Laser fields

Baumert T, Engel V, Meier Ch and Gerber G 1992 High laser field effects in multiphoton ionization of Na2 -experiment and quantum calculations Chem. Rhys. Lett. 200 488... 

Tannor D J 1994 Design of femtosecond optical pulses to control photochemical products Molecules in Laser Fields ed A Bandrauk (New York Dekker) p 403... 

Such electronic excitation processes can be made very fast with sufficiently intense laser fields. For example, if one considers monochromatic excitation with a wavenumber in the UV region (60 000 cm ) and a coupling strength / he 4000 (e.g. 1 Debye in equation (A3.13.59), / 50 TW cm ),... 

Seideman T 1995 Rotational excitation and molecular alignment in intense laser fields J. Chem. Phys. 103 7887-96... 

Friedrich B and Herschbach D 1995 Alignment and trapping of molecules in intense laser fields Phys. Rev. Lett. 74 4623-6... 

Mukamel S and Shan K 1985 On the selective elimination of intramolecular vibrational redistribution using strong resonant laser fields Chem. Rhys. Lett. 5 489-94... 

Marquardt R and Quack M 1989 Molecular motion under the Influence of a coherent Infrared-laser field infrared Phys. 29 485-501... 

For CW applieations of optieal-heterodyne eonversion, two laser fields are applied to the optoeleetronie material. The non-linear nature of the eleetro-optie effeet strongly suppresses eontimious emission relative to ultrashort pulse exeitation, and so most of the CW researeh earried out to date has used photoeonduetive anteimae. The CW mixing proeess is eharaeterized by the average drift veloeity t and earrier lifetime Xq of the mixing material, typieally... 

The conceptually simplest approach towards controlling systems by laser field is by teaching the field [188. 191. 192 and 193]. Typically, tire field is experimentally prepared as, for example, a sum of Gaussian pulses with variable height and positions. Each experiment gives an outcome which can be quantified. Consider, for example, an A + BC reaction where the possible products are AB + C and AC + B if the AB + C product is preferred one would seek to optimize the branching ratio... 

Friedrich B and Herschbach D 1996 Alignment enhanced spectra of molecules in intense non-resonant laser fields Chem. Phys. Lett. 262 41... 

Figure C 1.5.10. Nonnalized fluorescence intensity correlation function for a single terrylene molecule in p-terjDhenyl at 2 K. The solid line is tire tlieoretical curve. Regions of deviation from tire long-time value of unity due to photon antibunching (the finite lifetime of tire excited singlet state), Rabi oscillations (absorjDtion-stimulated emission cycles driven by tire laser field) and photon bunching (dark periods caused by intersystem crossing to tire triplet state) are indicated. Reproduced witli pennission from Plakhotnik et al [66], adapted from [118].

D. J. Tannor, Design of Femtosecond Optical Pdlse Sequences to Control Photochemical Products, in A. D. Bandrark, ed., MoleciM.es in Laser Fields, Dekker, New York, 1994. 

Multiphoton processes are also undoubtedly involved in the photodegradation of polymers in intense laser fields, eg, using excimer lasers (13). Moreover, multiphoton excitation during pumping can become a significant loss factor in operation of dye lasers (26,27). The photochemically reactive species may or may not be capable of absorption of the individual photons which cooperate to produce multiphoton excitation, but must be capable of utilising a quantum of energy equal to that of the combined photons. Multiphoton excitation thus may be viewed as an exception to the Bunsen-Roscoe law. 

Approximating any external fields (electric or magnetic) by considering only their lineal components. For normal conditions, this will be quite a good approxunation, however, this is not the case for intense laser fields for example. 

Codling K, Frasinski LJ (1996) Molecules in Intense Laser Fields an Experimental Viewpoint. 86 1-26... 

Kulander KC, Schafer KJ (1996) Time-Dependent Calculations of Electron and Photon Emission from an Atom in an Intense Laser Field 86 149-172 Kiinzel FM, see Buchler JW (1995) 84 1-70 Kurad D, see also Tytko KH (1999) 93 1-64 Kustin K, see Epstein IR (1984) 56 1-33... 

