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Provided for non-commercial research and educational use only. Not for reproduction, distribution or commercial use. This chapter was originally published in the book Advances in Quantum Chemistry, Vol.64, published by Elsevier, and the attached copy is provided by Elsevier for the author's benefit and for the benefit of the author's institution, for non-commercial research and educational use including without limitation use in instruction at your institution, sending it to specific colleagues who know you, and providing a copy to your institution’s administrator. All other uses, reproduction and distribution, including without limitation commercial reprints, selling or licensing copies or access, or posting on open internet sites, your personal or institution’s website or repository, are prohibited. For exceptions, permission may be sought for such use through Elsevier's permissions site at: http://www.elsevier.com/locate/permissionusematerial From: Svetlana A. Malinovskaya, Tom Collins, and Vishesha Patel. Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence Using Chirped Pulses and Optical Frequency Combs. In John R. Sabin and Erkki J. Brändas, editors: Advances in Quantum Chemistry, Vol.64, Burlington: Academic Press, 2012, pp. 211-258. ISBN: 978-0-12-396498-4 © Copyright 2012 Elsevier Inc. Academic Press Author's personal copy CHAPTER 7 Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence Using Chirped Pulses and Optical Frequency Combs Svetlana A. Malinovskaya, Tom Collins, and Vishesha Patel Contents 1. Introduction 2. Chirped Pulse Control in Coherent Anti-Stokes Raman Scattering for Imaging of Biological Structure and Dynamics 2.1. Theory 2.2. On the role of the ac Stark shifts in the nature of two-photon Raman transitions 2.3. The ‘‘roof’’ method 3. The Impact of Phase and Coupling Between the Vibrational Modes on Selective Excitation in CARS Microscopy 3.1. Theory 3.2. Numerical results and discussion 4. Selective Excitation of Raman Transitions by Two Chirped Pulse Trains in the Presence of Decoherence 4.1. Theoretical model 4.2. Numerical results: The effects of the vibrational energy relaxation and collisional dephasing on coherence loss 4.3. Prevention of decoherence 5. Feshbach-to-Ultracold Molecular State Raman Transitions via a Femtosecond Optical Frequency Comb 5.1. Theoretical model 212 216 217 219 223 227 227 229 238 239 242 243 246 247 Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, New Jersey, USA E-mail address: [email protected] Advances in Quantum Chemistry, Volume 64 ISSN 0065-3276, http://dx.doi.org/10.1016/B978-0-12-396498-4.00007-7 # 2012 Elsevier Inc. All rights reserved. 211 Author's personal copy 212 Svetlana A. Malinovskaya et al. 5.2. Numerical results: Creation of ultracold molecules using a standard optical frequency comb and a phase-modulated optical frequency comb 6. Conclusions Acknowledgments References Abstract 249 254 255 255 Femtosecond, chirped laser pulse-based methods are presented to control coherence in vibrational degrees of freedom in molecules via Raman transitions. The controllability of the methods is analyzed in the presence of fast decoherence and the coupling between vibrational modes. Applications are diverse and include the development of novel imaging techniques based on the coherent anti-Stokes Raman scattering, and the implementation of the optical frequency combs for internal state cooling from Feshbach molecules aiming at creation of deeply bound ultracold polar molecules. 1. INTRODUCTION Coherent optical light interacting with atoms and molecules is at the heart of the atomic, molecular, and optical physics research that has been attracting scientific minds already for a half-century. Multiphoton transitions that result from light–matter interactions occur between discrete levels in atoms and molecules and appeal to quantum properties of both the light and matter. Atoms and molecules, when excited, in response modulate the coherent light during its propagation through the medium; this modulation may be significant and measurable. One of the exciting examples is twophoton resonances induced in molecules followed by a generation of Raman fields whose frequencies differ from the carrier frequency of an incident light by exactly the amount of vibrational frequency of a specific vibrational mode. This phenomenon makes Raman spectroscopy an attractive and a practical tool for investigations of the structure and ultrafast dynamics since it possesses a signature of molecules or molecular groups involved. Studies of the nature of Raman scattering include an investigation of fundamental phenomena of light–matter interactions: light-induced population transfer and coherence, as well as intrinsic effects taking place in molecules, such as the coupling between vibrational modes via external fields. The amount of coherence created between the vibrational levels serves as a source for a generation of the Raman fields and provides with the Raman gain. Quantum coherence between two states, which are, within our interests, the vibrational states of the ground electronic state, is measured by the quantity jai*ajj, where ai and aj are the probability amplitudes of these states in the total wave function of the system. The investigation of coherence between states induced by the two-photon Raman transitions is one of the main topics of Author's personal copy Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence 213 our interest and is discussed here. We also present our results on newly developed quantum control methods aiming at creation of a maximum coherence between predetermined states in atoms and molecules.1–4 A fundamental interest of creating a maximum coherence may be understood from the Maxwell equations that show that the amplitude of the Raman fields is proportional to the induced coherence in a system.5 We also look into the effects of the coupling between vibrational modes via external fields on controllability of the proposed methods.6 Nowadays, femtosecond pulse techniques are admittedly prevailing over those using continuous-wave beams to probe and control ultrafast dynamics in atoms and molecules. Furthermore, the implementation of femtosecond, chirped laser pulses is known to be advantageous for these purposes because of their robustness in a broad range of pulse parameters, such as the pulse intensity, chirp rate, and the pulse duration. Two-photon Raman transitions induced by chirped femtosecond pulses have been implemented in solving a variety of physical, chemical, and biomedical problems. One of the leading techniques for the imaging of molecular specific structure is based on the induction of twophoton resonances in samples and is known as the coherent anti-Stokes Raman scattering (CARS) microscopy. We have dedicated our goals to the development of quantum control methods to advance bioimaging techniques based on CARS microscopy. One of the critical issues related to coherence property in condense phase is its loss. Decoherence is the scourge of many processes that scientists are eager to take a control over within the research areas that span from biomedicine to quantum computation. We present our results of investigations on optimizing the quantum response of a system in the presence of fast decoherence.7–9 We study decoherence phenomenologically with two main channels taken into account, the spontaneous emission and collisional dephasing. We find that an application of the chirped pulse trains may be very advantageous. One of the topics of our great interest is the usefulness of Raman transitions in application to ultracold gases manipulation.10–12 We analyze a possibility of using the phase-locked pulse trains that form the frequency combs in cooling of the internal degrees of freedom in molecules and creating them in the fundamentally cold state—the ground electronic state with zero vibrational and rotational quantum number. The hallmark of a chirped pulse is a frequency which is time dependent. So, as time evolves the frequency of the light of the laser pulse changes, thus allowing for a range of frequencies to be generated over the time of lasing. In a linear chirp, the instantaneous frequency is o(t) ¼ o0 þ at, which is linear in t. The coefficient a is the chirp parameter and determines how quickly the pulse’s frequency will change as time evolves. A pulse is said to be up-chirped if a > 0 and said to be down-chirped if a < 0. The chirp parameter can be controlled in an experiment by use of optical devices.13 The instantaneous frequency is the time derivative of the phase of the field F(t) ¼ o0t þ at2/2 þ f, where f is an arbitrary phase constant. The electric field at a point in space through which the pulse travels is given by Author's personal copy 214 Svetlana A. Malinovskaya et al. EðtÞ ¼ E0 et 2 =t2 at2 þf : cos o0 t þ 2 (1) Here, the exponential term, if plotted on its own, forms the Gaussian pulse envelope. The t is the pulse duration and the E0 is the peak amplitude of the field. The value o0 is the carrier frequency of the pulse. In the laboratory, not just one but many time-delayed pulses are utilized. Pulses are generated at a set time interval T and may be described as ! X aðt nTÞ2 ðtnTÞ2 =t2 þf : (2) E0 e cos o0 ðt nTÞ þ EðtÞ ¼ 2 n Such pulse trains may be generated from mode-locked lasers. A modelocked laser is one in which we have a large number of oscillating modes. These modes vibrate with different frequencies that are spaced by the same amount, Dn. This even spacing of the frequencies of the various components yields a final waveform that, even though it may appear chaotic, is periodic in time, forming a pulse train. The relative phase between successive modes is locked. This means that for two successive modes, l and l 1, the difference in phase obeys the relation fl fl 1 ¼ f.14 Now, let us consider what this pulse train looks like when we switch to the frequency domain. Assume we have the case of 2n þ 1 oscillators, each separated in the frequency domain by Dn and phases that obey the modelocking condition. If each oscillator contributing to the pulse has the same maximum amplitude, plotting the pulse in the frequency domain will give us a series of Dirac delta functions F(o) ¼ Sld[o (o0 2plDn)]. The function F(o) is the frequency comb of the field E(t). Its spectral bandwidth is determined by t 1 and the spacing between the high intensity peaks is given by Dn ¼ T 1. Modern mode-locked lasers typically produce pulse trains at the repetition rate frep about 100 MHz. Then, the optical frequencies nn of the comb lines can be written as nn ¼ nfrep þ f0, where n is a large integer number of the order 106 and f0 is the offset frequency due to carrier–envelope phase shift.15 The pulse train described by Eq. (2) will produce zero offset frequency since the carrier–envelope phase difference is zero. An optical frequency comb has been recognized as a new and unique tool for high-resolution spectroscopic analysis of internal energy structure and dynamics as well as for controlling ultrafast phenomena in atomic and molecular physics.16–19 Owing to its broadband spectrum, the frequency comb may efficiently interact with the medium inducing one-photon, two-photon, and multiphoton resonances between finely structured energy levels. A unique ability of the frequency comb is provided by the presence of about a million optical modes in its spectrum with very narrow bandwidth and exact frequency positions.20 During the past few years, the investigations have been carried out on implementation of a femtosecond frequency comb to manipulate ultracold gases. The pioneering works in quantum control in ultracold Author's personal copy Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence 215 temperatures include the two-photon excitation of specific atomic levels forming four-level diamond configuration in cold 87Rb using a phase-modulated, femtosecond optical frequency comb21 and a theory on piecewise stimulated Raman adiabatic passage performed with two coherent pulse trains possessing the pulse-to-pulse amplitude and chirped phase variation.22,23 In these papers, the authors reported on creation of ultracold KRb molecules from Feshbach states, the highest excited vibrational states of the ground electronic state, using the pump–dump stepwise technique that coherently accumulates population in the ground vibrational state. Experimentally, a dense quantum gas of ultracold KRb polar molecules was produced from the Feshbach molecules using the STIRAP scheme with two microsecond pulses.24 Coherent population transfer was demonstrated to rovibrational ground state of the triplet and singlet electronic ground potential. We describe how to use a single optical frequency comb to induce twophoton Raman transitions from the Feshbach state to a deeply bound molecular state to create ultracold molecules. Nowadays, chirped femtosecond pulses are widely used in investigations of ultrafast phenomena in natural sciences. In the past decade, CARS microscopy has developed as a promising technique for imaging of various objects of biological interest, for example, living cells, cancerous cells, and also for combustion diagnostics, and monitoring of molecules. The recent advances in shaping of ultrafast femtosecond laser pulses25 along with demonstration of different experimental and theoretical techniques26–35,1,36,37 of driving a system to the desired quantum yield have made CARS a major tool of investigation of biological structures. Experimental configurations such as Box-CARS,38 FM-CARS,39 EPI-CARS,40 heterodyne CARS,41 polarization CARS,42 FT-CARS,43 and interferometric CARS44 are among the most promising ones. A major drawback of CARS technique is the nonresonant background signal. Implementation of the femtosecond pulses in combination with the quantum control methods makes it possible to selectively