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arXiv:2308.11661 · Scientific Reports

Quantum control operations with fuzzy evolution trajectories based on polyharmonic magnetic fields

Jesús Fuentes

Published Dec 17, 2020• 5 citations
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Abstract

We explore a class of quantum control operations based on a wide family of harmonic magnetic fields that vary softly in time. Depending on the magnetic field amplitudes taking part, these control operations can produce either squeezing or loop (orbit) effects, and even parametric resonances, on the canonical variables. For these purposes we focus our attention on the evolution of observables whose dynamical picture is ascribed to a quadratic Hamiltonian that depends explicitly on time. In the first part of this work we survey such operations in terms of biharmonic magnetic fields. The dynamical analysis is simplified using a stability diagram in the amplitude space, where the points of each region will characterise a specific control operation. We discuss how the evolution loop effects are formed by fuzzy (non-commutative) trajectories that can be closed or open, in the latter case, even hiding some features that can be used to manipulate the operational time. In the second part, we generalise the case of biharmonic fields and translate the discussion to the case of polyharmonic fields. Using elementary properties of the Toeplitz matrices, we can derive exact solutions of the problem in a symmetric evolution interval, leading to the temporal profile of those magnetic fields suitable to achieve specific control operations. Some of the resulting fuzzy orbits can be destroyed by the influence of external forces, while others simply remain stable.

