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On the control of non-linear optical processes in strong laser fields (*)

J. Z. KAMIN´SKI(1) and F. EHLOTZKY(2)

(1) Institute of Theoretical Physics, Warsaw University - Hoz.a 69, 00-681 Warszawa, Poland

(2_{) Institute for Theoretical Physics, University of Innsbruck}

Technikerstrasse 25, A-6020 Innsbruck, Austria

(ricevuto il 19 Febbraio 1996; revisionato il 26 Marzo 1997; approvato il 7 Aprile 1997)

Summary. — Avoided crossings of quasi-energy levels are one of the most important

features of adiabatic approaches to quantum systems. We show that by studying these phenomena we gain deep insight into adiabatic dynamics, allowing one to reveal the mechanisms of non-adiabatic transitions. These non-adiabatic processes can be used to populate a desired excited state with significant probabilities and they permit to formulate necessary conditions for controlling such excitations.

PACS 32.80.Rm – Multiphoton ionization and excitation to highly excited states (e.g. Rydberg states).

PACS 34.50.Rk – Laser-modified scattering and reactions.

PACS 42.65.Re – Ultrafast processes; optical pulse generation and pulse compression.

1. – Introduction

Controlling processes that are described by quantum-mechanical equations of motion become more and more interesting recently, although this problem attracted a lot of attention already in the early days of quantum mechanics. With the advent of lasers and the rapid development of laser technology there were developed new methods which permit to control quantum systems (see, e.g., [1]). First of all, these investigations are connected with the necessity of achieving an optimum and stable action of lasers. On the other hand, the interaction of strong and coherent light pulses with matter uncovers new perspectives in technology. In this case we can achieve a desired transformation of matter, change its content, structure or state. From the point of view of the control process this task can be regarded as the problem of preparing on the macrolevel the form of the interaction of the electromagnetic field with atoms, molecules or solids in order to obtain suitable structures, states or transition probabilities on the microlevel. The famous Stern-Gerlach experiment can be considered as one of the earliest examples of such processes; this experiment allows—by applying an inhomogeneous magnetic field—to select atoms that have a definite projection of spin on a given axis.

(*) The authors of this paper have agreed to not receive the proofs for correction.

Controlling quantum-mechanical processes is different than the corresponding problem in classical mechanics. This is due to the fact that the basic dynamic equation of quantum theory, the time-dependent Schrödinger equation, contains the imaginary number i 4k_{21 , which nontrivially couples the real and imaginary parts of the wave}

function and thus, together with Born’s probabilistic interpretation of quantum mechanics, introduces into the theory an unobservable phase [2]. This fact immediately leads to the conclusion that the interference of waves plays the crucial role not only in obtaining the classical limit of quantum mechanics, but also in quantum processes themselves. This means that even small changes of macroscopic control parameters can induce large changes on the microlevel. The aim of this paper is to study such changes. We shall discuss how different forms of a time-dependent electromagnetic field modify quantum-mechanical processes.

The plan of this paper is as follows. In sect. 2 we discuss an adiabatic theorem and Landau-Zener transitions in the context of a quantum system interacting with short laser pulses. In sect. 3 we study a selective population of excited states, that are modeled by a general n-level quantum system interacting with an oscillating electric field. We demonstrate the necessary conditions to be fulfilled in order to achieve this goal. In particular, we show how the final population of an excited state depends on both the shape of the laser pulse and the form of the avoided crossings of quasi-energies, which can be modeled by changing the electric dipole transition elements between the states. We also summarize the discussions of the results and we suggest prospects for further investigations.

2. – Adiabatic theorem and Landau-Zener transitions

In this section we consider, from the practical point of view, probably one of the most important problems of controlling quantum processes. This problem can be formulated as follows. Let us assume our quantum system is prepared in a given initial state Ncib, which usually is the ground state. We require to find such a form of a laser

field which after a given period of time transforms the initial state into a desired final state Ncfb with a given—usually maximum—probability. Such a problem has two

important aspects. First of all, we have to determine the necessary conditions for the existence of solutions to our problem which allow to check which final states can be arrived at. Secondly, we have to construct an explicit form of these laser pulses which realize our task. It may well happen that such a form is not defined uniquely and one can ask for additional restrictions. For instance, one can look for such allowed pulses that have the smallest total energy or the shortest time of duration. Such an extension of the control problems is rather difficult to analyze and we are not going to discuss it here. The aim of this section is to study conditions under which one can populate a given excited state of an n-level system interacting with a laser pulse. Since such an excitation is due—as will be shown shortly—to non-adiabatic transitions between stationary states, we shall, for the sake of clarity, first collect briefly some basic facts about the adiabatic theorem.

