We consider the simultaneous control of the relative phase and populations of two-level quantum systems by an external field. We apply a reverse engineering approach, obtaining an analytical expression for the control field depending upon two user-defined functions that dictate the population and the relative phase dynamics. We show that, in general, the prescribed functions for the dynamics cannot be chosen arbitrarily, but still there is plenty of room for choice. We illustrate the reverse engineering technique for several target states using different kinds of functions to specify the system dynamics. We show that by adjusting these dynamical functions, we produce different kinds of control fields. These controls can be easily built, needing, apart from the dynamical function themselves, only their first derivatives. The methodology presented here will certainly find many applications that go beyond simple two-level systems.
In this paper, based on the idea of reverse engineering, we propose a scheme to find analytical expressions of a laser pulse that allows for controlling the evolution of two-level systems as desired, regardless of the time-dependent phase of the system. The pulse relates to particular functions that predefine the evolution of a system. For illustration, we present concrete examples of driving the system from an arbitrary initial state to a final state along the user-prescribed particular way in both closed and open quantum systems. Simulation results indicate that the desired evolutionary path is accurately obtained when the rotating wave approximation (RWA) is satisfied. Most importantly, the scheme allows us to maintain system coherence and populations even in the presence of certain dephasing and thermal noises. Thus it may naturally find applications in quantum computing and quantum memories.
We address the control of the dynamics of both population and coherence phase in an open two-level quantum system employing a single external control field. The system dynamics is described by a Markovian master equation that takes into account dephasing and thermal noise. The control is engineered by inverting the underlying equations of motion, which yields an analytical expression for the control field in terms of user-specified time-dependent functions for the population and coherence phase. Our approach allows us to dictate not only the initial and final populations and phases, but the full dynamics of these quantities. The chosen functions for population and phase have to conform to certain constraints indicated in our analysis. Our methodology also reveals the possible transitions for given initial conditions and environmental noise parameters.
Abstract We present an analytical electric field expression to simultaneously control the population and phase dynamics of two-level quantum systems. The presented electric field is obtained by a reverse engineering technique after the desired population and phase evolution pathways of the system have been specified. Upon application of this laser field, the system is driven from an arbitrary quantum state superposition to the desired population distribution; meanwhile the phase of one of the states proceeds according to a predefined path. The robustness of the engineered electric field is demonstrated by numerical simulations on the example of the 3s → 3p transition of atomic sodium. Furthermore, the limitations of the presented technique, which arise due to the application of the rotating wave approximation, are thoroughly analyzed and discussed.
We apply reverse engineering to design laser pulse shapes for controlling the time evolution of strong-field transitions under environmental effects. Our effective two-state representation of the full quantum system involves dynamic Stark shifts and multiphoton Rabi frequency. Furthermore, the environment is modeled with dephasing and dissipative terms. Solving the master equation of the two-level density matrix inversely for the field, we obtain quasianalytic laser pulse shapes that simultaneously drive the populations and relative phase according to user-defined quantum pathways. The validity of the proposed scheme is demonstrated with the example of the $3s\ensuremath{\rightarrow}\ensuremath{\rightarrow}4s$ two-photon transition of atomic sodium by solving the master equation of the multilevel system that incorporates effects beyond the adiabatic elimination and the rotating wave approximation. Due to the open character of the system, the proposed control is not always possible. We explore these situations and point out the conditions for creating steady states with large coherence.
A quantum control landscape is defined by the value of a physical observable as a functional of the time-dependent control field
E(t) for a given quantum-mechanical system. Level sets through this landscape are prescribed by a particular value of the target observable at the final dynamical time
T, regardless of the intervening dynamics. We present a technique for exploring a landscape level set, where a scalar variable
s is introduced to characterize trajectories along these level sets. The control fields
E(s,t) accomplishing this exploration (i.e., that produce the same value of the target observable for a given system) are determined by solving a differential equation over
s in conjunction with the time-dependent Schr\"odinger equation. There is full freedom to traverse a level set, and a particular trajectory is realized by making an a priori choice for a continuous function
f(s,t) that appears in the differential equation for the control field. The continuous function
f(s,t) can assume an arbitrary form, and thus a level set generally contains a family of controls, where each control takes the quantum system to the same final target value, but produces a distinct control mechanism. In addition, although the observable value remains invariant over the level set, other dynamical properties (e.g., the degree of robustness to control noise) are not specifically preserved and can vary greatly. Examples are presented to illustrate the continuous nature of level-set controls and their associated induced dynamical features, including continuously morphing mechanisms for population control in model quantum systems.
We propose a systematic scheme to engineer quantum states of a quantum system governed by a time-convolutionless non-Markovian master equation. According to the idea of reverse engineering, the general algebraic equation to determine the control parameters, such as coherent and incoherent control fields, is presented. Without artificially engineering the time-dependent decay rates and retaining the environment-induced Lamb shifts, the quantum state can still be transferred into the target state in a finite period of time along an arbitrary designed trajectory strictly in Hilbert space. As an application, we apply our scheme to a driven two-level non-Markovian system and realize instantaneous-steady-state tracking and a complete population inversion with control parameters which are available in experimental settings.