drive a predetermined Raman transition and effectively suppress the background signal. It has been shown recently4 that an application of femtosecond, chirped laser pulses induces adiabatic passage in a system and provides an optimal CARS signal. CARS is a spectroscopic method which relies on three photons in order to probe a multilevel quantum system formed by Raman active vibrational modes in a molecule. The process utilizes the two-photon transitions in the following manner: a pump pulse of frequency op interacts with the system in its initial, usually ground electronic and vibrational state exciting it to a virtual level. Next or simultaneously, a Stokes pulse of frequency os interacts with the system that is now in the virtual state and causes it to decay to one of its excited vibrational states of the ground electronic state with creation of a coherent superposition state. At this point, a probe pulse of frequency opr interacts with the induced coherence in the system and excites the system further to another virtual state, from which the system decays back to its Author's personal copy 216 Svetlana A. Malinovskaya et al. ground electronic and vibrational state, emitting an anti-Stokes photon of frequency oAS ¼ (op os) þ opr. If the resonance condition is satisfied, the quantity (op os) is equal to the frequency between the ground and the excited vibrational state of the system. The emitted anti-Stokes photons are coherent, and thus this method of spectroscopy is more efficient than spontaneous Raman spectroscopy, in which only a single frequency source, the pump pulse, excites the system to a virtual level and then allows spontaneous decay of the excited vibrational states down to the ground. The signal from the spontaneous emission is orders of magnitude weaker than that from the stimulated Raman scattering. Spontaneous Raman spectroscopy is often regarded in combination with the surface enhancement techniques. As with any spectroscopic method, the spectral resolution is of key importance. In order to identify the presence of certain chemical agents in a sample, it is desirable to have as fine a ‘‘tunable’’ spectrum as possible so that it could be set to resonance with the energy difference between certain states of the agent to be detected, such as the vibrational state of the molecule.13,45 The use of chirped pulses allows for selective Raman excitations which are necessary. Presently, using shaped femtosecond pulse techniques a spectral resolution can be obtained within about 3 cm 1.45 Described in this chapter, new control methods impose no limitation on the magnitude of the frequency difference between Raman active vibrational modes. The chapter is organized as follows. Section 2 is devoted to the description of the quantum control methods aiming at creation of a maximum coherence in a predetermined mode in CARS. Methods are distinguished depending on the scheme of chirped pulse implementation and the nature of the induced dynamics. In Section 3, the impact of the coupling between vibrational modes via external fields is investigated on controllability of selective excitation of Raman transitions. In Section 4, we describe a method to create a maximum coherence in the presence of fast decoherence by using chirped pulse trains. The method utilizes a series of chirped pulses with locked phase that form a frequency comb. Section 5 is dedicated to the application of optical frequency combs to control the dynamics in ultracold temperatures, namely, to perform rovibrational cooling in molecules from the Feshbach state to the ground electronic singlet state with zero rotational and vibrational quantum number. Section 6 has conclusions. 2. CHIRPED PULSE CONTROL IN COHERENT ANTI-STOKES RAMAN SCATTERING FOR IMAGING OF BIOLOGICAL STRUCTURE AND DYNAMICS Selective excitation of predetermined molecular vibrations in a complex biological sample allows one to picture biological structure and ultrafast dynamics using CARS. Recently, selective excitation of vibrational modes in Author's personal copy Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence 217 vital samples such as live cells has been achieved by means of picosecond laser pulses possessing a narrow spectral bandwidth.40 A series of articles, both theoretical and experimental, demonstrate feasibility of modern laser control methods for selective excitation of two-photon transitions.1,29–35 Coherent control methods may be used to execute vibrational excitations in biological samples in order to achieve high chemical selectivity and maximum intensity of the CARS signal. Implementation of laser control methods to noninvasive biological imaging may bring investigations of complex biological systems to a new level. In this chapter, we present quantum control methods for selective excitation of vibrational modes among many, a large number of them having close frequencies, by making use of ultrafast chirped laser pulses.1–4 The central method, the chirped pulse adiabatic passage method, addresses vibrational levels within a single electronic state via simultaneously applied chirped pump and Stokes laser pulses; pulse parameters satisfy specific conditions for the ac Stark shifts. These conditions provide fractional adiabatic passage resulting in a maximum coherence between vibrational states of a predetermined mode and zero coherence between vibrational states of other Raman active modes. This method is complementary to methods described in Refs. 46,47,26. In CARS, the amplitude of the anti-Stokes field, which is generated at frequency 2op os, is determined by the induced coherence according to the Maxwell–Bloch equations. Thus, the goal for a manipulation with chirped pump and Stokes pulses is to create a maximum coherence between the desired vibrational states to get a strong CARS signal from a predetermined molecular vibration and to suppress coherence in all other vibrational states. We discuss three methods that make use of chirped pulses and provide selective excitation of vibrational modes in nonadiabatic and adiabatic regime. 2.1. Theory A semiclassical model of light–matter interaction is developed, in which vibrational states are described by two-level systems (TLSs), and chirped pump and Stokes pulses are described by classical electromagnetic wave packets, having phase linearly depending on chirp. The schematic of the model is shown in Figure 7.1. We consider two, TLSs having slightly different transition frequencies o21 and o43, such that o43 o21 ¼ d o21. Initially, only ground states j1i and j3i of the TLSs are evenly populated. One TLS represents a vibrational mode to be selectively excited and another TLS represents an off-resonant vibrational mode or background to be suppressed. TLSs are uncoupled, meaning the probability for population transfer from one TLS to another via interaction with electromagnetic fields is zero. The intense pump and Stokes pulses having central frequencies op and os are defined as Author's personal copy 218 Svetlana A. Malinovskaya et al. ws wAS =2 w p −ws wp |4ñ d |2ñ w43 w 21 |1ñ |3ñ Figure 7.1 Schematic of two-level systems representing Raman active vibrational modes that interact with the intense pump, Stokes, and probe pulses, to generate an anti-Stokes signal at frequency 2op os. 0 1 0 1 2 2 at bt Ep ðtÞ ¼ Ep0 ðtÞcos @op t A; Es ðtÞ ¼ Es0 ðtÞcos @os t A; 2 2 Ep0 ðtÞ ¼ 0 E0 02 11=4 e t2 =2t2 Es0 ðtÞ ¼ 0 ; E0 02 1 et 2 =2t2 ; (3) @1 þ b A t40 @1 þ a A t40 where Ep0(t) and Es0(t) are the time-dependent pump and Stokes field envelope, a, a0 and b, b0 are the linear temporal and spectral chirps of the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pump and Stokes pulses, respectively, and t ¼ t0 1 þ a0 2 =t40 is the chirpdependent pulse duration assumed to be the same for the pump and Stokes pulses in our model. For zero chirp, pulse frequency difference (op os) is in resonance with the o21 frequency and the pump and Stokes pulse duration is t0. This specifies the value of the maximum temporal linear chirp to be equal to 1/(2t20) and the related spectral linear chirp to be equal to t20.26 The interaction Hamiltonian for a single TLS reads 0 1 1 O3 ðtÞ ðd þ ða bÞt þ O1 ðtÞ O2 ðtÞÞ B 2 C B C H¼B C; (4) 1 @ A ðd þ ða bÞt þ O1 ðtÞ O2 ðtÞÞ O3 ðtÞ 2 with d ¼ 0 for the j1i–j2i TLS and d ¼ 6 0 for the j3i–j4i TLS. Here, 2 E20 02 1=2 =t40 et 2 =t2 2 E20 t =t are the ac Stark 1=2 e ð1þa Þ ð1þb =t40 Þ shifts originated from the two-photon transition, where m is the dipole moment and D is the one-photon detuning from the excited state; O1 ðtÞ ¼ 4ℏm2 D ; O2 ðtÞ ¼ 4ℏm2 D 02 2 2 Author's personal copy Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence 2 O3 ðtÞ ¼ 4ℏm2 D " ! E20 !#1=4 et 2 219 =t2 is the effective Rabi frequency. We a b 1þ 1þ t40 t40 define O3 as the peak effective Rabi frequency. The pump and Stokes pulse bandwidth at zero chirp is chosen to be about the difference between transition frequencies of the TLSs and the effective Rabi frequency to be larger than that. Diagonalization of the Hamiltonian in Eq. (4) gives energy separation of the dressed states, called the generalized Rabi frequency: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (5) OðtÞ ¼ ðd þ ða bÞt þ Ot ðtÞ O2 ðtÞÞ2 þ 4O23 ðtÞ: 02 02 The generalized Rabi frequency depends on the chirps, the ac Stark shifts, and the effective Rabi frequency. The time-dependent probability amplitudes of the dressed states for two uncoupled TLSs are described by the coupled differential equations i _ a_ 1ð3Þ ¼ OðtÞa1ð3Þ þ YðtÞa 2ð4Þ ; 2 i _ a_ 2ð4Þ ¼ OðtÞa2ð4Þ þ YðtÞa 1ð3Þ : 2 (6) _ Here, O(t) is the generalized Rabi frequency and YðtÞ is the coupling parameter, which determines the nonadiabatic coupling between the dressed _ states. The coupling parameter YðtÞ reads _ 3 ðtÞ LO_ 3 ðtÞ LO _ ; YðtÞ ¼ 2 OðtÞ (7) where L ¼ d þ (a b)t þ O1(t) O2(t). Note that the degree of adiabaticity of a process may be evaluated by the Massey parameter M(t) defined as a ratio of the energy splitting of the dressed states O(t) to the coupling parameter _ .48 If M(t) 1, the light–matter interaction is essentially adiabatic, and YðtÞ if M(t) 1, nonadiabatic effects take place. Notably, the adiabaticity of light–matter interaction is dependent on properties of the ac Stark shifts present in the interaction Hamiltonian described by Eq. (4). In the following, we reveal the role of the ac Stark shifts in the nature of two-photon Raman transitions and demonstrate how the choice of the pump and Stokes pulse chirps determines the properties of the ac Stark shifts and adiabaticity of the dynamics in TLSs. 2.2. On the role of the ac Stark shifts in the nature of two-photon Raman transitions Generally, it is convenient to analyze the adiabatic effects in the dressed state picture. Assume that a transform-limited pump pulse (e.g., a ¼ 0) and a linearly chirped Stokes pulse (e.g., b ¼ constant) are applied simultaneously Author's personal copy 220 Svetlana A. Malinovskaya et al. in the CARS scheme shown in Figure 7.1. In this case, the ac Stark shifts (O1(t) and O2(t)) are essentially different, resulting in the generalized Rabi _ frequency O(t) and the coupling parameter YðtÞ to be dependent on the ac Stark shift difference (O1(t) O2(t)). In order to evaluate the adiabatic effects via Massey parameter, we have calculated the energy of the dressed states _ as a function of time. In the calcu 12 OðtÞ and the coupling parameter YðtÞ lations, we used the following values of the system and the field parameters: transition frequency o21 was chosen to be equal to one frequency unit and the o43 to be equal to 1.1, giving d ¼ 0.1. This model is valid for description of, for example, symmetric and asymmetric stretch modes in liquid methanol,49 or symmetric CH2 vibrational mode in lipids and the CH3 stretch mode in the proteins, both constituting the myelin sheath.50 All the frequency param1 eters given in the section are in [o21] units, temporal parameters are in [o 21 ] 2 2 units, frequency chirps are in [o21 ] units, and temporal chirps are in [o21] units. Duration of a transform-limited pulse was taken to be equal to 15, giving 1/t0 ’ d. In Figure 7.2, the dressed state energies and the coupling parameter are shown as a function of time for the peak effective Rabi frequency equal to 0.7, indicating a strong field regime, the spectral linear chirp equal to 270, which is slightly above b0 jbmax ¼ 225, and d ¼ 0.1 (a) and d ¼ 0 (b). Black and red curves show the energies of the upper and lower dressed states of a TLS, and green curve shows the coupling parameter as a function of time. For particular times, the energy separation of the dressed states O(t) decreases _ with a significant increase in the coupling strength YðtÞ. As a result, the Massey parameter decreases to much less than unity, implying nonadiabatic A 0.4 d = 0.1 0 -0.4 B 0.4 d=0 0 -0.4 0 50 100 150 tw Figure 7.2 The dressed state energies and the coupling parameter as a function of time for the peak effective Rabi frequency equal to 0.7, the spectral linear chirp equal to 270, and d ¼ 0.1 (a), and d ¼ 0 (b) for the CARS scheme implementing a transform-limited pump pulse and a linearly chirped Stokes pulse. Author's personal copy Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence 221 nature of the light–matter interaction. The nonadiabatic coupling between the dressed states takes place due to the ac Stark shifts. In the vicinity of _ small energy separation of the dressed states, the coupling parameter YðtÞ is large enough to provide a population transfer between the dressed states. It is known that the dressed states are a linear symmetric and antisymmetric combination of the bare states j1i and j2i (or j3i and j4i). Population transfer from the lower to the upper dressed state results in a final population of both bare states. It may be only partial depending on the coupling strength. The _ values of the O(t) and the coupling parameter YðtÞ, that