Quantum interference between multiple excitation pathways can be used to cancel the couplings to the unwanted, nonradiative channels resulting in robustly controlling decoherence through adiabatic coherent control approaches. We propose a useful quantification of the two-level character in a multilevel system by considering the evolution of the coherent character in the quantum system as represented by the off-diagonal density matrix elements, which switches from real to imaginary as the excitation process changes from being resonant to completely adiabatic. Such counterintuitive results can be explained in terms of continuous population exchange in comparison to no population exchange under the adiabatic condition.

## I. INTRODUCTION

A major development in the implementation of quantum computing has been in addressing an atomic or molecular ensemble, such as the one which addresses nuclear spin in bulk molecular systems.^{1} Typical problems in experimental implementation of quantum computing exist in the complexity of the experimental setup and in scaling the number of qubits.^{2} Other implementation approaches are being pursued to find ways to circumvent the existing problems. Optical approaches could be attractive but for the rapid decoherence time scales involved. Minimizing decoherence is also an important challenge 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 contributor to decoherence even in isolated molecules. Restricting IVR by optical schemes^{3–6} can be an attractive route towards selective excitation in large molecular systems. Although attractive, most of the photon-mediated theoretical approaches towards restricting IVR (also called “photon locking”) use complicated pulse shapes that are yet to be demonstrated in the laboratory due to stringent requirements of intensity and precision.

Use of adiabatic evolution for optical quantum computation has recently become an attractive approach due to its inherent robustness.^{7–11} In this framework, logical implementation of quantum gates uses the language of ground states, spectral gaps, and Hamiltonians wherein a quantum gate represents a device which performs a unitary transformation on selected qubits within a fixed period of time. Thus, a computational procedure in the adiabatic quantum computation model is a continuous time evolution of a time-dependent Hamiltonian with limited energetic resources.

Our previous paper^{11} showed that an important aspect of the adiabatic quantum computation model lies in addressing an atomic or molecular ensemble and hence in robust implementation. These were based on our earlier experimental demonstration on selective population transfer in a two-level and a three-level system.^{12,13} However, the issue of coherence aspects of such control processes has not been addressed before and this is the main topic of our present paper since coherence issues from a critical aspect in the quantum computing paradigm. With the rise in interest in quantum-control spectroscopy,^{14} it is an important interdisciplinary development to be pursued. Previous efforts^{15–19} in coherent control areas have explored coherence issues by exploring the population evolution aspects in the presence of decoherences. We show here that a study of the evolution of the off-diagonal elements in the density matrix is more critical and complete in providing insights which we explore for the first time. We include the case of fast uniform electronic relaxation which extends these results to cases beyond the pure states and justifies the density matrix formalism as a distinct advantage.

## II. FORMALISM

An ultrafast laser pulse can be represented as a coherent superposition of many monochromatic light waves within a range of frequencies that is inversely proportional to the duration of the pulse. Thus, for instance, a $30fs$ Gaussian pulse at $800nm$ that is available from a commercial laser has a spectrum as broad as $31nm$. Possibilities of manipulating such an ultrafast coherent bandwidth are nontrivial as it lasts for such an ultrashort duration wherein no modulators work. A creative solution to the problem of slow modulators is the indirect pulse shaping in the frequency domain. Typically, the time domain output pulse $Eout(t)$ from an input pulse $Ein(t)$ results as a convolution with a time domain filter characterized by a time response function $g(t)$, such that $Eout(t)=Ein(t)\u2297g(t)$. In the frequency domain, the filter is characterized by its frequency response $G(\omega )$, i.e., $Eout(\omega )=Ein(\omega )\xd7G(\omega )$, where $Ein(t)$, $Eout(t)$, and $g(t)$ and $Ein(\omega )$, $Eout(\omega )$, and $G(\omega )$ are Fourier transform pairs.^{19} Given a delta function input pulse, the input spectrum $Ein(\omega )$ is unity while the output is equal to the frequency response of the filter, and due to Fourier transform relationship, generation of a desired output wave form can be accomplished by implementing a filter with the required frequency response.