The adiabatic theorem of quantum mechanics deals with systems governed by slowly varying time-dependent Hamiltonians. More precisely, if the typical time scale of the Hamiltonian H(t) is T, the adiabatic theorem describes the singular limit T KQ of the Schrödinger equation

i¯tNc(t)b 4 H(t) Nc(t)b .

The original formulation, due to Born and Fock [3] (see also [4-6]), states that, if we prepare the system at t 4t0 in an eigenstate Nc1(t0)b associated with the eigenvalue

E1(t0) of the Hamiltonian H(t0), then the solution of the wave equation at any time

t1F t0has the form

Nc(t1)b 4exp

y



t0

t1

E1(t) dt

z

provided E1(t) remains isolated in the spectrum for all t0G t G t1. As a consequence, the transition probability P21(T) to any other eigenstate Nc2(t1)b of H(t1) is of the order

T22 _{and vanishes in the limit T KQ. When the Hamiltonian H(t) is analytic in time,} the transition probability takes the exponential form

P21(T) Abe2gT, b , g D0 . (2.3)

This formula has been obtained by Zener [7] in the particular case of a two-level Hamiltonian for which he found an analytic solution of the Schrödinger equation. It was also derived independently by Landau [8], who introduced the idea of integrating the Schrödinger equation in the complex plane, making explicit use of the analyticity of the Hamiltonian (see, e.g., [9]). Hence this formula is often called the Landau-Zener formula or the exponential approach to the adiabatic limit. Independently Stueckelberg [10] studied this problem by the WKB method, but his paper received little attention, presumably due to its relative length and complexity. A mathematically rigorous approach to this problem is presented in an article by Jaksˇic and Segert [11].

The adiabatic theorem and the Landau-Zener formula can straightforwardly be generalized in order to account for the interaction of laser pulses with matter. In this case, however, the laser field introduces two time scales, namely the period of oscillations and the time between switching on and off of the pulse. The first time is usually very short and cannot be considered long if compared with a typical time of a quantum system which is of the order of 1 ODE, with DE being a typical energy scale of the system. On the other hand, the time between switching on and off is usually much longer and can be considered as the “T ” which the adiabatic theorem deals with. This means that fast oscillations have to be treated exactly whereas the process of switching on and off can be accounted for adiabatically. Hence, we have to know which states take over the role of stationary states in the limit T KQ, i.e. when the laser pulse becomes a plane wave. It turns out that these are the Floquet states [12-14].

It is well known that Floquet states and quasi-energies describe very well the evolution of quantum systems interacting with a quasi-monochromatic plane wave. Surprisingly, this description remains valid even for very short and intense laser pulses provided that the quasi-energies do not meet avoided crossings. This fully agrees with the adiabatic theorem which states, in particular, that during adiabatic changes of H(t) a given quantum system evolves according to (2.2) provided that E1(t) does not approach too closely by another temporary eigenenergy of H(t) (apart from some exceptions which are not going to be discussed here). Applying arguments similar to those of Zener, Breuer and Holthaus [15] were able to show that at the avoided crossing of quasi-energies the probability P of jumping from one quasi-energy surface

to another is P 4exp

k

l

N

where dl is equal to that change of the intensity of the electric field for which the distance between the corresponding quasi-energy levels within a given Brillouin zone increases by the factork2, whereas de is the distance between these quasi-energies at the avoided crossing. Moreover, lNis the time derivative of l(t) (defined below) and lcis the position of the considered avoided crossing. It follows from this expression that the system does not jump from the one quasi-energy surface to another provided that

g

l.

h

N

c1 . (2.5)

It is possible to derive a generalization of this formula applying a method developed in [16], but for the analysis presented here we shall not need it.

3. – Stueckelberg oscillations in n-level systems

In order to quantify our discussion, let us consider a n-state atom interacting with a strong laser pulse the dynamics of which is governed by the Hamiltonian

H(t) 4

!

!

where l(t) is an envelope function of the oscillating electric field, slowly changing in time, which is supposed to describe the laser pulse, Ei are the energy levels in the

absence of the external field, and dijare the moments of the electric dipole transitions

between these levels. Let us note at this moment that we put the diagonal elements of

dij equal to zero, because such a self-interaction does not change significantly the

transition probabilities between different states and the necessary conditions for these to occur, which is the main topic of this section. Since the laser field is considered in the dipole approximation, the absorption or emission of laser photons does not change the momenta of the electrons. This means that this model can be applied not only to bound-bound transitions in atoms or molecules, but also to the analysis of transitions between energy bands of crystals if the quasi-momenta are conserved. Of course, such models do not account for relaxation processes (which usually are described by some phenomenological decay constants) as, for instance, in the interaction of electrons with phonons. Since, however, we are interested here in short laser pulses, it is therefore justified to neglect, at least in a first step, such relaxation processes.

