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Recent Research Topics and Developments in Chemical Physics : From Quantum Scale to Macroscale, 2008: 75-99 ISBN: 978-81-7895-316-8 Editors: Andreas F. Terzis and Emmanuel Paspalakis

4 Enhanced frequency conversion in coherently prepared media

Thomas Halfmann Institute for Applied Physics, TU Darmstadt, Schlossgartenstr. 7 64289 Darmstadt, Germany

Abstract Coherent interactions between light and matter provide powerful tools to manipulate and control properties and processes in atoms, molecules and solids. Adiabatic passage processes are a particular class of techniques, which offer pronounced robustnessand stability with respect to variations in the experimental parameters. We will discuss the basic concepts and experimental implementations of specificadiabatic interactions, i.e. rapid adiabatic passage (RAP) and Stark chirped rapid adiabatic passage (SCRAP), to prepare population inversion and drive large coherences in atomic, molecular and solid media. We will focus in particular on applications in

Correspondence/Reprint request: Dr. T. Halfmann, Institute for Applied Physics, TU Darmstadt Schlossgartenstr. 7, 64289 Darmstadt, Germany. E-mail: [email protected]

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nonlinear optics, e.g. to enhance the efficiency of amplification processes and frequency conversion to the regime of vacuum- and extreme-ultraviolet radiation, and to generate intense, coherent spectra with ultra-broad bandwidth, i.e. high-order Raman sidebands. The presented experimental data are striking proof for the significant advantages of coherently prepared nonlinear optical media, in order to dramatically enhance frequency conversion processes compared to conventional techniques.

I. Introduction and motivation For many decades lasers have been most extensively used to study and investigate quantum systems and processes [1]. As such studies focused mainly on the understanding of the system or process under observation, lasers were commonly used as a probe. Going far beyond these applications, in the last decade laser-matter interactions were also found to permit the efficient control of quantum systems, which raised considerable attraction to a large scientific community. The control scenarios do not exclusively rely on the strength of the laser-matter interaction only, i.e. the amplitude of the driving radiation field. A rich variety of surprising phenomena arises when the coherence of the exciting radiation field is taken into account [2]. The field of coherent laser-matter interaction exhibits a variety of different approaches to steer atomic and molecular processes. Among these are techniques based on quantum interference between different excitation pathways [3]. These methods are often termed phase control techniques, as constructive or destructive interference is usually controlled by the relative phase of the driving radiation fields. If these techniques are extended to interactions with radiation fields of highest intensities, approaches based on pulse-shaping of ultra-short (fs) laser pulses also provide very powerful tools to control the media [4]. In contrast to phase control or pulse shaping approaches the research projects presented in the following focus on the development and application of laser-matter interactions, based on adiabatic passage processes [5]. As a main advantage, adiabatic passage processes offer significant robustness with respect to variations or fluctuations in the experimental parameters, e.g. laser intensities, detunings, pulse delays or shapes, provided some limits are kept in view. Applications of coherent, adiabatic interactions are numerous, e.g. including the preparation of population distributions [5, 6], the control of fragmentation processes [7] and the manipulation of optical properties [8, 19, 21-23]. While most of the experiments have been conducted in the gas phase, a few experiments on adiabatic interactions have also been conducted in selected solid state systems, e.g. doped solids [9, 10] or nanostructures [12]. In this article, we will draw our attention to applications in nonlinear optics. The latter is at the very heart of every laser-based science, as it serves

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to expand the accessible wavelength regime of coherent radiation sources. Thus, we will review our recent investigations on frequency conversion in coherently prepared media. We will discuss novel and efficient techniques, based on adiabatic interactions, to support and enhance frequency conversion processes. In particular, we will study nonlinear optical processes, aiming at the efficient generation of short-wavelength radiation, i.e. vacuum- or extreme-ultraviolet radiation. The latter finds applications in laser lithography, high-resolution microscopy, or spectroscopy. Moreover we will discuss efficient techniques to provide intense coherent spectra with ultra-broad bandwidth. Such spectra may serve as efficient sources for the generation of ultra-short radiation pulses, providing sub-femtosecond resolution. In the following sections, first we will briefly introduce the basics of coherent, adiabatic interactions between light and matter. We will discuss quite simple adiabatic processes, which permit the manipulation of populations distributions in low-dimensional quantum systems and present experimental data, which clearly demonstrate the advantages of adiabatic interactions. Based on these considerations, we will then discuss possibilities to control nonlinear optical properties by adiabatic interactions. Finally, a number of experimental implementations will be reviewed, which demonstrate the significant enhancement of frequency conversion processes in coherently prepared media. As this article focusses on experimental work, the theoretical treatment is simplified as much as possible. Thus the author hopes, that also readers, who are no experts in the field, should be enabled to easily follow the arguments. II. Basics of coherent interactions Consider a two-level quantum system of a ground state and an excited state , driven by a single, incoherent pump radiation field. Commonly, such an incoherently driven medium is described by rate equations. These equations implicitly include averages over phase fluctuations, e.g. in the driving radiation field. The rate equations involve the fundamental processes of stimulated absorption and stimulated emission as well as spontaneous emission. The relevant variables in the rate equations are the Einstein coefficients B12 for stimulated absorption, B21 for stimulated emission and A21 for spontaneous emission of photons. Spontaneous emission can be neglected, if the lifetime of the excited state is much longer than the interaction time with the radiation field. If this holds true, and provided the radiation field is of appropriate strength or the interaction time is long enough, a maximum of 50 % of the population in the ground state can be transferred to the excited state . This is termed saturation of the transition. The equal distribution of population at the end of the strong interactions mirrors the equal probabilities for stimulated absorption and emission, i.e. B12 = B21. Thus, in the case of incoherent

