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J Opt B Quantum Semiclassical Opt. Author manuscript; available in PMC 2007 March 29.

Published in final edited form as:

J Opt B Quantum Semiclassical Opt. 2005 October; 7(10): S265–S269.

doi: 10.1088/1464-4266/7/10/009PMCID: PMC1839073

EMSID: UKMS5501

Indian Institute of Technology, Kanpur 208016, India

E-mail: ni.ca.ktii@imawsogd

See other articles in PMC that cite the published article.

Control of multiphoton transitions is demonstrated for a multilevel system by generalizing the instantaneous phase of any chirped pulse as individual terms of a Taylor series expansion. In the case of a simple two-level system, all odd terms in the series lead to population inversion while the even terms lead to self-induced transparency. The results hold for multiphoton transitions that do not have any lower-order photon resonance or any intermediate virtual state dynamics within the laser pulse width.

Most of the models for coherent control deal with light–matter interaction at the single-photon level, which is fairly well understood with various theoretical models [1]. Actual photochemical processes often involve multiphoton interactions. Multiphoton interactions typically induce additional complications. In essentially every large molecule, energy can be deposited in a specific bond only for a short period. The energy then randomizes due to the typically strong coupling amongst the molecular degrees of freedom. Minimizing decoherence is an important challenge on the route towards the realization of the decades-old dream of using photons as ‘reagents’ in chemical reaction and has doomed the attempts to do ‘laser selective chemistry’. The intramolecular relaxation processes, such as intramolecular vibrational relaxation (IVR), are the most important contributors to decoherence even in isolated molecules [2]. Controlling IVR also has important prospects for decoherence issues in quantum computing, which has led to a further resurgence of activity in this area [3]. Restriction of IVR by optical schemes [4] can be an attractive route towards selective excitation in large molecular systems. Albeit attractive, most of the photon-mediated approaches towards restricting IVR (also called ‘photon locking’) have used complicated pulse shapes that are yet to be demonstrated in the laboratory due to stringent requirements of intensity and precision.

In a recent paper [5], we have proposed the use of simple chirped pulses, which, by contrast, have been produced routinely at very high intensities and at various different wavelengths for many applications, including selective excitation of molecules in coherent control. We have shown that even simple linearly chirped pulses could restrict IVR in systems at least as complicated as those investigated earlier [5]. We proved that higher-order chirps that distort the linear chirps experimentally result, in fact, in better performance in restricting IVR. In the case of two-level systems, such shaped pulses result either in population inversion or self-induced transparency. Similar results hold even in the extreme case of a two-photon or multiphoton transition occurring with a chirped pulse, where the lower-order photon processes are non-resonant [6, 7]. This makes these results more attractive, since the intensity of the laser fields needed to reach adiabatic limits often leads to multiphoton processes.

In this current paper, we discuss the generalized situations for complex systems, starting from simple two-level systems and going to *N* -level systems in the multiphoton as well as single-photon scenario. We will establish that in spite of the inherent complexity of the systems and in spite of including multiphoton interactions, it is possible to come up with controllable parameters and rules, which are generic in nature. Such generalizations would enable us to treat real systems with our model framework developed in this paper.

We apply a linearly polarized laser pulse of the form *E* (*t*) = (*t*)e^{i[ (}^{t}^{)}^{t}^{+(}^{t}^{)]} to a simple two-level system with a |1 → |2transition, where |1 and |2 represent the ground and excited eigenlevels, respectively, of the field-free Hamiltonian. The laser carrier frequency or the centre frequency for pulsed lasers is ω. We have (*t*) and (*t*) as the instantaneous amplitude and phase. We can define the rate of change of the instantaneous phase,
$\dot{\phi}(t)$, as the frequency sweep. If we expand the instantaneous phase function of *E* (*t*) as a Taylor series with constants *b** _{n}* , we have

$$\begin{array}{c}\phi (t)={b}_{0}+{b}_{1}t+{b}_{2}{t}^{2}+{b}_{3}{t}^{3}+{b}_{4}{t}^{4}+{b}_{5}{t}^{5}+\dots \\ \dot{\phi}(t)={b}_{1}+2{b}_{2}t+3{b}_{3}{t}^{2}+4{b}_{4}{t}^{3}+5{b}_{5}{t}^{4}+\dots \\ \dot{\phi}(t)=\sum _{n=1}n{b}_{n}{t}^{(n-1)}.\end{array}$$

(1)

