Open Access Article

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A.
García-Vela

Instituto de Física Fundamental, Consejo Superior de Investigaciones Científicas, Serrano 123, 28006 Madrid, Spain. E-mail: garciavela@iff.csic.es

Received
23rd January 2020
, Accepted 5th June 2020

First published on 5th June 2020

It is demonstrated both numerically and mathematically that the dynamical behavior of an isolated resonance state, which comprises the resonance decay lifetime and the asymptotic fragment state distribution produced upon resonance decay, can be extensively controlled by means of quantum interference induced by a laser field in the weak-field regime. The control scheme applied is designed to induce interference between amplitudes excited at two different energies of the resonance line shape, namely the resonance energy and an additional energy. This scheme exploits the resonance property of possessing a nonzero energy width, which makes it possible that a resonance state may interfere with itself, and thus allows interference between the amplitudes excited at the two energies of the resonance width. The application of this scheme opens the possibility of a universal control of both the duration and the fragment product distribution outcome of any resonance-mediated molecular process.

A variety of molecular processes (among them photodissociation and reactive and non-reactive collisional processes) are governed by resonance states (either isolated or overlapping ones).^{20–34} A strategy used to control those processes under weak-field conditions has been to modify the decay behaviour of the resonances involved by inducing quantum interference between them. In this sense, inducing interference between overlapping resonances excited within a superposition state has been successfully used to delay significantly radiationless transitions and intramolecular vibrational redistribution processes in different molecules.^{21,23} Vibrational cooling was achieved by inducing resonance coalescence with a laser field.^{26} Also in a framework of overlapping resonances, it has been shown that by means of interference between the resonances, it is possible to strongly enhance the lifetime of individual resonances within a superposition,^{14,15,35,36} as well as to modify the fragment state distribution produced upon resonance decay.^{30,36,37} Control over the resonance decay lifetime and over the fragment distribution provides control over both, the duration and the outcome, respectively, of the resonance-mediated molecular process of interest. In this latter case the control scheme applied was a simple but efficient one using a laser field that consisted of two pulses delayed in time, each pulse exciting a different energy at which several resonances overlap. Excitation of the two different energies is what induces interference between the overlapping resonances.

The possibility of modifying a resonance decay behaviour through interference between overlapping resonances has been thus widely demonstrated, and it allows for control of resonance-mediated molecular processes where such overlapping resonances are present. In addition to the processes mediated by overlapping resonances there are, however, other molecular processes mediated by isolated resonance states. The question thus arises whether it is possible to design similar control schemes that can be applied to these isolated-resonance processes. If so, control of resonance-mediated molecular processes would become universal, for any molecular system featuring either isolated or overlapping resonances. To the best of the author knowledge, such a control over isolated resonances has not yet been demonstrated.

Resonance states are intriguing quantum objects with very interesting properties. A well-known property of a resonance state is that it possesses a nonzero energy width. Such a property makes it possible that an isolated resonance state can interfere with itself, which can be exploited in order to modify its decay behaviour, similar to that performed with overlapping resonances. In this work it is shown numerically and demonstrated formally that interference of an isolated resonance state with itself can be induced by applying a laser field. By controlling this interference, both the resonance lifetime and the asymptotic fragment state distribution produced upon resonance decay can be modified, allowing the control of any (isolated) resonance-mediated molecular process of interest.

Laser field excitation of Ne–Br_{2}(B,v′,n′) and the subsequent predissociation was simulated with a full three-dimensional wave packet method described in detail elsewhere.^{14,40} A Chebychev propagator, which is both accurate and efficient for the present purposes, was used. In order to assess the quality of the model applied, it is noted that the lifetime calculated with the present theoretical model for the decay of the -Br_{2}(B,v′ = 16) ground intermolecular resonance has been found to be 69 ps,^{42} while the corresponding lifetime estimated experimentally is 68 ± 3 ps.^{39} This good agreement with the experimental lifetime implies that both the three-dimensional wave packet method and the potential surfaces used in the present simulations are realistic enough in order to describe this resonance decay process.

