Chapter VI: Appendices
6.1 Cluster Expansion
In this section, we derive the Floquet-Redfield equation (FRE) and simpler Floquet-Boltzmann (FBE) using the cluster expansion approach discussed in the textbook Semiconductor Quantum Opticsby Mackillo Kira and Stephan Koch (2012).
Clusters
The cluster expansion is a systematic decomposition of an N “object” expectation value hNi. An object is defined a single boson operator (e.g. b,b†obeying canonical commutation relations) or a fermion bilinear operator (e.g. c†cwhere c,c† obey canonical anti-commutation relations). This definition is consistent with spin-statistics in that a fermion bilinear has bosonic properties. Fermion occupation numbers/polarizationshci†cjior single boson expectation valueshbki,hb†ki(wherei,j,k index relevant degrees of freedom in the problem) are examples of singlets (single objects) which are schematically denotedh1i. Doublets (2 objects), schematically denotedh2i, include expectation values such as fermion density-density hci†cic†jcji, boson bilinears hb†jbki, and mixed terms like hci†cjb†ki, etc. Triplets (3 objects), schematically denoted as h3i, include expectation values such ashb†ibjbki, hci†c†jc†kclcmcni, hck†clb†ibji, and so on.
With this definition, one can define a “correlated cluster” ofN objects,∆hNi, which measures the uniquecorrelation among theNobjects that is not decomposable into correlations between smaller sub-clusters ofN−1,N−2, ...etc. Formally, we write
hNi =
N−1Õ
J=1
hNiJ +∆hNi (6.1)
wherehNiJ denotes factorization ofhNiinto clusters of max sizeJ. For example,
h1i ≡ h1i1 ≡∆h1i
h2i = h2i1+∆h2i ∼ h1ih1i+∆h2i
h3i = h3i1+h3i2+∆h3i ∼ h1ih1ih1i+h1i∆h2i+∆h3i hNi = hNi1+[hN−2i1∆h2i+hN−4i1∆h2i∆h2i+...]2
+[hN−3i1∆h3i+ hN−5i1∆h3i∆h2i+...]3+
N−1
Õ
J=4
hNiJ +∆hNi
where[·]Jcontains terms which have max sizeJ. In terms of increasing complexity, singlets encode single-particle physics for fermions and coherent (classical) states of bosons, doublets encode interacting pairs/2-particle bound states, triplets encode three-particle correlated objects/bound states, and so on for highern-droplets.
Explicitly,h·i1denotes factorization of the expectation value into singlets,h·i2denotes factorization of the expectation values into singlets and doublets, and so on. In this light, hNi1is also denoted hNiSfor a singlet factorization, hNi2is also denotedhNiSDfor single/doublet factorization and so on. For example, fermion/boson doublets, can be broken into singlets h2i1 ∼ h1ih1iin a manner that respects their antisymmetry/symmetry as follows
hck†
1c†k
2ck3ck4i1 = hc†k
1ck4ihc†k
2ck3i − hc†k
1ck3ihc†k
2ck4i hbi†b†ji1 = hb†iihb†ji
hc†k
1ck2bii1 = hc†k
1ck2ihbii
In general, a fully singlet factorizationhNi1= h1i...h1i(Nterms ofh1i) for bosons is just a product
hbi1...biNi1 = hbi1i...hbiNi (6.2) and for fermions is given by a Slater determinant for fermions
hc†k
1...c†k
Nck˜N...ck˜1i1 = Det[M] Mjl = hc†k
jckli (6.3)
For mixed fermion/boson expectation values, one just fully anti-symmetrizes the fermion bilinears upon normal ordering as per Eq.6.3 after extracting the boson singlets as in Eq.6.2.
Cluster Dynamics
Given a many-body quantum Hamiltonian of interestH, anN object expectation value (we just use N as the operator representing this below), evolves under the Ehrenfest theorem as
i d
dthNi = h[N,H]i+h∂tNi
i∂thNi = L[hNi]+Y[hN +1i] (6.4) where L[hNi] is the functional that contains terms with same number of objects andY[hN +1i]
contains terms with an additional object, generated by any interaction/non-quadratic terms in H.
By iterating the equations of motion for each subsequent correlation, hNi,hN +1i,hN + 2i, ..., an infinite (for thermodynamically large systems) hierarchy of differential equations appears. In general, a many-body system has nonzero initial expectation values for sets of observables in each object number sector. For example, a free-fermion system is fully determined by Slater determinant (fully-antisymmetrized) single particle states which have nonzero expectation values for any sector; higher sectors are factorizable by Wick’s theorem into single particle expectations.
