Coherent states , second-harmonic

The ability to create and observe coherent dynamics in heterostructures offers the intriguing possibility to control the dynamics of the charge carriers. Recent experiments have shown that control in such systems is indeed possible. For example, phase-locked laser pulses can be used to coherently amplify or suppress THz radiation in a coupled quantum well [5]. The direction of a photocurrent can be controlled by exciting a structure with a laser field and its second harmonic, and then varying the phase difference between the two fields [8,9]. Phase-locked pulses tuned to excitonic resonances allow population control and coherent destruction of heavy hole wave packets [10]. Complex filters can be designed to enhance specific characteristics of the THz emission [11,12]. These experiments are impressive demonstrations of the ability to control the microscopic and macroscopic dynamics of solid-state systems. 

The same equations, albeit with damping and coherent external driving field, were studied by Drummond et al. [104] as a particular case of sub/second-harmonic generation. They proved that below a critical pump intensity, the system can reach a stable state (field of constant amplitude). However, beyond the critical intensity, the steady state is unstable. They predicted the existence of various instabilities as well as both first- and second-order phase transition-like behavior. For certain sets of parameters they found an amplitude self-modula-tion of the second harmonic and of the fundamental field in the cavity as well as new bifurcation solutions. Mandel and Erneux [105] constructed explicitly and analytically new time-periodic solutions and proved their stability in the vicinity of the transition points. 

Now, assuming that the two modes are not correlated at time x = 0, it is straightforward to calculate the variances of the quadrature field operators and check, according to the definition (12), whether the field is in a squeezed state. If the initial state of the field is a coherent state of the fundamental mode and a vacuum for the second-harmonic mode, /0) = wa(0)) 0), for which we have... 

The typical initial conditions for the second-harmonic generation are a coherent state of the fundamental mode and the vacuum of the second-harmonic mode. The initial state of the field can thus be written as... 

Equation (127) describing the evolution of the system is our starting point for further discussion of the second-harmonic generation. If the initial state of the fundamental mode is not a coherent state but has a decomposition into a number states of the form (125) with different bn, equation (127) is still valid when corresponding bn are taken. It is true, for example, for the initially squeezed state of the fundamental mode. 

Figure 5. Intensity of the second-harmonic (a) initial coherent state with Na —2 (solid line) and initial number state with two photons (dashed line) (b) initial coherent state with Na = 10 (solid line), Na = 40 (dashed line), and Na — 100 (dashed-dotted line). Dotted line marks the classical solution.

Let us start with the short-time approximation in which we can use the symbolic manipulation computer program described in Appendix A to find the corrections coming from the quantum fluctuations of the fields. The operator formulas (94) and (95) are valid also for the degenerate downconversion because the two processes are governed by the same Hamiltonian, but now initially the second-harmonic mode is populated while the fundamental mode is initially in the vacuum state. Assuming that the pump mode at the frequency 2oo is in a coherent state fi0) (p0 = /Ni,exp(k )h)), we have... 

Figure 1. Fano factors of the fundamental, Ff, and the second-harmonic mode, / V. in the long-time interaction for initial coherent states with real amplitudes (a) ai = 6,0C2 — 1, and (b) ai — 6,0C2 = 3. Case a is a typical example of super-Poissonian behavior in both modes outside the no-energy-transfer regime. In case b, the harmonic mode exhibits stable sub-Poissonian statistics with F — 0.83. It is a charactersitc example of the sub-Poissonian behavior within the no-energy-transfer regime along the line ai = 2 ct21-...

Figure 2. Global Fano factor,, of the second-harmonic mode as a function of the initial coherent state amplitudes 04 and < 2 with 0 = 0. It is seen that the harmonic mode exhibits globally sub-Poissonian behavior (Fj3 < 1) near the diagonal oci = 2 a2 and 0 = 0.

Figure 4. Quantum evolution of the Q function for the fundamental (outer contour plots) and the second-harmonic mode (inner plots) at six time moments for initial coherent states with 0(1 = 6,0(2 - 5.0 0. Solution obtained by quantum numerical method.

Besides the phase of the fundamental mode, strictly speaking, the preferred phase, many other characteristics have been studied in [226]. Because a large mismatch was chosen, they have lacked any trend, but an interesting oscillatory behavior has been discovered for the initial two-mode coherent state. Within each period, the phase-matched second-harmonic and second-subharmonic generation processes can be prepared. The model of an ideal Kerr-like medium [223] have been considered for a comparison with cascaded quadratic non-linearities. It follows that these nonlinearities exhibit not only self-phase modulation in the fundamental mode but also a cross-phase modulation of the modes that can be considered for a nondemolition measurement. 

The high intensity and coherence of laser radiation can lead to more elaborate photon scattering processes than those involved in the conventional Raman effect. The simplest example is second harmonic generation (hyper-Rayleigh scattering) and the associated hyper-Raman effect in which two laser photons of frequency interact simultaneously with the molecule to produce a scattered photon at frequency (hyper-Rayleigh), or at XiOi.-tO (Stokes hyper-Raman) or at (antiStokes hyper-Raman). As illustrated in figure 1.3, these processes involve two virtual intermediate excited states. 

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. 

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