Representation weak coupling

The early approaches to this model used perturbative expansion for weak coupling [Silbey and Harris 1983]. Generally speaking, perturbation theory allows one to consider a TLS coupled to an arbitrary bath via the term where / is an operator that acts on the bath variables. The equations of motion in the Heisenberg representation for the a operators, 8c/8t = ih [H, d], have the form... 

Figure 7.17 (a) Magnetic properties of [LaTb] and [Tb2] in the form of yT versus T plot per mole of Tb(lll). (b) Schematic representation of the qubit definition, weak coupling and asymmetry, as derived from magnetic and heat capacity data. 

Tang et al. [20] have examined the population dynamics in a three-level system, and its representation in a surrogate two-level system, to test the scheme outlined above. In the model system considered state 3 is weakly coupled with states 1 and 2, so that population transfer between states 1 and 2 should dominate the dynamics, with only a small contribution from population transfer to and from state 3. The coupling of state 3 with states 1 and 2 was taken to be one-tenth of the coupling between states 1 and 2, that is, M)3 = M23 = Mn/10 = -1/10. Using the formalism sketched above, the exact system dynamics is governed by the coupled equations of motion for the three states,... 

Fig. 5. Schematic representation of various energy curves due to weak (w) and strong (s) interactions between two harmonic oscillators. Energy in arbitrary units with the following values (k/2)% = 3, E = 0 for strong coupling, U = 7, for weak coupling, U = 5. The dotted curves indicate the intermediate states for weak coupling case, [W (w)].

When the system is weakly coupled to the leads, the polaron representation (154), (162) is a convenient starting point. Here we consider how the sequential tunneling is modified by vibrons. 

There are some situations when this is the most convenient representation of the dipolar coupling, for example, when S and I are very strongly coupled to each other but weakly coupled to the molecular rotation, as in the H ion. However, an alternative form which is often more suitable is... 

If the oscillator is weakly coupled to the bath, in canonical thermal equilibrium the probability of finding the oscillator in the nth state is of course P q = e / En/Zq, where ft = I/kT and the oscillator s canonical partition function is Zq = e In addition, the oscillator s off-diagonal (in this energy representation) density matrix elements are zero. The average oscillator energy (in thermal equilibrium) is Eeq = n13nPnq-... 

The spectroscopic factors are critical quantities in determining the accuracy of a configuration-interaction calculation of the structure of the ion. The ion state /) is written in the weak-coupling representation as... 

The spectroscopic factor (11.16) is the absolute square of the coefficient of the one-hole state a) in this expansion. Determination of one coefficient in each of several eigenstates of Hi in the representation is a strong constraint on the calculation. Furthermore the determination of spectroscopic factors depends only on the validity of the weak-coupling... 

Fig. 11.6 shows the noncoplanar-symmetric differential cross sections at 1200 eV for the Is state and the unresolved n=2 states, normalised to theory for the low-momentum Is points. Here the structure amplitude is calculated from the overlap of a converged configuration-interaction representation of helium (McCarthy and Mitroy, 1986) with the observed helium ion state. The distorted-wave impulse approximation describes the Is momentum profile accurately. The summed n=2 profile does not have the shape expected on the basis of the weak-coupling approximation (long-dashed curve). Its shape and magnitude are given quite well by... 

Let us start with the E e JT polaron in the weak-coupling region (or for the case of small gEigie), in which the perturbation approach in momentum representation is useful. The thermal one-electron Green s function Gkya(ico ) with co the fermion Matsubara frequency is defined at temperature T by... 

The Redfield equation, Eq. (10.155) has resulted from combining a weak system-bath coupling approximation, a timescale separation assumption, and the energy state representation. Equivalent time evolution equations valid under similar weak coupling and timescale separation conditions can be obtained in other representations. In particular, the position space representation cr(r, r ) and the phase space representation obtained from it by the Wigner transform... 

Fig. 1 Schematic representation of the pressure effect on the singlet (S(r)=log<(s ip(r)) ) and triplet (/-(r)=log ) content of the excited molecular state. Crosses, collision-free conditions, points and solid line, increasing inert-gas pressure, (a) statistical-limit (b) strongcoupling case (incoherent excitation) (c) strong coupling case (coherent excitation) (d) weak-coupling case (small polyatomics) (e) weak-coupling case (CO,N2).

E and are the energy and the width of the useful part of the continuum (doorway state) [22, 33]. The two-dimensional non-Hermitian effective Hamiltonian (30) is the simplest matrix representation linking the microscopic level characterized by the complex energy E — iFc/2 to the macroscopic level of interest (the resonance). In Eq. (30), the energy of the resonance El is real. We will see below that if the resonance is weakly coupled to the microscopic level (AE F ), the complex part of energy can be uncovered by... 

Fig. 5.6.2. Schematic representation of the molecular arrangement in (a) the weakly coupled incommensurate smectic A, and b) the strongly coupled incommensurate smectic A. (After Frost and Barois. " )...

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