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Modern Physics Letters B

Geometric phase for “dark” subspaces in coherent population trapping

T. S. Yakovleva, A. M. Rostom, V. A. Tomilin, Leonid V. Il'ichov

Published Sep 18, 2020• 0 citations
Publisher

Abstract

The concept of geometric phase acquired by a quantum state during its evolution is generalized to the subspace of states of evolving quantum system. Physical motivation of this generalization comes from the effect of coherent population trapping. Under certain conditions, there exists a two-dimensional “dark” subspace of atomic ground states that does not interact with the external radiation. The structure of this “dark” subspace depends on the local field amplitude. We show that certain mathematical difficulties in finding the geometric phase for “dark” subspaces can be circumvented with the help of the orthogonal “bright” subspace.

DOI
10.1142/s0217984921500214

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The concept of geometric phase acquired by a quantum state during its evolution is generalized to the subspace of states of evolving quantum system. Physical motivation of this generalization comes from the effect of coherent population trapping. Under certain conditions, there exists a two-dimensional “dark” subspace of atomic ground states that does not interact with the external radiation. The structure of this “dark” subspace depends on the local field amplitude. We show that certain mathematical difficulties in finding the geometric phase for “dark” subspaces can be circumvented with the help of the orthogonal “bright” subspace.
Description of states of quantum systems in terms of bivectors (in the general case, polyvectors) is proposed as an alternative to vectors in a Hilbert space. This approach is developed for slow atoms under coherent population trapping conditions (in a dark state). The field of local two-dimensional dark subspaces is naturally associated with a local bivector field. An analysis in terms of bivectors is performed to recover the state of an atom or an atomic ensemble by using the condition for its being in some unknown dark state. This approach exposes natural correlations between atoms. The corresponding many-particle states belong to the class of states with positive partial transpose.
I present a feasible scheme to investigate the geometric phase for an atom trapped in an optical cavity induced by the effective decay process due to cavity-photon loss. The cavity mode, together with the external driving fields, acts as the engineered environment of the atom. When the parameters of the reservoir are adiabatically and cyclically changed, the system initially in the nontrivial dark state of the effective Lindblad operator undergoes a cyclic evolution and acquires a geometric phase. The geometric phase can be observed with the atomic-Ramsey-interference oscillation in the decoherence-free subspace.
A full quantum treatment of coherent population trapping (CPT) is given for a system of resonantly coupled atoms and electromagnetic field. We develop a regular analytical method of the construction of generalized dark states (GDS). It turns out that GDS do exist for all optical transitions FgtoFeF_g\\to F_eFg​toFe​, including bright transitions FtoF+1F\\to F+1FtoF+1 and F′′toF′′F''\\to F''F′′toF′′ with F′′F''F′′ a half-integer, for which the CPT effect is absent in a classical field.
We present a feasible scheme to investigate the geometric phase for an atom trapped in an optical cavity induced by the effective decay process due to cavity photon loss. The cavity mode, together with the external driving fields, acts as the engineered environment of the atom. When the parameters of the reservoir is adiabatically and cyclically changed, the system initially in the nontrivial dark state of the effective Lindblad operator undergoes a cyclic evolution and acquires a geometric phase. The geometric phase can be observed with the atomic Ramsey interference oscillation in the decoherence-free subspace.
A full quantum treatment of coherent population trapping (CPT) is given for a system of resonantly coupled atoms and electromagnetic field. We develop a regular analytical method of the construction of generalized dark states (GDS). It turns out that GDS do exist for all optical transitions F g → F e , including bright transitions F → F + 1 and F '' → F '' with F '' a half-integer, for which the CPT effect is absent in a classical field. We propose an idea to use an optically thick medium with a transition F '' → F '' with F '' ⩾ 3/2 a half-integer as a "quantum filter", which transmits only a quantum light with limited number of photons. Such a quantum filter can be used as a deterministic source of Fock's state of light.