Quantum Distribution-function Transport Equations in Non-normal Systems and in Ultra-fast Dynamics of Optically-excited Semiconductors

The derivation of the quantum distribution-function transport equations combines the 
Liouvillian super-Green's function technique and the lattice Weyl-Wigner formulation 
of the quantum theory of solids. A generating super-functional is constructed which 
allows an algebraic and straightforward application of quantum field-theoretical 
techniques in real time to derive coupled quantum-transport, condensate, and pairwavefunction 
equations. In optically-excited semiconductors, quantum distributionfunction 
transport equations are given for phonons, plasmons, photons, and electron-hole 
pairs and excitons by transforming the Bethe-Salpeter equation into a multi-time 
evolution equation. The virtue of quantum distribution function is that it allows easy 
application of ‘device-inflow’ subsidiary boundary conditions for simulating femtosecond 
device-switching phenomena.


INTRODUCTION
There is a need for generalized quantum distribu- tion-function transport equations, valid for non- normal, non-uniform, and ultra-fast systems, as bases for large-scale computer simulations.This becomes urgent with advances in material science, ultra-fast laser probes, nanofabrication, and the development of more powerful energy beams.The drive to produce systems which are functionally more dense and have wider bandwidths will lead nanostructure devices to atomic-scale dimensions with different materials: insulators, semiconduc- tors, metals, and superconductors.
The nonequilibrium quantum transport theory including pairing dynamics is formulated in terms of the Liouville-space (L-space) quantum-field theory [1-2] and lattice Weyl transform technique [3, 4].For normal systems, this reduces to the nonequilibrium Green's function technique of Schwinger [5], Kadanoff and Baym [6], and Keldysh [7], coupled with the lattice Weyl-Wigner formula- tion of the quantum theory of solids [3,4].Several new results are derived with the present approach.F.A. BUOT   This L-space approach has provided the action principle for a multi-variable functional theory of nonequilibrium condensed-matter systems [8][9][10].Thus, the method set forth here may open doors for the investigation of ultra-fast dynamics in quantum nanostructures.So far, only the distribution-function approach has characterized, in time-domain, a highly-nonlinear and highly- nonequilibrium quantum behavior [1 4].

QUANTUM DYNAMICS IN LIOUVILLE SPACE
The density-matrix equation of quantum statistical dynamics in Hilbert space (H-space) becomes a super-Schrodinger equation for the super-state vector in L-space as p(t) is the density-matrix operator for the whole many-body system in H-space, and Ip(t))) is the corresponding super-state vector in L-space.The super-operator corresponds to the commutator [oCt, p], and is referred to in this paper as the Liouvillian.Thus, we may write the Liouvillian as 5e-07g_ f, which define and 3f.These have the property that lp(t)))= IWp(t))), and lp(t))) Ip(t)Ygt)).These relations are valid for fermions and bosons.For number-conserving fermion operator o, 07f 3gt.The quantum field super-operators, ()and t(t), are defined through their commutation relations in H-Space.
where 1)) is a unit super-vector.We have where T ac denotes anti-chronological time oraer- ing and {ifo' } The "transition probability" obeys the equality where W is identified as the effective action.It can be shown that 0 W- This relation forms the basis of a time-dependent functional theory of condensed matter discussed by Rajagopal and Buot in a series of papers [8-1 0]. 4. GENERALIZED QUANTUM DISTRIBUTION FUNCTIONS
The field super-operator averages can be written in terms of the S-matrix, e.g., if(1 2) We obtained the following (e= for bosons, e-for fermions), where the superscript T indicates the taking of the transpose, and, expanded about the condensate for Wick's theo- rem to be applicable) to evaluate the aJ's in terms of diagrams or graphs, (1,2,... n) represents the topologically distinct 'connected' subset of graphs.We have 6" ln((11S(oc, 0C(1,2,..., n Similar functional derivative relations can be obtained between GQDF with even number of indices by using the variation with respect to the external Schwinger source term u(1,2).Since u (1,2) is an ordinary c-number, the order of the u(i, j)'s is not critical in taking the functional derivatives.

