Using a set of general methods developed by Krotov [A. I. Konnov and V. A. Krotov, Automation and Remote Control 60, 1427 (1999)], we extend the capabilities of optimal control theory to the nonlinear Schr\"odinger equation (NLSE). The paper begins with a general review of the Krotov approach to optimization. Although the linearized version of the method is sufficient for the linear Schr\"odinger equation, the full flexibility of the general method is required for treatment of the nonlinear Schr\"odinger equation. Formal equations for the optimization of the NLSE, as well as a concrete algorithm are presented. As an illustration, we consider a Bose-Einstein condensate (BEC) initially at rest in a harmonic trap. A phase develops across the BEC when an optical lattice potential is turned on. The goal is to counter this effect and keep the phase flat by adjusting the trap strength. The problem is formulated in the language of optimal control theory (OCT) and solved using the above methodology. To our knowledge, this is the first rigorous application of OCT to the nonlinear Schr\"odinger equation, a capability that is bound to have numerous other applications.
Using a set of general methods developed by Krotov [A. I. Konnov and V. A. Krotov, Automation and Remote Control 60, 1427 (1999)], we extend the capabilities of optimal control theory to the nonlinear Schr"odinger equation (NLSE). The paper begins with a general review of the Krotov approach to optimization. Although the linearized version of the method is sufficient for the linear Schr"odinger equation, the full flexibility of the general method is required for treatment of the nonlinear Schr"odinger equation. Formal equations for the optimization of the NLSE, as well as a concrete algorithm are presented. As an illustration, we consider a Bose-Einstein condensate (BEC) initially at rest in a harmonic trap. A phase develops across the BEC when an optical lattice potential is turned on. The goal is to counter this effect and keep the phase flat by adjusting the trap strength. The problem is formulated in the language of optimal control theory (OCT) and solved using the above methodology. To our knowledge, this is the first rigorous application of OCT to the nonlinear Schr"odinger equation, a capability that is bound to have numerous other applications.
Mathematical problems of optimal control of quantum systems attract high interest in connection with both fundamental questions and existing and prospective applications. An important problem is the development of methods for constructing controls for quantum systems. One of the widely used methods is the Krotov method. The article is the first part of the review devoted to mathematical aspects of this method for optimal control of quantum systems. Beyond quantum control this method was suggested in the articles [V.F. Krotov, I.N. Feldman, Eng. Cybern. 1983. Vol. 21, 123], [A.I. Konnov, V.F. Krotov, Automat. Remote Control, 1999], and then adopted to optimal quantum control in papers by V.A. Kazakov, V.F. Krotov, R. Kosloff, C.P. Koch, D.J. Tannor, and others. The review considers this method, including cases with constraints on control spectrum and on states of the quantum system, various ways for choosing the improvement function, in particular, linear or linear-quadratic form, etc. We also discuss GRAPE (GRadient Ascent Pulse Engineering) and CRAB (Chopped Random-Basis) methods, as well as combined methods such as the method suggested by I. Maday and G. Turinici for generalizing the 1st order Krotov method and the Zhu-Rabitz method. Nowadays, various versions of the Krotov method are used as a key tool in optimal quantum control, and the review describes applications of this method to mathematical control problems for molecular dynamics, Bose-Einstein condensate, nuclear-magnetic resonance, quantum gates generation, etc. The first part of the review is devoted to closed quantum systems.
It has been proposed that the adiabatic loading of a Bose-Einstein condensate (BEC) into an optical lattice via the Mott-insulator transition can be used to initialize a quantum computer [D. Jaksch et al., Phys. Rev. Lett. 81, 3108 (1998)]. The loading of a BEC into the lattice without causing band excitation is readily achievable; however, unless one switches on an optical lattice very slowly, the optical lattice causes a phase to accumulate across the condensate. We show analytically and numerically that a cancellation of this effect is possible by adjusting the harmonic trap force constant of the magnetic trap appropriately, thereby facilitating quick loading of an optical lattice for quantum computing purposes. A simple analytical theory is developed for a nonstationary BEC in a harmonic trap.