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excitation, population inversion is not possible in a two-level system of bound quantum states. In contrast, if we now consider the two-level system, driven by a strong, coherent pump radiation field, the model presented above is no longer valid. While the rate equations are based on perturbation theory, i.e. the description of the system in the basis of the bare states (the eigenstates of the system without the interaction with the radiation field), the correct treatment for strong, coherent interaction needs to go beyond perturbation theory. As the radiation field strongly affects the system, new eigenstates, i.e. the dressed states, arise from the Hamiltonian involving the coherent laser-matter interaction. As we will see later, these dressed states are coherent super-positions of the bare atomic states. The Hamiltonian for a two-level atom interacting with a strong, coherent radiation field, after rotating wave approximation and transformation to Dirac representation, reads [2] :

(1)

with the Rabi frequency the electric field strength E, the transition dipole moment µ and the detuning from resonance ∆. The dressed states, i.e. the eigenstates of the Hamiltonian including the interaction with the radiation field, read [2] :

(2)

(3) with the wavefunctions of the bare states and the mixing angle given by

(4) In contrast to incoherent excitation the (diabatic) population dynamics for coherent interaction exhibit Rabi oscillations of the bare state populations as a prominent feature. The time scale of the oscillation is determined by the Rabi frequency Ω (t). If the radiation field is tuned to resonance, the population in the states and oscillates between 0 % and 100 % [2]. In the case of fluctuations in the laser intensity, frequency or phase, the Rabi oscillations average to an equal (50:50) population distribution, as expected also for incoherent excitation.


III. Rapid adiabatic passage (RAP) To study the possibility of complete and robust population transfer between the ground and the excited state, we will now consider the dressed state dynamics. Complete population transfer in a two-level system, e.g. via the dressed state Φ+ is possible, if the following requirements are met: (i) At the beginning of the interaction all population is prepared in the dressed state Φ+, thus the initial bare state aligns parallel to Φ+ for early times (t → − ∞). (ii) During the interaction the state vector Ψ of the system must follow the dressed state Φ+ adiabatically, i.e. no population is exchanged between the dressed states Φ±. (iii) At the end of the interaction (t → + ∞) Φ+ must align parallel to the excited state in order to project all the population from the dressed state onto the target bare state. Regarding the definitions of the dressed state Φ+ and the mixing angle θ (see eqn. 2,3,4), the conditions (i) and (iii) require the detuning ∆ to change from a large positive to a large negative value during the interaction. Thus a frequency chirp, i.e. the laser frequency driven through the atomic resonance, is required. Provided the transfer process is adiabatic, i.e. no population is exchanged between the two dressed states during the process, all population can be transferred by a chirped pulse from the ground state to the excited state (see fig. 1). This technique is called rapid adiabatic passage (RAP) [5]. The specification "rapid" indicates, that the interaction time has to be short with respect to the lifetime of the excited state, otherwise population will decay from the excited state during the process. Besides the requirements for the chirp, as deduced above, a full theoretical treatment of RAP also reveals conditions for adiabatic following. The adiabaticity criterion essentially demands a minimum peak Rabi frequency or pulse area A = Ω⋅τ, i.e. the product of the interaction strength and the interaction time should be large. In terms of incoherent excitation this is equivalent to saturation of the transition. Moreover, a minimum chirp rate is required [2, 5]. Small variations in the exact temporal evolution of the chirp do not change the adiabatic population dynamics, i.e. it does not matter too much, whether the chirp follows a linear or any other dependence in time. While RAP has been extensively investigated in media in the gas phase (see [5] and references therein), only few investigations were performed in solids [10, 11]. Due to their high density and scalability, solid state media are attractive for applications, e.g. for optical data storage and quantum information processing. Strong decoherence in solids is an obstacle, but special systems combining the advantages of solids and the coherence properties of atoms have been identified, e.g. quantum dots and rare earth ion doped inorganic crystals. In particular the latter are promising candidates for applications in high-density