To our knowledge, this approach of formulating a general expression for the continuously frequency modulated (‘chirped’) pulses is the first of its kind [2, 5]. Establishing this generalization enables us to treat all possible chirped pulse cases by exploring the effects of each of the terms in equation (1) initially for a simple two-level system and then extend it to the multilevel situation for a model five-level system of the anthracene molecule, which has been previously investigated with complicated shaped pulses [4]. We use a density matrix approach by numerically integrating the Liouville equation
${\scriptstyle \frac{\text{d}\rho (t)}{\text{d}t}}={\scriptstyle \frac{\text{i}}{\hslash}}[\rho (t),{H}^{\text{FM}}(t)]$ for a Hamiltonian in the rotating frequency modulated (FM) frame of reference. (*t*) is a 2 × 2 density matrix whose diagonal elements represent populations in the ground and excited states and off-diagonal elements represent the coherent superposition of states. The Hamiltonian for the simple case of a two-level system under the influence of an applied laser field can be written in the FM frame for the *N* -photon transition [7] as
${H}^{\text{FM}}=\hslash \left(\begin{array}{cc}\Delta +N\dot{\phi}(t)& {\scriptstyle \frac{{\Omega}_{1}}{2}}\\ {\scriptstyle \frac{{\Omega}_{1}^{*}}{2}}& 0\end{array}\right).$ The time derivative of the phase function,
$\dot{\phi}(t)$, appears as an additional resonance offset over and above the time-independent detuning Δ = ω_{R} − *N*ω, while the direction of the field in the orthogonal plane remains fixed. We define the multiphoton Rabi frequencies as complex conjugate pairs: Ω_{1}(*t*) = *k*(μ_{eff} (*t*))^{N}*/* and
${\Omega}_{1}^{*}(t)=k{({\mu}_{\text{eff}}{\epsilon}^{*}(t))}^{N}/\hslash $, where *k* is a proportionality constant having dimensions of (energy)^{(1−}^{N}^{)}, which in SI units would be joule^{(1−}^{N}^{)}. For the |1 → |2 transition, ω_{R} = ω_{2} − ω_{1} is the single-photon resonance frequency. We have assumed that the transient dipole moments of the individual intermediate virtual states in the multiphoton ladder result in an effective transition dipole moment,
${\mu}_{\text{eff}}^{N}$, which is a product of the individual *N* virtual state dipole moments, μ_{N} (i.e.,
${\mu}_{\text{eff}}^{N}={\prod}_{n}^{N}{\mu}_{n}$). This approximation is particularly valid when intermediate virtual level dynamics for multiphoton interaction can be neglected [6, 7].

Let us extend the two-level formalism to the multilevel situation involving IVR. In the conventional zeroth-order description of intramolecular dynamics, the system can be factored into an excited state that is radiatively coupled to the ground state, and nonradiatively to other bath states that are optically inactive (figure 1(a)). These ‘dark’ states have no radiative transition moment for going from the ground state as determined by optical selection rules. They can belong to very different vibrational modes in the same electronic state as the ‘bright’ state, or to different electronic manifolds. These dark states can be coupled to the bright state through anharmonic or vibronic couplings. Energy flows through these couplings and the apparent bright state population disappears. Equivalently, the oscillator strength is distributed among many eigenstates. The general multilevel Hamiltonian in the FM frame for an *N* -photon transition (*N* 1), expressed in the zero-order basis set, is

$$H=\hslash \left(\begin{array}{cccccc}|0\rangle & |1\rangle & |2\rangle & |3\rangle & |4\rangle & \dots \\ 0& {\Omega}_{1}(t)& 0& 0& 0& \dots \\ {\Omega}_{1}^{*}(t)& {\delta}_{1}(t)& {V}_{12}& {V}_{13}& {V}_{14}& \dots \\ 0& {V}_{12}& {\delta}_{2}(t)& {V}_{23}& {V}_{24}& \dots \\ 0& {V}_{13}& {V}_{23}& {\delta}_{3}(t)& {V}_{34}& \dots \\ 0& {V}_{14}& {V}_{24}& {V}_{34}& {\delta}_{4}(t)& \dots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \end{array}\right)$$

(2)

where Ω_{1}(*t*) (and its complex conjugate,
${\Omega}_{1}^{*}(t)$) is the transition matrix element expressed in Rabi frequency units, for the transition between the ground state |0 and the excited state |1. The background levels |2*,* |3*, . . .* are coupled to |1 through the matrix elements *V*_{12}, *V*_{23} etc. Both the Rabi frequency Ω_{1}(*t*) and the detuning frequency
$[{\delta}_{1,2,\dots}={\Delta}_{1,2,\dots}+N\dot{\phi}(t)]$ are time dependent (the time dependence is completely controlled by the experimenter). In general, the applied field would couple some of the dark states together, or would couple |1 to dark states, and thus the *V** _{i j}* terms would have both an intramolecular, time-independent component and a field-dependent component. As an alternative to equation (2), the excited states’ submatrix containing the bright state |1 and the bath states |2, |3