In this work, quantum control of trapped Bose-Einstein-Condensates (BEC) is considered at matter surface. For particles at BEC status, quantum system is described by Gross-Pitaevskii equation, experimental control of BEC is happened at physics field, and achieved at laboratory. At theoretic aspect, control of trapped condensates is not sufficiently investigated at academic level. What we interest is applying control theory to BEC trapped on the surface (metallic, crystal). At optical lattice, particles are trapping by constrained forces at cooling technique, and temporally take the same quantum states, such kind of condensates phenomena had already been surveyed at a variety of areas. The most works are reported on free BEC particles, quite natural question is arising on the surface science: BEC particles created,detected, and placed on a certain chemical surface, control of trapped particles is difference or not? We are curious about optical and mechanical constraints take action together on particles. In the viewpoint of quantum control realm, our purpose is to apply optimal control theory (OCT) to trapped Bose-Einstein-Condensates as they are occurred at surface. In the framework of variational theory at complex Hilbert spaces, prove the existence of quantum optimal control, and characterize optimal control using optimality (Euler-Lagrange) system. Control variables for trapped BEC contain three functions: one is electro-magnetic force; another is external constraint from optical equipment (optical frequency, lattice number); third is quantum mechanics against gravitational force, which making BEC particles stay at surface stationary. Review the literatures, electro-magnetic-optical controls are extremely considered at last couple of years. Gravitational control is rarely considered. Further extension of the work is to do real-time computer-aided BEC control at matter surface. Computational approach for simulation of BEC control at two and three dimensions would be a promise direction.
In this work, quantum control of trapped Bose-Einstein-Condensates (BEC) is considered at matter surface. For particles at BEC status, quantum system is described by Gross-Pitaevskii equation, experimental control of BEC is happened at physics field, and achieved at laboratory. At theoretic aspect, control of trapped condensates is not sufficiently investigated at academic level. What we interest is applying control theory to BEC trapped on the surface (metallic, crystal). At optical lattice, particles are trapping by constrained forces at cooling technique, and temporally take the same quantum states, such kind of condensates phenomena had already been surveyed at a variety of areas. The most works are reported on free BEC particles, quite natural question is arising on the surface science: BEC particles created,detected, and placed on a certain chemical surface, control of trapped particles is difference or not? We are curious about optical and mechanical constraints take action together on particles. In the viewpoint of quantum control realm, our purpose is to apply optimal control theory (OCT) to trapped Bose-Einstein-Condensates as they are occurred at surface. In the framework of variational theory at complex Hilbert spaces, prove the existence of quantum optimal control, and characterize optimal control using optimality (Euler-Lagrange) system. Control variables for trapped BEC contain three functions: one is electro-magnetic force; another is external constraint from optical equipment (optical frequency, lattice number); third is quantum mechanics against gravitational force, which making BEC particles stay at surface stationary. Review the literatures, electro-magnetic-optical controls are extremely considered at last couple of years. Gravitational control is rarely considered. Further extension of the work is to do real-time computer-aided BEC control at matter surface. Computational approach for simulation of BEC control at two and three dimensions would be a promise direction.
This thesis develops a numerical optimal-control framework for investigating the manipulation of one-dimensional Bose–Einstein condensates governed by the Gross–Pitaevskii equation. The control problem is formulated as a constrained optimization problem in which time-dependent external potentials are optimized to steer condensate dynamics while minimizing unwanted excitations.The first part considers the compression of a condensate containing a darksoliton confined in a box potential. Optimal trajectories of the trap boundaries are obtained that preserve the solitonic structure while strongly reducing den-sity and phase excitations compared with conventional linear protocols. Thesecond part addresses the opposite problem of soliton absorption. By optimizing the motion of the trapping boundaries, moving gray solitons can be efficiently removed and the condensate driven toward its ground state. The third part investigates the merging of two condensates separated by a potential barrier. Optimal barrier-removal protocols are computed for different initialphase differences between the condensates. The dependence of the optimal control strategy on the phase mismatch is analyzed, including the formation of adark soliton in the case of a PI phase difference and the construction of robustprotocols applicable when the phase difference is unknown.Finally, the optimal-control framework is extended to tunnel-coupled con-densates. The coupled Gross–Pitaevskii equations are formulated as a con-strained optimization problem, and the corresponding adjoint equations arederived. This framework provides the foundation for future investigations of optimal control in coupled condensate systems.Together, these results demonstrate how optimal control can be used both to preserve and to manipulate nonlinear excitations and phase structures in Bose–Einstein condensates, providing tools for quantum-state engineering inultracold atomic systems.
The work in this article extends the optimal control framework of variational calculus to optimize the conversion of a Bose-Einstein condensate of atoms to one of molecules. It represents the derivation of the closed form optimal control equations for a system governed by a nonlinear Schr\"odinger equation and its successful application. It was necessary to derive a density matrix formulation of the coupled Gross-Pitaevskii equations to optimize STIRAP-like Raman light fields, to overcome dissipation.