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[1] and it can be applied to generalize the semiconductor Bloch equations and semiconductor luminescence equations.
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.[2] 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[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[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 very suitable approach for semiconductor quantum optics.
\dagger | |
\hat{B} | |
q |
\hat{B}q
\hbarq
B
\dagger | |
\hat{a} | |
λ,k |
\hat{a}λ,k
\hbark
λ
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
\langle\hat{N}\rangle\equiv\langle
\dagger | |
\hat{B} | |
1 |
…
\dagger | |
\hat{B} | |
K |
\dagger | |
\hat{a} | |
1 |
…
\dagger | |
\hat{a} | |
N\hat{a |
where
N=N\hat{B
N\hat{B
Once the system Hamiltonian is known, one can use the Heisenberg equation of motion to generate the dynamics of a given
N
N
(N+1)
i\hbar
\partial | |
\partialt |
\langle\hat{N}\rangle=T\left[\langle\hat{N}\rangle\right]+Hi\left[\langle\hat{N}+1\rangle\right]
T
Hi[\langle\hat{N}+1\rangle]
The hierarchy problem can be systematically truncated after identifying correlated clusters. The simplest definitions follow after one identifies the clusters recursively. At the lowest level, one finds the class of single-particle expectation values (singlets) that are symbolized by
\langle1\rangle
\langle2\rangle
\langle2\rangleS=\langle1\rangle\langle1\rangle
\langle1\rangle
\langleN\rangleS
N
\hat{B} → \langle\hat{B}\rangle
The correlated part of
\langle2\rangle
\langle2\rangle
\langle2\rangleS
\langle2\rangle=\langle2\rangleS+\Delta\langle2\rangle
where the
\Delta
\Delta\langle2\rangle=\langle2\rangle-\langle2\rangleS
\begin{align} \langle3\rangle&=\langle3\rangleS+\langle1\rangle \Delta\langle2\rangle+\Delta\langle3\rangle,\\ \langleN\rangle&=\langleN\rangleS\\ & +\langleN-2\rangleS \Delta\langle2\rangle\\ & +\langleN-4\rangleS \Delta\langle2\rangle \Delta\langle2\rangle+...\\ & +\langleN-3\rangleS \Delta\langle3\rangle\\ & +\langleN-5\rangleS \Delta\langle3\rangle \Delta\langle2\rangle+...\\ & +\Delta\langleN\rangle, \end{align}
where each product term represents one factorization symbolically and implicitly includes a sum over all factorizations within the class of terms identified. The purely correlated part is denoted by
\Delta\langleN\rangle
\Delta\langle2\rangle
\Delta\langle3\rangle
As this identification is applied recursively, one may directly identify which correlations appear in the hierarchy problem. One then determines the quantum dynamics of the correlations, yielding
i\hbar
\partial | |
\partialt |
\Delta\langle\hat{N}\rangle=T\left[\Delta\langle\hat{N}\rangle\right]+NL\left[\langle\hat{1}\rangle,\Delta\langle\hat{2}\rangle, … ,\Delta\langle\hat{N}\rangle\right] + Hi\left[\Delta\langle\hat{N}+1\rangle\right],
where the factorizations produce a nonlinear coupling
NL\left[ … \right]
However, as a major difference to a direct expectation-value approach, both many-body and quantum-optical interactions generate correlations sequentially.[8] In several relevant problems, one indeed has a situation where only the lowest-order clusters are initially nonvanishing while the higher-order clusters build up slowly. In this situation, one can omit the hierarchical coupling,
Hi\left[\Delta\langle\hat{C}+1\rangle\right]
C
C
C
Besides describing quantum dynamics, one can naturally apply the cluster-expansion approach to represent the quantum distributions. One possibility is to represent the quantum fluctuations of a quantized light mode
\hat{B}
\langle[\hat{B}\dagger]J\hat{B}K\rangle
\langle[\hat{B}\dagger]J\hat{B}K\rangle
This completely mathematical problem has a direct physical application. One can apply the cluster-expansion transformation to robustly project classical measurement into a quantum-optical measurement.[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. This technique is already being used to access and derive quantum-optical spectroscopy from a set of classical spectroscopy measurements, which can be performed using high-quality lasers.