Quantum Distribution Functions
We have gC gea (1,2)   eG< ) ea ac (1,2), Ga (1,2)   ( G(1,2) corresponds to the Keldysh nonequilibrium Green's function.We refer to the aj(1,2) simply as moment quantum distribution function.Moments are defined for time-ordered quantum field superoperators.We also define quantum correlation functions or quantum cumulants, g(, analogous to the classical statistical theory.This distinction is important in treating the quantum transport of superfluids and quanta of real classical fields.We will also refer to both as generalized quantum distribution functions (GQDF).
In the application of Wick's theorem (for superfluid Bose system it is assumed that 0 is r(12) -1 (12) ih w h e r e o(,,, 2)-1_ (7.g)-1/(,,, 2)O/Ot.-((', 2), where the (T)-matrix arise from the "'s ((' commutation relation of the (i, j) ,2) is a one-body potential matrix, and E(12) is the particle super self-energy matrix.For fermions, (1,2)-5U(1,2).E (12) is expressed in terms of Yg's (up to second-order cumulants) and vertex functions.These vertex functions obey equations similar to the Dyson equation, involving functional derivative of the self-energy with respect to second- order GQDF hence decoupling the BBKGY hierarchy.The self-energy due to e-e interaction includes the electron-plasmon vertex function.

QUANTUM TRANSPORT EQUATIONS
The time-evolution equation for Yg(i, j) is obtained.We write the resulting equations for F.A. BUOT the 2x2 matrix elements of (i, j) as (13) (14) (15) (16) where A is the pair potential or gap function, and G -1 is a diagonal matrix with elements propor- tional to ih6( 12)O/Ot2-, (12) with re(12) propor- tional to a one-body external potential.These equations were also given by Aronov, et al. [17a] and formally resemble the well-known Gorkov equations [17b] for superconductors at thermal equilibrium.
Solving quantum transport problems [11] cen- ters on the evolution of p<(12)-eihG <(12), which happens to be one of the matrix elements of the nonequilibrium matrix Green's function of Eq. ( 9). (>,< contains all information about the statistical aspects of the field intensity.This is coupled to the 'advanced' and 'retarded' propaga- tors often directly related to the experiment and contain all the energetics and dynamical informa- tion of the system.We obtain the transport equation for G >,< given by the following expres- Equations for the pair wavefunctions, g>'<ee geer, ghh>'<, grhh, etc, are also obtained.Using lattice Weyl transformation, we can transform the above total time evolution equations, Eqs.(17) (18) into quantum transport equations in (p, q, E, t) phase space for superconductive, systems.
The multi-component (in the "hat" and "tilde" indices) quantum field super-operators for electro- magnetic field (transverse and longitudinal) and lattice vibrations are given, respectively, by It is helpful to derive from the Bethe-Salpeter equation, the Schr6dinger-Wannier equation for the electron-hole pair wavefunction.Neglecting the self-energy terms and retaining only the Coulomb interaction terms, we obtain after setting tl t2 We used a "composite field operator" as the fourth field, (1,2), since coupling to fermions only occur through bilinear product of fermion fields.This is made up of ( 12), (12), er( 12), and t-( 12), consisting of different combination of the "hat" and "tilde" indices.We have, The time-ordered averages of the components give the familiar quantities, namely, The scalar potential field super-operator O() is not an independent field [25].

ELECTRON-HOLE AND EXCITON TIME EVOLUTION EQUATION
The super propagator for electron-hole and exciton is defined by g3(( (22) (23; 12) fext( 12fext(12) includes the matrix element of the dipole moment containing the selfconsistent transverse electromagnetic field in the microcavity, for example.In the matrix definition of 3((12; 34), the nonequilibrium pair propagator is identified as one of the matrix elements, namely, the '(1,4th)'component.0t( 12)- -mV2+mzV22 [rl r2----l (12)   (23) This is the familiar Schr6dinger equation for a two-particle system consisting of an electron and a hole.Using a dielectric function to screen the potential, the resulting bound states correspond to exciton states [23]. 7.2.Time Evolution Equations for C (12; 34) The total time derivative of 3((12; 34) is obtained from the Bethe-Salpeter equation, the details will be published elsewhere.The '(1,4th)' element describing the electron-hole pair propagator is denoted by (ffexciton (12; 34), which in turn contains in its ' (1,4)th' element the e-h pair density matrix < CNexciton (12; 34)  What needs to be done is to single out the equation for the electron-hole pair density matrix, which we < denote by CNexciton(12; 34).The result is very long and will be published elsewhere.Interested readers are encouraged to contact the authors for this detail.The final transport equation is obtained by setting l---t2, and t3 t4 and taking the lattice Weyl transform of the above transport equation [in general double lattice Weyl transform in spatial variables].To demonstrate this, let us take the simplest approximation of this equation, which we Upon taking the lattice Weyl transform and taking the gradient expansion the result is (25) The last equation is the appropriate Wigner distribution transport equation of a two-particle distribution corresponding to the Schr6dinger-Wannier equation of Eq. ( 23)