Another common situation with short pulses is a ladder excitation situation where the individual excited states undergo dephasing through a coupled energy structure with states |0, |1, |2 etc in the zero-order basis as shown in figure 1(b). Such a model of IVR is often referred to as a tier model and is in common use for polyatomic molecules and in most rovibrational states [8]; it can be represented by the following Hamiltonian:

$${H}^{\text{FM}}=\hslash \times \left(\begin{array}{ccccc}|0\rangle & |1\rangle & |2\rangle & |3\rangle & |4\rangle \\ 0& {\Omega}_{1}(t)& {\Omega}_{2}(t)& {\Omega}_{3}(t)& 0\\ {\Omega}_{1}^{*}(t)& {\delta}_{1}(t)& {V}_{12}& {V}_{13}& {V}_{14}\\ {\Omega}_{2}^{*}(t)& {V}_{12}& {\delta}_{2}(t)& {V}_{23}& {V}_{24}\\ {\Omega}_{3}^{*}(t)& {V}_{13}& {V}_{23}& {\delta}_{3}(t)& 0\\ 0& {V}_{14}& {V}_{24}& 0& {\delta}_{4}(t)\\ 0& {V}_{15}& {V}_{25}& 0& 0\\ 0& 0& {V}_{26}& {V}_{36}& 0\\ 0& 0& {V}_{27}& {V}_{37}& 0\\ 0& 0& 0& {V}_{38}& 0\\ 0& 0& 0& {V}_{39}& 0\end{array}\begin{array}{ccccc}|5\rangle & |6\rangle & |7\rangle & |8\rangle & |9\rangle \\ 0& 0& 0& 0& 0\\ {V}_{15}& 0& 0& 0& 0\\ {V}_{25}& {V}_{26}& {V}_{27}& 0& 0\\ 0& {V}_{36}& {V}_{37}& {V}_{38}& {V}_{39}\\ 0& 0& 0& 0& 0\\ {\delta}_{5}(t)& 0& 0& 0& 0\\ 0& {\delta}_{6}(t)& 0& 0& 0\\ 0& 0& {\delta}_{7}(t)& 0& 0\\ 0& 0& 0& {\delta}_{8}(t)& 0\\ 0& 0& 0& 0& {\delta}_{9}(t)\end{array}\right).$$

(3)

A short pulse laser can optically couple the states |1, |2, |3 etc to the ground state |0 with respective transition matrix elements expressed in Rabi units as Ω_{1}(*t*), Ω_{2}(*t*), Ω_{3}(*t*) etc and their corresponding complex conjugates. The background levels |4, |5, |6 etc are coupled to the optically excited states |1, |2 and |3 through the matrix elements *V*_{14}, *V*_{15}, *V*_{24} etc. Such a molecular system can become useful for realizing qubits [9] effectively if this large number of optically coupled states can be accessed simultaneously as has been the case for the atomic system of Rydberg states of caesium [10]. However, the difficulty in extending this scheme to the molecular system starts at the very first step of initializing the qubits due to high decoherence of the possible qubit states as in the gedanken system Hamiltonian presented in equation (3).

From experimental results on the fluorescence quantum beats in jet-cooled anthracene [11], the respective values (in GHz) of Δ_{1}_{,}_{2}_{,...,}_{4} are 3.23, 1.7, 7.57 and 3.7; and *V*_{12} = −0.28, *V*_{13} = − 4.24, *V*_{14} = −1.86, *V*_{23} = 0.29, *V*_{24} = 1.82, *V*_{34} = 0.94. When these values are incorporated in equation (2), we obtain the full zero-order Hamiltonian matrix that can simulate the experimental quantum beats (figure 2(a)) upon excitation with a transform-limited Gaussian pulse (i.e.,
$\dot{\phi}(t)=0$). Since |0 and |1 do not form a closed two-level system, considerable dephasing occurs during the second half of the Gaussian pulse. Thus, in a coupled multilevel system, simple unchirped pulses cannot be used to generate sequences of π*/*2 and π pulses, as in NMR. The dark states start contributing to the dressed states well before the pulse reaches its peak, and this results in redistribution of the population from the bright state (|1) into the dark states (figure 2(a)). The situation is worse when we use the tier model Hamiltonian in equation (3) as the redistribution occurs within the bright states through the participation of the dark states (figure 2(b)).