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Figure 1. Numerical simulation of RAP. Coupling scheme, population dynamics and electric field of the pump laser vs. time (schematically only). In the simulation the pump laser frequency is linearly chirped. The transition frequency ω12 is met at the peak intensity of the radiation pulse. optical data storage and information processing. We studied the implementation of rapid adiabatic passage in a Pr3+:Y2SiO5 crystal [11], i.e. Praseodymium ions doped in a Y2SiO5 host crystal. To reduce homogeneous broadening, the crystal is cooled to cryogenic temperatures of approximately T = 4 K. Due to the remaining, still substantial, inhomogeneous bandwidth, Pr3+:Y2SiO5 is an efficient medium for optical data storage by spectral hole burning. The multi-level structure of the doped Praseodymium ions and the large inhomogeneous bandwidth demand specific optical preparation, prior to the RAP experiment. Essentially, the preparation pulse sequence generates a two-level system with a single, well defined absorption line in the Praseodymium ions. In this two-level system a pump laser pulse drives a RAP process. The pump laser frequency is linearly chirped across the transition frequency of the prepared two-level system. A weak and short probe pulse, well delayed with respect to the pump pulse and tuned to resonance in the two-level system, serves to


monitor the absorption in the medium after the interaction with the pump pulse. The absorption coefficient α is proportional to the population difference between the ground state and the excited state. If the major part of the population is in the ground state, the absorption coefficient is α > 0. If the medium is inverted, the absorption coefficient becomes α < 0. Thus α serves as a measure for the population distribution. The experimental data strikingly confirm these expectations: When the pump laser is switched off, all population remains in the ground state. The delayed probe laser experiences absorption, i.e. α > 0 (see fig. 2, upper trace). When the pump laser is switched on and chirped through resonance, all population flows to the excited state. Now the delayed probe pulse experiences significant amplification in the inverted medium, i.e. α < 0 (see fig. 2, lower trace) [11]. In contrast, incoherent excitation would not permit the generation of population inversion nor amplification of the probe pulse at all. In the same experimental setup, as discussed above, we also monitored the dynamics of RAP in the Pr3+:Y2SiO5 crystal [11]. To permit temporal resolution of the dynamics, we monitored the transmission of the short, weak probe pulse vs. the delay of the latter with respect to the pump pulse. The measured absorption coefficient α for a certain delay translates via Beer's law to the population distribution in the medium. Thus α serves as a measure for the transfer efficiency to the excited state, at a certain time defined by the probe pulse. The results of the experiment are shown in fig. 3. Here negative

Figure 2. RAP in a Pr3+:Y2SiO5 crystal. Absorption of a probe laser pulse.

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Figure 3. Population dynamics of RAP in a Pr3+:Y2SiO5 crystal. pulse delay corresponds to the probe pulse preceding the pump pulse. The rising edge of the transfer efficiency, at delays around zero, exhibits the typical dynamics of RAP (see fig. 3). The risetime is mainly determined by the pump pulse duration, i.e. in the order of 10 µs. Population flows smoothly from the ground state to the excited state. The transfer reaches a maximum of 100 %, i.e. complete population inversion. The slow decay of the signal for large pulse delays is due to fluorescence from the excited state, well after the interaction with the pump pulse. The data clearly confirm the smooth, adiabatic dynamics of RAP (compare fig. 1), in our experiment implemented in the environment of a solid. The experiments, discussed above [11], as well as related work by other authors [9, 10], serve as first steps towards the implementation of adiabatic processes in solids. Future investigations will yield new insight towards the implementation of efficient and robust adiabatic techniques, applied to high-density optical data storage and processing in solids. IV. Stark chirped rapid adiabatic passage (SCRAP) RAP exhibits a robust tool for efficient population transfer, provided the excited transition is strongly driven, and chirping of the pump laser is possible. Chirps are easily implemented for very long interaction times by acousto-optical modulation of the laser frequency, or for short (picosecond or femtosecond) radiation pulses by methods based on dispersion. However, the


best combination of both large pulse energy and long interaction time is usually provided by rather long (nanosecond) pulses. In this case, the pulse area Ω⋅τ (see above) becomes large. On the other hand, due to the small bandwidth of nanosecond pulses, chirping is usually hard, if not impossible to implement. To overcome these difficulties, we suggested an extension of RAP, i.e. Stark chirped rapid adiabatic passage (SCRAP) [5, 13]. The basic idea of SCRAP is to replace the frequency chirp of the pump pulse by a modulation of the atomic resonance frequency itself. Thus, the atomic resonance is chirped rather than the laser pulse. The modulation of the atomic resonance is implemented by an additional strong off-resonant radiation pulse, inducing dynamic Stark-shifts (see fig.4). The coupling scheme is as follows : A pump pulse excites a (single- or multi-photon) transition from the ground state to an excited state . The pump laser frequency ωP is slightly detuned by ∆P from the atomic resonance frequency ω12. No chirp is applied to the pump laser. An intense Stark laser pulse enters the medium, either preceding or following the pump pulse. The time delay ∆τ is chosen such, that the pulses still overlap partially in time. In the following we consider the case of the Stark pulse following the pump pulse. However, our basic arguments hold true also for the Stark pulse preceding the pump pulse. The frequency of the Stark laser must not overlap with any transition frequency in the medium. This requirement is usually very easy to fulfil. Only incidently the frequency of a high-power, fixed frequency Stark laser would overlap with atomic resonances. Though the Stark laser does not excite resonant transitions, off-resonant interactions with all states in the medium lead to a Stark shift of the transition frequency in the two-level system. The Stark shifts S(t) are proportional to the Stark laser intensity ISt(t). The sign of the Stark shifts depends upon the level structure. We note, that in most quantum systems the Stark shift of the ground state is negligible. Thus in good approximation the modulation of the transition frequency is equal to the Stark shift of the excited state. As the rising edge of the Stark pulse overlaps partially with the pump pulse, the modulated transition frequency ω12 + S(t) will be driven through resonance with the pump laser frequency ωP , provided the peak Stark shift is larger than the detuning ∆P . This is equivalent to RAP, when the pump laser frequency is driven through the atomic resonance. As in RAP, population flows in a adiabatic passage process from the ground state to the excited state. The transfer is completed in the rising edge of the Stark pulse. The dynamic Stark shifts in the trailing edge of the Stark pulse drive the transition frequency a second time through resonance with the pump laser frequency. As the pump pulse is essentially off now, the second pass through resonance does not effect the medium any more. We also note, that the population dynamics in SCRAP,