determine the quantum yield, are dependent on the choice of the pulse chirp and the field amplitude. The nonadiabatic coupling gives rise to an oscillatory dependence of the coherence in the TLSs on the spectral linear chirp and the peak effective Rabi frequency, see Figure 7.3. Here, the coherence density plot of the resonant r12 (yellow) and the detuned r34 (black) TLSs is shown as a function of the peak effective Rabi frequency O3 and the dimensionless chirp parameter b0 =t0 2 . Regions where lines of different color coincide are the regions of selective excitation of molecular vibrations. Projection of any point from such a region onto abscise and ordinate axes gives values of the field parameters required for selective excitation. The plot shows that the ac Stark shifts, different for the lower and upper levels of the resonant and detuned TLSs due to interaction with a transformlimited pump pulse and linearly chirped Stokes pulse, lead to nonadiabatic 4 W3 3 2 1 -20 -15 -10 -5 0 2 b¢/t0 Figure 7.3 The coherence density plot of the resonant (yellow) and the detuned (black) TLSs as a function of the peak effective Rabi frequency and the dimensionless chirp parameter b0 =t0 2 for the CARS scheme implementing a transform-limited pump pulse and a linearly chirped Stokes pulse. Author's personal copy 222 Svetlana A. Malinovskaya et al. coupling, whose value could be manipulated by variation of the pulse chirp, giving a desired dynamics and the quantum yield (see also Ref. 1). Understanding of the role of the ac Stark shifts suggests an approach for selective excitation based on the adiabatic passage: choosing totally overlapped pulses may provide identical ac Stark shifts O1 and O2, resulting in their cancellation in the interaction Hamiltonian, Eq. (4), and, thus, removing their influence on the final quantum yield. First, we consider equally chirped laser pulses in the CARS scheme, which implies a ¼ b in Eqs. (3) giving the pump and Stokes pulse frequency difference op os being in resonance with the o21 frequency. Then, in the Eqs. (4) and (5) the ac Stark shift difference (O1(t) O2(t)) is equal to zero. In this case, the interaction Hamiltonian and the generalized Rabi frequency read 0 1 1 d O3 ðtÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B 2 C B C (8) H¼B ; OðtÞ ¼ d2 þ 4O23 ðtÞ: C 1 A @ d O3 ðtÞ 2 _ For d ¼ 0, the nonadiabatic coupling parameter YðtÞ is exactly zero, according to Eq. (7), two dressed states are uncoupled and the system dynamics takes place within a single dressed state having lower energy. In this case, the probability amplitude of the bare states, following the ðt Rabi oscillations, depends on the pulse area O3 ðt0 Þdt0 . 0 For d 6¼ 0, the nonadiabatic coupling between the dressed states is negligibly small, as shown by numerical calculations of Eq. (7), resulting in the system dynamics to be also within a single dressed state. The solution for the probability amplitudes of the bare states is the Rabi oscillations having amplitude proportional to the ratio of the effective Rabi frequency O3(t) and the generalized Rabi frequency O(t), resulting in low-amplitude oscillations vanishing with the increase in the chirp parameter. The chirp dependence of the effective Rabi frequency O3(t) establishes selectivity of excitation of the resonant TLS in the framework of the Rabi solution. In Figure 7.4a, the coherence density plot of the resonant TLS r12 is depicted as a function of the peak effective Rabi frequency O3 and dimensionless frequency chirp parameter b0 =t0 2 . According to the Rabi solution, the coherence oscillates as a function of O3, independent of b0 =t0 2 . The coherence density plot of the off-resonant TLS r34 is shown in Figure 7.4b. The coherence is zero in the large region of chirp parameters, b0 =t0 2 , for any value of O3. It is the adiabatic dynamics in the TLS that gives zero population transfer to the upper level. In the vicinity of zero chirp, the off-resonant Rabi oscillations of the coherence are observed as a function of O3. Thus, control of coherence in TLSs, having close frequencies, is achieved by means of the _ chirp parameters that provide the conditions for adiabaticity, OðtÞ=YðtÞ 1, and optimally chosen pulse area. Author's personal copy 223 Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence 1.6 1.1 1.1 W3 W3 1.6 0.6 0.1 -20 0.6 -10 0 10 0.1 -20 20 -10 0 10 20 b¢ / t02 2 b¢ / t0 Figure 7.4 The coherence density plot of the resonant TLS (a) and the detuned TLS (b) is depicted as a function of the peak effective Rabi frequency and dimensionless frequency chirp parameter b0 =t0 2 for the CARS scheme implementing equally chirped pump and Stokes pulses. Maximum coherence is shown by blue color and minimum coherence, by red color. 2.3. The ‘‘roof’’ method Now, we consider the adiabatic passage control method implementing chirped pulses for selective excitation of spectrally close Raman transitions within the scheme in Figure 7.1. This method implies the chirp parameter of the Stokes pulse b to be constant, giving monotonous change of the Stokes pulse frequency, see Eq. (3), and chirp parameter of the pump pulse a to have same magnitude and opposite sign before the central time t0, when the pulse amplitude reaches maximum, and then to flip the sign of the pump pulse chirp. At t0, the difference of the pump and Stokes pulse frequencies comes into resonance with the o21 frequency and stays in resonance for the rest of the time. We call this method the ‘‘roof’’ method in accordance with the temporal profile of the pump pulse instantaneous frequency. The interaction Hamiltonian of a TLS and the generalized Rabi frequency reads 0 1 d at O3 ðtÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B 2 C B C (9) ; OðtÞ ¼ ðd þ 2atÞ2 þ 4O23 ðtÞ; t < t0 ; H ¼ B C d @ A O3 ðtÞ þ at 2 0 B B t t0 ; H ¼ B @ d 2 O3 ðtÞ O3 ðtÞ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C C ; OðtÞ ¼ d2 þ 4O23 ðtÞ: C d A 2 Author's personal copy 224 Svetlana A. Malinovskaya et al. In accordance with the Eq. (9), the roof method induces qualitatively different dynamics in the TLSs exposed to the pump and Stokes laser fields with respect to previously considered methods. Because the ac Stark shifts are the same and cancel each other in Eq. (4) giving the interaction Hamiltonian described in Eq. (9), the adiabatic passage takes place leading to the maximum coherence in the resonant TLS, Figure 7.5a (blue), and minimum coherence in the detuned TLS, Figure 7.5b (red), in a broad range of the peak effective Rabi frequency O3 and dimensionless frequency chirp b0 =t0 2 . We refer once again to the dressed state picture to understand the adiabatic features of the system dynamics. The energies of the bare (dashed curves) and the dressed (solid curves) states in the field-interaction representation are shown as a function of time for the resonant TLS in Figure 7.6a. Bare states come to resonance at central time t0 and further remain unchanged. Since the nonadiabatic coupling is negligibly small, the systems time evolution takes place within a single dressed state. Initially, the lower dressed state coincides with the populated bare ground state. As time approaches the central time t0, both bare states acquire equal probability amplitudes and remain unchanged giving maximum final coherence r12. Coherence is more sensitive to the field amplitude in the detuned TLS and depends on the sign of the chirp in weak fields. For the positive chirp, Figure 7.6b, the crossing of the bare states takes place far before t0, when the field amplitude is rather weak and therefore dressed states have small energy separation. At this time, the nonadiabatic coupling (green curve) is strong enough to couple them and to partly promote population to the upper dressed state, leading to nonzero coherence r34. For the negative chirp, the crossing does not occur and evolution is essentially adiabatic, giving zero 1.6 1.1 1.1 W3 W3 1.6 0.6 0.6 0.1 -20 -10 0 10 2 b¢ / t0 20 0.1 -20 -10 0 10 20 2 b¢ / t0 Figure 7.5 The coherence density plot of the resonant TLS (a) and the detuned TLS (b) as a function of the peak effective Rabi frequency and the dimensionless chirp parameter b0 =t0 2 for the roof method. Maximum coherence is shown by blue color and minimum coherence, by red color. Author's personal copy Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence 225 0.1 0.05 0.05 0 0 -0.05 -0.05 -0.1 300 400 500 600 tc 700 800 900 300 400 500 600 tc 700 800 900 tw Figure 7.6 The energies of the bare (dashed lines) and the dressed (solid lines) states in the field-interaction representation are shown as a function of time for the resonant TLS, for the peak effective Rabi frequency (before chirping) equal to 1.1 and b0 =t0 2 ¼ 10, (a) and the detuned TLS, for the peak effective Rabi frequency (before chirping) equal to 0.2 and b0 =t0 2 ¼ 10 (b). Blue curve shows the effective Rabi frequency as a function of time and green curve shows the nonadiabatic coupling parameter _ðtÞ. population transfer and zero coherence in the detuned system. In strong fields, the dressed states have large energy separation all the time, resulting in negligible nonadiabatic coupling and complete (zero) population transfer to the upper level and zero coherence for a0 > 0 (a0 < 0) (see also Ref. 4). Numerical calculations have shown that the conditions for the selectivity of excitations 1/t d must be satisfied. The roof method suggests the robust way to obtain a noninvasive image of a biological structure. Various biological tissues contain molecular groups, having CH vibrations which span from 2800 to 3100 cm 1 and may be selectively excited to provide noninvasive image with high chemical sensitivity. For example, in order to get CARS imaging of axonal myelin in live spinal tissues by obtaining a strong signal from the symmetric CH2 stretch vibration that appears at 2840 cm 1, the chirped pulse adiabatic passage method suggests to use femtosecond pulses generated from Ti:sapphire oscillators, having pulse duration t0 ¼ 176 fs, linear spectral chirp a ¼ 30 10 5 cm2, and field intensity in the range of 1012 W/cm2. Similar to the multiplex CARS method,51–53 the adiabatic passage control method may be applied by fixing the pump central frequency and scanning the Stokes frequency to obtain the vibrational spectrum of unknown molecular species. However, in contrast to the multiplex CARS method, for each instantaneous magnitude of the Stokes central frequency, the maximum intensity of the CARS signal and the efficient suppression of the background signal will be provided. Recently, somewhat related idea of using chirped pulses in CARS spectroscopy has been demonstrated experimentally,13 where the two equally Author's personal copy 226 Svetlana A. Malinovskaya et al. chirped pulses were used. Time delay between pulses has been utilized as an effective way to tune two-photon detuning into resonance. That allowed the authors to achieve high spectral resolution using broadband pulses. One can see from the present analysis that imposing temporal delay gives rise to ac Stark shifts. In turn, these time-dependent Stark shifts result in an effective nonlinear avoided crossing and substantial nonadiabatic coupling.1 Optimization of the selective excitation of the vibrational coherence under the condition of nonlinear avoided crossing could be a challenge even in the adiabatic limit. Therefore, the method13 is sensitive to the pulse parameters, and excitation selectivity is limited because of the nonadiabatic coupling induced by the Stark shifts. In contrast, our methods employ full advantage of adiabatic passage technique and provide a clear and simple way of controlling predetermined vibrational coherence. The robustness of the methods should facilitate experimental implementations. To reach adiabatic regime, one can use 100 fs transform-limited pulses at a repetition rate 1 MHz with averaged beam power of 25 mW similar to that in Ref. 40. Assuming dipole moments to be equal to 1 D, we estimate Oef R 200 THz giving the Landau–Zener parameter to be well in the adiabatic range, 2 (Oef R ) /a 40. In summary, we have shown that selective excitation of spectrally close vibrational modes may be achieved using chirped pump and Stokes pulses in CARS spectroscopy and microscopy within the nonadiabatic and adiabatic regimes of light–matter interaction. By simultaneously applying a transform-limited pump pulse and a linearly chirped Stokes pulse, selective excitation is achieved as a result of strong nonadiabatic coupling of the dressed states originated from the induced ac Stark shifts having substantially different values. A careful choice of a spectral chirp of the Stokes pulse results in a desirable final quantum state of a multimode system. Two methods implementing chirped pump and Stokes pulses provide selective excitation of predetermined Raman transitions based on adiabatic regime of light–matter interaction. Choosing totally overlapped chirped pulses, we removed the ac Stark shifts influence on the final quantum yield. Implementation of two equal chirps in one scheme provides the Rabi oscillation type of coherence control. The roof method makes use of a constant chirp in the pump (Stokes) pulse and the flipping sign of the chirp in the Stokes (pump) pulse. This method provides the adiabatic passage type of coherent control and gives a robust solution for a wide range of the field amplitude and the chirp values. Both methods can be used for noninvasive imaging of biological structure and femtosecond dynamics studies, and as an efficient tool for suppression of the background signal in the CARS spectroscopy and microscopy. All three methods demonstrate the possibility of coherent control over spectrally close modes with frequencies larger than the bandwidth of the pulse. Methods may be applied to control an induced polarization, decoherence, and quantum gates. Author's personal