Time-ordering operator

Let the time-ordering operator 16 S, when applied to the product... 

If the nucleus in the emitter is under the influence of time-dependent forces, it is convenient to modify Eq. (99) by introducing, instead of Eq. (96), the time-ordered operator ... 

The lesser and greater functions are not time-ordered and arguments of the operators are not affected by time-ordering operator. Nevertheless we can write such functions in the same form (283). The trick is to use one time argument from the forward contour and the other from the backward contour, for example... 

Here, P is the Dyson time-ordering operator and Q(l)IP is given by the Heisenberg transformation Eq. (G.6), that is,... 

The different situations illustrated in Fig. 6 correspond more and more to approximate models when passing up to down. The top of the figure given the case described by Eq. (126) and corresponds to the reference quantum indirect damping. Below, is depicted the situation described by Eq. (146), where the Dyson time ordering operator is ignored in the quantum model. Further below is given the behavior that corresponds to both Eqs. (152) and (174), that is, to weak approximations on the classical limit of the quantum model [Eq. (152)] and to the semiclassical model [Eq. (174)]. At last, at the bottom we visualize the semiclassical model of Robertson and Yarwood [described by Eq. (185)]. 

Here, P is the Dyson time-ordering operator [57], Q (t)IP is the coordinate in the interaction picture with respect to the thermal bath and to the diagonal part of the Hamiltonian of the H-bond bridge, and the notation (( )e)siow has the meaning of a partial trace on the thermal bath and on the H-bond bridge coordinates. 

Besides P, which showed not be confused with the momentum operator P of the H-bond bridge, is the Dyson time-ordering operator [57] acting on the Taylor expansion terms of the exponential operator in such a way so that the time arguments involved in the different integrals will be t > t > t". 

This is the reason for the use of the Dyson time-ordering operator. 

This expansion lies also at the basis of the perturbation expansion given by Eq. (66). The time-ordering operator Torders the (imaginary) times rt, r . For brevity we have momentarily stopped indicating the dependence of all quantities on the coordinates u/> and co/>. Using the notation p F = p(// F) and (A )mf = Tr(A p F), we readily derive... 

T+ being the time-ordering operator. In the derivation of Eq. (168) we assumed that B(t)) = 0. It needs to be stressed that Eq. (168) generalizes previously known master equations to arbitrary time-dependent hamiltonians, Hs t) for the system and Hi(t) for system-bath coupling, [Cohen-Tannoudji 1992], Henceforth, we explicitly consider a driven TLS undergoing decay, whose resonant frequency and dipolar coupling to the reservoir are dynamically modulated, so that... 

Because the time order of the incremental evolution operators is destroyed by the integral in the exponent, the so-called Dyson time ordering operator is introduced to reestablish the lost time order. 

Here T is the time-ordering operator. In consequence, the mean value of the vector potential in the case, when external currents are accounted for, takes the form 50... 

Note that the upper integration limits form a reverse chronological order. Identical upper limits are obtained upon the introduction of the time ordering operator... 

The effect of the time ordering operator on a product of time-dependent operators is to place them in reverse chronological order ti > t2 >. . > t as if they commuted (operators that do not conserve particle number is treated as if they anticommute), e.g. 

The commutator and anticommutator operations in Hilbert space can thus be implemented with a single multiplication by a and + superoperator, respectively. We further introduce the Liouville space-time ordering operator T. This is a key ingredient for extending NEGFT to superoperators when applied to a product of superoperators it reorders them so that time increases from right to left. We define (A(t)) = Tr A(f)Peq where peq = p(t = 0) represents the equilibrium density matrix of the electron-phonon system. It is straightforward to see that for any two operators A and B we have... 

These are known as T (T) is the Hilbert space-time (anti-time) ordering operator when applied to a product of operators, it reorders them in ascending (descending) times from right to left. 

T is the Liouville space-time ordering operator that rearranges all superoperators in increasing order of time from right to left. 

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