantum key distribution protocols, such as BB84, enhancing security against photon number splitting attacks. In quantum computing and quantum simulation, they are used as logical basis states in bosonic encoding schemes and for simulating complex many-body physics in systems like the Bose–Hubbard model. Their non-classical statistics are crucial for surpassing the standard quantum limit in quantum metrology and quantum sensing, enabling more precise measurements of quantities like phase or amplitude in LIGO-style interferometers.

Relationship to other quantum states

Fock states are a specific point in a broader landscape of quantum states. A coherent state, as described by Roy Glauber, is a superposition of Fock states with a Poisson distribution of photon numbers and represents the classical limit of a laser field. A squeezed coherent state reduces noise in one quadrature below the vacuum level at the expense of the other, a phenomenon studied by Carlton Caves. The thermal state or chaotic light, described by a Bose–Einstein distribution over Fock states, represents blackbody radiation. Furthermore, Schrödinger cat states are macroscopic superpositions of coherent states, which can also be expressed in the Fock basis, highlighting their connection to foundational tests of quantum decoherence and the quantum measurement problem.

Category:Quantum mechanics Category:Quantum optics Category:Quantum field theory