copy Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence 227 3. THE IMPACT OF PHASE AND COUPLING BETWEEN THE VIBRATIONAL MODES ON SELECTIVE EXCITATION IN CARS MICROSCOPY In this chapter, we analyze the impact of the coupling between Raman active vibrational modes on selectivity of their excitation in CARS microscopy using method proposed in Ref. 4 and discussed in Section 2. We study two theoretical models demonstrating control over the quantum yield and optimization of the CARS signal.6 Each model consists of two effective TLSs with (a) degenerate ground states and (b) nondegenerate ground states, which interact with two chirped femtosecond pulses within the Raman configuration in accordance with Ref. 4. The first model, the one with degenerate lower states, may be used to describe the induced dipole moments coupled via dipole–dipole interactions and subject to interaction with external electromagnetic fields,54 while the second one with nondegenerate lower states is useful for the description of Raman modes present in a molecule and interacting with light.1 The nonlinear nature of CARS is due to four-wave mixing. When the frequency difference between the pump and Stokes beams is in resonance with a molecular vibration, it excites the molecule to a higher vibrational level creating a coherence on corresponding transition. On de-excitation by the probe pulse, the anti-Stokes frequency light is emitted. It contains vibrational signature of the molecule which is unique in its nature. Thus, a large amount of information can be extracted from the CARS spectrum. By applying femtosecond, chirped laser pulses as in Ref. 4 within two models, and performing relative phase dependence studies, we analyze the role of coupling between vibrational modes as well as relative phase dependence on optimizing the coherence in the desired vibrational mode and suppressing the unwanted one. We would like to point out that our results are related to the so-called strong field control regime when perturbation theory with respect to the external field amplitude is not valid and the exact solution of the Schrödinger equation must be obtained to describe the excitation dynamics correctly. 3.1. Theory We consider a semiclassical model of light–matter interaction, where strong femtosecond laser pulses interact with two coupled TLSs representing two Raman active vibrational modes in a molecule. A molecular medium of interest may be considered as an ensemble of TLSs,4,26 with no relaxation or collisional dephasing effects taken into account. We investigate two models: one with degenerate ground states (Figure 7.7a) and the other with nondegenerate ground states (Figure 7.7b). Here, the j1i–j2i TLS has transition frequency o21 and the j3i–j4i TLS has transition frequency o43. These two modes are coupled by an external field, meaning that all states are effectively coupled. In addition, we take into account the phase relation Author's personal copy 228 Svetlana A. Malinovskaya et al. |bñ |bñ D D ws ws wp wp |4ñ |4ñ d |2ñ |1ñ |2ñ |3ñ (a) 3d /2 d /2 |1ñ |3ñ (b) Figure 7.7 Schematic of the energy levels in the degenerate model (a) and nondegenerate model (b). between different modes, assuming that the relative phase between initially populated states j1i and j3i is random in a large ensemble of molecules. Therefore, most of the presented results are phase averaged. Note that the bandwidth of the applied pulses is larger than the frequency mode difference. Our goal is to suppress coherence in the j3i–j4i TLS and to create a maximum coherence in the j1i–j2i TLS. The frequency chirped pump and Stokes pulses having central frequencies op and os are defined as Ep;s ðtÞ ¼ Ep0;s0 ðtÞ cos½op;s ðt t0 Þ þ ap;s ðt t0 Þ2 =2; E0 exp½ðt t0 Þ2 =2t2 ; Ep0;s0 ðtÞ ¼ ð1 þ a0 2p;s =t40 Þ1=4 (10) where Ep0(t) and Es0(t) are the time-dependent pump and Stokes field envelopes, ap,s and a0 p, s are the linear temporal and spectral chirps of the pump qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and Stokes pulses, respectively, and tp;s ¼ t0 1 þ a0 2p;s =t40 is the chirp-dependent pulse duration. For zero chirp, the frequency difference op os is in resonance with the frequency o21, and the pump and Stokes pulse duration before chirping is t0. Interaction of vibrational modes with ultrafast chirped laser pulses is described in the rotating wave approximation by a semiclassical Hamiltonian obtained using adiabatic elimination of the virtual state jbi. Within the fieldinteraction representation, the Hamiltonian for the degenerate model reads 0 1 O3 ðtÞ O1 ðtÞ O3 ðtÞ dðtÞ Od ðtÞ B O3 ðtÞ C dðtÞ þ Od ðtÞ O3 ðtÞ O2 ðtÞ C: (11) H¼B @ O1 ðtÞ A O3 ðtÞ dðtÞ Od ðtÞ O3 ðtÞ O2 ðtÞ O3 ðtÞ d þ dðtÞ þ Od ðtÞ O3 ðtÞ Author's personal copy Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence 229 The interaction Hamiltonian for the nondegenerate model reads 0 1 dðtÞ Od ðtÞ O3 ðtÞ O1 ðtÞ O3 ðtÞ B O3 ðtÞ C dðtÞ þ Od ðtÞ O3 ðtÞ O2 ðtÞ C: H¼B @ O1 ðtÞ A O3 ðtÞ d=2 dðtÞ Od ðtÞ O3 ðtÞ O2 ðtÞ O3 ðtÞ 3d=2 þ dðtÞ þ Od ðtÞ O3 ðtÞ (12) Here, d(t) ¼ (as ap)(t t0)/2, Od(t) ¼ (O1(t) O2(t))/2, and O1, 2(t) ¼ m2E2p0, s0(t)/(4ℏ2D) are the ac Stark shifts originated from the two-photon transitions, m mij is the dipole moment (for simplicity we consider all the dipole moments to be equal to 1 D), D is the single photon detuning from the excited state jbi (assumed equal for both pump and Stokes pulse frequencies), and O3(t) ¼ m2Ep0(t)Es0(t)/(4ℏ2D) is the effective Rabi frequency. The diagonal elements of the Hamiltonian describe bare state energies in the field-interaction representation, and they depend on the chirp parameters ap,s and detuning d ¼ o43 o21. The off-diagonal elements represent coupling of the bare states through the effective Rabi frequency and the ac Stark shifts. Here, we consider the case when pump and Stokes chirp rates have equal value, japj ¼ jasj ¼ a. The control in the TLSs is achieved by linearly chirped pulses with chirp parameters such that the frequency difference of the pump and Stokes pulses first reduces at 2a rate and comes to resonance with o21 at the central time t0 without further change till the end of the pulse. This method is known as the roof method (see Chapter 2 and Ref. 4). In the case of two uncoupled TLSs, studied in Ref. 4, the proposed scheme resulted in a creation of a maximum coherence in the resonant TLS and zero coherence in the off-resonant TLS. Now, the model is modified by switching on the coupling between two TLSs via external fields in order to analyze the impact of the coupling on controllability of excitation and also the attendant relative phase effects. 3.2. Numerical results and discussion To analyze the dynamics of the population and coherence, we solve the timedependent Schrödinger equation with the Hamiltonians in Eqs. (11) and (12) for the total wave function jC(t)i ¼ S4i ¼ 1 ai(t)jii, where ai(t) are the timedependent probability amplitudes. Calculations were performed using the Runge–Kutta method55 under the initial conditions of equally populated ground states, j1i and j3i, which is likely to be the case for molecules at room temperature, particularly, for the nondegenerate model, because the energies of the two nondegenerate ground states are very close compared to the kT expected under all experimental conditions on biological samples. The population of each state was chosen to be 0.5, giving the total population in the system equal to unity and coherence value ranging from zero to the Author's personal copy 230 Svetlana A. Malinovskaya et al. maximum value 0.5. Below we use a standard density matrix notation for the population rii ¼ jai(t)j2 and coherence rij ¼ jai*(t)aj(t)j, i, j ¼ 1, 2, 3, 4. The parameters of the fields and the systems used in numerical calculations correlate with experimental conditions discussed in Refs. 33,49 and also used in Ref. 35. They are also equally good fit to address different vibrational modes of CHn molecular species in biological samples.40 We chose o21 ¼ 84.9 THz (2840 cm 1), o43 ¼ 87.6 THz (2930 cm 1). The range of intensity of the laser fields is from 1011 to 1013 W/cm2, and the transform-limited pulse duration is t0 ¼ 176 fs. The spectral chirps used in calculations are a0 p,s ¼ 31 10 5 cm 2, giving the chirped pulse duration t ¼ 1.8 ps. Let us discuss phase averaging in detail to gain more understanding of its importance in making a connection with experimentally measurable CARS signal. Phase is embedded as the complex part in the probability amplitude of the state. When states are coupled by external fields, the relative phase between them just before the fields strike the medium is of key importance since it determines the evolution of the population and coherence in the TLSs. Obviously, the quantum yield at the end of the pulse is phase dependent. One can prepare a particular relative phase between initially populated states by optical pumping56 into state j1i and creating a j1i–j3i state coherence using a Raman scheme. Some values of the initial relative phase are known to bring the system to an optimal quantum yield.57 In the bulk gas or liquid medium, molecules have all possible relative phases between vibrational states at the instant when pulses strike the molecular medium. The Raman signal measured from such a macroscopic ensemble of molecules is phase averaged. To include this idea in our theoretical approach, we take into account the effect of relative phase between initially populated states by performing phase averaging. The procedure consists of calculating state physical quantities at the end of the pulse for 100 initial relative phases between initially populated states j1i and j3i ranging from zero to 2p and averaging over the results obtained. 3.2.1. Degenerate model The coherence density plots of the resonant and detuned Ðsystems are depicted in Figure 7.8 as a function of effective pulse area A ¼ O3(t)dt and dimensionless frequency chirp parameter a0 =t0 2 . The sign of the chirp parameter, a0 , determines the direction of the pump chirp before the central time when the sign changes. The Stokes chirp is the linear chirp and it has an opposite sign to the pump chirp before the central time. The transformlimited pulse duration is t0 ¼ 15 [o 1], where o is the unit frequency equal to o21. This value of t0 corresponds to 176 fs. The figure shows that there is a broad parameter region providing maximum coherence r12 in the resonant TLS (shown in blue color) for the positive chirp values. In the off-resonant TLS, coherence is zero in the same region of field parameters. Negative values of the chirp parameter do not provide selective excitation of coherence in the coupled TLSs. Note that the effective pulse area of 0.85p Author's personal copy Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence 0.18 8 7 7 0 5 4 5 4 3 3 2 2 1 -20 -10 0 a ¢/t02 (a) 10 20 0 6 A [p] A [p] 0.18 8 6 231 1 -20 -10 0 a ¢/t02 (b) 10 20 Figure 7.8 Degenerate model: Phase-averaged coherence density plot for the resonant (a) and off-resonant (b) systems as a function of the effective pulse area, A, and chirp parameter, a0 =t0 2 . (corresponding to the chirped pulse peak intensity in the range of 1011 W/cm2) shows coherent control of the vibrational excitations in molecules. This is a useful addition to previously obtained results on selective excitation in a strong field regime when the field intensity ranges from 1012 to 1013 W/cm2.4 Thus, chirped pulse adiabatic passage method—the roof method—is feasible in achieving a maximum coherence in a predetermined vibrational mode in the presence of strong coupling between many Raman active vibrational modes via external fields. The understanding of the mechanism of selective excitation under the condition of the coupling between vibrational modes is gained through the dressed state picture analysis. We refer in this case only to a particular value of initial relative phase. To do so, we first consider the dependence of coherence and population on the initial relative phase between the ground states j1i and j3i at final time, which is shown in Figure 7.9. In the case of the negative Stokes chirp, Figure 7.9b, for most values of the relative phase between 0 and 2p, there is a strong coherence in the resonant TLS, while zero coherence in the nonresonant TLS. The relative phase p can be inferred as the one creating the dark state, as the coherences r12 and r34 both come to zero and all population is collected in the ground states of TLSs, meaning that the TLSs stay uncoupled from the external fields. In the case of the positive Stokes chirp, Figure 7.9a, both coherences r12 and r34 have some value and the relative phase dependence is almost symmetrical with respect to phase p, so that averaging over the phase gives almost zero as shown in Figure 7.8 (see left side of the plots (a) and (b)). The asymmetry in the coherence value at final time with respect to the chirping direction (positive or negative) Author's personal copy 232 Svetlana A. Malinovskaya et al. 0.8 r11 r44 Population, coherence Population, coherence 0.8 0.6 r11 r33 0.4 r34 0.2 r12 r22 0 r33 0.6 r12 0.4 r22 0.2 r44 0 0 p/4 p/2 3p/4 p 5p/4 3p/2 7p/4 2p 0 p/4 p/2 3p/4 p r34 5p/4 3p/2 7p/4 Relative phase Relative phase (a) (b) 2p Figure 7.9 Degenerate model: Relative phase dependence of population, rii, and coherence, rij, at final time for positive (a) and negative (b) chirp of the Stokes pulse; O3(t0) ¼ 0.35 [o21] (respective effective pulse area A ffi 2.95p), ja0 j/t02 ¼ 10. might be clearly observed using the dressed states. As usual, the dressed states can be obtained by diagonalizing the 4 by 4 Hamiltonians (Eq. (2) or (3)) by solving the corresponding eigenvalue problem. In the present situation, the expressions for the dressed state energies and corresponding dressed vectors are too complicated to be presented in a compact form in the chapter. In general, each dressed state is a liner superposition of all four bare states, jii, with