Since it is difficult to study even such a simple model with the help of purely analytic methods, if we want to investigate non-perturbative effects, we shall thus concentrate on the numerical analysis of this model and on the graphical presentation of the results. To this end, let us consider a three-level systems characterized by the energies Ei,

i 41, 2, 3, and the non-vanishing electric dipole moments dij. Moreover, the laser field

choose as l(t) 4./ ´ l exp [2t2 O2 s12] , l exp [2t2 O2 s22] , t E0 , t D0 , (3.2)

where s1bs2. The asymmetric form of this envelope is to describe a real laser pulse, which usually is rapidly switched on and then slowly levels off. Our goal is to answer the following question: is it possible to prepare the laser pulse in such a form that after it got switched off, the second excited state is populated with a significant probability, if only the ground state was populated initially? The positive answer to this question has practical implications as is discussed, for instance, in [17]. Before attacking this problem, however, we have to ask ourselves what are the conditions under which such an excitation can take place at all. Certainly, we can attain our goal if the ground state and the second excited state are at resonance, i.e. E32 E1 is very close to v. This situation, however, cannot always be satisfied experimentally, because sometimes the excited states are lying so high that there are no lasers available which generate pulses of such high frequencies. A way of bypassing this obstacle could be to prepare a quantum system such that all three states are at resonance. This method, however, has two disadvantages. Firstly, it is limited to particular quantum systems which have three states at resonance. Secondly, what is more important, after turning off the laser pulse, the first excited state is populated with a significant probability, as will follow shortly. Another proposal would be, to use the consequences of the adiabatic theorem; namely, that transitions between Floquet states are possible if quasi-energies meet

Fig. 1. – Two Brillouin zones of the Floquet spectrum(quasi-energies Ei(l) modulo v)for a model

defined by the energies E14 54 meV, E24 135 meV, E34 216 meV, and the oscillator strengths

f124 0.59, f134 1.23. Quantum systems with similar parameters can be fabricated

experimentally [18, 19]. The quantities Ei(l) Ov are plotted as functions of the dimensionless

parameter lsc4 lv23 O2for v 481 meV. Let us note that these quasi-energies meet for lsc4 0,

whereas for a non-vanishing laser field the corresponding Floquet states are a priori unknown linear combinations of the unperturbed states. Hence, in this case, we cannot say which quasi-energies correspond to which initial states. To the numerical value of the dimensionless parameter lv23 O2_{4 1 there corresponds for the above frequency an intensity with an electric}

field amplitude l 41.631024_{(in atomic units), which can be attained by present day free electron}

Fig. 2. – Two Brillouin zones of the Floquet spectrum(quasi-energies Ei(l) modulo v)for a model

defined by the energies E14 54 meV, E24 135 meV, E34 216 meV, and the oscillator strengths

f124 0.59, f134 1.23. The quantities Ei(l) Ov are plotted as functions of the dimensionless

parameter lsc4 lv23 O2 for v 450 meV. We have marked which quasi-energies correspond to

which initial states. By the capital letter A we indicate the avoided crossing between the first and the third states. The dimensionless parameter lv23 O2_{4 1 for the present frequency corresponds}

to an intensity with an electric field amplitude l 40.831024_{(in atomic units).}

each other. Therefore, it is helpful to know how quasi-energies evolve with increasing intensity of the plane wave. To this end, let us consider a semiconductor quantum well with three bound states and plot two Brillouin zones of the spectrum of quasi-energies as functions of the dimensionless parameter lsc4 lv23 O2 (in units ˇ 4c4m41) as

shown in figs. 1, 2 for two different laser field frequencies. Our model is characterized by three bound-state energies Eiand by dimensionless oscillator strengths fijwhich are

related to the dipole elements dijby means of the equation

fij4 2 m * NEi2 EjNdij2,

(3.3)

where m * is the effective electron mass which for the GaAs semiconductor is equal to 0.067 (in atomic units). Such a quantum well can easily be fabricated, see e.g. [18, 19], where in general asymmetric wells were considered. In this paper we shall only analyze symmetric quantum wells for which one of the oscillator strengths, f13, vanishes. Let us emphasize, however, that the conclusions drawn here also apply to more general cases.

