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Physica Scripta

Fifteen-dimensional optical Bloch dynamics and subspace decomposition in a four-level diamond system

Dawit Hiluf Hailu

Published Jul 7, 2026• 0 citations
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Abstract

Abstract This paper investigates the dynamics of a four-level diamond-type quantum system using the Alhassid–Levine (AL) and Hioe–Eberly (HE) formalisms. In the configuration considered, the Stokes fields β j ( t ) ( j = 1 , 2 ) drive the transitions | 1 ⟩ ↔ | 4 ⟩ and | 2 ⟩ ↔ | 3 ⟩ , while the pump fields α j ( t ) couple the transitions | 1 ⟩ ↔ | 2 ⟩ and | 3 ⟩ ↔ | 4 ⟩ . We first establish the correspondence between the AL and HE approaches and then derive the complete coherence-vector equations of motion within the su ( 4 ) Lie algebra using all fifteen generators subject to the normalization constraint. A central result of this work is the demonstration that when all driving fields share an identical temporal profile, differing only in amplitude, the fifteen-dimensional dynamics decomposes into six invariant subspaces with dimensions 1 , 1 , 2 , 3 , 4 , and 4 . Each subspace evolves independently and possesses its own conserved norm in the coherent limit. This decomposi

DOI
10.1088/1402-4896/ae86ed

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Abstract This paper investigates the dynamics of a four-level diamond-type quantum system using the Alhassid–Levine (AL) and Hioe–Eberly (HE) formalisms. In the configuration considered, the Stokes fields β j ( t ) ( j = 1 , 2 ) drive the transitions | 1 ⟩ ↔ | 4 ⟩ and | 2 ⟩ ↔ | 3 ⟩ , while the pump fields α j ( t ) couple the transitions | 1 ⟩ ↔ | 2 ⟩ and | 3 ⟩ ↔ | 4 ⟩ . We first establish the correspondence between the AL and HE approaches and then derive the complete coherence-vector equations of motion within the su ( 4 ) Lie algebra using all fifteen generators subject to the normalization constraint. A central result of this work is the demonstration that when all driving fields share an identical temporal profile, differing only in amplitude, the fifteen-dimensional dynamics decomposes into six invariant subspaces with dimensions 1 , 1 , 2 , 3 , 4 , and 4 . Each subspace evolves independently and possesses its own conserved norm in the coherent limit. This decomposi
I study the coherent interaction of a three-level atom in the lambda configuration with pump, Stoke, and anti-Stokes laser fields. The pump laser field drives transitions from the two non-degenerate lower levels to the excited state which is highly detuned for either transition. This excited state is also coupled to each of the lower states by Stokes or anti-Stokes transitions so that this system may represent a model of four-wave mixing. Assuming large detunings, the excited state is adiabatically eliminated from obtaining an effective two-level system with Raman coupling between the levels. Optical Bloch equations are obtained for this effective two-level system, and phenomenological damping terms are added for collisional dephasing of the Raman transitions. The equations are solved for the steady-state as well as by the method of adiabatic following. From these solutions we obtain the polarization.
There are many distinguishable subtypes of diamond-type four-level optical quantum coherent(OQC) systems,which come with high structural diversity.We selected transition channels properly according to the rules for dipole transition selection.With a corresponding laser field,we used Boolean algebra methods to classify diamond-type four-level OQC systems into nine real-world channel subtypes(RCSs).Also,we went further by classifying those RCSs into eighty so-called intensity subtypes(ISs) exactly,depending on the excitation intensity of the laser for different transition channels.We focused our discussion on the structural diversity of diamond-type four-level systems,while pointing out which substypes have been studied in some aspects and which ones have not even been theoretically studied.
The SU(3) dynamical evolution of three-level systems at two-photon resonance induced by two strong linearly polarized monochromatic fields is studied exactly by means of the semiclassical many-mode Floquet theory (MMFT) recently developed by the authors. Within the rotating-wave approximation (RWA), Hioe and Eberly have recently shown that the eight-dimensional SU(3) coherent vector S\ensuremath{\rightarrow} characterizing the time evolution of three-level systems can be factored into three independent vectors of dimensions three, four, and one, at appropriate two-photon resonance conditions. In practice, however, if the laser-atom interactions occur away from the two-photon resonance, or if the RWA is not valid, etc., this Gell-Mann--type SU(3) dynamical symmetry will be broken. It is shown in this paper that instead of solving the time-dependent generalized Bloch equations, the SU(3) dynamical evolution of the coherent vector S\ensuremath{\rightarrow} as well as various symmetry-breaking effects can be expediently studied by the use of the MMFT and expressed in terms of a few time-independent quasienergy eigenvalues and eigenvectors. Furthermore, we have extended the generalized Van Vleck (GVV) nearly degenerate perturbation theory to an analytical treatment of the two-mode Floquet Hamiltonian. This reduces the infinite-dimensional time-independent Floquet Hamiltonian into a 3\ifmmode\times\else\texttimes\fi{}3 effective Hamiltonian, from which useful analytical properties of the SU(3) coherent vector can be easily obtained. The combination of the MMFT and the GVV method thus greatly facilitates the study of the dynamical evolution. Pictorial comparison of the exact and the RWA results of the time evolution of the eight-dimensional coherent vector under several different physical conditions is presented and discussed at length.
We present a comprehensive analytical study of the optical Bloch equations for a closed two-level atom. We obtain a map in phase space of the Rabi frequency and the laser frequency detuning, discriminating the characteristics of solutions to the optical Bloch equations separated by a boundary in phase space. Outside the boundary, oscillatory behavior is shown, while simple decaying behavior is shown inside the boundary. The behavior of the solutions is also found to be dependent on the relative magnitude of the decay rates of the optical coherence and the excited state population.
Link between Alhassid-Levine and Hioe-Eberly formalisms of SU(N)SU(N)SU(N) equation of motion
Geometric representations of solutions provides intuitive physical insights. To which end studying dynamics of Quantum systems via su(n)su (n)su(n) Lie algebra proves to be convenient way of obtaining geometric solution. In this paper link is established between two formalisms that made use of Lie algebra to describe equation of motion for quantum system. In both approaches the Hamiltonian and the density matrix are expressed as a linear combination of the Lie group. To exemplify the approach we consider a very well studied two level system coupled by a laser pulse. Beyond establishing link between these two formalism we obtained two constants of motion by assuming time dependent detuning whose time profile is assumed to be same as the laser pulse. Consequently we have shown how one can have two disjoint subspaces whose evolution vector is independent of each other.
Aspects of coherence and decoherence are analyzed within the optical Bloch equations. By rewriting the analytic solution in an alternate form, we are able to emphasize a number of unusual features: (a) despite the Markovian nature of the bath, coherence at long times can be retained; (b) the long-time asymptotic degree of coherence in the system is intertwined with the asymptotic difference in level populations; (c) the traditional population-relaxation and decoherence times, T1 and T2, lose their meaning when the system is in the presence of an external field, and are replaced by more general overall time scales; (d) increasing the field strength, quantified by the Rabi frequency Omega, increases the rate of decoherence rather than reducing it, as one might expect; and (e) maximum asymptotic coherence is reached when the system parameters satisfy Omega2=1(T1T2).