The Interaction Picture

Before we can start with the discussion of time-dependent perturbation theory in the form of response theory, we need to introduce an alternative formulation of quantum mechanics, called the interaction or Dirac representation. In general, several representations of the wavefunctions or state vectors and of the operators of quantum mechanics are equivalent, i.e. valid, as long as the expectation values of operators ( 0 I d I o) or inner products of the wavefunctions ( o n) are always the same. Measurable quantities and thus the physics are contained in the expectation values or inner products, whereas operators and wavefunctions are mathematical constructs used in a particular formulation of the theory. One example of this was already discussed in Section 2.9 on gauge transformations of the vector and scalar potentials. In the present section we want to look at a transformation that is related to the time dependence of the wavefunctions and operators. 

The operators H and O and the wavefunctions o(i)) that we have used so far are said to be in the Schrodinger picture, where the wavefunctions carry the time dependence and the operators are time independent apart from the case of an explicit time dependence due to a time-dependent perturbation. The Schrodinger representation is the natural choice for time-independent systems. 

The interaction or Dirac representation becomes, on the other hand, useful, if one deals with a system that is described by a time-dependent Hamiltonian such as 

The essence of the interaction picture can be illustrated when we assume for a moment that the time-dependent perturbation Hamiltonian vanishes. The 

As the time dependence of the unperturbed wavefunctions is simply a rotation in the complex plane, Eq. (2.6), we can say that in the interaction picture this rotation is frozen out or that we have switched to a rotating frame that rotates with the time dependence of the unperturbed wavefunctions. The time evolution of the perturbed wavefunction 4 o( )) i the interaction picture thus governed by the perturbation Hamiltonian alone, as we will see in the following section. This will greatly 

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Further evaluation of Eq. (2.35) requires an expression connecting 0(g)) (assumed to be nondegenerate) with Ox 5 ) (also assumed to be nondegenerate). This link is established via the interaction-picture time-evolution operator i.e. by an adiabatic switching oiHi ... 

To derive a universal ME for both AN and PN scenarios, we move to the interaction picture and rotate to the appropriate diagonalizing basis, where the appropriate basis for the AN case of Eq. (4.91) is... 

Equation (4.150) implies that Ap and with it P are small. Hence Ap must refer to the interaction picture and a weak interaction, while P[p(t)] should not be affected by the internal dynamics (so that no separate time dependence emerges in Eq. (4.150), which is not included in the chain-rule derivative). In the examples above, this is obvious for state purity, whereas an observable Q might be thought of coevolving with the internal dynamics. 

In the language of control theory, Tr[p(0)P] is a kinematic critical point [87] if Eq. (4.159) holds, since Tr[e p(0)e- P] = Tr[p(0)P] + Tr(7/[p(0),P]) + O(H ) for a small arbitrary system Hamiltonian H. Since we consider p in the interaction picture, Eq. (4.159) means that the score is insensitive (in first order) to a bath-induced unitary evolution (i.e., a generalized Lamb shift) [88]. The purpose of this assumption is only to simplify the expressions, but it is not essential. Physically, one may think of a fast auxiliary unitary transformation that is applied initially in order to diagonalize the initial state in the eigenbasis of P. 

To this end, we resort to a novel general approach to the control of arbitrary multidimensional quantum operations in open systems described by the reduced density matrix p(t) if the desired operation is disturbed by linear couplings to a bath, via operators S B (where S is the traceless system operator and B is the bath operator), one can choose controls to maximize the operation fidelity according to the following recipe, which holds to second order in the system-bath coupling (i) The control (modulation) transforms the system-bath coupling operators to the time-dependent form S t) (S) B(t) in the interaction picture, via the rotation matrix e,(t) a set of time-dependent coefficients in the operator basis, (Pauli matrices in the case of a qubit), such that ... 

The parameter is the damping constant, and (n) is the mean number of reservoir photons. The quantum theory of damping assumes that the reservoir spectrum is flat, so the mean number of reservoir oscillators (n) = ( (O)bj(O j) = ( (1 / ) — 1) 1 in the yth mode is independent of j. Thus the reservoir oscillators form a thermal system. The case ( ) = 0 corresponds to vacuum fluctuations (zero-temperature heat bath). It is convenient to consider the quantum dynamics of the system (56)-(59) in the interaction picture. Then the master equation for the density operator p is given by... 

