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arXiv:0901.3248

Bloch space structure of cascade, lambda and vee type of three-level systems and qutrit wave function

Surajit Sen, Mihir Ranjan Nath, Dey, Tushar Kanti, Gautam Gangopadhyay

Published Jan 21, 2009• 0 citations
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Abstract

The cascade, lambda and vee type of three-level systems are shown to be described by three different Hamiltonians in the SU(3) basis. We investigate the Bloch space structure of each configuration by solving the corresponding Bloch equation and show that at resonance, the seven-dimensional Bloch sphere ${\mathcal S}^7$ is broken into two distinct subspaces ${\mathcal S}^2{\times}{\mathcal S}^4$ due to the existence of a pair of quadratic constants. We also give a possible representation of the qutrit wave function and discuss its equivalence with the three-level system.

DOI
10.48550/arxiv.0901.3248

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The cascade, lambda and vee type of three-level systems are shown to be described by three different Hamiltonians in the SU(3) basis. We investigate the Bloch space structure of each configuration by solving the corresponding Bloch equation and show that at resonance, the seven-dimensional Bloch sphere S7{\mathcal S}^7S7 is broken into two distinct subspaces S2×S4{\mathcal S}^2{\times}{\mathcal S}^4S2×S4 due to the existence of a pair of quadratic constants. We also give a possible representation of the qutrit wave function and discuss its equivalence with the three-level system.
Abstract We investigate the structure of the Bloch space for the qutrit system by analyzing its properties and phase portraits. Within the framework of the SU(3) representation theory and transition features between various states, the qutrit system is categorized as lambda, vee, and cascade type three-level systems. Each of these systems has its own Bloch equation, resulting in distinct dynamics with unique dynamical trajectories. Expanding upon our earlier work (S. Sen, M. R. Nath, T. K. Dey and G. Gangopadhyay, Ann. Phys. 327 (2012) 224–252), which demonstrated the splitting of the Bloch sphere into two distinct sectors, in this paper we provide their explicit derivation from the dynamical perspective and examine the phase portraits of two sectors separately. These phase portraits, ranging from simple to complex patterns of curves, reveal the characteristic pattern of evolution for each sub-sector of the qutrit Bloch space.
We present a novel method to study the Bloch space of the qutrit system by examining the Bloch trajectories in it. Since such system is inherently a three-level quantum system, therefore we use the SU(3) group as the basis group to obtain the Bloch vectors of different configurations of it. The norm of the Bloch space is evaluated from the geometric consideration and also from the dynamics of the Bloch vectors and both results are found to be identical. The analysis of the dynamical evolution of the Bloch vectors reveals an additional feature that, under resonant conditions, the Bloch sphere S7\mathbb{S}^{7}S7 splits into two parts, a four-sphere S4\mathbb{S}^{4}S4 and a two-sphere S2\mathbb{S}^{2}S2. The Bloch trajectories of the two sectors across different configurations exhibit a range of simple to complex curves, highlighting the non-trivial structure of the Bloch space of the qutrit system.
We present a surprisingly simple three-dimensional Bloch sphere representation of a qutrit, i.e., a single three-level quantum system. We start with a symmetric state of a two-qubit system and relate it to the spin-1 representation. Using this representation we associate each qutrit state with a three-dimensional vector a\mathbf{a}a and a metric tensor $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}$ which satisfy $\mathbf{a}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{a}\ensuremath{\le}1$. This resembles the well known condition for qubit Bloch vectors in which case $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}=\mathbb{1}$. In our case the vector a\mathbf{a}a corresponds to spin-1 polarization, whereas the tensor $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}$ is a function of polarization uncertainties. Alternatively, a\mathbf{a}a is a local Bloch vector of a symmetric two-qubit state and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}$ is a function of the corresponding correlation tensor.
Abstract We study an analogous Bloch sphere representation of higher-level quantum systems using the Heisenberg-Weyl operator basis. We introduce a parametrization method that will allow us to identify a real-valued Bloch vector for an arbitrary density operator. Before going into arbitrary d-level (d ≥ 3) quantum systems (qudits), we start our analysis with three-level ones (qutrits). It is well known that we need at least eight real parameters in the Bloch vector to describe arbitrary three-level quantum systems (qutrits). However, using our method we can divide these parameters into four weight, and four angular parameters, and find that the weight parameters are inducing a unit sphere in four-dimension. And, the four angular parameters determine whether a Bloch vector is physical. Therefore, unlike its qubit counterpart, the qutrit Bloch sphere does not exhibit a solid structure. Importantly, this construction allows us to define different properties of qutrits in terms of Bloch vector components. We also examine the two and three-dimensional sections of the sphere, which reveal a non-convex yet closed structure for physical qutrit states. Further, we apply our representation to derive mutually unbiased bases (MUBs), characterize unital maps for qutrits, and assess ensembles using the Hilbert-Schmidt and Bures metrics. Moreover, we extend this construction to qudits, showcasing its potential applicability beyond the qutrit scenario.
Abstract We theoretically study the resonance fluorescence spectra of the lambda (Λ), vee (V) and cascade (Ξ) type three-level configurations. It is shown that each system with two detuning frequencies can be modelled using the SU(3) symmetry group to derive a generalized optical Bloch equation. For each configuration, this equation is solved to calculate the two-time correlation function by invoking the quantum regression theorem. The incoherent part of the power spectra gives the characteristic multi-peak fluorescence profiles which are different for different configurations. We also discuss how the dressed-state structure of such system can explain the origin of quintuplet profile of the fluorescent spectrum.
Geometric intuition is a crucial tool to obtain deeper insight into many concepts of physics. A paradigmatic example of its power is the Bloch ball, the geometrical representation for the state space of the simplest possible quantum system, a two-level system (or qubit). However, already for a three-level system (qutrit) the state space has eight dimensions, so that its complexity exceeds the grasp of our three-dimensional space of experience. This is unfortunate, given that the geometric object describing the state space of a qutrit has a much richer structure and is in many ways more representative for a general quantum system than a qubit. In this work we demonstrate that, based on the Bloch representation of quantum states, it is possible to construct a three dimensional model for the qutrit state space that captures most of the essential geometric features of the latter. Besides being of indisputable theoretical value, this opens the door to a new type of representation, thus extending our geometric intuition beyond the simplest quantum systems.