Hence, truncating the hierarchy in Eq.6.4 arbitrarily at some ¯N is an uncontrolled and inaccurate approximation that does not allow for physical initial conditions.
In order to overcome this, one should consider the dynamics of correlated clusters which evolve as
i∂t∆hNi = L[∆hNi]+Y[∆hN +1i]
+
N
Õ
n=1
V1[∆hni∆hN −ni]+
N
Õ
n=1
V2[∆hni∆hN +1−ni]
+
N−1
Õ
n=1 N−n
Õ
m=1
V3[∆hni∆hmi∆hN+1−n−mi] (6.5) whereL,Y are functionals as in Eq.6.4 andV1,V2,V3are functionals containingnonlinearterms of products of various two or three cluster sizes. Clusters have initial conditions that are markedly different fromNobject expectation values. In general, a physical system, such as weakly-interacting fermions, can begin with a physical initial condition of just singlet correlations (a gas or plasma).
At this point there are no higher order correlations and soY terms do not contribute. It is the nonlinear termsV1,V2,V3that source doublets. After some time horizon, doublets become sizeable and their nonlinear terms source triplets and so on. Higher order clusters/correlations are developed
sequentially in time and hence truncating the cluster expansion encodes, physically, the evolution of the system upto a finite time horizon correctly described by the clusters upto the truncation size. In many physical cases, such as a weakly-interacting gas, singlets and possibly doublets accurately describe the physics even at very long times, and so the cluster expansion becomes a useful description for arbitrarily long time horizons. Note that the cluster expansion is a variant of the cumulant expansion where, in the usual scenario, cumulants are the connected correlation functions of various sizes. The cluster expansion enjoys the same spirit as the Martin-Schwinger hierarchy for nonequilibrium Green’s functions.
Scattering Approximation
Suppose we are interested in the dynamics of singlets and doublets while excluding the formation of triplets.
i∂t∆h1i = L[∆h1i]+Y[∆h2i]+V2[∆h1i∆h1i]
i∂t∆h2i = L[∆h2i]+V1[∆h1i∆h1i]+V2[∆h1i∆h2i]+V3[∆h1i∆h1i∆h1i]
where we have ignoredY[∆h3i] by the assumption of no triplets. Introducing an approximation that assumes doublets are weak for the time horizon of interest,
L[∆h2i]+V2[∆h1i∆h2i] → (EM F −iγ)∆h2i
whereEM F is some mean-field energy andγis a phenomenological dephasing constant for doublets (presumably due to the formation of triplets), we get
i∂t∆h2i = (EM F −iγ)∆h2i+V1[∆h1i∆h1i]+V3[∆h1i∆h1i∆h1i]
which is easily solved
∆h2i(t) = 1 i~
∫ t
−∞
dze~i(EM F−iγ)(z−t)(V1[∆h1i∆h1i]+V3[∆h1i∆h1i∆h1i]) Under the standard Markov approximation (see below sections for more detail),
∆h2iscat ≈ −V1[∆h1i∆h1i]+V3[∆h1i∆h1i∆h1i]
EM F −iγ
This expression can be substituted into the singlet dynamics
i∂t∆h1i = L[∆h1i]+Y[∆h2iscat]+V2[∆h1i∆h1i] (6.6) to acheive a closed differential equation in terms of pure singlets. This level of approximation, which treats doublets at the scattering level, is between the pure singlet level, and the full singlet- doublet dynamics. It takes into account the contribution of 2-singlet scattering to singlet dynamics but cannot capture the formation of true 2-object correlations (e.g. bound states).
Alternatively, one may write, using the the definition of a cluster,
i∂t∆h2i = i∂th2i −i∂th2iS
≈ (i∂th2i)S− (i∂th2iS)S (6.7) where in the second line, we have factorized i∂t∆h2i in terms of singlets and so this can be integrated formally to obtain ∆h2i. We will also denote this as ∆h2iscat as it acheives a similar closed differential equation for singlets upon insertion intoi∂t∆h1ias before in Eq.6.6. Equation 6.6 may also be rewritten as
i∂th1i = L[h1i]+Y[∆h2iscat+h2iS] (6.8) using Eq.6.4. We will use this result to compute the Floquet singlet dynamics at the scattering level.