Another attractive approach is 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 and we discuss further on that now for various physical models.

### A. Two-level model

The simplest model describing a molecular system is an isolated two-level system or an ensemble without relaxation or inhomogeneities. This simple model often turns out to be a very practical model for most systems interacting with ultrashort laser pulses as the magnitude of the relaxation processes are immensely large as compared to the light-matter interaction time. Let us apply a linearly polarized laser pulse of the form $E(t)=\epsilon (t)ei[\omega t+\varphi (t)]$ to a simple two-level system with $\u22230\u27e9\u2192\u22231\u27e9$ transition, where ∣0⟩ and ∣1⟩ represent the ground and excited eigenlevels, respectively, of the field-free Hamiltonian (Fig. 1).

The laser carrier frequency or the center frequency for pulsed lasers is $\omega $. We have $\epsilon (t)$ and $\varphi (t)$ as the instantaneous amplitude and phase. We can define the rate of change of instantaneous phase $\varphi \u0307(t)$ as the frequency sweep. If we expand the instantaneous phase function of $E(t)$ as a Taylor series with constants $bn$, we have

^{20}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. Establishing this generalization enables us to treat all possible chirped pulse cases by exploring the effects of each of the terms in Eq. (1) initially for a simple two-level system and then extend it to the multilevel situation for a model six-level system, similar to the five-level model of the anthracene molecule, which has been previously investigated with complicated shaped pulses.

^{7}

We use a density matrix approach by numerically integrating the Liouville equation, $d\rho (t)\u2215dt=(i\u2215\u210f)[\rho (t),HFM(t)]$ for a Hamiltonian in the rotating frequency modulated (FM) frame of reference. $\rho (t)$ is a $2\xd72$ density matrix whose diagonal elements represent populations in the ground and excited states and off-diagonal elements represent coherent superposition of states. The Hamiltonian for the simple case of a two-level system under the effect of an applied laser field can be written in the FM frame for $N$-photon transition^{21} as

The time derivative of the phase function, $\varphi (t)$, appears as an additional resonance offset over and above the time-independent detuning $\Delta =\omega R\u2212N\omega $, while the direction of the field in the orthogonal plane remains fixed. We define the multiphoton Rabi frequencies as complex conjugate pairs, $\Omega 1(t)=k(\mu eff\epsilon (t))N\u2215\u210f$ and $\Omega 1*(t)=k(\mu eff\epsilon *(t))N\u2215\u210f$, where $k$ is a proportionality constant having dimensions of $(energy)(1\u2212N)$, which in SI units would be $J(1\u2212N)$. For the $\u22231\u27e9\u2192\u22232\u27e9$ transition, $\omega R=\omega 2\u2212\omega 1$ is the single-photon resonance frequency. We have assumed that the transient dipole moment of the individual intermediate virtual states in the multiphoton ladder results in an effective transition dipole moment $\mu effN$, which is a product of the individual $N$ virtual-state dipole moments $\mu N$ (i.e., $\mu effN=\Pi nN\mu n$). This approximation is particularly valid when intermediate virtual-level dynamics for multiphoton interaction can be neglected.^{22} Such approaches have helped us to look at the population dynamics. Here, we also use the off-diagonal elements of the density matrix evolution and the underlying dressed states to discuss the coherence aspects of the associated dynamics.

### B. Multilevel IVR model

We now extend the two-level formalism into a multilevel situation involving IVR. In the conventional zeroth order description of intramolecular dynamics,^{23} 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 (Fig. 2).

