The cluster-expansion approach is a technique in quantum mechanics that systematically truncates the BBGKY hierarchy problem that arises when quantum dynamics of interacting systems is solved. This method is well suited for producing a closed set of numerically computable equations that can be applied to analyze a great variety of many-body and/or quantum-optical problems. For example, it is widely applied in semiconductor quantum optics and it can be applied to generalize the semiconductor Bloch equations and semiconductor luminescence equations.
Background
Quantum theory essentially replaces classically accurate values by a probabilistic distribution that can be formulated using, e.g., a wavefunction, a density matrix, or a phase-space distribution. Conceptually, there is always, at least formally, probability distribution behind each observable that is measured. Already in 1889, a long time before quantum physics was formulated, Thorvald N. Thiele proposed the cumulants that describe probabilistic distributions with as few quantities as possible; he called them half-invariants. The cumulants form a sequence of quantities such as mean, variance, skewness, kurtosis, and so on, that identify the distribution with increasing accuracy as more cumulants are used.
The idea of cumulants was converted into quantum physics by Fritz Coester
and Hermann Kümmel
with the intention of studying nuclear many-body phenomena. Later, Jiři Čížek and Josef Paldus extended the approach for quantum chemistry in order to describe many-body phenomena in complex atoms and molecules. This work introduced the basis for the coupled-cluster approach that mainly operates with many-body wavefunctions. The coupled-clusters approach is one of the most successful methods to solve quantum states of complex molecules.
In solids, the many-body wavefunction has an overwhelmingly complicated structure such that the direct wave-function-solution techniques are intractable. The cluster expansion is a variant of the coupled-clusters approach and it solves the dynamical equations of correlations instead of attempting to solve the quantum dynamics of an approximated wavefunction or density matrix. It is equally well suited to treat properties of many-body systems and quantum-optical correlations, which has made it very suitable approach for semiconductor quantum optics.
Like almost always in many-body physics or quantum optics, it is most convenient to apply the second-quantization formalism to describe the physics involved. For example, a light field is then described through Boson creation and annihilation operators and , respectively, where defines the momentum of a photon. The "hat" over signifies the operator nature of the quantity. When the many-body state consists of electronic excitations of matter, it is fully defined by Fermion creation and annihilation operators and , respectively, where refers to particle's momentum while is some internal degree of freedom, such as spin or band index.
Classification of N-particle contributions
When the many-body system is studied together with its quantum-optical properties, all measurable expectation values can be expressed in the form of an N-particle expectation value
where and while the explicit momentum indices are suppressed for the sake of briefness. These quantities are normally ordered, which means that all creation operators are on the left-hand side while all annihilation operators are on the right-hand side in the expectation value. It is straight forward to show that this expectation value vanishes if the amount of Fermion creation and annihilation operators are not equal.
Once the system Hamiltonian is known, one can use the Heisenberg equation of motion to generate the dynamics of a given -particle operator. However, the many-body as well as quantum-optical interactions couple the -particle quantities to -particle expectation values, which is known as the Bogolyubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy problem. More mathematically, all particles interact with each other leading to an equation structure
where functional symbolizes contributions without hierarchy problem and the functional for hierarchical (Hi) coupling is symbolized by . Since all levels of expectation values can be nonzero, up to the actual particle number, this equation cannot be directly truncated without further considerations.