Rotating-wave approximation system

The theory has been generalized by us to finite temperatures and to qubits driven by an arbitrary time-dependent field, which may cause the failure of the rotating-wave approximation (RWA) [11]. It has also been extended to the analysis of multilevel systems, where quantum interference between the levels may either inhibit or accelerate the decay [19]. 

Eq. (6.26) is the TDSE in the Schrodinger picture. In general, it proves more convenient to discuss the time evolution of the driven system in a rotating frame, such as the frame rotating with the laser carrier frequency q- After transformation into the carrier frequency picture and application of the rotating wave approximation (RWA), the TDSE takes the form [92]... 

Figure 1- Liouville space diagram corresponding to the only term that contributes to the spontaneous light emission from a two-level system within the rotating-wave approximation [Eq. (2.7)]. Here ]g) and e) denote the ground and the excited states, respectively.

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... 

The coupled channels expansion can be further simplified by introducing the (number state) rotating-wave approximation (RWA), valid only when the field is jjsfif moderate intensity and the system is near resonance. As pointed out above, igtyen an initial photon number state [JVf), the components of E, n, N — 1") of. . greatest interest for a one-photon transition are (JV, , n",JV, —1 ) and y (Nj dt l[Ji, n, Nj — 1 ). If [ , ) is the ground material state, then the (Nj+m,... 

The theory of the saturable absorption effect in single-wall carbon nanotubes has been elaborated. The kinetic equations for density matrix of n-electrons in a single-wall carbon nanotube have been formulated and solved analytically within the rotating wave approximation. The dependence of the carbon nanotube absorption coefficient on the driving field intensity has been shown to be different from the absorption coefficient behavior predicted forthe case of two level systems. 

We consider a two-level atom with excited and ground states e) and g) when in a photonic crystal coupled to the field of a discrete (or defect) mode and to the photonic band structure in the vacuum state. The hamiltonian of the system in the rotating-wave approximation assumes the form [Kofman 1994]... 

Here we discuss in detail a model for measurement-induced decay modification in a multilevel system. The system with energies frwn, 1 < n < N, is coupled to a zero-temperature bath of harmonic oscillators with frequencies uj. The corresponding Hamiltonian, in the rotating-wave approximation, is... 

These results leave several basic questions open How to derive a non-Markovian master equation (ME) for arbitrary time-dependent driving and modulation of a thermally relaxing two-level system Would the two-level system (TLS) model hold at all for modulation rates, that are comparable to the TLS transition frequency u)a (between its states e) and g)) which may invalidate the standard rotating-wave approximation (RWA), [to hen-Tannoudji 1992] Would temperature effects, which are known to incur upward g) —> e) transitions, [Lifshitz 1980], further complicate the dynamics and perhaps hinder the suppression of decay How to control decay in an efficient, optimal fashion We address these questions by outlining the derivation of a ME of a TLS that is coupled to an arbitrary bath and is driven by an arbitrary time-dependent field. 

Here, [in is the transition dipole moment for the transition between V22 and Vn. In the adiabatic approximation (see, e.g.. Ret. [Stenholm 1994]), the coupled bare potentials V22 and V33 can be replaced by uncoupled adiabatic potentials. Using the rotating-wave approximation, the dynamics of the system is then described by the Schrodinger equation... 

Finally, it is also interesting to compare the result (9.65) to the result (8.106) ofthe very different semiclassical formalism presented in Section (8.3.3). If we identify y of the present treatment with the factor Zgoi/< Eq. (8.96) the two results are identical for e = hco ksT. The rotating wave approximation used in the model (9.44) cannot reproduce the correct result in the opposite, classical, limit. Most studies of vibrational relaxation in molecular systems are done at temperatures considerably lower than s/ks, where both approaches predict temperature-independent relaxation. We will see in Chapter 13 that temperature-dependent rates that are often observed experimentally are associated with anhannonic interactions that often dominate molecular vibrational relaxation. 

The total Hamiltonian describing the energies of the systems, electromagnetic field and interactions, in the electric dipole and RWA (rotating-wave approximation) approximations [21], is composed of four terms... 

As we stressed above, the rotating wave approximation consists of taking into account only the terms varying with time at approximately the same frequency, that is, according to the resonance condition (9.11). Then we neglect in eqn (9.18) all interaction terms except A B oc exp[—i(w6 — lo0)L and in eqn (9.19) all interaction terms except A2 oc exp(— 2iujat). As a result, we arrive at the following simple system of equations... 

The dynamics of populations of the electronic states in a 4,4 -bis(dimethylamino) stilbene molecule (two-photon absorption) was studied against the frequency, intensity, and shape of the laser pulse [52]. Complete breakdown of the standard rotating wave for a two-photon absorption process was observed. An analytical solution for the interaction of a pulse with a three-level system beyond the rotating wave approximation was obtained in close agreement with the strict numerical solution of the amplitude equations. Calculations showed the strong role of the anisotropy of photoexcitation in the coherent control of populations that can affect the anisotropy of photobleaching. The two-photon absorption cross section of an ethanol solution of a trans-stilbene and its derivatives exposed to radiation of the second harmonic of a Nd YAG laser (532 nm) of nanosecond duration has been detected [53]. In experiments, the method based on the measurement of the photochemical decomposition of examined molecules was used. The quantum yield of the photoreaction (y266) of dyes under one-photon excitation (fourth harmonic Nd YAG laser 266 nm) was detected by absorption method. 

Applying the electric dipole and rotating-wave approximation, the equations of motion for the density matrix elements ay of the three-level system are [34],... 

The advantage of the rotating wave approximation at the instantaneous frequency is that it leads to the more familiar picture of a two-level system, as described in Section 7.3.3. The coupling term then has a slow time dependence determined by the envelope of the electric field. 

For Eb > Ea the term a>ba = (Eb — Ea)lh is positive. In the transition Ea Eb, the atomic system absorbs energy from the radiation field. Noticeable absorption occurs, however, only if the field frequency a> is close to the eigenfrequency (i>ba- In the optical frequency range this implies that o)ba — < < ( ba The second term in (2.70b) is then small compared to the first one and may be neglected. This is called the rotating-wave approximation for only that term is kept in which the atomic wave functions and the field waves with the phasors exp(—icu / t) = txpi+icobJ) and exp(— cot) rotate together. 

In the rotating-wave approximation we obtain from (2.70b) for the probability b(t) that the system is at time t in the upper level Eb... 

Here aj(aj) are creation and annihilation operators of the bath mode of frequency ojj and bc(bt) are annihilation operators of a coupling and a tuning mode, respectively. The so-called rotating-wave approximation (RWA) has been invoked in Eq. (17), neglecting terms of the type ba and b a. The effect of the bath on the system dynamics is completely determined by the... 

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