coefficients depending on the Rabi frequency, detuning, and pulse chirps. The energies of the dressed states are shown in Figure 7.10. Within the adiabatic approximation, the system dynamics starts in dressed states I and III that initially correlates with the bare state j1i and j3i correspondingly, independent of the chirping direction. Consider first the positive Stokes chirp (Figure 7.10a). As time evolves and the pulse intensity increases, dressed state III reaches the region of avoiding crossing, with dressed state IV following immediately after that by diabatic crossing with state I. It is clear that in adiabatic approximation, the dressed state III correlates with state j4i at final time and will provide population transfer to this bare state. The dressed state I has several diabatic crossings with states III and IV, and it correlates with state j1i at a later time. In fact, the dressed states I, II, and IV coincide in energy at a later time; therefore (if follow this route), the system will end up in coherent superposition of all three bare states, j1i, j2i, and j3i, at final time. Population of the state j3i at final time depends strongly on the adiabaticity of the first avoiding crossing between the dressed states III and IV: the higher the adiabaticity parameter, the less the population will be in state j3i at the end. Author's personal copy 0.2 0.1 |1ñ |3ñ Dressed states energy Dressed states energy Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence |Iñ |IIIñ 0.0 -0.1 -0.2 |4ñ |IVñ |2ñ |IIñ 800 1000 1200 tw (a) 1400 0.3 233 |IVñ 0.2 |4ñ 0.1 |2ñ |IIñ 0.0 -0.1 -0.2 |1ñ |3ñ |IIIñ |Iñ 800 1000 1200 1400 tw (b) Figure 7.10 Degenerate model: Dressed states energies (solid lines) and bare states energies (dashed lines) as a function of time for the positive (a) and negative (b) Stokes chirp; O3(t0) ¼ 0.35 (respective effective pulse area A ’ 2.95p), ja0 j/t02 ¼ 10. In the case of the negative Stokes chirp, Figure 7.10b, the dressed state energies are much different. Now, the dressed state I effectively has only one avoiding crossing at central time and it is located far below all other dressed states. In adiabatic approximation, the wave function dynamics of states j1i and j2i is very close to the case of TLS considered in Ref. 4. This is the case of most robust solution which is supported by the results presented in Figure 7.8 (see right side of the panel (a)), where maximum coherence in the resonant mode can be prepared in a wide range of the chirp rates and pulse areas. Of course, averaging over the relative phase reduces the maximum value of the coherence. Figure 7.11 shows the dynamics of the state population and coherence as a function of time for zero initial relative phase between states j1i and j3i. In essence, this figure confirms the discussion of the dressed state analysis presented above. For the positive Stokes chirp, Figure 7.11a, population is mostly transferred to the state j4i because of adiabatic following in dressed state III which correlates with the bare state j4i (Figure 7.9a). At final time, the coherences r12 and r34 are of order 0.05. For the negative Stokes chirp, Figure 7.11b, population dynamics follows the dressed state picture presented in Figure 7.9a. At final time, the state j4i is empty and whole population is almost equally distributed between states j1i, j2i, and j3i, which gives maximum coherence for the resonant mode, r12 0.33. The result additionally demonstrates a good correlation between the dressed state picture and the exact solution, showing that by preparing a molecular system in the initial state with a particular relative phase between vibrational modes we can achieve a high value of coherence for the resonant mode and zero excitation in the off-resonant mode. From the discussion above, it follows that there is a near adiabatic solution for achieving a maximum coherence in a predetermined TLS in the presence Author's personal copy 234 Svetlana A. Malinovskaya et al. 0.5 r11 r44 0.6 Population, coherence Population, coherence 0.8 W3(t) . 20 r11 r33 0.4 r34 0.2 r22 r12 r33 0.4 r22 0.3 r12 0.2 W3(t) . 10 r34 0.1 r44 0 0 0.8 1 1.2 1.4 10-3 tw (a) 1.6 0.8 1 1.2 1.4 1.6 10-3 tw (b) Figure 7.11 Degenerate model: Coherences and state populations as a function of time for the positive (a) and negative (b) Stokes chirp; O3(t0) ¼ 0.35 [o21] (respective effective pulse area A ’ 2.95p), ja0 j/t02 ¼ 10. of its coupling with another TLS via external fields. This solution may be achieved in a relatively strong field regime. The results demonstrate that coherence in both modes is sensitive to the initial relative phase between originally populated states and to the field parameters such as intensity and the chirp sign. 3.2.2. Nondegenerate model In this section, we discuss the nondegenerate model in which the ground states j1i and j3i are shifted by d/2 (Figure 7.7b). The dynamics of the system is governed by the time-dependent Schrödinger equation with the Hamiltonian in Eq. (12). Figure 7.12 shows the phase-averaged coherence density plots r12 and r34 as a function of the chirp parameter and effective pulse area. The range of the effective pulse area corresponds to the peak intensity of a transform-limited pulse in the range of 32 1011–32 1012 W/cm2. The chirp parameter spans between a0 =t0 2 ¼ 20 corresponding to 62 10 5 cm 2. Here, essentially for both coherences r12 and r34, the region of the positive chirp (of the pump pulse before the central time) shows zero value in strong fields, where one could expect adiabatic solution. However, there is a relatively large area of the moderate chirp rates, which provides r12 coherence of order 0.15 (green area in Figure 7.12a, left side), while r34 is almost zero (Figure 7.12b, left side). In the negative chirp region, the topology of the coherence density plots is almost identical, meaning that there is no selectivity of the mode excitation. It is clear that there is a dependence of the coherence on the initial relative phase between states, and that a specific relative phase may provide the time evolution leading to a maximum coherence in the j1i–j2i TLS and zero Author's personal copy 235 Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence 0.25 0.21 8 8 7 7 0 5 4 5 4 3 3 2 2 1 -20 -10 10 0 0 6 A [p] A [p] 6 1 -20 20 -10 0 10 a ¢/t02 a ¢/t02 (a) (b) 20 Figure 7.12 Nondegenerate model: Phase-averaged coherence density plot for the resonant (a) and off-resonant (b) systems as a function of the effective pulse area, A, and chirp parameter, a0 =t0 2 . r22 0.8 r33 r44 Population, coherence Population, coherence 0.8 0.6 0.4 r34 0.2 0.6 r12 0.4 r11 0.2 r33 r11 r12 r22 0 r44 0 0 p/4 p/2 3p/4 p 5p/4 3p/2 7p/4 2p 0 p/4 p/2 3p/4 p r34 5p/4 3p/2 7p/4 Relative phase Relative phase (a) (b) 2p Figure 7.13 Nondegenerate model: Relative phase dependence of population, rii, and coherence, rij, at final time for positive (a) and negative (b) chirp of the Stokes pulse; O3(t0) ¼ 0.7 [o21] (respective effective pulse area A ’ 5.9p), ja0 j/t02 ¼ 10. coherence in the j3i–j4i TLS. Phase dependence of r12 and r34 at final time is demonstrated in Figure 7.13. Notably, coherence r12 is nonzero and significant, while coherence r34 is negligible for any relative initial phase in the case of the negative Stokes chirp (Figure 7.13b). The positive Stokes chirp case shows relatively high values of the r34 coherence, while r12 is very small (Figure 7.13a). Author's personal copy 236 Svetlana A. Malinovskaya et al. Let us discuss the case of zero relative phase in more detail. In Figure 7.14, the density plots of r12 and r34 are presented as a function of effective pulse area and the spectral chirp parameter when the initial relative phase is zero. Note that the field conditions are the same as those given in Figure 7.12. Blue regions of a maximum coherence r12 and red regions of zero coherence r34 are observed for the both positive and negative chirps. It is interesting that the selectivity of excitation of TLSs resulting in optimal values of coherence is achieved for zero relative phase in a relatively weak field regime (0.85p pulse area). This is an important observation that highlights the coupling as an additional channel of controllability. In Ref. 4, where a model of two uncoupled TLSs was investigated, it was demonstrated that no selectivity can be achieved in the weak field regime. However, when coupling between TLSs is present, it opens an additional channel for population transfer and, depending on the initial phase conditions, leads to the desired selective excitation with optimal value of coherence r12 and r34 in the moderate fields. To see the adiabatic effects of light–matter interaction within the current model, we analyzed the dressed state picture. We numerically diagonalized the Hamiltonian in Eq. (12) and obtained the time-dependent energy of the dressed states and eigenvectors. The diagonalization was carried out under the conditions that (a) the ac Stark shifts are equal, O1(t) ¼ O2(t), due to identical pulse envelopes of the pump and Stokes pulses, and (b) the chirp parameter ap changes the sign at the central time, t0, while the absolute value of the pump and Stokes pulse chirps is preserved. Figure 7.15 shows the 0.45 0.5 8 8 7 7 0 5 4 5 4 3 3 2 2 1 -20 -10 0 10 20 0 6 A [p] A [p] 6 1 -20 -10 0 a ¢/t02 a ¢/t02 (a) (b) 10 20 Figure 7.14 Nondegenerate model: Coherence density plot for the resonant (a) and offresonant (b) systems as a function of the effective pulse area, A, and chirp parameter, a0 =t0 2 , for zero initial relative phase between states j1i and j3i. Author's personal copy 0.2 |3ñ |IIIñ 0.1 |1ñ 0 -0.1 -0.2 Dressed states energy Dressed states energy Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence |Iñ |IVñ |4ñ |2ñ |IIñ 800 1000 1200 1400 237 0.3 0.2 |4ñ |IVñ 0.1 |2ñ |IIñ 0.0 -0.1 -0.2 |3ñ |IIIñ |1ñ |Iñ 800 1000 1200 tw tw (a) (b) 1400 Figure 7.15 Nondegenerate model: Dressed states energies (solid lines) and bare states energies (dashed lines) as a function of time for the positive (a) and negative (b) Stokes chirp; O3(t0) ¼ 0.7 (respective effective pulse area A ’ 5.9p), ja0 j/t02 ¼ 10. dressed state energies (solid lines) and the bare state energies (dashed lines) as a function of time. For the case of the positive Stokes chirp, three dressed states are involved in the time evolution of the system: dressed states I, III, and IV. Initially, dressed states I and III are populated due to initial population of the bare states j1i and j3i. As time evolves, dressed states I and IV approach avoiding crossing, which is followed by the avoiding crossing between states I and III. These two avoiding crossings are not really separated in time; most probably, they cannot be treated independently. This complex situation results in essentially nonadiabatic population transfer between dressed states I, III, and IV. In turn, populations and coherences in the bare state basis show no sign of adiabatic control. However, there are areas of the parameters (blue area in Figure 7.14b, left side) where r34 ¼ 0.5, which means all population is now distributed between states j3i and j4i. In the dressed state, these regions correspond to the case when only two dressed states I and III are populated, and the first avoiding crossing between I and IV is adiabatic; at a later time, the state I correlates with bare state j3i while the dressed state III correlates with bare state j4i. In the case of the negative Stokes chirp, Figure 7.15b, the dressed state picture looks much better for realizing adiabatic control at least of the resonant mode. Here, the dressed states I, II, and III are involved in the system dynamics and the dressed state IV is well separated from all other states. However, there is the avoiding crossing between states III and II, which effectively involves the bare state j3i into evolution. In fact, the oscillations in the coherence r12 at final time (Figure 7.14a, left side) as a function of the effective pulse area at the fixed chirp rate demonstrate the importance of the dynamical phase, meaning that several dressed states provide a contribution to the bare state populations; in this case, they are Author's personal copy 238 Svetlana A. Malinovskaya et al. the states I, II, and III. Note that at some values of the effective pulse area the population is only in the states j1i and j2i which provide r12 ¼ 0.5 at the fixed relative phase. In summary, we investigate the impact of the coupling between Raman active vibrational modes on the controllability of their excitation. We also analyze a possibility for optimizing the CARS signal in the case of coupled Raman active vibrational modes for enhanced imaging. Selective excitation in a predetermined vibrational mode among many of them having close transitional frequencies may be achieved by using the chirped pump and Stokes laser pulses within CARS. A theory developed in this Section implements the roof method4 to a system of two coupled TLSs and gives us a broader essence of the method implementation in laboratory. The use of femtosecond, chirped laser pulses with chirp sign variation at central time provides adiabatic or near adiabatic passage in two coupled TLSs leading to a significant coherence in the resonant TLS and zero coherence in nonresonant TLS in the presence of the coupling between them via external electric fields. The results show that by applying the roof method, one can stay in low-intensity regime and gain coherent control over the system. The positive chirp is desirable for the excitation in the resonant TLS for degenerate model. Also, single phase calculations support this idea for optimizing coherence among the states of interest. For the nondegenerate model, the phaseaveraged solution gives population transfer among all states in TLSs. Near adiabatic passage resulting in substantial coherence in the resonant mode is observed for a single, fixed phase between initially populated states in the coupled TLSs. The dressed states analysis supports this conclusion by showing an optimal population transfer between the ground and excited states in the resonant TLS, which is a desirable condition for having a high value of coherence. Thus, the roof method can be used for noninvasive imaging of the biological specimens in the presence of the coupling between vibrational modes and be an efficient tool to suppress the contribution of the nonresonant background and, thus, improve the selectivity and chemical sensitivity of the CARS signals. 