Figure 3 shows the final populations of all states of the three-level system represented by the first model of fig. 1. All three levels are at resonance with the laser frequency. We observe that the populations of all three states oscillate rapidly with increasing top intensity of the laser pulse. The second excited state can be populated with a maximum probability approaching 0.8, which appears to be remarkable. However, one has to remember that it is very difficult to prepare the laser pulse exactly in a form suitable for our problem. It also follows from this figure that even for small deviations of the pulse from the optimum shape the final populations can decrease

Fig. 3. – Final occupation probabilities piof the unperturbed states Nib for i41, 2, 3 (p1, p2and

p3, respectively) as functions of the top scaled laser-field amplitude lsc4 lv23 O2for the model of

fig. 1. Observe the irregular changes of the probabilities as the intensity increases. The more regular pattern developing around lsc4 0.5 is due to the avoided crossing A in fig. 1.

significantly. Therefore, with respect to small fluctuations of the radiation, one has to find a more stable method of excitation. Such a method can be based on the non-adiabatic optical Landau-Zener transitions, which are related to avoided crossings of quasi-energies.

Figure 2 exhibits the avoided crossings of the quasi-energies E1(l) and E3(l) marked by the capital letter A. For our further analysis we shall choose two shapes of the laser pulse, namely, the first one with s14 10 T, s24 10 s1, and the second one with

s14 7 T, s24 10 s1, where T 42pOv. At the avoided crossing A the time derivatives l . are different in both cases and, therefore, we can expect that the non-adiabatic Landau-Zener transition probability between the above states is larger for the second pulse, as follows from (2.4). This leads to a larger quantum interference between the corresponding Floquet states and, consequently, we should observe different populations of the excited state N3b and the ground state N1b after the interaction of our quantum systems with a laser pulse. Of course, the first excited state N2b can only be populated with marginally small probabilities. These probabilities are presented in fig. 4 as functions of the peak intensity of the laser pulse. Before reaching the avoided crossing the excitation of the state N3b is very small, then it rises rapidly for larger intensities and starts oscillating. These are the Stueckelberg oscillations. It is clearly seen that the form of these oscillations depends on the shape of the laser pulse, but also, which is not considered here, on the width of the avoided crossing (such a width can be changed by modifying the oscillator strengths of our model which nowadays can be achieved in practice). Evidently, by decreasing the duration time of a laser pulse both, the period and the amplitude of the Stueckelberg oscillations are increasing. This means that for a given laser pulse the largest population of the second excited state can be achieved for quantum systems which exhibit avoided crossings as sharp as possible or interact with laser pulses as short as possible. On the other hand, one must have in mind that present day lasers generate light pulses the shapes of which are not very well defined and can fluctuate. Hence, for systems with too sharp avoided crossings

Fig. 4. – Final occupation probabilities p1 and p3 of the unperturbed states N1b and N3b,

respectively, as functions of the top scaled laser-field amplitude lsc4 lv23 O2 for the model of

fig. 2. The first excited state N2b is populated marginally. The upper row corresponds to a pulse with s14 10 T, s24 10 s1, whereas the lower row to one with s14 7 T, s24 10 s1, with T 42pOv.

Observe that both, the period and the amplitude of Stueckelberg oscillations, can be controlled by the shape of a laser pulse.

even small fluctuations of the pulses can induce significant changes in the final populations of the eigenstates. Therefore, in order to populate effectively the second excited state, one has to balance between the shapes of the laser pulses and the shapes of the avoided crossings.

4. – Conclusions

In this paper we discussed a generalization of the adiabatic theorem by Born and Fock to also include the interaction of a quantum system with a laser pulse and we investigated the selective population of excited states by this field, by considering as an example a quantum system, which is modeled by a general n-level system. We showed which necessary conditions have to be fulfilled in order to arrive at this goal of selective population. Of particular interest was to show how the final population of an excited state depends on both, the shape of the laser pulse and the form of the avoided crossings of quasi-energies which can be varied by changing the elements of the

electric dipole transitions of the quantum systems. The appearance of Stueckelberg oscillations were discussed in this connection. In our investigations we have neglected laser field fluctuations as well as relaxation processes induced by phonon interactions in semiconductors. In order to account for these effects one has to apply the density matrix approach instead of using the Schrödinger equation as starting point, considered in this paper. Such generalizations are under consideration now.

* * *

We would like to thank Prof. E. HEIRERand PROF. G. WANNERfor providing us with their FORTRAN codes. This work has been supported by the East-West-Program of the Austrian Academy of Sciences and by the Austrian Ministry of Science, Transportation and Art under project No. 45.372O2-IVO6O97. One of us (JZK) acknowledges the support by the Polish Committee for Scientific Research (grant KBN 2 P302 070 07).

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