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arXiv:0911.2961 · Physical Review A

Universality in exact quantum state population dynamics and control

Lian-Ao Wu, Dvira Segal, I. L. Egusquiza, Paul W. Brumer

Published Sep 9, 2010• 3 citations
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Abstract

We consider an exact population transition, defined as the probability of finding a state at a final time that is exactly equal to the probability of another state at the initial time. We prove that, given a Hamiltonian, there always exists a complete set of orthogonal states that can be employed as time-zero states for which this exact population transition occurs. The result is general: It holds for arbitrary systems, arbitrary pairs of initial and final states, and for any time interval. The proposition is illustrated with several analytic models. In particular, we demonstrate that in some cases, by tuning the control parameters, a complete transition might occur, where a target state, vacant at $t=0$, is fully populated at time $\ensuremath{\tau}$.

DOI
10.1103/physreva.82.032307

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We consider an exact population transition, defined as the probability of finding a state at a final time that is exactly equal to the probability of another state at the initial time. We prove that, given a Hamiltonian, there always exists a complete set of orthogonal states that can be employed as time-zero states for which this exact population transition occurs. The result is general: It holds for arbitrary systems, arbitrary pairs of initial and final states, and for any time interval. The proposition is illustrated with several analytic models. In particular, we demonstrate that in some cases, by tuning the control parameters, a complete transition might occur, where a target state, vacant at t=0t=0t=0, is fully populated at time \ensuremathτ\ensuremath{\tau}\ensuremathτ.
We consider an exact state transmission, where a density matrix in one information processor A at time t=0t=0t=0 is exactly equal to that in another processor B at a later time. We demonstrate that there always exists a complete set of orthogonal states, which can be employed to perform the exact state transmission. Our result is very general in the sense that it holds for arbitrary media between the two processors and for any time interval. We illustrate our results in terms of models of spin, fermionic, and bosonic chains. This complete set can be used as a basis to study the perfect state transfer which is associated with degenerate subspaces of this set of states. Interestingly, this formalism leads to a proposal of perfect state transfer via adiabatic passage, which does not depend on the specific form of the driving Hamiltonian.
As a paradigm for describing open-quantum-system dynamics, the non-Hermitian Schrödinger equation generally modifies state norms, complicating coherent control over the system's Hilbert space. With a general instantaneous frame spanned by biorthogonal ancillary bases, we develop a biorthogonal quantum-control theory for discrete-variable systems governed by time-dependent non-Hermitian Hamiltonian. In particular, we derive a necessary and sufficient condition for extracting the transitionless pathways that transport parallel in ket and bra states. When this condition is satisfied by a set of projectors constructed with biorthogonal basis states, the non-Hermitian Hamiltonian becomes diagonal in the relevant biorthogonal subspace, enabling arbitrary initial state to evolve as a superposition of controlled transitionless passages. We further identify the parametric constraints under which the transformation linking the initial and final states becomes unitary. As a result, the final system state is automatically normalized even though the transitional dynamics is nonunitary. Our theory is justified by the bidirectional population transfer and a holomonic quantum gate implemented by a non-Hermitian three-level system in ΛΛΛ type. For the population transfer, controlled gain and loss reduce the required coherent pulse area, enabling shorter transfer times than the Hermitian benchmark with the same average Rabi frequency. Our work integrates the universal quantum control (UQC) into the biorthogonal representation, displaying full-rank passage engineering with coherent operations in non-Hermitian systems.
We find conditions required to achieve complete population transfer, via coherent population trapping, from an initial state to a designated final state at a designated time in a degenerate nnn-state atom, where transitions are caused by an external interaction. In systems with degenerate states there is no time ordering. Analytic expressions have been found for transition probabilities in a degenerate nnn-state atom interacting with a strong external field that gives a common time dependence to all of the transition matrix elements. Except for solving a simple nthn^{th}nth order equation to determine eigenvalues of dressed states, the method is entirely analytic. These expressions may be used to control electron populations in degenerate nnn-state atoms. Examples are given for n=2n=2n=2 and n=3n=3n=3.
Conventional manipulations over quantum systems for, such as coherent population trapping and unidirectional transfer, focus on Hamiltonian engineering while regarding the system's manifold geometry and constraint equation as secondary causes. Here, we treat them on equal footing in controlling a finite-dimensional quantum system under a time-dependent non-Hermitian Hamiltonian, which is inspired by the D'Alembert principle of regarding active force, constraint force, and inertial force in an unbiased way. Under the biorthogonal condition, the non-Hermitian Hamiltonian could be triangularized in a constraint picture spanned by a set of completed and orthonormal basis states, which is found to be a sufficient condition to construct at least one universal nonadiabatic passage in both bra and ket spaces. The passage ends up with a desired target state that is automatically normalized without artificial normalization in the existing treatments for non-Hermitian quantum systems. Moreover, the passage is found to be robust against the parametric deviation when the real part of its global phase is rapidly varying with time. Our protocol is explicitly verified for the perfect population transfer in the two-level system and the chiral population transfer in the three-level system. It generalizes our framework of universal quantum control to the field of the biorthogonal quantum mechanics.
A zero-area pulse produces no excitation in a two-state quantum system on exact resonance. We show analytically and numerically that off resonance the transition probability may acquire significant values, and as the Rabi frequency increases, it approaches steadily unity, a feature reminiscent of adiabatic passage. The nature of the transition is not, however, of adiabatic nature. We show that complete population transfer occurs as a result of a quasiresonant transition between the adiabatic states due to a $\ensuremath{\delta}$-function-type interaction with an area of $\ensuremath{\pi}$.
We consider a scenario where we wish to bring a closed system of known Hilbert space dimension d S (the target), subject to an unknown Hamiltonian evolution, back to its quantum state at a past time t 0 .The target is out of our control: This means that we ignore both its free Hamiltonian and how the system interacts with other quantum systems we may use to influence it.Under these conditions, we prove that there exist protocols within the framework of nonrelativistic quantum physics that reset the target system to its exact quantum state at t 0 .Each "resetting protocol" is successful with nonzero probability for all possible free Hamiltonians and interaction unitaries, save a subset of zero measure.When the target is a qubit and the interaction is sampled from the Haar measure, the simplest resetting circuits have a significant average probability of success and their implementation is within reach of current quantum technologies.Finally, we find that, in case the resetting protocol fails, it is possible to run a further protocol that, if successful, undoes both the natural evolution of the target and the effects of the failed protocol over the latter.By chaining in this fashion several such protocols, one can substantially increase the overall probability of a successful resetting.