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Figure 4. Numerical simulation of SCRAP. Couplings, populations dynamics and pulse sequence. implemented with a Stark laser pulse of Gaussian temporal profile (see fig. 4), are almost the same as in the case of RAP, implemented with a linearly chirped pump pulse (compare fig. 1). This feature also mirrors the robustness of adiabatic passage processes. As in the case of RAP the criterion for adiabatic following [5, 13] yields conditions for the peak pump Rabi frequency and the peak Stark shift. These conditions are very similar to RAP, i.e. the essential requirements are strong interaction, i.e. large pulse area Ω⋅τ, and sufficient Stark shift, i.e. both a large peak Stark shift and a large shift rate. We note, that in SCRAP the pump laser frequency is detuned from the atomic resonance. This offers significant advantages for excitation of inhomogeneously broadened media. In such media, the resonance frequency changes for different atomic ensembles. Resonant excitation selects a specific ensemble, while other ensembles are unaffected. In contrast, SCRAP excites all ensembles off resonance, provided the laser-induced dynamic Stark shifts are larger than the inhomogeneous broadening. SCRAP offers also an advantage, if the pump laser is chosen to drive a multi-photon transition rather than a single-photon transition. In this case, also the pump laser will induce Stark shifts. These shifts are detrimental for techniques, relying on resonant


excitation [15]. In contrast, SCRAP still permits complete population transfer, provided the shifts induced by the Stark laser exceed the shifts induced by the pump laser. We implemented the first experiment on SCRAP with radiation pulses in the nanosecond time domain [13]. A pump pulse excites a (two-photon) transition between the initial state 2s 1S0 and the target state 3s 1S0 in a supersonic jet of metastable Helium atoms. Prior to the SCRAP experiment, the metastable state 2s 1S0 was efficiently populated in a pulsed, electron injection-seeded gas discharge [14]. The Stark pulse is provided by a small fraction of the radiation from a solid-state laser system, operating at fixed wavelength. The pump laser pulse with tunable frequency is deduced from a more complex laser system. The Stark pulse is delayed with respect to the pump pulse. Thus the rising edge of the Stark pulse drives the SCRAP process. The population in the excited state is monitored by photoionization, driven by a probe laser pulse. The probe pulse is well delayed with respect to the pump and Stark pulse. Fig. 5 shows the transfer efficiency vs. the detuning of the pump laser from two-photon resonance. When the pump laser frequency is slightly detuned by approximately ∆P = 3 GHz from the atomic resonance, the SCRAP process reaches maximum efficiency. In this case, the data indicate complete population inversion, i.e. a transfer efficiency of 100 %. The asymmetric shape of the spectral line and the shift of the maximum with respect to the atomic resonance is typical for SCRAP. The shift of the maximum permits a first estimation of the average Stark shift during the

Figure 5. SCRAP in metastable Helium atoms. Coupling scheme and transfer efficiency vs. detuning of the pump laser frequency. Experimental data and numerical simulation. Here and in all following figures vertical numbers in the coupling scheme indicate laser wavelengths.

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interaction. We confirmed experimentally the robustness of SCRAP with respect to variations in the experimental parameters, e.g. pulse delay, pump Rabi frequency and peak Stark shift [13].