These “dark” states have no radiative transition moment from the ground state as determined by optical selection rules.^{24} They can belong to very different vibrational modes in the same electronic state as the “bright” state, or can belong 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

where $\Omega 1(t)$ [and its complex conjugate pair, $\Omega 1*(t)$] is the transition matrix element expressed in Rabi frequency units, between the ground state ∣0⟩ and the excited state ∣1⟩. The background levels ∣2⟩,∣3⟩,… are coupled to ∣1⟩ through the matrix elements $V12$, $V13$, etc. Both Rabi frequency $\Omega 1(t)$ and detuning frequency $[\delta 1,2,\u2026=\Delta 1,2,\u2026+N\varphi \u0307(t)]$ 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 $Vij$ terms would have both an intramolecular, time-independent component and a field-dependent component. As an alternative to Eq. (2), the excited states’ submatrix containing the bright state ∣1⟩ and the bath states ∣2⟩,∣3⟩,… can be diagonalized to give the eigenstate representation containing a set of $\Delta i\u2032$ as diagonal elements and corresponding $\Omega i\u2032$ as off-diagonal elements. The eigenvalues of such a time-dependent Hamiltonian representation is often referred to as the dressed states of the system. Such a representation corresponds closely to what is observed in conventional absorption spectroscopy. As long as the intensity of the field is very low $(\u2223\Omega i\u2032\u2223\u2aa1\Delta i\u2032)$, the oscillator strength from the ground state (and hence the intensity of the transition, which is proportional to $\u2223\Omega i\u2032\u22232$, is distributed over the eigenstates, and the spectrum mirrors the distribution of the dipole moment. On the other hand, a pulsed excitation creates a coherent superposition of the eigenstates within the pulse bandwidth. Physically, in fact, the presence of the dark states has been a key to the loss of selectivity of excitation to a specified bright state. Interestingly, such a process essentially is a Hadamard operation in quantum computing language as this enables us to produce equal superposition between the ground and excited states which form the qubits.^{11} 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 Fig. 2(b). Such a model of IVR is often referred to as the tier model and is common in polyatomic molecules and in most rovibrational states^{25} and can be represented by the following Hamiltonian:

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 $\Omega 1(t)$, $\Omega 2(t)$, $\Omega 3(t)$, 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 $V14$, $V15$, $V24$, etc. Such a molecular system can become useful for realizing qubits effectively if these large number of optically coupled states can be accessed simultaneously, as has been demonstrated in the model Rydberg atom case of cesium.^{26} 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 Eq. (3). Thus an adiabatic scheme is necessary. Such adiabatic schemes have been extremely successful with the stimulated raman adiabatic passage (STIRAP) (Ref. 27) schemes also. Our density matrix approach also elucidates these aspects.

## III. COHERENCE CORRELATIONS

Effects of the pulse profile and the frequency sweep on the population are usually interpreted in the framework of adiabatic rapid passage (ARP).^{12,19} In ARP, the barriers to total population inversion in real systems are removed by either modulating the laser pulse so that its spectrum “sweeps” through the distribution of resonances of the atoms or molecules in the ensemble or by perturbing the atomic or molecular energy levels via an electric field (i.e., a Stark shift) so that they shift through resonance with the irradiating laser field. If this frequency sweep is sufficiently slow, the transition is said to be “adiabatic.” If the rate at which the frequency of the laser pulse changes with time is slow compared with the precession rate of $\mu $ (and fast compared with the relaxation rates of the material system), $\mu $ is said to “adiabatically follow” the pseudofield vector, thereby giving rise to complete population transfer. Herein, with the help of the models discussed in the earlier sections, we probe the real part, the imaginary part, the phase angle, and the absolute values of the off-diagonal elements of the density matrix model propagations and compare it with the expected population dynamics that our model has been efficiently successful in corresponding to the experimental results.^{7,11,19,20}