DOI
10.1038/s41598-020-79309-8

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•Jan 26, 2018•1 citation
We explore a class of quantum control operations based on a wide family of harmonic magnetic fields that vary softly in time. Depending on the magnetic field amplitudes taking part, these control operations can produce either squeezing or loop (orbit) effects, and even parametric resonances, on the canonical variables. For these purposes we focus our attention on the evolution of observables whose dynamical picture is ascribed to a quadratic Hamiltonian that depends explicitly on time. In the first part of this work we survey such operations in terms of biharmonic magnetic fields. The dynamical analysis is simplified using a stability diagram in the amplitude space, where the points of each region will characterise a specific control operation. We discuss how the evolution loop effects are formed by fuzzy (non-commutative) trajectories that can be closed or open, in the latter case, even hiding some features that can be used to manipulate the operational time. In the second part, we generalise the case of biharmonic fields and translate the discussion to the case of polyharmonic fields. Using elementary properties of the Toeplitz matrices, we can derive exact solutions of the problem in a symmetric evolution interval, leading to the temporal profile of those magnetic fields suitable to achieve specific control operations. Some of the resulting fuzzy orbits can be destroyed by the influence of external forces, while others simply remain stable.
We consider the most general Time-Dependent (TD) quadratic Hamiltonian written in terms of the bosonic operators a and a + , which may represent either a charged particle subjected to a harmonic motion, immersed in a TD uniform magnetic field, or a single mode photon field going through a squeezing medium. We solve the TD Schrödinger equation by a method that uses, sequentially, a TD unitary transformation and the diagonalization of a TD invariant, and we verify that the exact solution is a complete set of TD states. We also obtain the evolution operator which is essential to express operators in the Heisenberg picture. The variances of the quadratures are calculated and a phase space of parameters introduced, in which we identify squeezing regions. The results for some special cases are presented and as an illustrative example the parametric oscillator is revisited and the trajectories in phase space drawn.
We look for new steps on the dynamical operations that may squeeze simultaneously some families of quantum states, independently of their initial shape, induced by softly acting external fields which might produce the squeezing of the canonical observables q,pq,pq,p of charged particles. Also, we present some exactly solvable cases of the problem which appear in the symmetric evolution intervals permitting to find explicitly the time dependence of the external fields needed to generate the required evolution operators. Curiously, our results are interrelated with a simple, non--trivial, anti--commuting algebra of Toeplitz which describes the problem more easily than the frequently used Ermakov--Milne invariants.
The time-dependent Schrödinger equation describing a generalized two-dimensional quantum parametric oscillator in the presence of time-variable external fields is solved using the evolution operator method. For this, the evolution operator is found as a product of exponential operators through the Wei–Norman Lie algebraic approach. Then, the propagator and time-evolution of eigenstates and coherent states are derived explicitly in terms of solutions to the corresponding system of coupled classical equations of motion. In addition, using the evolution operator formalism, we construct linear and quadratic quantum dynamical invariants that provide connection of the present results with those obtained via the Malkin–Man’ko–Trifonov and the Lewis–Riesenfeld approaches. Finally, as an exactly solvable model, we introduce a Cauchy–Euler type quantum oscillator with increasing mass and decreasing frequency in time-dependent magnetic and electric fields. Based on the explicit results for the uncertainties and expectations, squeezing properties of the wave packets and their trajectories in the two-dimensional configuration space are discussed according to the influence of the time-variable parameters and external fields.
In the studies of the squeezing it is customary to focus more attention on the particular squeezed states and their evolution than on the dynamical operations that could squeeze simultaneously some wider families of quantum states, independently of their initial shape. We look for new steps in this direction, carried out by softly acting external fields which might produce the squeezing of the canonical observables q,pq,pq,p of charged particles. The works on these problems collect so many valuable results that the question is whether something more is indeed something more in our knowledge. Yet we decided to present some exactly solvable cases of the problem which appear in the symmetric evolution intervals permitting to find explicitly the time dependence of the external fields needed to generate the required evolution operators. Curiously, our results are interrelated with a simple anti-commuting algebra of Toeplitz which describes the problem more easily than the frequently used Ermakov--Milne invariants. Some trending topics, as well as some fundamental problems in quantum theory, might also be involved.
The evolution operator of a Caldirola-Kanai type quantum parametric oscillator with a generalized quadratic Hamiltonian is obtained using the Wei-Norman Lie algebraic approach, and time evolution of the eigenstates of a harmonic oscillator and Glauber coherent states is found explicitly. Behavior of this oscillator is investigated under the influence of the external mixed term B(t)(q^p^+p^q^)/2, which affects the squeezing properties of the wave packets, and linear terms D0(t)q^, E0(t)p^ responsible for their displacement in time. According to this, we construct all exact quantum models with different parameters B(t), for which the structure of the Caldirola-Kanai oscillator in position space is preserved. Then, for each model, we obtain explicit solutions and analyze the squeezing and displacement properties of the wave packets according to the frequency modification by B(t) and periodic forces in the corresponding classical equation of motion.
We investigate the controllability of quantum systems in two different settings: the standard "closed" setting, in which a quantum system is seen as isolated and the control problem is formulated on the Schrödinger equation; the open setting that describes a quantum system in interaction with a larger one, of which just qualitative parameters are known, by means of the Lindblad equation on states. In the context of closed systems we focus our attention to an interesting class of models, namely the spin-boson models. The latter describe the interaction between a 2-level quantum system and finitely many distinguished modes of a bosonic field. We discuss two prototypical examples, the Rabi model and the Jaynes-Cummings model, which despite their age are still very popular in several fields of quantum physics. Notably, in the context of cavity Quantum Electro Dynamics (C-QED) they provide an approximate yet accurate description of the dynamics of a 2-level atom in a resonant microwave cavity, as in recent experiments of S. Haroche. We investigate the controllability properties of these models, analyzing two different types of control operators acting on the bosonic part, corresponding -in the application to cavity QED- to an external electric and magnetic field, respectively. We review some recent results and prove the approximate controllability of the Jaynes-Cummings model with these controls. This result is based on a spectral analysis exploiting the non-resonances of the spectrum. As far as the relation between the Rabi and the Jaynes-Cummings Hamiltonians concerns, we treat the so called rotating wave approximation in a rigorous framework. We formulate the problem as an adiabatic limit in which the detuning frequency and the interaction strength parameter goes to zero, known as the weak-coupling regime. We prove that, under certain hypothesis on the ratio between the detuning and the coupling, the Jaynes-Cumming and the Rabi dynamics exhibit the same behaviour, more precisely the evolution operators they generate are close in norm. In the framework of open quantum systems we investigate the controllability of the Lindblad equation. We consider a control acting adiabatically on the internal part of the system, which we see as a degree of freedom that can be used to contrast the action of the environment. The adiabatic action of the control is chosen to produce a robust transition. We prove, in the prototype case of a two-level system, that the system approach a set of equilibrium points determined by the environment, i.e. the parameters that specify the Lindblad operator. On that set the system can be adiabatically steered choosing a suitable control. The analysis is based on the application of geometrical singular perturbation methods.