In the simulations the wave packet is represented in Jacobian coordinates (R,r,θ), where R is the distance between the Ne atom and the Br_{2} center of mass, r is the Br–Br internuclear distance, and θ is the angle between the vectors associated with R and r. In this representation the rovibrational eigenstates associated with the Br_{2}(B,v,j) fragment are χ^{(j)}_{v}(r)P_{j}(cosθ), where χ^{(j)}_{v}(r) is the vibrational eigenfunctions of Br_{2}(B) with associated energies E_{v,j} and P_{j}(cosθ) is a Legendre polynomial, with v and j being the Br_{2} vibrational and rotational quantum numbers, respectively. The energy-resolved Br_{2}(B,v,j) fragment state population is computed along time by projecting out the wave packet onto the corresponding states

(1) |

k_{v,j} = [2μ(E − E_{v,j})]^{1/2}, | (2) |

(3) |

The goal is to modify the decay behaviour of the Ne–Br_{2}(B,v′ = 21,n′ = 4) resonance, that is, its decay lifetime and the energy-resolved (at the energy E = −38.90 cm^{−1}) asymptotic Br_{2}(B,v_{f} < v′) fragment vibrational state distribution (v_{f} = v′ − 1, v′ − 2, v′ − 3,…) produced upon resonance decay. To this purpose, the control strategy adopted is similar to that used with overlapping resonances,^{15,35–37} namely to excite two different energies E_{a} and E_{b} of the spectrum of Fig. 1 in order to induce interference between them. These two energies are excited by two pulses delayed in time. As previously shown, by varying the delay time between the pulses the interference is controlled, and the enhancement of the resonance lifetime can be optimized.^{15,35,36} Modification of the asymptotic fragment distribution only requires a sufficiently long delay time between the pulses, without optimization.

The two energies chosen to induce interference are the resonance energy E_{a} = −38.90 cm^{−1} and E_{b} = −38.50 cm^{−1}, both indicated in Fig. 1. The control scheme applies a pump laser field that combines three Gaussian-shaped pulses with the form

ε_{3}(t) = A_{1}e^{−(t−t1)2/2σ2}cos[ω_{1}(t − t_{1}) + ϕ_{1}] + A_{2}e^{−(t−t2)2/2σ2}cos[ω_{2}(t − t_{2}) + ϕ_{2}] + A_{3}e^{−(t−t3)2/2σ3}cos[ω_{2}(t − t_{3}) + ϕ_{3}], | (4) |

Regarding the specific parameters used in ε_{3}(t) in the simulations, for simplicity it is assumed that ϕ_{1} = ϕ_{2} = ϕ_{3} = 0. The amplitudes of the pulses are A_{1} = A_{2} = 1.0 × 10^{−6} a.u., and A_{3} = 3A_{1}, which correspond to a maximum pulse intensity of about 3.5 × 10^{4} W cm^{−2} and 3.2 × 10^{5} W cm^{−2}, respectively, within the weak-field regime. In practice t_{1} is fixed at a value t_{1} = 0, and t_{2} and t_{3} are varied. Thus, the delay time between the pulses becomes Δt_{12} = t_{2} − t_{1} = t_{2} and Δt_{13} = t_{3} − t_{1} = t_{3}. The temporal width of all the pulses (related to σ) is the same, and corresponds to a full width at half maximum of FWHM = 200 ps. The spectral profiles of these pulses are shown in Fig. 1 for the two energies E_{a} and E_{b}. They are rather narrow and do not overlap in the energy domain. In addition to the simulations applying the ε_{3}(t) field, simulations using a single-pulse field ε_{1}(t) = A_{1}e^{−(t−t1)2/2σ2}cos[ω_{1}(t − t_{1}) + ϕ_{1}] to excite only the E_{a} resonance energy were carried out in order to obtain the resonance lifetime and the Br_{2}(B,v_{f} < v′) distribution in the absence of interference. The same values given above were used for the parameters A_{1}, t_{1}, σ, ω_{1}, and ϕ_{1}.

Control of the resonance lifetime is achieved by applying the two first pulses of ε_{3}(t) with different delay times Δt_{12} in the range −500 ps ≤ Δt_{12} ≤ 500 ps. For each value of Δt_{12} the resonance survival probability I_{n′=4}(t) = |〈ψ_{n′=4}(t)|Φ(t)〉|^{2} is computed, where ψ_{n′=4}(t) is the resonance wave function and Φ(t) is the wave packet created by the two first pulses of ε_{3}(t). Now the corresponding lifetime, τ, is obtained by fitting I_{n′=4}(t) to the function^{38}