(a) A transform-limited Gaussian pulse interacts with a model anthracene molecule in a single-photon mode or in a multiphoton condition. (b) Due to strong coupling of the excited states, |1, |2 and |3 to |4, |5 **...**

A linear sweep in frequency of the laser pulse (i.e.,
$\dot{\phi}(t)=2{b}_{2}t$) can be generated by sweeping from far above resonance to far below resonance (blue to red sweeps), or its opposite. For a sufficiently slow frequency sweep, the irradiated system evolves with the applied sweep and the transitions are ‘adiabatic’. If this adiabatic process is faster than the characteristic relaxation time of the system, such a laser pulse leads to a smooth population inversion, i.e., an adiabatic rapid passage (ARP) [12]. If the frequency sweeps from below resonance to exact resonance with increasing power, and then remains constant, adiabatic *half* -passage occurs and photon locking is achieved with no sudden phase shift. However, even under adiabatic *full* passage conditions, figure 3 shows that there is enough slowing down of the ** E** field to result in photon locking over the FWHM of the pulse. These results hold even under certain multiphoton conditions where only an

A linearly swept Gaussian pulse can generate ‘photon locking’. The evolution of the dressed state character is unchanged while locking occurs, but as the pulse is turned off, the eigenenergy curves cross and the bright state population **...**

The quadratic chirp, i.e.,
$\dot{\phi}(t)=3{b}_{3}{t}^{2},$ is the most efficient in decoupling the bright and dark states as long as the Stark shifting of these states prevails at the peak of the pulse. As the pulse is turned off, the system smoothly returns to its original unperturbed condition (figure 4(a)). This would be a very practical approach to controlling the coupling of the states with realistic pulse shapes. In figure 4(b), we show that it is possible to initialize the qubits to equal superposition, as is required for further quantum operations, only with the help of decoherence controlling shaped pulses even when the intramolecular couplings between the states are strong (i.e., for large values of *V*_{14}, *V*_{15}, *V*_{24} etc). This is possible since the time dependence can be completely controlled by the experimenter when a shaped pulse is being used.

(a) The quadratic chirped Gaussian pulse is the most efficient in decoupling the bright and dark states during the pulse. The eigenenergy curves and the corresponding evolution of the dressed state character show that the entire process is highly adiabatic. **...**

The cubic term, i.e., $\dot{\phi}(t)=4{b}_{4}{t}^{3}$ behaves more like the linear term (figure 5). It also decouples the bright and dark states as long as the Stark shifting of these states prevails at the peak of the pulse. However, the oscillatory nature of the ‘photon locking’ shows that the higher-order terms in the Taylor series involve more rapid changes and there is failure to achieve perfect adiabatic conditions. As the pulse is turned off, there is an attempt to invert the bright state population, which quickly dephases, analogously to the linear chirp case. Thus, in an isolated two-level system that does not suffer from the population dephasing, the linear, cubic and all the higher odd-order terms of the Taylor series (equation (1)) yield inversion of population, while the even-order terms produce self-induced transparency.

For a multilevel system, the optical ac Stark shift induced by the frequency swept pulse moves the off-resonant coupled levels far from the resonant state leading to an effective decoupling. Under the perfectly adiabatic condition, pulses with the even terms in the Taylor series return the system to its unperturbed condition at the end. In fact, all higher-order odd terms behave in one identical fashion and the even terms behave in another identical fashion. It is only during the pulse that the Stark shifts of the dark states are decoupled and IVR restriction is possible in the multilevel situation. In the present calculations, we have used values equal to *b** _{n}* in equation (1) to bring out the effects of the higher-order terms in the series. In practice, since equation (1) represents a convergent series, only lower-order terms are important, and since all higher-order terms produce the same qualitative results as the lower-order terms, one needs to consider only up to the quadratic term.

The results are generic and illustrate that the intramolecular dephasing can be kept to a minimum for the duration of the ‘locking’ period under adiabatic conditions. Since the effect occurs under an adiabatic condition in all these frequency swept pulses, it is insensitive to the inhomogeneity in Rabi frequency. The simulations have been performed with laser pulses with Gaussian, hyperbolic secant and cosine-squared intensity profiles over a range of intensities. They show identical results of ‘locking’ the population in the chosen excited state of a multilevel system, conforming to the adiabatic argument that there is hardly any effect of the actual envelope profile. Promoting novel chemical reactions during photon locking, or generating several quantum computing accessible states (multiple qubits) can, thus, be accomplished within the pulse before dephasing randomizes the initially prepared state.

The author is supported through the Wellcome Trust International Senior Research Fellows programme of the Wellcome Trust Foundation (UK) and the Swarnajayanti Fellowship scheme under the Department of Science and Technology, Government of India. He also wishes to thank the Ministry of Information Technology, Government of India, for partial funding of the research results presented here.

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