V. Efficient excitation of molecular vibrations by SCRAP In this section, we will discuss an extension of SCRAP, which serves to prepare high lying vibrational states in molecules, i.e. to store a large amount of vibrational energy in the system. The key idea of this extension is to drive a multiple SCRAP process in a three-level lambda-type level scheme (see inset in fig. 6) [17]. In a simple approach, population in the vibronic ground state is transferred by a pump pulse in the rising edge of a Stark pulse to an intermediate, excited state . Then a Stokes pulse in the trailing edge of the Stark pulse drives the population in the excited state completely to a metastable, vibrationally highly excited state (see fig. 6(a)). This process is a SCRAP-variant of stimulated emission pumping (SEP) [16], involving a pump laser driven SCRAP process and a Stokes laser driven SCRAP process. However, if the excited state is subject to radiation decay during the interaction, i.e. if the lifetime of is shorter than the pulse delay between pump pulse and Stokes pulse, a large amount of the population will be lost. Thus this simple version of a double-SCRAP process would be very inefficient (see fig. 6(a)). In a better approach, the pump and Stokes pulse are coincident, both in the rising edge of the Stark pulse. The detunings of the pump laser ∆P and of the Stokes laser ∆S from the relevant transitions must be different, i.e . ⎢∆S ⎢ > ⎢∆P⎢. In a simplified picture, the transfer process in this three-state SCRAP process can be envisioned as follows : The rising edge of the Stark pulse induces Stark shifts, which increase with the laser intensity. As ⎢∆S ⎢ > ⎢∆P⎢, the pump laser will be driven first through resonance, i.e. population flows from the ground state to the intermediate state . Very shortly after the first SCRAP process the Stark shift approaches the value of ⎢∆S ⎢. Thus, the rising edge of the Stark laser also drives the Stokes laser through resonance, i.e. population flows from the intermediate state to the target state . The transfer time from the ground state to the target state is determined by the derivative of the Stark shift vs. time. If this derivative is large enough, the storage time and the accumulated population in the intermediate state are very small. Thus three-state SCRAP leads only to minor losses due to decay from the intermediate state (see fig. 6(b)). We like to note, that the simplified picture of two independent SCRAP processes, as presumed above, is not fully correct for three-state SCRAP.


Figure 6. Numerical simulations of (a) double-SCRAP and (b) three-state SCRAP. Pulse sequences and population dynamics. The inset shows the principle coupling scheme. In the simulation, the lifetime of the intermediate state was set as comparable to the pulse durations. The laser detunings are chosen such, that the transfer efficiency reaches a maximum. Instead, we must consider the dynamics of a coherently driven three-state system rather than two independent two-level systems [17]. However, the complete treatment yields the same result, as determined from the simplified approach : If pump pulse and Stokes pulse are coincident in the rising wing of the Stark laser, population is very quickly transferred via the intermediate state. Losses due to radiative decay are negligible in the case of three-state SCRAP. Fig. 7 shows data from the experimental implementation of three-state SCRAP in nitric oxide (NO) molecules. SCRAP served to transfer population from the vibrational ground state X 2Π1/2 (v" = 0) via the intermediate state A 2∑1/2 (v' = 0) to the vibrationally highly excited state X 2Π1/2 (v" = 6) [18]. The population in the target state was monitored by resonantly enhanced multi-photon ionization, driven by a probe laser pulse. The experiment was implemented with nanosecond (ns) laser pulses. The pump and Stokes pulse were coincident. The Stark laser pulse was slightly delayed with respect to the

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pump and Stokes pulse. The probe pulse was well delayed with respect to the other laser pulses. The data show a maximum transfer efficiency approaching 100 % for appropriate detunings of the pump and Stokes laser frequency from the molecular resonances. As the extended plateau of maximum population transfer in fig. 7 indicates, three-state SCRAP is robust with respect to fluctuations in the experimental parameters, e.g. laser detunings. Thus the technique permits a robust tool to store a large amount of vibrational energy in the NO molecules, both in a highly efficient and selective way. We note, that we calibrated and compared three-state SCRAP to conventional techniques for population transfer from a ground state via an intermediate state to a target state, e.g. SEP [16]. In SEP the coincident pump pulse and Stokes pulse are tuned to resonance with the relevant transitions. The Stark pulse is switched off. SEP permits a maximum transfer efficiency of 33 % only. Moreover, the selectivity in SEP is low, as population decays from the intermediate state to the multitude of excited vibrational states. Three-state SCRAP offers also a promising alternative to coherent techniques, e.g. stimulated Raman adiabatic passage (STIRAP) [5]. In contrast to STIRAP, SCRAP does not rely on resonant interaction. Thus three-state SCRAP is also applicable in systems, involving multi-photon excitations or inhomogeneous broadenings (see above).

Figure 7. Three-state SCRAP in NO molecules. Transfer efficiency vs. laser detunings.


VI. Frequency conversion, enhanced by SCRAP The projects, discussed above, dealt with the manipulation of population distributions in quantum systems. Thus, so far we focussed on the properties of the medium itself and neglected its feedback on incoming or generated radiation fields. Only very briefly we discussed the effect of adiabatic interactions on a resonant probe laser pulse, propagating through the coherently driven medium. Thus, the modulation of absorption by RAP (see fig. 2), served as a first, but also very obvious example for the manipulation of optical properties by adiabatic interactions. This experiment already demonstrated the close connection between population distributions and optical properties. In what follows, we will discuss consequences of this connection also with regard to the manipulation of nonlinear optical processes, in particular frequency conversion processes. The population distributions in a quantum system determine any property, also any optical property. Thus, coherent preparation is expected to permit the control of nonlinear optical processes. If the aim is to enhance nonlinear optical processes, i.e. frequency conversion processes, the wave equation for the propagation of a radiation field through a medium determines the conditions for appropriate control mechanisms. The wave equation in slowly varying envelope approximation, for propagation in the direction z, with the frequency components of the electric field and the corresponding components of the polarization reads:

(5)

The wave equation identifies the polarization as the physical property, which permits control of any optical response of the medium. In a two-level system the polarization reads with the number of atoms N, the dipole operator and the state vector

