Cluster-expansion approach
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.
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, a 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 idea of cumulants was converted into quantum physics by Fritz Coester[3] and Hermann Kümmel[4] 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[1][5] 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 a very suitable approach for semiconductor quantum optics.
For example, a light field is then described through Boson creation and annihilation operators
When the many-body state consists of electronic excitations of matter, it is fully defined by Fermion creation and annihilation operators
It is straight forward to show that this expectation value vanishes if the amount of Fermion creation and annihilation operators are not equal.
[6][7] Once the system Hamiltonian is known, one can use the Heisenberg equation of motion to generate the dynamics of a given
Since all levels of expectation values can be nonzero, up to the actual particle number, this equation cannot be directly truncated without further considerations.
The hierarchy problem can be systematically truncated after identifying correlated clusters.
At the lowest level, one finds the class of single-particle expectation values (singlets) that are symbolized by
that contains a formal sum over all possible products of single-particle expectation values.
Physically, the singlet factorization among Fermions produces the Hartree–Fock approximation while for Bosons it yields the classical approximation where Boson operators are formally replaced by a coherent amplitude, i.e.,
As this identification is applied recursively, one may directly identify which correlations appear in the hierarchy problem.
Obviously, introducing clusters cannot remove the hierarchy problem of the direct approach because the hierarchical contributions remains in the dynamics.
This property and the appearance of the nonlinear terms seem to suggest complications for the applicability of the cluster-expansion approach.
However, as a major difference to a direct expectation-value approach, both many-body and quantum-optical interactions generate correlations sequentially.
is typically much smaller than the overall particle number, the cluster-expansion approach yields a pragmatic and systematic solution scheme for many-body and quantum-optics investigations.
One possibility is to represent the quantum fluctuations of a quantized light mode
to the density matrix is unique but can result in a numerically diverging series.
This problem can be solved by introducing a cluster-expansion transformation (CET)[9] that represents the distribution in terms of a Gaussian, defined by the singlet–doublet contributions, multiplied by a polynomial, defined by the higher-order clusters.
[10] This property is largely based on CET's ability to describe any distribution in the form where a Gaussian is multiplied by a polynomial factor.