(5) |

The I_{n′=4}(t) curves obtained for several values of Δt_{12} are displayed in Fig. 2(a), along with the survival probability computed when only the resonance energy E_{a} is excited with the single-pulse field ε_{1}(t). In Fig. 2(b) a typical fit obtained using eqn (5) is also shown. As expected, the single-energy I_{n′=4}(t) curve displays no structure, since interference is not possible. The lifetime obtained for this curve with eqn (5) is τ_{sing} = 16.0 ps. The two-pulse curves, however, display a pronounced structure of peaks or undulations, which are the signature of quantum interference between the amplitudes excited to the energies E_{a} and E_{b}. Actually the different peaks of each curve are separated by the same constant amount of time, which is proportional to the inverse of the energy separation E_{b} − E_{a}, as expected from an interference event.

Fig. 2 (a) Resonance survival probability I_{n′=4}(t) computed when ε_{3}(t) is applied with different delay times Δt_{12}, from Δt_{12} = −150 ps to Δt_{12} = 200 ps, between the pulses exciting the E_{a} and E_{b} energies. The corresponding I_{n′=4}(t) curve obtained when the single-pulse field ε_{1}(t) is applied to excite only the E_{a} energy is also displayed. (b) I_{n′=4}(t) curve calculated for the delay time Δt_{12} = 40 ps (red line), along with the corresponding fit (green line) obtained by using eqn (5). |

Interference between the amplitudes at E_{a} and E_{b} requires their simultaneous excitation, and therefore some temporal overlap between the two first pulses of ε_{3}(t).^{15,36} Thus, the basis of the control scheme applied is the variation of Δt_{12}, because it modifies the temporal overlap between the two first pulses of ε_{3}(t). Varying this overlap implies the variation of the relative amplitudes that are excited to both E_{a} and E_{b}, and therefore their mechanism of interference. When interference between the amplitudes at E_{a} and E_{b} is modified, the shape of the I_{n′=4}(t) curve changes as well, as shown in Fig. 2(a), which leads to the variation of the associated resonance lifetime.

By applying eqn (5) the resonance lifetime is calculated for the different values of Δt_{12}, and the results are plotted in Fig. 3. The figure shows that for very large delay times |Δt_{12}| = 500 ps the lifetime found is τ = 16.0 ps, the same value obtained when the single-pulse field ε_{1}(t) is applied. For large |Δt_{12}| there is no temporal overlap between the two first pulses of ε_{3}(t), and thus no interference between E_{a} and E_{b} is possible, leading to the same τ obtained with ε_{1}(t). However, when |Δt_{12}| decreases, the overlap between the pulses becomes nonzero and interference between the amplitudes at E_{a} and E_{b} takes place. The result is a gradual enhancement of the resonance lifetime, which increases from τ = 16.0 ps to τ = 31.0 ps at Δt_{12} = 110 ps, nearly twice the value obtained in the absence of interference.

In a previous study^{35} the variation of the resonance lifetime was analyzed by changing both the laser fields and the delay times between the pulses, in a framework of overlapping resonances, and such analysis provided very useful insight about how the interference mechanism works. The shape of the curve of Fig. 3 is similar to those found for overlapping resonances,^{15,35} indicating that the mechanism of interference operates similarly in the lifetime enhancement. In this sense, the value of Δt_{12} at which maximum lifetime enhancement is achieved is determined by the maximization of the intensity of interference between the amplitudes of E_{a} and E_{b}.^{35} And the achievement of maximum interference intensity depends on reaching enough temporal overlap between the two pulses (albeit not necessarily the maximum overlap, occurring at Δt_{12} = 0), but such that the mechanism of interference between the amplitudes excited at E_{a} and E_{b} is optimized. Such optimization of the interference is what determines the maximum enhancement of the resonance lifetime achieved (τ = 31.0 ps) and the value of Δt_{12} at which it takes place (Δt_{12} = 110 ps in this case).^{35,36} It is noted, however, that complete optimization of the laser field (involving going beyond just varying Δt_{12}, and changing the Gaussian shape of the pulses) in order to fully maximize the resonance lifetime enhancement has not been pursued, and thus the enhancement currently achieved could be increased further.

