By combining the theories of optimal control and coherent control, we derive an analytical formulation to evaluate the optimal 1-photon and n -photon fields together with their relative phase and intensity in the optimal “1 + n ”-coherent control scheme. The optimal coherent control exploits tailored light pulses to produce the best overlap with a predefined nuclear target in an electronically degenerate region. It also exploits explicitly the quantum interference between two optimal excitation pathways to achieve the product electronic selectivity. Numerical demonstrations are carried out to selectively control a minimum-uncertainty outgoing wave packet target in a molecular system. The electronically degenerate byproduct is shown to be completely discriminated against via the destructive interference between two optimal excitation pathways. The target considered in this paper would neither be achievable via the conventional continuous-wave coherent control nor via a single optimal control pathway in the weak response regime. Proposed is also a time-dependent excitation frequency degeneracy condition that incorporates the evolution of carrier frequencies in the optimal coherent control with tailored light fields.
By combining the theories of optimal control and coherent control, we derive an analytical formulation to evaluate the optimal 1-photon and n -photon fields together with their relative phase and intensity in the optimal “1 + n ”-coherent control scheme. The optimal coherent control exploits tailored light pulses to produce the best overlap with a predefined nuclear target in an electronically degenerate region. It also exploits explicitly the quantum interference between two optimal excitation pathways to achieve the product electronic selectivity. Numerical demonstrations are carried out to selectively control a minimum-uncertainty outgoing wave packet target in a molecular system. The electronically degenerate byproduct is shown to be completely discriminated against via the destructive interference between two optimal excitation pathways. The target considered in this paper would neither be achievable via the conventional continuous-wave coherent control nor via a single optimal control pathway in the weak response regime. Proposed is also a time-dependent excitation frequency degeneracy condition that incorporates the evolution of carrier frequencies in the optimal coherent control with tailored light fields.
Quantum mechanics, though a probabilistic theory, gives a 'deterministic' answer to the question of how the present determines the future. In essence, in order to predict future probabilities, we need to (numerically) propagate the time-dependent Schrödinger equation from the present to the future. It is interesting to note that classical mechanics of macroscopic bodies, though reputed to be a deterministic theory, does not allow, due to chaos (which unfortunately is more prevalent than integrability), such clear insights into the future. In contrast, small (e.g., atomic, molecular and photonic) systems which are best understood using the tools of quantum mechanics, do not suffer from chaos, rendering the prediction of the probability-distributions of future events possible. The field of quantum control deals with an important modification of this task, namely, it asks: given a wave function in the present, what dynamics, i.e. what Hamiltonian, guarantees a desired outcome or 'objective' in the future? In practice one may achieve this goal of modifying and finding the desired Hamiltonian by introducing external fields, e.g. laser light. It is then possible to reach the objective in a 'trial-and-error' fashion, performed either numerically or in the laboratory. We can guess or build a Hamiltonian, do an experiment, or propagate the initial wave function to the future, compare the result with the desirable objective, and correct the guess for the Hamiltonian until satisfactory agreement with the objective is reached. A systematic way of executing this procedure is the sub-field called 'optimal control'. The trial-and-error method is often very time consuming and rarely provides mechanistic insight. There are situations where analytical solutions exist, rendering the control strategies more transparent. This is especially so when one can identify quantum interferences as the heart of quantum control, the essence of the field called 'coherent control'. The experience accumulated so far by many groups is that quantum interferences are the mechanism underlying most successful control scenarios and control experiments. The identification of the control mechanism is especially facile when the time-dependence of the Hamiltonian merely serves to prepare a state which then evolves in the absence of external fields. Analytical theories also exist when the explicit time dependence of the Hamiltonian can be treated adiabatically. Under these circumstances one can identify the quantum interfering pathways that lead to control, allowing for the simultaneous exploration of entire 'landscapes' (and not just a single 'desirable' objective) resulting from varying all the experimental 'knobs' at our disposal, e.g., relative- and envelope-phases, intensities, polarizations, and pulse-sequencing. Such 'pulse-shaping' knobs are now available down to the sub-femtosecond regime in broad spectral regions ranging from the UV to the IR. We expect that comprehensive investigations of the quantum interference required to control dynamical processes will result in a more detailed understanding of the dynamics themselves. Together with experimental progress---the stage is now set for many exciting applications in fundamental and applied sciences by 'tailoring' light--matter interactions essentially at will.
Coherent control of the atomic two-photon absorption with shaped single-cycle pulses is examined theoretically in the weak-field regime. The control over the stabilized carrier-envelope phase (CEP) of the pulses is determined as a key parameter allowing the full utilization of the ultrabroad pulse spectrum. This bandwidth is sufficient to provide besides the sequential two-photon pathways, additional Raman-type pathways coupling between the ground and the excited states. The interference between the different pathways is efficiently controlled via the control over the CEP. Simplified two-level model is analyzed in the frequency domain where the rational pulse shaping is applied. The developed intuition is applied to atomic Cs and verified by the exact numerical solution of time-dependent Schrodinger equation.