We first discuss the simplest two-level model as depicted in Fig. 1 to understand the behavior of the off-diagonal elements of the density matrix. We use $31nm$ bandwidth equivalent transform-limited (unchirped) pulses with $30fs$ full width at half maximum (FWHM) versus $1ps$ FWHM chirped pulses. We show that, for an unchirped pulse excitation, only the imaginary part of the off-diagonal elements has nonzero values [Fig. 3(a)]. On the other hand, in the adiabatic case of either complete inversion (ARP) or coherent superposition (half-ARP), the evolution is smoothly characterized by the real part of the off-diagonal elements [Figs. 3(b) and 3(c)]. The maximum possible absolute value of the off-diagonal elements of the density matrix is always 0.5 or less and while it is possible to have similar absolute values for both the chirped versus unchirped cases, their complex character are in quadrature [Fig. 3(a) versus Figs. 3(b) and 3(c)]. The corresponding dressed state character is always an equal mix of the ground and excited states of the two-level system with a raw value of 0.5 throughout its time evolution, while in case of ARP, it follows one or the other state and its value remains as either 1 or zero throughout the time evolution [Fig. 3(d)]. Thus, we find that for the classic case of the “half-pi” pulse versus the adiabatic half-passage scenario, the absolute value of the final coherence is the same for the two cases, namely, $1\u22152$, i.e., there is complete coherence between the two concerned states. This difference can be correlated to the fact that the precession angle cone under the adiabatic condition remains very small as it undergoes the half-passage while the precession cone for the half-pi pulse essentially is along the equator in the Bloch or Feynman vector picture representing the essential differences in the two coherent phases. We are, therefore, able to correlate the phase of the complex coherence term in the density matrix to the precession angle cone. Essentially, all the real part of the coherence term (off-diagonal element term of the density matrix) corresponds to the refractive index while the imaginary part is the absorption coefficient of the two coupled states. When it is only the imaginary part of the off-diagonal element terms, it means that we are having real absorption and emission between the two coupled states which correlates to the Rabi oscillations where there is perfect mixing of the states and the dressed states have diagonal density matrix elements of 0.5 exactly. When we talk about the adiabatic passage condition, the dressed states are well separated (0 and 1) and there is no absorption possible between the states. The interaction is all refractive and, hence, is represented by the real term in the off-diagonal element.

With this correlation let us now look into the density matrix characteristics of the multilevel system. In all the different IVR models discussed in the previous section, essentially, the population of the initial bright state gets redistributed to the dark states as shown in Fig. 4(a). Under adiabatic conditions it is possible to restrict this IVR process to either “photon lock” the ground and the bright states with no participation of the dark states [Fig. 4(b)] or to generate an equally populated bright state distribution which can be represented as a Hadamard transform [Fig. 4(c)]. The population evolution of the multilevel system is dictated by the dressed state character, their corresponding eigenenergy evolution, and their crossings, as shown in Fig. 4. At low powers where almost no population transfer occur [Fig. 4(d)], there is minimal interaction of the eigenenergies corresponding to the dark states although dressed state character switching between DS1, DS2, DS3, DS4, DS5, and DS6 corresponding to the six level model is already evident [Figs. 5(a) and 5(b)]. As soon as the power is high enough to induce population transfer, the dressed state interaction increases and their characters undergo multiple switching and several eigenenergies interact [Fig. 4(a), inset] which is evidenced in terms of population redistribution and IVR [Fig. 4(a)]. However, under the adiabatic condition, the dressed states due to the ground and the bright states never interact with the dressed states corresponding to the dark states [insets in Figs. 4(b) and 4(c)] and as such there is no population leakage to the dark states. The eigenenergy distribution also reaffirms these results.

Now let us probe into the off-diagonal elements of the density matrix for the six-level IVR model which we discuss in detail in Fig. 5. We focus on the adiabatically decoupled model with quadratic chirped Gaussian pulse excitation which has the simplest dressed state distribution and noncrossing eigenenergies [insets in Figs. 4(b) and 4(c)], where we find in our two-level generalization that the real part has most of the contribution to the off-diagonal elements of density matrix under the adiabatic condition holds (Fig. 5).

While the two-level characteristic of the multilevel system is generated due to half-ARP chirped pulses and the contribution of the dark states are minimized, the major contribution of the real part in the values of the off-diagonal elements of density matrix has been shown in Fig. 5. This further reinforces the argument that the approach to control IVR through half-ARP essentially succeeds in creating a coherent superposition of a two-level system from a multilevel system. Using the same arguments, the Hadamard gate generating adiabatic pulse [Fig. 4(b)] manages to generate identical set of coherently coupled two-level systems.