Electron Nuclear Dynamics (48) departs from a variational form where the state vector is both explicitly and implicitly time-dependent. A coherent state formulation for electron and nuclear motion is given and the relevant parameters are determined as functions of time from the Euler equations that define the stationary point of the functional. Yngve and his group have currently implemented the method for a determinantal electronic wave function and products of wave packets for the nuclei in the limit of zero width, a "classical" limit. Results are coming forth protons on methane (49), diatoms in laser fields (50), protons on water (51), and charge transfer (52) between oxygen and protons. 

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 simplified theory is adequate to obtain qualitative agreement with experiment [1,16]. Comparisons between the simplified and more advanced versions of the theory show excellent agreement for the dominant (electronic) contribution to the time-dependent dipole moment, except during the initial excitation, where the k states are coupled by the laser field [17]. The contributions to the dipole from the heavy holes and light holes are not included in the simplified approach. This causes no difficulty in the ADQW because the holes are trapped and do not make a major contribution to the dynamics [1]. This assumption may not be valid in the more general case of superlattices, as discussed below. 

The simplified theory allows the time-dependent wave function to be calculated rapidly for any specified laser field. However, controlling the dynamics of the charge carriers requires the answer to an inverse question [18-22]. That is, given a specific target or objective, what is the laser field that best drives the system to that objective Several methods have been developed to address this question. This section sketches one method, valid in the weak response (perturbative) regime in which most experiments on semiconductors are performed. 

In some cases, calculating the optimal fields explicitly is inconvenient, either for computational reasons, or because the quantity to be optimized cannot be expressed in terms of a simple control functional. In such situations alternative procedures can be applied. One method is to express the laser field E t) as a function of a small number of parameters, and then vary the parameters to maximize the yield. For example, the laser field can be written... 

Figure 1. DC-biased, ten-well superlattice. A shaped laser field excites a wave packet localized initially in the injection well. The objective is to created the maximum density possible in the detection well, at a chosen target time.

The globally optimal laser field for this example is presented in Fig. 2. The field is relatively simple with structure at early times, followed by a large peak with a nearly Gaussian profile. Note that the control formalism enforces no specific structure on the field a priori. That is, the form of the field is totally unconstrained during the allotted time interval, so simple solutions are not guaranteed. Also shown in Fig. 2 is the locally optimal... 

Figure 2. Optimal laser fields for the control scenario in Fig. 1. The solid line is the globally optimal laser field. The dashed line is the locally optimal Gaussian field.

In summary, this work has shown that superlattices are promising systems for investigation of quantum control in the solid state. The examples presented here show that the dynamics of charge carriers can be controlled using relatively simple, experimentally laser fields. Superlattices are ideal candidates for quantum control precisely because their complexity does not allow for simple, intuition-guided experiments, and because their dynamics are largely unknown. 

Optimal control theory, as discussed in Sections II-IV, involves the algorithmic design of laser pulses to achieve a specified control objective. However, through the application of certain approximations, analytic methods can be formulated and then utilized within the optimal control theory framework to predict and interpret the laser fields required. These analytic approaches will be discussed in Section VI. 

Perhaps the most straightforward method of solving the time-dependent Schrodinger equation and of propagating the wave function forward in time is to expand the wave function in the set of eigenfunctions of the unperturbed Hamiltonian [41], Hq, which is the Hamiltonian in the absence of the interaction with the laser field. 

One way in which we can solve the problem of propagating the wave function forward in time in the presence of the laser field is to utilize the above knowledge. In order to solve the time-dependent Schrodinger equation, we normally divide the time period into small time intervals. Within each of these intervals we assume that the electric field and the time-dependent interaction potential is constant. The matrix elements of the interaction potential in the basis of the zeroth-order eigenfunctions y i Vij = (t t T(e(t)) / ) are then evaluated and we can use an eigenvector routine to compute the eigenvectors, = S) ... 

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