, including the probability amplitudes c1 and c2. As the expression for the polarization shows, reaches a maximum, if the product of the amplitudes i.e. the coherence of the medium, is maximized. The coherence, and also the polarization, are maximal for equal amplitudes

In this case, the system is prepared in the state of maximal coherence, i.e. a coherent superposition of ground state and excited state with equal amplitudes. Every optical process will be most efficiently driven at maximal polarization, i.e. at maximal coherence. As the coherences in a medium depend upon the population distributions, we must draw our attention now to techniques, which permit the efficient manipulation of quantum state populations. Indeed, the techniques, which we already studied in the sections

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above, will turn out as very appropriate for this purpose. Thus, in the following we will discuss the application of SCRAP [5, 13] to drive a quantum system to the state of maximal coherence and to permit efficient frequency conversion processes. Consider the population dynamics in a two-level system, driven by SCRAP. In SCRAP all population is transferred from a ground state to an excited state , e.g. via a pump laser driven two-photon excitation at ω12 = 2ωP. Since SCRAP inverts the population distribution, during the process an equal population distribution is prepared. The coherence reaches the maximum value of 1/2, i.e. a transient maximal coherent superposition of the ground state

and the excited state is established (see fig. 8). As the transfer process is adiabatic, i.e. it shows a smooth temporal profile, also the coherence evolves smoothly in time, without any fast oscillations. The coherence can be monitored by a frequency conversion process, e.g. driven by an additional probe laser pulse at frequency ωPr. The probe pulse must be timed appropriately with the maximal coherence in order to generate a four-wave mixing signal at frequencies ωS = 2ωP ± ωPr. The latter equation indicates both sum- and difference frequency mixing, if possible. When the wavelength of the probe laser is in the visible regime, sum frequency mixing with an ultraviolet pump laser generates a signal wave in the vacuum- or extreme ultraviolet spectral region. As the system is prepared by SCRAP in the state of maximal coherence, the conversion efficiency will be significantly enhanced with respect to conventional four-wave mixing [19-21]. We implemented the experimental configuration, sketched above, in a first experiment on frequency tripling (see fig. 9), driven by nanosecond (ns) radiation pulses [21]. In contrast to the coupling scheme in fig. 8, the same laser pulse was used to pump and probe the coherence. Thus, the four-wave mixing process with ωS = 2ωP ± ωPr degenerates to third-harmonic generation with ωS = 3ωP. The experiment was performed in a dense, supersonic jet of Krypton atoms, which serve as a typical and efficient medium for the generation of short-wavelength radiation. An ultraviolet pump laser pulse at λP = 213 nm is frequency tripled in Krypton to generate extreme-ultraviolet radiation at λS = 71 nm. First, let us consider conventional frequency tripling. This corresponds to the case of the Stark laser pulse switched off. Only the pump pulse interacts with the medium. Atomic resonances can be used to enhance the efficiency of such conventional frequency conversion, since resonant (diabatic) excitation slightly enhances the atomic coherence. Thus, when the pump laser is tuned close to two-photon resonance between the ground state 4p 1S0 and the excited state 5p [1/2]0 in Krypton, the signal intensity exhibits a resonance enhancement (see fig. 9, lower trace in the third-harmonic yield). However, the


Figure 8. Numerical simulation of SCRAP. Coupling scheme involving probing by four-wave mixing. Populations and coherence vs. time. (compare fig. 4).

Figure 9. Enhancement of third-harmonic generation in Krypton atoms, driven in SCRAP. Coupling scheme and third-harmonic yield vs. pump laser detuning. Conventional case of frequency conversion (lower trace, open triangles) and frequency conversion, enhanced by SCRAP (upper trace, solid squares). For better visibility the trace for conventional frequency conversion is shifted downwards.

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absolute conversion efficiency for conventional frequency tripling is still very small. In contrast, when a Stark laser pulse at λSt = 1064 nm is switched on and the pulse delay as well as the pump laser frequency are appropriately chosen, the medium is driven in SCRAP. All population flows from the ground state 4p 1S0 to the excited state 5p [1/2]0. During the process, a maximal coherence on the two-photon resonance is established. The maximal coherence enhances the generation of extreme-ultraviolet radiation by more than one order of magnitude, i.e. a factor of 25, with respect to conventional frequency conversion (see fig. 9, upper trace in the third-harmonic yield) [21]. The shift and the asymmetry in the spectral line shape is due to the laser-induced Stark shift (compare also fig. 5). In a second experiment, we investigated the enhancement of four-wave mixing by SCRAP in a cell with Mercury atoms at high density [19]. The experiment was also implemented with nanosecond (ns) radiation pulses. A pump laser pulse at λP = 313 nm drives a two-photon transition in a dense cell of Mercury vapour (see fig. 10). The Mercury atoms exhibit a broad isotope distribution, covering a range of approximately 25 GHz for the most abundant species. When a photon from a probe laser field at λPr = 626 nm is added, signal radiation in the vacuum-ultraviolet spectral regime at λS = 125 nm is generated. An off-resonant Stark laser pulse at λSt = 1064 nm serves to induce dynamic Stark shifts. First, to demonstrate the conventional case of four-wave mixing, the Stark laser is switched off. In the conventional case the conversion