The next goal is to modify the other resonance properties that determine the outcome of a resonance-mediated molecular process, namely the energy-resolved asymptotic fragment distribution. In the present case it corresponds to the Br_{2}(B,v_{f} < v′) fragment distribution produced upon predissociation at the resonance energy E_{a} = −38.9 cm^{−1}. To this end, the third pulse of ε_{3}(t) is used to excite the E_{b} energy, similar to that performed previously with the second pulse. The difference now is that the first and the third pulse will not overlap in time, and the delay time Δt_{13} between them will be much longer than Δt_{12}. The reason for a longer Δt_{13} is to allow enough time for the first amplitude excited at E_{a} to decay completely and to reach the asymptotic regime of the fragment distribution produced. Thus, by exciting amplitude to E_{b}, quantum interference is induced between this and the asymptotic decayed amplitude initially excited to E_{a} by the first pulse of ε_{3}(t).^{36,37} In the present simulations a long enough delay time Δt_{13} = 1500 ps has been chosen, and in Fig. 4(a) the temporal profile of the ε_{3}(t) field applied is displayed, with Δt_{12} = 110 ps, A_{1} = A_{2}, and A_{3} = 3A_{1}.

Fig. 4 (a) Temporal profile (red line) along with its envelope (green line) of the ε_{3}(t) laser field applied to excite the Ne–Br_{2}(B,v′ = 21, n′ = 4) resonance, with Δt_{12} = 110 ps in this case, and Δt_{13} = 1500 ps. (b) Energy-resolved Br_{2}(B,v_{f}) fragment vibrational populations in the v_{f} = v′ − 1,…, v′ − 4 final vibrational state produced upon predissociation of Ne–Br_{2}(B,v′ = 21, n′ = 4), associated with the resonance energy E_{a} = −38.9 cm^{−1}, when the ε_{3}(t) field of Fig. 4(a) is applied. All the vibrational v_{f} populations are labeled in the figure except v_{f} = v′ − 4, which is very small. |

In Fig. 4(b) the energy-resolved Br_{2}(B,v_{f}) fragment vibrational populations in the v_{f} = v′ − 1,…,v′ − 4 final vibrational state associated with the E_{a} resonance energy are shown. The different populations display a clear modification in the asymptotic time regime when the third pulse of ε_{3}(t) is applied to excite the E_{b} energy. Such a modification manifests itself in the form of undulations that reflect the interference taking place between the amplitudes excited at both energies. This interference occurs between the asymptotic amplitude at the E_{a} energy and the amplitude excited at the E_{b} energy, which temporarily populates the continuum fragment states at E_{a}.^{36,37} The interference effect is increasingly more intense as the v_{f} population is larger in magnitude, because the larger is the asymptotic amplitude the more intense will be the interference terms. In this sense it is noted that the second pulse of ε_{3}(t) also causes an interference effect on the vibrational populations of Fig. 4(b) around Δt_{12} = 110 ps, although being much weaker since A_{2} = A_{3}/3. Once the third pulse is over and the amplitude excited at E_{b} has decayed completely, the asymptotic populations converge back to the values previous to the application of the third pulse. The implication is that the modifications caused by interference in the fragment distribution cannot be observed asymptotically in the same vibronic state (Br_{2}(B,v_{f}) in our case) where they are produced. This, however, does not prevent an effective control of the fragment distribution and its observation, if the fragments are detected or moved to other vibronic states of interest (applying a further laser pulse) while the interference effect takes place.

The above results demonstrate numerically that by inducing quantum interference between amplitudes at two different energies by applying a simple laser field like ε_{3}(t) of eqn (4), extensive control over the decay lifetime and the asymptotic fragment distribution produced upon decay of an isolated resonance can be achieved in the weak-field regime. In the following the formal theory underlying those results and the present control scheme is developed.

Let Ĥ be the total Hamiltonian of a general molecular system that supports isolated resonances (Ne–Br_{2}(B,v′) would be an example of such a general system). Following the discussion on the decay of a resonance state of Cohen-Tannoudji et al.,^{43} we can write Ĥ as Ĥ = Ĥ_{0} + W, where Ĥ_{0} is a zeroth-order Hamiltonian and W is a coupling. The spectrum of Ĥ_{0} consists of a set of discrete bound states χ_{i} (located in the interaction region) with associated energies E_{i}, and a set of continuum states φ_{E,m} (associated with the product fragments in the asymptotic region) with associated energies E, and m being a global label for the fragment internal states. When W = 0 the χ_{i} states are true bound states, but when W ≠ 0, χ_{i} become resonances ψ_{i} that decay to the continuum of φ_{E,m} states. These states fulfill the orthogonality relationship