However, the situation is not as simple in case of the unchirped pulses, as shown in Fig. 6. At low enough power, wherein the population transfer is minimal, some evidence of the more imaginary part characteristic of the off-diagonal elements of density matrix can still be argued [Figs. 6(a) and 6(b)], since it is reminiscent of the two-level system interacting with unchirped pulses. As the power increases, the two-level character is further reduced as the coupling to the dark states overwhelms and there is no preference of the real versus the imaginary part of the off-diagonal elements of density matrix, as shown in Figs. 6(d) and 6(f), indicating efficient IVR process.

Thus, the imaginary component of the off-diagonal elements of density matrix can be treated as a measure of the two-level character in a multilevel system for unchirped pulse excitation and the real component of the off-diagonal elements of density matrix can be treated as a measure of the two-level character in a multilevel system for chirped pulse excitation.

As a part of our use of the density matrix formulation we have also explored the use of damping terms that allow us to follow situations beyond the pure-state regime. We find that as the pure-state character decreases with the increase in the contribution of the damping terms, coherent control becomes more difficult at lower energies of the laser field (Rabi frequencies), irrespective of the choice of chirp and intensity [Figs. 7(a)–7(d)]. As the pure-state character decreases, it becomes increasingly difficult to attain an effective two-level character with lower driving fields and chirps. At higher powers and frequency chirps, however, it is possible to reach the adiabatic conditions and an effective two-level character [Fig. 7(e)] for moderate damping conditions. However, if there is uniform dampening term in the last tier which mimics the fast quenching by intermolecular electronic relaxation, bath states, etc., it will eventually leak out all the population and much more effectively when the adiabatic condition is attained between the ground state and the bright state [Fig. 7(f)].

## IV. DISCUSSION

Quite a bit of effort has gone into the study of pure states of multilevel systems for recent developments in coherent control^{28–31} and for quantum computing.^{9,11,32} However, there has been almost no effort to explore the dynamics as one deviates from the pure-state picture though there have been previous attempts that utilized the dressed state formalism.^{20} As discussed here, we have elucidated these effects particularly taking advantage of the off-diagonal elements of the density matrix formalism and have shown how the deviation from the two-level character due to the influence of the damping terms compromises the effectiveness of the shaped pulses for either quantum computational applications or coherent control issues. It is, thus, important to look into the density matrix formalism and effectively utilize the rich information contained in the off-diagonal elements so that the imperfections due to the deviation from the pure states can be compensated with careful choice of laser pulse shaping parameters.

The present work has also indicated that a controlled two state character in a multilevel system can be attained even in the presence of fast quenching by intermolecular electronic relaxation, bath states, etc. over a limited time window. However, under the adiabatic condition as explored in this paper, this window is somewhat controllable in the hands of the experimenter with the choice of frequency sweep and intensity. A typical nonadiabatic situation would be completely limited by the time scales offered by the system in question.^{31}

## V. CONCLUSIONS

Evolution of the off-diagonal elements of density matrix can be an important parameter for determining the coherence prevailing in the system. The change in the nature of the off-diagonal elements of density matrix between unchirped and adiabatic chirped pulse case is an important indication of the change in the mechanism of the coherence process underlying the evolution of the system. For a multilevel system, it is only the adiabatic pulses that can generate and maintain coherence between states as is demonstrated here through the evolution of the off-diagonal elements of density matrix and as such forms an attractive approach for both robust quantum computing processing as well as coherent control for multilevel systems.

## ACKNOWLEDGMENTS

The author thanks the funding support of the Ministry of Communication and Information Technology, the Swarnajayanti Fellow scheme under the Dept. of Science and Technology, Govt. of India, and the Wellcome Trust Senior Research Fellowship program of the Wellcome Trust Foundation (UK) for the work presented here.