Figure 10. Enhancement of four-wave mixing in Mercury atoms, driven by SCRAP. Coupling scheme and four-wave mixing yield vs. pump laser tuning (see also caption of fig. 9). The zero in the frequency scale was set to the resonance frequency of the most abundant Mercury isotope.


efficiency reaches maxima for pump laser frequencies, tuned to two-photon resonance for different Mercury isotopes (see fig. 10, lower trace in the four-wave mixing yield). Depending on the pump laser frequency, only a fraction of the full isotope distribution contributes to the conventional frequency conversion process. When the Stark laser is switched on and the pump laser is tuned slightly off the two-photon resonance for the most abundant isotopes, the medium is driven in SCRAP. As in our experiment the peak Stark shift exceeds the isotope shifts, all Mercury isotopes are now driven by SCRAP and participate in the frequency conversion process. A maximal coherence is established between the states 6s 1S0 and 7s 1S0. In this case of adiabatically driven four-wave mixing, SCRAP enhances the conversion efficiency by more than one order of magnitude with respect to the conventional case (see fig. 10, upper trace of the four-wave mixing yield) [19]. The main results from our experiments [19, 21] as well as from analytical studies [20] of frequency conversion processes in SCRAP-prepared media are : (i) SCRAP considerably enhances the efficiency of frequency conversion processes; (ii) The preparation of a maximal coherence by SCRAP is not effected by inhomogeneous broadening in the medium, provided the laser-induced Stark shifts are larger than the inhomogeneous linewidth. This is a very interesting feature with regard to applications, as usually dense, gaseous media are used for efficient frequency conversion. These media suffer from inhomogeneous broadenings. SCRAP overcomes this problem. (iii) SCRAP provides the largest enhancement for quite moderate laser intensities, i.e. at the edge of saturation. There is no need for larger laser intensities to exploit the advantages of SCRAP. We like to note, that highly efficient frequency conversion at maximal coherence was also demonstrated in multi-level quantum systems, prepared by electromagnetically-induced transparency (EIT) [8]. The latter exhibits another powerful adiabatic technique, which we will not discuss here - but must not fail to mention. The experiments on efficient frequency conversion, driven either by EIT or SCRAP, demonstrate the huge potential of adiabatic interactions in the field of nonlinear optics. A. High-order Raman sideband generation The preparation of a maximal coherence, as discussed in the previous section, was implemented with rather long laser pulses in order to fulfill the condition for adiabatic following. This condition essentially gives lower limits for the pump laser intensity and the Stark shift [5, 13]. For conventional laser systems and excitations on single- or two-photon transitions, the best combination of laser intensity and pulse duration, i.e. interaction time, is provided by nanosecond (ns) or long picosecond (ps) pulses. Thus these pulse

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durations are, in general, best to drive adiabatic processes. Laser systems providing ultra-short (fs) radiation pulses are in most cases not appropriate for adiabatic preparation. On the other hand, due to the large intensity, ultra-short (fs) pulses are favorable to drive frequency conversion processes. Thus, coherent preparation of a nonlinear optical medium by ultra-short (fs) pulses would offer considerable advantages. In the following we will show, that even if proper preparation of a maximal coherence is not possible, also less efficient enhancement of a coherence by ultra-short (fs) pulses significantly supports nonlinear optical processes. The effect of enhanced molecular coherences to drive Raman scattering processes was investigated in a cooperative project, directed by J.P. Marangos at Imperial College (London) [22, 23]. Raman sidebands show up, when an intense pump laser at frequency ωP drives a molecular medium far off any resonance with excited electronic states. Interaction via a virtual state generates weak sidebands. The frequencies of the sidebands are separated by the vibrational quanta ωυ in the electronic ground state. To enhance the generation efficiency of the first sideband, a Stokes laser pulse, coincident with the pump laser pulse, is introduced in the medium (see fig. 11 (a)). The frequency difference of the laser pulses is tuned close to the two-photon Raman resonance between the vibrational ground and the first vibrationally excited state. The quantum system acts now like a molecular modulator, oscillating at the frequency of the Raman transition. The interaction of the molecular modulator and the radiation fields leads to the generation of sidebands. The interaction is mediated via a process, closely related to four-wave mixing. In a simplified picture, the generation of high-order Raman sidebands may be understood as a sequence of frequency conversion processes : Initially, the pump pulse and the Stokes pulse enhance the molecular coherence between

Figure 11. High-order Raman sideband generation in molecular media. Basic coupling schemes.