4. SELECTIVE EXCITATION OF RAMAN TRANSITIONS BY TWO CHIRPED PULSE TRAINS IN THE PRESENCE OF DECOHERENCE Maximum coherence between predetermined states in atoms and molecules is an important aspect of many problems in quantum optics. One possible example is the generation of optimal CARS signal in the CARS microscopy aiming to reveal structures consisting of particular molecular groups. Presently, in experiments on biological imaging, the CARS signal is generated with implementation of picosecond laser pulses,58 a narrow spectral bandwidth of which allows one to address a particular molecular group with a Author's personal copy Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence 239 good accuracy. Aiming at improving chemical sensitivity and obtaining an optimal CARS signal for a given pulse intensity, we propose to use the quantum control methods for selective excitation of required vibrations in complex molecular systems. We developed a novel method that allows one to achieve a maximum coherence in a predetermined vibrational mode in a multimode molecular system via the Raman transitions induced by the pump and Stokes chirped pulses discussed in Section 2. According to the Maxwell–Bloch equations, the maximum coherence is the condition for the CARS signal to acquire the highest intensity upon propagation through a molecular medium.5 The mechanism of creation of the maximum coherence underlying the proposed method is based on the adiabatic passage. An important aspect in the investigations associated with a condense phase is decoherence. Two main sources of decoherence, the vibrational energy relaxation and collisional dephasing, may significantly affect the CARS signal intensity. The phase control of molecular processes in the presence of collisions was demonstrated in Ref. 59. In this Section, we show that the use of two chirped pulse trains, prepared in accordance with the proposed method of chirped pulse adiabatic passage, allows one to control coherence between selected molecular states in the presence of decoherence caused by the spontaneous emission, which occurs on the timescale close to an incident pulse duration.7–9 Femtosecond pulse trains have been recently utilized in the frequency comb spectroscopy to control atomic and molecular systems.60–62 In Ref. 61, a coherent train of weak pump–dump pulse pairs is implemented to perform the narrow-band Raman transitions. A piecewise adiabatic passage is demonstrated in Ref. 61 on an example of the STIRAP scheme using a series of femtosecond transform-limited pulses. The experiments on the molecular trace presence, spectroscopic parameters evaluation, and realtime population dynamics implementing the frequency comb are described in Ref. 62. 4.1. Theoretical model In our model, we make use of a series of femtosecond, chirped pulses with the phase changing in each pulse as f(t) ¼ o0t þ ap,st2/2. We describe molecular vibrational modes by TLSs with the lower and upper levels representing the ground and excited molecular vibrational states of an individual mode (Figure 7.16). Transition frequencies of the TLSs o21 and o43 are chosen to be very close to model a typically dense vibrational frequency spectrum, for example, o43 o21 ¼ d o21. Initially, only ground states j1i and j3i are evenly populated. Let our goal be to generate and sustain high level coherence in the j1i–j2i TLS and zero coherence in the j3i–j4i TLS in the presence of decoherence caused by vibrational energy relaxation and collisions. Under these conditions, a strong CARS signal will be generated from a molecular group represented by Author's personal copy 240 Svetlana A. Malinovskaya et al. ws was = 2wp - ws wp |4ñ d |2ñ w21 |1ñ w43 |3ñ Figure 7.16 A scheme of the two, two-level systems representing Raman active vibrational modes that interact with the intense chirped pump and Stokes pulses to generate maximum coherence in the j1i–j2i TLS and zero coherence in the j3i–j4i TLS. the j1i–j2i TLS, and there will be no signal from any other molecular group, particularly, having close vibrational frequency or from the background. To that end, two-photon Raman transitions are realized by a series of femtosecond pump and Stokes pulses having central frequencies op and os, which (0) 2 are described as Ep, s(t) ¼ E(0) p, s(t)cos(op, s(t t0) þ ap, s(t t0) /2). Here, Ep, s(t) ¼ 2 2 E0 exp((t t0) /2t ) are the pump and Stokes pulse envelopes, and t ¼ t0[1þ a0 p, s/t40]1/2 is the chirp-dependent pulse duration.26 We choose that for ap,s ¼ 0, op os ¼ o21. The temporal (a) and spectral (a0 ) chirps are related as 4 4 02 ap, s ¼ a0 p, st 0 /(1 þ a p, s/t0). The Hamiltonian that describes the interaction of the pump and Stokes pulse pair with a single TLS with no decoherence taken into account, reads as1 1 0 1 O3 ðtÞ ðd þ ðap as Þt þ O1 ðtÞ O2 ðtÞÞ C B2 C B H ¼B C; 1 A @ ðd þ ðap as Þt þ O1 ðtÞ O2 ðtÞÞ O3 ðtÞ 2 O1 ðtÞ ¼ m2 E20 m2 E20 2 2 t2 =t2 e ; O ðtÞ ¼ et =t ; 2 2 4 1=2 2 4 1=2 2 2 0 0 4ℏ D ð1 þ a p =t0 Þ 4ℏ D ð1 þ a s =t0 Þ O3 ðtÞ ¼ m2 E20 2 2 et =t ; 2 2 4ℏ D ½ð1 þ a0 p =t40 Þð1 þ a0 2s =t40 Þ1=4 ð13Þ with d ¼ 0 for the resonant, j1i–j2i TLS and d 6¼ 0 for the detuned, j3i–j4i TLS. Here, O1(t) and O2(t) are the ac Stark shifts originated from two-photon Raman transitions, and O3(t) is the effective Rabi frequency. Author's personal copy Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence 241 Intensity wp t0 Time wp ws t0 Time Frequency Frequency Intensity The proposed method suggests to take a constant value of as, to provide monotonous change of the Stokes pulse frequency, and to choose ap to have same magnitude and opposite sign before the central time t0, when the pump and Stokes pulses reach maximum amplitude, and then to flip the sign. Under these conditions, at t0, the difference of the pump and Stokes pulse frequencies comes into resonance with the frequency o21 and does not change further on. The adiabatic passage takes place in both TLSs and results in zero coherence r34 in the detuned, j3i–j4i TLS and maximum coherence r12 in the resonant, j1i–j2i TLS in a wide range of the field parameters satisfying condition t ’ 1/d (see also Ref. 63). The pump and Stokes pulses are described by the Wigner diagrams in Figure 7.17. In the following, we analyze the vibrational energy relaxation and collisional dephasing as factors that cause decoherence in a selectively excited Raman mode for characteristic times of these processes close to the pulse duration. Then, we analyze a possibility of counteracting decoherence by applying two chirped pulse trains, prepared in accordance with the method stated above. Using the Liouville von Neumann equation and including relaxation terms, we derived a set of differential equations for the density matrix evolution which reads as ws as t0 Time t0 Time Figure 7.17 The Wigner presentation of the pump (left panel) and Stokes (right panel) pulses. The positive and negative slopes of white dashed lines on the density plots correspond to an upward and a downward frequency chirps. Author's personal copy 242 Svetlana A. Malinovskaya et al. r_ 11 ¼ 2O3 ðtÞIm½r21 þ g2 r22 ; r_ 22 ¼ 2O3 ðtÞIm½r21 g2 r22 ; 0 1 g r_ 12 ¼ iðd þ ðas ap Þt þ O1 ðtÞ O2 ðtÞÞr12 þ iO3 ðtÞðr22 r11 Þ @ 2 þ GAr12 ; 2 0 1 g r_ 21 ¼ iðd þ ðas ap Þt þ O1 ðtÞ O2 ðtÞÞr21 iO3 ðtÞðr22 r11 Þ @ 2 þ GAr21 : 2 (14) Parameter g2 ¼ 1/T1 is the vibrational energy relaxation rate, and G ¼ 1/T2* is the collisional dephasing rate. Parameters g2 and G can originate from the inelastic and elastic collisions, respectively. In Eq. (14), we used indexes 1 and 2 that label states of the resonant, j1i–j2i TLS, in which case d ¼ 0. To get the equations for the detuned, j3i–j4i TLS, the former indexes have to be substituted by 3 and 4. This strategy has to be applied to all following equations. 4.2. Numerical results: The effects of the vibrational energy relaxation and collisional dephasing on coherence loss The values of parameters used in numerical calculations are as follows. Raman transition frequencies are o21 ¼ 85.35 THz (2845 cm 1) and o43 ¼ 88.68 THz (2956 cm 1). These are the frequencies of CH2 and CH molecular groups, which are abundantly present in various biological tissues. Field parameters are O3 ¼ 28.4 1012 W/cm2 and t0 ¼ 177 fs, a0 s ¼ 2.8 10 4 cm 2, giving t ¼ 1.58 ps. These parameters of the fields are chosen in accordance with the data published in Ref. 63 to provide the adiabatic passage in the TLSs resulting in a maximum value of r12 and zero r34. The parameters that determine decoherence g2 and G vary in the region from 0 to 0.085 THz, allowing the time of vibrational energy relaxation and collisional dephasing to change from infinity to 11.76 ps. At first, we calculated r12 using Eq. (14) in the case when a single pair of pump and Stokes laser pulses interacts with the TLS. In Figure 7.18, dotted lines show the dependence of coherence r12 on g2 for various fixed values of G: from G ¼ 0 (the upper curve) to G ¼ 10 3 (the lowest curve), solid lines show the dependence of coherence r12 on G for analogous fixed values of g2. The parameters g2 and G are given in the units of frequency o21. Given o21 ¼ 85.35 THz, the value g2 ¼ G ¼ 10 3 corresponds to 0.085 THz. For g2 ¼ 0, increasing G from 0 to 0.085 THz causes significant decrease in r12 from the maximum value (0.25) to 0.065. While for G ¼ 0, increasing g2 causes twice less decrease in r12 in agreement with Eq. (14). This demonstrates that the dynamics of induced coherence is very sensitive to collisional dephasing, and to less extent to the vibrational energy relaxation. When decoherence time is one order of magnitude larger than the pulse duration, coherence in the resonant TLS is small, yielding a small amplitude of the generated Author's personal copy Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence 243 0.25 r12 0.2 0.15 0.1 0.05 0 0 0.0002 0.0004 0.0006 0.0008 0.001 g 2/w, G /w Figure 7.18 Dependence of coherence r12 on g2 and G. Dotted lines show the dependence of coherence r12 on g2 for various fixed values of G: from G ¼ 0 (the upper curve) to G ¼ 10 3 (the lowest curve), solid lines show dependence of coherence r12 on G for analogous fixed values of g2. The parameters g2 and G are given in the units of frequency o21. Raman fields and a weak CARS signal. For the detuned TLS, numerical calculations show minor changes in this picture, preserving characteristic features of the behavior of the coherence as a function of the g2 and G parameters. 4.3. Prevention of decoherence In the framework of the dressed state picture in the adiabatic approximation, valid for g2, G O3, the balance equations for the population of the dressed states read as r_ d11 ¼ ðg2 sin2 Y þ 1=2ðG g2 =2Þ sin2 2YÞrd11 þ ðg2 cos4 Y þ 1=2G sin2 2YÞrd22 ; r_ d22 ¼ ðg2 sin4 Y þ 1=2G sin2 2YÞrd11 ðg2 cos2 Y þ 1=2ðG g2 =2Þ sin2 2YÞrd22 ; (15) where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinY ¼ 1=2ð1 R0 =RÞ; cosY ¼ 1=2ð1 þ R0 =RÞ; R0 ¼ d þ ðas ap Þt þ O1 ðtÞ O2 ðtÞ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ ðd þ ðas ap Þt þ O1 ðtÞ O2 ðtÞÞ2 þ 4O3 ðtÞ2 : (16) These equations demonstrate the time evolution of the population of the dressed states assuming rd12 ¼ 0. The correlation between the dressed states and the bare states in the fieldinteraction representation is as follows: Author's personal copy 244 Svetlana A. Malinovskaya et al. rd11 ¼ cos2 Yr11 þ sin2 Yr22 2sinY cosYRe½r12 ; rd22 ¼ sin2 Yr11 þ cos2 Yr22 þ 2sinY cosYRe½r12 : (17) According to Eq. (17), at t ! 1, for the detuned TLS, when d 6¼ 0, the coefficients are sin2 Y ¼ 0, and cos2 Y ¼ 1. It means that the dressed state population rd11 relaxes to the bare state population r11, and dressed state population rd22 relaxes to the bare state population r22. In the resonant TLS, the asymptotic solution is qualitatively different. When d ¼ 0, it gives sin2 Y ¼ cos2 Y ¼ 1/2, and asymptotically the dressed states populations are rd11 ¼ 1=2ðr11 þ r22 Þ Re½r12 ; rd22 ¼ 1=2ðr11 þ r22 Þ þ Re½r12 ; (18) which means that as t ! 