Quantum control refers to our ability to manipulate quantum systems. This tutorial-style chapter focuses on the use of classical electromagnetic fields to steer the system dynamics. In this approach, the quantum nature of the control stems solely from the underlying dynamics, through the exploitation of destructive and constructive interference to reach the control target. We first discuss two basic control principles -- coherent control which uses manipulation in frequency or time to design these interferences, and adiabatic following where access to the control target is enabled by tracking the time-dependent ground state. For complex control targets and system dynamics that exceed the scope of these basic principles, optimal control theory provides a powerful suite of tools to design the necessary protocols. A key consideration for the successful application of optimal control theory is a proper choice of the optimization functional. All concepts are illustrated using recent work from my research group, with a focus on controlling atoms and superconducting qubits. The chapter concludes with an outlook on integrating coherent control with engineered dissipation and a discussion of open questions in the field.
Weak-field coherent phase control of atomic non-resonant multiphoton excitation induced by shaped femtosecond pulses is studied theoretically in the single-cycle regime. The carrier-envelope phase (CEP) of the pulse, which in the multi-cycle regime does not play any control role, is shown here to be a new effective control parameter that its effect is highly sensitive to the spectral position of the ultrabroad spectrum. Rationally chosen position of the ultrabroadband spectrum coherently induces several groups of multiphoton transitions from the ground state to the excited state of the system: transitions involving only absorbed photons as well as Raman transitions involving both absorbed and emitted photons. The intra-group interference is controlled by the relative spectral phase of the different frequency components of the pulse, while the inter-group interference is controlled jointly by the CEP and the relative spectral phase. Specifically, non-resonant two- and three-photon excitation is studied in a simple model system within the perturbative frequency-domain framework. The developed intuition is then applied to weak-field multiphoton excitation of atomic cesium (Cs), where the simplified model is verified by non-perturbative numerical solution of the time-dependent Schrödinger equation. We expect this work to serve as a basis for a new line of femtosecond coherent control experiments.
The subtle and fundamental issue of indistinguishability and interference between independent pathways to the same target state is examined in the context of coherent control of atomic and molecular processes, with emphasis placed on possible "which-way" information due to quantum entanglement established in the quantum dynamics. Because quantum interference between independent pathways to the same target state occurs only when the independent pathways are indistinguishable, it is first shown that creating useful coherence between nondegenerate states of a molecule for subsequent quantum interference manipulation cannot be achieved by collisions between atoms or molecules that are prepared in momentum and energy eigenstates. Coherence can, however, be transferred from light fields to atoms or molecules. Using a particular coherent control scenario, it is shown that this coherence transfer and the subsequent coherent phase control can be readily realized by the most classical states of light, i.e., coherent states of light. It is further demonstrated that quantum states of light may suppress the extent of phase-sensitive coherent control by leaking out some which-way information while "incoherent interference control" scenarios proposed in the literature have automatically ensured the indistinguishability of multiple excitation pathways. The possibility of quantum coherence in photodissociation product states is also understood in terms of the disentanglement between photodissociation fragments. Results offer deeper insights into quantum coherence generation in atomic and molecular processes.
Coherent control steers a quantum system from an initial state to a target state by controlling quantum interference phenomena via an external field, which is central to vast applications ranging from quantum information processing to attosecond physics. Here, we analyze the quantum pathways interference in two-color coherent control of photoemission using exact analytical solutions of the time-dependent Schr\"odinger equation. The theory includes all possible quantum pathways and their interference terms. Constructive (or destructive) interferences among the pathways leads to the maximum (or minimum) emission with varying phase delay of the two-color lasers. It is found that increasing the intensity ratio of the second harmonic $(2\ensuremath{\omega})$ to fundamental $(\ensuremath{\omega})$ lasers results in less contribution from the $\ensuremath{\omega}$ pathway (absorption of $\ensuremath{\omega}$ photons only) and more contribution from multicolor pathway (simultaneous absorption of both $\ensuremath{\omega}$ and $2\ensuremath{\omega}$ photons) and $2\ensuremath{\omega}$ pathway (absorption of $2\ensuremath{\omega}$ photons only), and therefore stronger pathways interference and increased visibility larger than 95%. Increasing bias voltages shifts the dominant emission to processes with lower-order photon absorption, which sequentially decreases the interference between the $\ensuremath{\omega}$ and the $2\ensuremath{\omega}$ pathways, and between single-color and multicolor pathways, leading to two peaks in the visibility.