two low vibrational states. The pump laser field beats with the enhanced coherence to generate a first anti-Stokes Raman sideband with high efficiency (see fig. 11 (b)). This first sideband again beats with the enhanced coherence to generate a second sideband (see fig. 11 (c)). Subsequent mixing processes proceed to higher order sidebands. As the coherence is enhanced, also the overall conversion efficiency is enhanced with respect to the case of conventional Raman sideband generation. We note, that the same holds true also for the Stokes sideband. A detailed theoretical investigation reveals, that the coherence reaches a maximum, when the difference frequency of the laser fields is slightly detuned from exact Raman resonance. This, on the first glance surprising effect, is a characteristic feature of electromagnetically induced transparency (EIT) [8] and coherent population return (CPR) [6]. Besides nonlinear optics, the latter technique also finds applications in trace isotope detection [6]. Fig. 12 shows the spectra, obtained in an experiment on stimulated Raman side band generation in Hydrogen molecules, driven by a pair of ultra-short (fs) laser pulses [22, 23]. The laser frequencies are tuned to match the transition frequency between the vibrational ground state and the first excited

Figure 12. High-order Raman sideband generation in Hydrogen molecules, driven by ultra-short (fs) radiation pulses. Sideband spectrum for a single Stokes pulse (open triangles) and for coincident pump and Stokes pulse (solid squares). Note the logarithmic scale for the sideband intensity.

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vibrational state in Hydrogen molecules, i.e. ∆ν = 4155 cm−1. Thus the wavelengths of the two laser pulses are chosen as λ1 = 600 nm and λ2 = 800 nm. When only the Stokes laser pulse alone drives the medium, the intensities of the high-order anti-Stokes sidebands are below the noise level. Only the fundamental Stokes laser frequency at λ2 = 800 nm shows up in the spectrum (see fig. 12, open triangles). When both laser pulses drive the system and enhance the molecular coherence, strong anti-Stokes sidebands show up (see fig. 12, solid squares). As expected (compare fig. 11) the frequencies of the sidebands are separated by ∆ν = 4155 cm−1. The total conversion efficiency reaches 10 %. The high-order Raman sidebands extend from the infrared to the ultraviolet spectral region, i.e. they cover a huge spectral regime. The sidebands, dispersed by a prism, are intense enough to be observed by the naked eye on a piece of cardboard (see inset in fig. 12). VII. Conclusions The experimental data, presented above, demonstrate the possibilities of robust, adiabatic interactions, e.g. rapid adiabatic passage (RAP) and Stark chirped rapid adiabatic passage (SCRAP), to manipulate population distributions and coherences in atoms, molecules and solids. The techniques are applied to significantly enhance nonlinear optical processes, e.g. frequency conversion processes. In particular, the main results are as follows : (i) While most experiments on adiabatic interactions have been performed in gaseous media, solids are of interest for applications in large-density optical data storage. Thus, we implemented RAP in a doped solid. RAP served to prepare the medium in a state of population inversion and drive amplification of probe laser pulses. We also presented data on the temporally resolved dynamics of the RAP process, which agree very well with expectations. (ii) We demonstrated efficient, coherent population transfer by SCRAP in metastable Helium atoms. In contrast to RAP, the SCRAP technique drives adiabatic passage via the modulation of transition frequencies by laser-induced Stark shifts rather than by the modulation of laser frequencies. We also demonstrate the implementation of an extension of SCRAP, i.e. SCRAP among three states, to prepare vibrationally highly excited molecules with large efficiency and selectivity. (iii) We applied the SCRAP technique to drive atomic media to a state of maximal coherence, i.e. maximal polarization. Due to the maximal coherence, third-harmonic generation to the regime of extreme-ultraviolet radiation in coherently prepared Krypton atoms was enhanced by far more than one order of magnitude with respect to conventional frequency conversion. (iv) In a similar way, also four-wave mixing to generate vacuum-ultraviolet radiation in a dense medium of Mercury atoms, prepared by SCRAP, was enhanced by more than one order of magnitude. The experiments in Krypton


and Mercury vapour convincingly demonstrate the significant advantages of SCRAP, applied to frequency conversion processes. (v) High-order Raman sideband generation in molecular media, driven by a pair of ultra-short (fs) radiation pulses, yielded very large conversion efficiencies in the order of 10 %. Also in this case, enhanced molecular coherences played a major role to support the nonlinear optical processes. All of these results reveal the tremendous potential of coherent, adiabatic interactions in the field of nonlinear optics. The expected impact in the field of applied optics, e.g. the development of novel and efficient radiation sources, based on coherent preparation of nonlinear optical media, as well as the generation of ultra-short (sub-fs) radiation pulses by combination of frequency components from intense spectra with ultra-broad bandwidth, offer exciting possibilities for future applications.

VIII. Acknowledgements The author thanks his coworkers Torsten Rickes, Jens Klein, Martin Oberst, Thorsten Peters, Holger Münch, and Fabian Beil who collected most of the data, presented above. The experiment on high-order Raman sideband generation, discussed above, was conducted at Imperial College (London). The author thanks his cooperation partner Jon Marangos (Imperial College, London) and the group members at Imperial College, in particular Emiliano Sali (now at LENS, Florence). The author also acknowledges most valuable cooperations and discussions with Klaas Bergmann (University of Kaiserslautern), Bruce W. Shore (Livermore, USA), Leonid Yatsenko (Ukrainian Academy of Sciences, Kiev), and Nikolay V. Vitanov (Bulgarian Academy of Sciences, Sofia).

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