1, rd11 and rd22 approach r11/2. This behavior of the dressed states populations may also be obtained from the numerical solution of Eq. (15). It follows from Eq. (18) that periodic restoration of the population of the upper level r22 in the resonant TLS would provide the asymptotic solution for the dressed state to be not r11/2 but a nonzero superposition of the lower and upper bare states and coherence. This is the essence of the mechanism of preventing decoherence. To do so in a selectively excited TLS, we propose to use 2-fs pulse trains chirped in accordance with the chirped pulse adiabatic passage. A series of pump and Stokes chirped pulses, when applied with the period comparable to the vibrational energy relaxation time, create and sustain high coherence in the selected vibrational mode. The values of g2 and G used in further calculations are equal to 0.085 THz. These values give the time of vibrational energy relaxation and collisional dephasing (decoherence time) to be 11.76 ps. Figure 7.19 shows results of interaction of the resonant TLS with two sequential pump and Stokes pulse pairs having the time delay approximately four times longer than the decoherence time, namely, 44 ps. The bold black curve shows the effective Rabi frequency O3, which is the same for both the pump and Stokes pulses. The time dependence of the state population and coherence, calculated in the presence of only vibrational energy relaxation, is marked by r11 ðerÞ , r22 ðerÞ , and r12 ðerÞ ; the state population and coherence, calculated in the presence of only collisional dephasing is marked by r11 ðcdÞ , r22 ðcdÞ , and r12 ðcdÞ . Figure 7.19 shows that r12 ðerÞ achieves the maximum value during the application of the first pulse pair and undergoes decay up to the instant when the second pulse pair arrives and restores population of the upper level and coherence. Notably, the second pair of pulses does not affect coherence r12 ðcdÞ because it is not coupled to the population distribution. The restoration of maximum coherence takes place because of the time delay between the first and the second pulse pairs being longer than the vibrational relaxation time. This results in almost complete relaxation of the population to the ground state in the resonant TLS and return of the system into its initial state. Figure 7.20 shows the time- Author's personal copy Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence 245 0.5 r (er) 11 0.4 0.3 (cd) (cd) r 22 r11 0.2 r (er) r(er) 12 22 0.1 W3 r (cd) 12 0 0 1 2 3 5 4 tw 21 ⫻ 10 3 Figure 7.19 Dynamics in the resonant TLS in the field-interaction representation. Figureshows the results of application of two sequential pump and Stokes pulse pairs with the time delay 44 ps. The parameters are g2 ¼ G ¼ 10 3 (0.085 THz). (er) 0.3 (er) r11, r22, r12 0.4 (er) 0.5 0.2 W3(t)/w 21 0.1 0.20 0.15 0.1 0.05 0 0 2 4 6 8 10 tw 21⫻103 Figure 7.20 Dynamics in the resonant TLS in the field-interaction representation. Figureshows the state population and coherence resulting from application of the pump and Stokes chirped pulse trains. The pulse trains period is 11.76 ps, g2 ¼ 10 3 (0.085 THz). dependent picture of the resonant TLS’s parameters resulting from its interaction with the sequence of pump and Stokes pulses. The pulse train period is 11.76 ps, which is the same as the time of vibrational energy relaxation. The upper panel shows the populations r11 (long-dashed line) and r22 (dotted line), and coherence r12 (solid line) in the TLS, and the lower panel Author's personal copy 246 Svetlana A. Malinovskaya et al. shows the dimensionless Rabi frequency O3(t)/o21. Prepared by the first pulse pair, the coherence r12 ðerÞ decreases from maximum (0.25) to 0.10 and varies within a 0.10–0.15 region, subject to vibrational energy relaxation and periodic interaction with the subsequent pulse pairs. The variation of coherence within a restricted region is determined by the population dynamics which is different from the previous case. Here, according to Figure 7.20, the population of the upper level in the TLS only partially relaxes to the lower level during the time between pulses. This population distribution determines the ‘‘initial’’ condition for each subsequent pulse pair and provides a partial adiabatic population transfer to the upper level resulting in coherence within indicated region. The choice of the period of the pulse train depends on one’s need to either maintain uninterruptedly substantial coherence in a TLS or get maximum coherence periodically. In the detuned TLS, the pump and Stokes chirped pulse trains suppress generation of the coherence by preserving the population in the ground state. Thus, the proposed method satisfies the goal of selective excitation and creation of the coherence in a vibrational mode having known frequency. In summary, we propose a novel method to control molecular vibrational excitations and sustain high coherence in a predetermined vibrational mode in the presence of decoherence by making use of the pump and Stokes femtosecond, chirped pulse trains. The method employs a constant chirp in the pump (Stokes) pulse train and the flipping sign of the chirp in the Stokes (pump) pulse train. The pulse train period must be close to the time of vibrational energy relaxation. High coherence is sustained in a selectively excited vibrational mode (modeled by a TLS) in the presence of decoherence because of a periodic, adiabatic pumping of the population to the upper level by the pump and Stokes chirped pulse trains. The method can be used for noninvasive imaging of biological structure and femtosecond dynamics studies, and as an efficient tool for suppression of the background signal in the CARS spectroscopy and microscopy. It demonstrates the possibility of coherent control over spectrally close vibrational modes in the nonimpulsive Raman scattering. 5. FESHBACH-TO-ULTRACOLD MOLECULAR STATE RAMAN TRANSITIONS VIA A FEMTOSECOND OPTICAL FREQUENCY COMB Magnetically tunable Feshbach resonances are used to create loosely bound Feshbach molecules by association of atom pairs in cold temperatures. The methodology is based on magnetic field sweep across zero-energy resonance between the diatomic vibrational bound state and the threshold for dissociation into an atom pair at rest.64 The technique was applied to produce diatomic molecules consisting of identical atoms, for example, K2,65 Li2,66 and Cs2,67 or of different species of atoms, resulting in creation of polar Author's personal copy Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence 247 molecules, for example, the KRb.20,68 The permanent dipole moment associated with polar molecules may be used for studies of long-range interactions in many-body quantum systems. Optically induced transitions in Feshbach molecules are an efficient route to reach the fundamentally lowest state in molecules. In Ref. 24, using two narrow-band pulses applied to weakly bound KRb molecules in accordance with the STIRAP scheme, the ultracold gas of polar KRb molecules in the ground singlet or triplet electronic configuration and rovibrational state was created. Recently, a theory on piecewise stimulated Raman adiabatic passage involving Feshbach states was performed using two coherent pulse trains.22,23 The ultracold KRb molecules were created using the pump–dump stepwise technique that coherently accumulated population in the ground electronic–vibrational state. An optical frequency comb is a unique tool for high-resolution spectroscopic analysis of internal energy structure and dynamics as well as for controlling ultrafast phenomena in atomic and molecular physics.16–19 Owing to its broadband spectrum, the frequency comb may efficiently interact with the medium inducing one-photon, two-photon, and multiphoton resonances in atoms and molecules. A unique ability of the frequency comb is provided by the presence of about a million optical modes in its spectrum with very narrow bandwidth and exact frequency positions.69 In this Section, aiming at creation of deeply bound ultracold molecules from Feshbach states, the control of population dynamics in a molecular system is investigated using a single optical frequency comb with zero offset frequency.10–12 The frequency comb is generated by phase-locked femtosecond pulse train having unmodulated or sinusoidally modulated phase across an individual pulse. We investigate the dynamics of rovibrational cooling on an example of the KRb molecule, which involves the interaction of loosely bound Feshbach KRb molecules with the femtosecond optical frequency comb resulting in the molecular transfer to the ground electronic and rovibrational state. The population dynamics takes place via two-photon Raman transitions between three energy levels separated by terahertz region frequencies. Coherent accumulation of population in the ultracold KRb state with a negligible population (for the case of the sinusoidally modulated optical frequency comb applied) of the excited state is accomplished by a series of sequential pulses with zero carrier–envelope phase and within the lifetime of the Feshbach KRb molecules, which is about 100 ms.68,70 5.1. Theoretical model A semi-classical model of two-photon Raman transitions induced by a femtosecond optical frequency comb in a l system, Figure 7.21, describes the cooling process of Feshbach KRb molecules. A frequency comb known to be characterized by two key parameters, the radio frequency ( fr), determined by the pulse repetition rate and specifying the spacing between modes, and the carrier–envelop offset frequency (f0 ¼ frDfce/(2p)), where Dfce is the Author's personal copy 248 Svetlana A. Malinovskaya et al. |2Ò d wL |1Ò fr |3Ò Figure 7.21 The three-level l-system modeling the molecular energy levels, involved in Raman transitions. State j1i is the Feshbach state, state j2i is the transitional, electronic excited state, and state j3i is the cold molecular state. carrier–envelope phase difference. Then, a mode in the comb has a frequency equal to f ¼ nfr þ f0. Both parameters, the f0 and the fr, were efficiently used to manipulate dynamics in, for example, Ref. 21, to resonantly enhance two-photon transitions in cold 87Rb atoms. Here, we make use of the radio frequency (fr) only, and put f0 equal to zero by implying zero value of Dfce. The pulse train that generates the optical frequency comb reads 2 2 EðtÞ ¼ SN1 k¼0 E0 expððt kTÞ =ð2t ÞÞ cosðoL ðt kTÞÞ: (19) Here, E0 is the peak field amplitude, T is the pulse train period, t is a single pulse duration, N is the number of the pulses in the pulse train, and oL is the carrier frequency. To get zero value of the Dfce, the temporal variation in the Gaussian envelope and in the harmonic wave is taken in the form (t kT). It guarantees the envelope maximum to coincide with the peak value of the amplitude of the electric field. The fine structure of the optical broadband comb is owing to the modes equally spaced by the radio frequency 1/T 1. In our model, a single optical frequency comb induces Raman resonances in the three-level l-system resulting in full population transfer from the initial state j1i through intermediate state j2i to the final state j3i. We assume that state j1i is the Feshbach state, state j2i is the transitional, electronic excited state, or state manifold, and state j3i is the ultracold molecular state. There are a large number of mode pairs in the frequency comb that differ by exactly the transition frequency o31 and, thus, satisfy the condition of the Raman resonance. These lead to an efficient stepwise population accumulation in the final state in the l-system. The evolution of the density matrix of the l-system is investigated via a set of coupled differential equations obtained from the Liouville von Neumann equation: r_ 11 ¼ 2Im½H12 r21 ; r_ 22 ¼ 2Im½H21 r12 þ H23 r32 ; r_ 33 ¼ 2Im½H32 r23 ; r_ 12 ¼ iH12 ðr22 r11 Þ þ iH32 r13 ; r_ 13 ¼ iH12 r23 þ iH23 r12 ; r_ 23 ¼ iH23 ðr33 r22 Þ iH21 r13 : (20) Author's personal copy Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence 249 Two nonzero matrix elements of the interaction Hamiltonian are Hij ¼ OR(t T) [exp{ i((oL þ oji)(t T))} þ exp{i((oL oji)(t T))}], where i and j are the indexes of the basis set, i ¼ 1, 2, and j ¼ i þ 1, OR(t T) ¼ OR exp (t T)2/(2t2) is the Rabi frequency, and OR is the peak value of the Rabi frequency. Calculations are done beyond the rotating wave approximation. 5.2. Numerical results: Creation of ultracold molecules using a standard optical frequency comb and a phase-modulated optical frequency comb For the l-system, we apply a set of parameters that corresponds to data on molecular cooling of loosely bound KRb molecules from the Feshbach states presented in Ref. 24. Experimental schemes used in Ref. 24 involve a transition to the ground electronic triplet or the ground electronic singlet state with zero rovibrational quantum number. These states are achieved by implementing the STIRAP control scheme with a pair of pulses having a narrow bandwidth and being in resonance with the transitions between three states involved in the process. These are the Feshbach state, the 23S electronically excited state, and the triplet or singlet ground electronic state with zero rotational and vibrational quantum number. In our model, we implement parameters that correspond to the experiment involving the singlet ground electronic state, thus addressing a fundamentally cold molecule. The parameters of the l-system are o21 ¼ 309.3 THz and o32 ¼ 434.8 THz, making the frequency of two-photon transition o31 to be equal to 125.5 THz. The carrier frequency of the pulse train oL is chosen to be in resonance with the one-photon transition frequency o32 in the l-system, and the multiples of the radio frequency provide two-photon resonances when the condition oL nfr ¼ o31 is satisfied, where n is integer number. The peak Rabi frequency is OR ¼ 0.01o31 ¼ 1.255 THz, the pulse duration t is 3 fs, and the pulse train period is T ¼ 6.4 105t (2 ns), giving the radio frequency 500 MHz. For a given peak Rabi frequency, the evaluated peak field amplitude is 106 –107 V/cm. The results of population transfer are presented in the Figure 7.22. A stepwise adiabatic population transfer is observed from the initial j1i (black) to the final j3i state (green) via the transitional state j2i (red), which gets populated up to 45%. Each pulse brings a fraction of population to the final state and contributes to the accumulative effect. Total population transfer occurs in 460 ns after 242 sequential pulses, which is within the lifetime of the Feshbach KRb molecules. Next 242 pulses return population to the initial state, the system returns to the initial conditions, and, then, the dynamics repeats. Within the lifetime of the Feshbach states, it is possible to transform the medium from highly vibrationally excited molecules to the ultracold state and back. Notably, the detuning of the carrier frequency off-resonance with the o32 gives very similar dynamics of population transfer to the ultracold molecular state and back as in the resonance case with a difference Author's personal copy 250 Svetlana A. Malinovskaya et al. 1 Population 0.8 0.6 0.4 0.2 0 0 5000 10000 tw ⫻ 104 Figure 7.22 Population dynamics in the three-level l-system using an optical frequency comb having fr ¼ 500 MHz, and zero offset frequency. Parameters of the pulse train are the carrier frequency oL ¼ 434.8 THz, the pulse duration t0 ¼ 3 fs, and the peak Rabi frequency OR ¼ 1.26 THz; the system parameters are o21 ¼ 309.3 THz, o32 ¼ 434.8 THz, and o31 ¼ 125.5 THz. Black line shows population of the ground state, red line, population of the transitional state, and green line, that of the final state. Time is given in the units of [o 1], where o ¼ o31 ¼ 125.5 THz. being some reduction in the number of pulses needed to accomplish this transfer. The increase in the strength of the electric field decreases the number of pulses and the overall time duration needed for the full population transfer, preserving the quality of the dynamics picture. Decreasing the pulse repetition rate elongates the duration of the coherent accumulation dynamics; however, it does not change the number of the required pulses (given other field parameters are preserved). Since a single pulse duration determines the bandwidth of the comb, the shorter it is the more efficient excitation of Raman transitions takes place. We performed the phase modulation across an individual pulse in the pulse train in the form of a sinusoidal function. In previous works, the sinusoidal modulation in the terahertz region was applied to the carrier frequency, for example, to study absorption resonances in I2 with high precision.71 A general form of a phase-modulated pulse train reads Eðt; zÞ ¼ N 1 X E0 expððt kTÞ2 =ð2t2 ÞÞ cosðoL ðt kTÞ þ F0 sinðOðt kTÞÞÞ: k¼0 (21) Here, O is the modulation frequency, and F0 is the modulation amplitude. The carrier–envelope phase is zero in the pulse train resulting in zero offset frequency of the generated optical frequency comb. The time-dependent Author's personal copy Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence 251 phase across each pulse in the form of the sin function enriches the frequency comb power spectrum with new peaks compared to a standard optical frequency comb discussed above. More specifically, laser frequency oL determines the center of the frequency comb, and the sinusoidal modulation forms the sidebands at multiples of O with the amplitude dictated by F0. Numerical analysis of the population dynamics in the l-system was done using Eq. (21) with the Hamiltonian having two nonzero matrix elements Hij ¼ OR(t T) [exp{ i((oL þ oji)(t T) þ M(t T))} þ exp{i((oL oji) (t T) þ M(t T))}], where i ¼ 1, 2 and j ¼ i þ 1, and M(t T) ¼ F0 sin O(t T) is the phase modulation in a single pulse. The carrier frequency of the pulse train oL is chosen to be equal to the o32, and the modulation frequency O, to the o21, so that the field frequencies are in resonance with the one-photon transitions in the three-level l-system which automatically satisfies the condition for the two-photon Raman resonance. Additionally, the modes that are multiples of the radio frequency fr provide with pairs of frequencies that differ by exactly the transition frequency o31. These lead to an efficient stepwise population accumulation in the final state in the l-system. Parameters of the l-system used in these calculations were taken from Ref. 23. The system levels addressed in Ref. 23 include the final state as the ground electronic singlet state with vibrational quantum number v ¼ 22, which suggests the following parameters for the l-system: the transitional frequency from the initial to the excited electronic state o21 ¼ 340.7 THz, and the transition frequency from the excited electronic state to the final state o32 ¼ 410.7 THz. It makes the initial-to-final state frequency difference o31 ¼ 70 THz. The carrier frequency and the modulation frequency of the pulse train are in resonance with the o32 and o21, respectively; the peak Rabi frequency is equal to o31, F0 is equal to 4 (the discussion of this choice will be given below), a single pulse duration t is 3 fs, and the pulse train period T is 6400t (about 20 ps). The modulation of the carrier frequency having value 410.7 THz is to be done at frequency 340.7 THz. To achieve the needed modulation, an approach, described in Ref. 72, to efficient generation of a Raman-type optical frequency comb in an enhancement cavity may be applied. The technique provides the whole comb bandwidth covering 300–900 THz. The results of the numerical calculations, Figure 7.23, show a smooth stepwise population transfer from the initial to the final state with a negligible population of the excited state, in analogy with the STIRAP stepwise scheme proposed by Shapiro,23 however, performed with a single optical frequency comb. In our opinion, making use of a single pulse train, in contrast to the two pulse trains scheme in Ref. 23, significantly simplifies the experimental conditions for cooling of rovibrational degrees of freedom in molecules. Total population transfer occurs after 109 sequential pulses and, thus, is accomplished in 2.5 ns. Notably, there is a strong dependence of the efficiency of population transfer on the value of the amplitude F0 of sinusoidal modulation of the Author's personal copy 252 Svetlana A. Malinovskaya et al. 1 Population 0.8 0.6 0.4 0.2 0 0 10 20 30 tw ⫻ 104 Figure 7.23 Population transfer in the three-level l-system, achieved via the resonant Raman transitions using a phase-modulated optical frequency comb described by Eq. (21). The values of the parameters are the carrier frequency oL ¼ 410.7 THz, the modulation frequency O ¼ 340.7 THz, the modulation amplitude F0 ¼ 4, and the peak Rabi frequency OR ¼ 70 THz. Stepwise, adiabatic accumulation of the population is observed in state j3i (green), which is the ultracold KRb state. The population of the Feshbach state j1i (black), comes gradually to zero, while the excited state manifold j2i (red), is slightly populated during the transitional time. Full population transfer is accomplished in 109 pulses. Time is given in the units of [o 1], where o ¼ o31 ¼ 70 THz. phase across an individual pulse in the pulse train. For the resonant excitation, population dynamics was calculated using different values of the parameter F0; however, for the values of F0 from 1 to 9, only F0 ¼ 4 gave the desired population transfer to the cold state. To get insight into the mechanism of the frequency comb–system interaction leading to a successful cooling, we made a Fourier transform both analytically and numerically to reveal its spectral properties linking to the molecular system resonances. The Fourier transform reads EðoÞ ¼ ðE0 tÞ=2Sn Jn ðF0 Þ expð1=2ðoL þ nO oÞ2 t2 ÞSk expðiokTÞ: (22) Here, Jn(F0) is the Bessel function of the order n and F0 is the modulation index. When multiplied by exp ( 1/2(oL þ nO o)2t2), it determines the shape of the power spectrum of the optical frequency comb. Depending on the value of F0, the power spectrum has different number of maxima as seen in Figure 7.24 showing the envelope of the power spectrum for F0 ¼ 3, 4, 5, 8. The increase in modulation index brings additional, intense peaks of modes into spectrum and broadens it. These maxima are located at different frequencies for different values of F0 affecting the population dynamics in the l-system. Author's personal copy Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence 253 0.8 0.4 E 2 WT 0.6 0.2 2 4 6 8 10 12 14 WT 2 E (wt) Figure 7.24 The envelope of the power spectrum of the optical frequency comb described by Eq. (21) (no dense radio-frequency comb lines are included). The pulse train parameters are the effective Rabi frequency OR ¼ 70 THz, the carrier frequency oL ¼ 410.7 THz, the modulation frequency O ¼ 340.7 THz, t ¼ 3 fs, and the modulation amplitudes F0 ¼ 3 (red), F0 ¼ 4 (green), F0 ¼ 5 (blue), and F0 ¼ 8 (black). 0 wt Figure 7.25 The fast Fourier transform of the phase-modulated optical frequency comb in Eq. (21). The pulse train parameters are the effective Rabi frequency OR ¼ 70 THz, the carrier frequency oL ¼ 410.7 THz, the modulation frequency O ¼ 340.7 THz, t ¼ 3 fs, and the modulation amplitude F0 ¼ 4. The power spectrum of the pulse train with the modulation amplitude F0 ¼ 4 has three maxima; the highest one is located at o ¼ 4.9 (which is in resonance with the o21), making the pulse train with F0 ¼ 4 an optimal one for coherent accumulation of the population in the final state of the l-system. It provides full population transfer to the ground electronic–vibrational state, thus cooling the KRb Feshbach molecule. The numerical result of the Fast Fourier Transform of Eq. (21) with F0 ¼ 4, Figure 7.25, shows a power Author's personal copy 254 Svetlana A. Malinovskaya et al. spectrum of the optical frequency comb with the fine structure of the radiofrequency spaced modes appearing as a solid color under the envelope. The parameters of the phase-locked pulse train, needed to accomplish molecular cooling from Feshbach states, have to be chosen based on the analysis of the power spectrum of the sin-phase-modulated optical frequency comb and the energy levels involved in dynamics of the molecular system. The offset frequency not necessarily has to be zero. Its nonzero value does not affect the two-photon resonance condition, while, for one-photon transitions, it induces a detuning. The field carrier frequency detuned off-resonance with the one-photon transition in the l-system leads to the same quantum yield on a shorter timescale. Alternatively, the detuning may be compensated by adjusting the radio frequency to keep the frequency comb modes in resonance with the l-system. In summary, we have demonstrated a coherent population transfer from a loosely bound Feshbach state to the ultracold molecular state using a single femtosecond, optical frequency comb. We studied the phenomenon on an example of the KRb molecule, modeled by a three-level l-system with the energy levels taken from Refs. 23,24. Coherent accumulation of the population in the ground electronic and rovibrational state is achieved by applying a standard optical frequency comb with zero offset frequency, or an optical frequency comb generated by a pulse train with a phase modulation in the form of the sinusoidal function across an individual pulse. The mechanism of the accumulative effect leading to full population transfer is based on the excitation of the two-photon Raman resonances by pairs of optical frequency modes with the frequency difference matching Feshbach-to-ultracold molecular state transition. In the case of sinusoidally modulated optical frequency comb, the Raman transitions are stimulated by the carrier and the modulation frequencies. A strong dependence of the population dynamics on the amplitude of the sinusoidal modulation is demonstrated, suggesting to choose this parameter based on the analysis of the envelope of the power spectrum of the optical frequency comb. 6. CONCLUSIONS Recent advancements in the application of ultrafast laser pulses to control Raman transitions have led to the development of new quantum control methods aiming at enhancement of CARS bioimaging techniques. The proposed methods utilize femtosecond, chirped pulses to induce two-photon excitations in Raman active vibrational modes and provide adiabatic, via the pulse area solution or by the roof method, fractional population transfer in a predetermined vibrational mode, resulting in a maximum coherence. At the same time, coherence in other Raman active vibrational modes is completely suppressed. Methods provide high chemical selectivity between spectrally close Raman modes, whose frequency difference d is on the order of t 1, the Author's personal copy Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence 255 inverse chirped pulse duration. To control molecular vibrational excitations and sustain high coherence in a predetermined vibrational mode in the presence of decoherence, the method implementing two chirped pulse trains was proposed with the chirp scheme as in the roof method. The pulse trains period must be close to the time of vibrational energy relaxation. High coherence is sustained because of periodic adiabatic pumping of the population to the upper level. A pair of chirped pulse trains periodically restores population in the excited state and, therefore, coherence between the ground and the excited vibrational states and, thus, prevents decoherence caused by spontaneous emission. We have also reviewed our achievements in a theory of ultracold gases manipulation aiming at creation of ultracold molecules from Feshbach states. For this purpose, we investigated applicability and robustness of an optical frequency comb, which may be a standard one, produced by a phase-locked pulse train with no modulation, or a frequency comb with a sinusoidal phase modulation. It was demonstrated that a single optical frequency comb, with no modulation or modulated, may be used to cool the internal degrees of freedom in molecules and create ultracold molecules from Feshbach states via two-photon Raman transitions. ACKNOWLEDGMENTS Authors acknowledge Philip H. Bucksbaum, Jun Ye, Paul R. Berman, Vladimir S. Malinovsky, John R. Sabin, N. Yngve Öhrn, Jeffrey L. Krause, Moshe Shapiro, and Pierre Meystre for many fruitful discussions. This research is partially supported by the National Science Foundation under Grant No. PHY-0855391 and DARPA under Grant No. HR0011-09-1-0008. REFERENCES 1. Malinovskaya, S.A. Mode selective excitation using ultrafast chirped laser pulses. Phys. Rev. A 2006, 73, 033416. 2. Malinovskaya, S.A. Chirped pulse adiabatic passage in CARS for imaging of biological structure and dynamics. In AIP Conference Proceedings 963, 2 Part A, 2007; p 216. 3. Malinovskaya, S.A. Chirped pulse control methods for imaging of biological structure and dynamics. Int. J. Quantum Chem. 2007, 107, 3151. 4. Malinovskaya, S.A.; Malinovsky, V.S. Chirped pulse adiabatic control in CARS for imaging of biological structure and dynamics. Opt. Lett. 2007, 32, 707. 5. Malinovskaya, S.A. 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