After transformation into the interaction picture and application of the rotating-wave approximation [46, SO, 54] the population dynamics can be calculated numerically by solving the time-dependent three-level Schrodinger equation or (if phenomenological relaxation rates are considered) by solving the density matrix equation (3) for the molecular system. The density matrix equation is given by... 

For a three-level system the Hamiltonian in the interaction picture //, in Rotating Wave Approximation is given in matrix representation by... 

Henriksen, N.E. and Heller, E.J. (1988). Gaussian wave packet dynamics and scattering in the interaction picture, Chem. Phys. Lett. 148, 567-571. 

Zhang, J.Z.H. (1990). New method in time-dependent quantum scattering theory Integrating the wave function in the interaction picture, J. Chem. Phys. 92, 324-331. 

Observe that Eq. (61) may be written within the interaction picture (IP) procedure according to... 

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. 

In the interaction picture with respect to the unperturbed part, the full-time evolution operator Uxot(f) is... 

Within the interaction picture, the time-evolution operator is... 

Here, Q(f)IP is the H-bond bridge coordinate in the interaction picture that is given by... 

Q(tf Coordinate operator describing the H-bond bridge in the interaction picture with respect to H°, in the absence of thermal bath. 

It is convenient to use the interaction picture to represent the time evolution of the wave function v /(f)). By writing the formal solution of the Schrodinger equation in the interaction picture and returning to the laboratory reference frame, we obtain... 

Give examples of amino acids that could give rise to the interactions pictured in Fig. 22.25 that maintain the tertiary structures of proteins. 

The standard approach to the calculation of the propagators (A.4, A.5) is perturbation theory with respect to the electron-electron coupling constant a = e /(hc) on the basis of the interaction picture. Technically this results in an expansion of expectation values of interacting field operators in powers of expectation values of the free (or noninteracting) field operators i o and Ag. The structure of this expansion can be summarised in a set of formal rules, the Feynman rules. For instance for the electron propagator one obtains ... 

The CFP (see Chapter III) requires the unique theoretical background behind Eq. (3.43) [which is dosely related to Eq. (3.52)], whereas the AEP (see also Chapter II) is basically founded on Eq. (2.13) (written in the interaction picture). Equations (3.54) and (2.13) have been derived using the same projection technique. These two equations are related to each other still more deeply. To demonstrate this, let us note that within the quantumlike formalism of Eq. (3.5), Eq. (2.23) reads... 

As pointed out in Chapter I, we have to apply the Zwanzig projection method to the interaction picture. For clarity let us rewrite the corresponding result (see Chapter I) ... 

As pointed out in Section III we have to give the conditions of applicability of our AEP to the case under study. The first condition should be T iy, where the mechanical time scale ty is of the order x/f(x). This constraint, however, can be bypassed, by transferring to the interaction picture, that is, by taking the integral of motion of (5.8) as a new variable. In the present form we must assume t c t. The second condition is essential r Tg, where t is a time scale somehow related to g x) and of the order /8(j )( )eq 11 lo " simplicity we choose g x) = x, then When these conditions are applied to the system of Eq. (5.7) with ( )eq fixed, Eq. (5.16) becomes an expansion with perturbation parameter r. 

V the interaction Hamiltonian describing the coupling between the system and the bath. An operator with a tilde means the one in the interaction picture. If we assume that V is small in some sense, we can carry out the perturbation expansion for V as... 

Starting from the gap Hamiltonian (33) and the interactions with the reservoirs (34) and (35), eliminating the reservoir degrees of freedom within the standard Bom-Markov approach we derive a master equation for the reduced density operator of the atomic ensemble. The calculation is lengthy but straight forward. Disregarding level shifts caused by the bath interaction we find for the populations in states cj ) the following density matrix equations in the interaction picture ... 

In the interaction picture, the Schrodinger equation is now (putting for simplicity h = l)... 

The rationale for this approximation can be seen in the interaction picture in which V becomes Vpt) = exp -(i/K)Hot)... 

Let us start the discussion of practical possibilities of the FD coherent-state generation from the simplest case, where only superpositions of vacuum and single-photon state are involved (the Hilbert space discussed is reduced to two dimensions). We consider the system governed by the following Hamiltonian defined in the interaction picture (in units of h = 1) to be... 

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