〈χ_{i}|χ_{i}〉 = δ_{ij},〈φ_{E′,m′}|φ_{E,m}〉 = δ_{m′m}δ(E′ − E),〈χ_{i}|φ_{E,m}〉 = 0, | (6) |

(7) |

Let us now focus on one of the isolated resonances of our general molecular system. By applying a single-pulse field like ε_{1}(t) to excite the resonance energy E_{a}, a wave packet ξ_{Ea}(t) is created

(8) |

Φ(t) = ξ_{Ea}(t) + ξ_{Eb}(t), | (9) |

(10) |

A resonance wave function ψ_{i} can also be expressed in terms of the stationary eigenstates ψ_{E} as

(11) |

Thus, the resonance survival probability I_{i}(t) is

(12) |

The term of eqn (12) is the survival probability that would be obtained if a single resonance energy E_{a} was excited with the single-pulse field E_{1}(t) (i.e., the plain curve of Fig. 2(a), with associated lifetime τ_{sing} = 16.0 ps). The three additional terms of eqn (12) arise from the excitation of amplitude at energy E_{b} by the second pulse of ε_{3}(t), and its interference with the resonance amplitude excited at E_{a}. Such terms associated with the interference are the ones that cause the undulations of the I_{n′=4}(t) curves of Fig. 2(a). As mentioned above, the requirement for these terms to be nonzero is that the amplitudes and must be generated simultaneously at E_{a} and E_{b}, respectively, which implies the temporal overlap to some extent of the two pulses exciting those energies. When the delay time Δt_{12} between the pulses is varied, the range of temporal overlap between them is modified, changing the relative and amplitudes excited simultaneously. This causes a variation of the interference terms of eqn (12) in a controlled manner, which leads to a change in the shape of I_{i}(t) (see Fig. 2(a)), and thus also in the associated lifetime (as shown in Fig. 3). In brief, interference induces a new decay mechanism with a longer lifetime that replaces the intrinsic decay mechanism, in which a transfer of amplitude back and forth between the two energies takes place.^{44}Eqn (11) reflects the fact that a resonance state ψ_{i} possesses a nonzero energy width. This finite width is what makes possible interference of the resonance with itself, when a wave packet Φ(t) containing different energies within this width (essentially E_{a} and E_{b}) is created with the two pulses of the field. This is the key aspect of the present weak-field control scheme of the isolated resonance behavior.

Regarding the fragment state distribution produced upon resonance decay, the asymptotic probability associated with the fragment state φ_{E,m} can be expressed as

(13) |

(14) |

(15) |

The mechanism of interference in this case is the following. After the amplitude excited to E_{a} has decayed to the continuum fragment states, becoming asymptotic, the third pulse of ε_{3}(t) pumps amplitude to the E_{b} energy. When this latter amplitude decays, it spreads and redistributes temporarily among all the φ_{E,m} continuum states accessible by the resonance state within its energy width, including those associated with the E_{a} energy, φ_{Ea,m}. This generates temporarily the amplitude appearing in the last three terms of eqn (15) that produce the interference effect in the fragment distribution.^{36,37} The temporary dispersion of the amplitude excited by the third pulse among different φ_{E,m} asymptotic states within a range of energy that includes the φ_{Ea,m} fragment states is due to the uncertainty principle. Once the third pulse of ε_{3}(t) is over and all the amplitude excited to E_{b} has decayed completely to the appropriate φ_{Eb,m} fragment states, producing a distribution at energy E_{b}, interference ceases and the asymptotic distribution at E_{a} converges again to . Since the interference terms of eqn (15) appear as long as the amplitude ξ_{Eb}(t) (or equivalently ) is created by the third pulse of ε_{3}(t), it becomes clear that this can be done at any asymptotic time as long as desired, and as many times as desired (using further successive pulses after the third one in the laser field).

The above general equations, and specifically eqn (12) and (15), which govern the resonance survival probability and product fragment distribution, respectively, provide the formal support to the results of the numerical simulations shown in Fig. 2–4. It is stressed that in the derivation of these equations no assumption is made on the nature or type of the molecular system that originates or supports the isolated resonance under control. Therefore these equations are valid for any isolated resonance, regardless of the origin of the system featuring the resonance. The consequence is that the application of the present control scheme behind these equations is general and universal to any molecular system featuring isolated resonance states, thus making possible the control of both the duration and the outcome of any resonance-mediated molecular process.

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