The Phonon Boltzmann Equation
More revealing equations can be seen by neglect- ing off-diagonal or "inter branch" terms, and expanding the equations in terms of the gradients.We defined a renormalized kinematic frequency by f (p, q, E, t) ,2 Rei-i (p, q, E, t).The Boltzmann equation for the distribution function of phonons, n (p, q, E, t), from vibration branch A readily follows by neglecting leading quantum corrections.We obtain the familiar interpretation [22] relating (i/2){hII(p,q,E,t)/E} as the scat- tering-out rate and (i/2){hII (p, q, E, t)/E} as the scattering-in rate.By taking the renormalized frequency to be given by the solution of the equality f (p, q, a; , with this solution denoted by a;(p, t), we finally obtain 0 O---t n (p, q, 9, t) fl-phCV (19, t) Vqn (p, q, co, t) I-I (p, q, a;, t) n,q,,t) mann equation is obtained by means of gradient expansion.We have 2 --tD o (p, q, E, t) + _ Re i_Iro (p, q, E, t) VqD '> (p, q, E, t) i(I-I>o'<(p,q,E,t) <,> ) =qz-hoRei_iro(p,q,E,t) Do (p,q,E,t) i{ I-i<o'>(p,q,E,t) >,< } where we have left out terms involving diffusion in momentum and energy space.We would like to point out the notable use of the renormali- zed plasmon group velocity given by (p-VpRel-Iro(p,q,E,t)) /oReI-[o(p,q,E,t).The use of the renormalized diffusion velocity arises from the same reason as that given in the derivation of the Boltzmann equation for phonons.The right-hand side are the familiar collision terms.The omitted terms describe the kinematics of the dynamical motion of plasmons.
The renormalized group velocity [26] emerges since phonons do not diffuse freely but interact with the environment as well as collide with each other.
Similar situation arises in deriving the plasmon and photon Boltzmann equations.We expect the leading term of Re I-I (P, q, E, t) to be independent of time.An example of the RHS of Eq. (33) for phonon-phonon interaction can be found in the work of the author [27] and in the study of phonon hydrodynamics and second sound [28]. 9. TRANSPORT EQUATIONS FOR PLASMONS 10.TRANSPORT EQUATIONS FOR PHOTONS We will also neglect the off-diagonal terms in polarization indices.The result to leading order is given by (35)

The Plasmon Boltzmann Equation
The plasmon super-propagator is given by Do((,(') 6(O('))/6pext(").The plasmon Boltz- where a2_x,x(p, q, E, t) c2p 2 ReP ,,x(P, q, E, t) and we have neglected the leading quantum corrections on the right-hand side.We note that p is related to the transverse dielectric tensor [29].We may F.A. BUOT consider fa(p, q, E, t) > 0 to constitute the valid- ity of the Boltzmann equation for photons with diffusive term.The use of renormalized group velocity corresponds to the use of refractive index well-known in optics.
A self-consistent Boltzmann transport equ- ation follows by setting, fZ2(p,q,03, t)-c2p -RePr(p, q, 03, t) 03 2 > 0. Upon substituting the solution for 03, which we will denote by 03 (p, ), we obtain 0 Ot Fa< (p' q' 03' t) + Vph03a (p, t) VqFA < (p, q, 03, t) 2 i { P (P' q' 03' t) } F (p' q' E' { Pa,q,a,t) }Fa,q,E,t) (36) + 11.CONCLUDING REMARKS A major challenge in analyzing the .ultra-fastdynamics of semiconductor gain material for investigating microcavity lasers is the fact that the electron-hole system properties are strongly affected by many-particle Coulomb interactions, and strong coupling to the light and crystal lattice- phonon fields [24].In device physics of highly excited semiconductor systems, involving all ranges of e-h pair densities, the exciton, e-h Cooper pairs, and e-h plasma energetics and their accompanying distributions greatly affect the polaritons, phonons, biexcitons, and higher-order 'pairing' (excitonic molecules) distributions, their wavefunctions and energies.For the bosons of real fields, the diffusion velocity is not equal to the group velocity of bare excitations but is defined only by its renormalized value.Quanta of these classical fields interact with the environment or collide with each Other as it diffuse in space.The condensate and normal excitation energetics and their accompanying distribution also influence the gap function and energy gap.The full transport equations will be published elsewhere.

7. 3 .
Transport Equation for the Nonequilibrium Pair Propagator