We provide a one-to-one map between the spin correlations and certain three-dimensional shapes, analogous to the map between single spins and Bloch vectors, and demonstrate its utility. Much as one can reason geometrically about dynamics using a Bloch vector---e.g., a magnetic field causes it to precess and dissipation causes it to shrink---one can reason similarly about the shapes we use to visualize correlations. This visualization demonstrates its usefulness by unveiling the hidden structure in the correlations. For example, seemingly complex correlation dynamics can be described as simple motions of the shapes. We demonstrate the simplicity of the dynamics, which is obscured in conventional analyses, by analyzing several physical systems of relevance to cold atoms.We provide a one-to-one map between the spin correlations and certain three-dimensional shapes, analogous to the map between single spins and Bloch vectors, and demonstrate its utility. Much as one can reason geometrically about dynamics using a Bloch vector---e.g., a magnetic field causes it to precess and dissipation causes it to shrink---one can reason similarly about the shapes we use to visualize correlations. This visualization demonstrates its usefulness by unveiling the hidden structure in the correlations. For example, seemingly complex correlation dynamics can be described as simple motions of the shapes. We demonstrate the simplicity of the dynamics, which is obscured in conventional analyses, by analyzing several physical systems of relevance to cold atoms.

arXiv:1612.06459Physical Review AApr 6, 20187 citations

Bloch Sphere of the Qutrit System

Surajit Sen, Dey, Tushar Kanti

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 $\mathbb{S}^{7}$ splits into two parts, a four-sphere $\mathbb{S}^{4}$ and a two-sphere $\mathbb{S}^{2}$ . 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 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 $\mathbb{S}^{7}$ splits into two parts, a four-sphere $\mathbb{S}^{4}$ and a two-sphere $\mathbb{S}^{2}$. 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.

arXiv:2411.16480Nov 25, 20240 citations

Bloch oscillations in free space?

John N. Churchill, F.E. Holmstrom

Physics Letters AJan 1, 19904 citations

A Quantum Dynamics Tutorial: Visualising Dynamical Decoupling Sequences on the Bloch Sphere

Charlie J. Patrickson, Isaac Bailey, Antonio Orazzo +2 more

Dynamical decoupling protocols provide a versatile toolbox for robust control of quantum systems, suppressing noise that would otherwise limit their performance. This tutorial aims to provide an intuitive interpretation of these protocols. Written from an experimentalist's perspective, we use the Bloch sphere picture to visualise quantum dynamics for a range of sequences. Here each dynamical decoupling operation can be viewed as an "effective field vector", whilst the phase and population of the two-level system is encoded into a "Bloch vector". The two-level dynamics are calculated by the cross product between these two vectors, such that the quantum state represented by the Bloch vector rotates around the effective field vector. Several prominent control schemes are simulated using this picture, starting with a Rabi oscillation, followed by the Ramsey sequence and a selection of pulsed and continuous dynamical decoupling techniques. These sequences are engineered to preserve control of the quantum system for as long as possible, with their efficacy often quantified by a coherence time. We clarify the definition of the longitudinal $T_1$ , transverse $T_2$ and inhomogeneous $T_2^*$ coherence times, which are used across the literature. Each simulation has an accompanying animation to illustrate the protocol's two-level dynamics on the Bloch sphere. The custom simulation and Bloch sphere plotting scripts are also provided in an open access repository to allow the reader to reproduce, modify, and explore these results. Dynamical decoupling sequences have established themselves as an invaluable tool across the entire quantum technologies remit; by providing an open framework alongside this tutorial, we aim to make these protocols more accessible to new researchers discovering this thriving field.Dynamical decoupling protocols provide a versatile toolbox for robust control of quantum systems, suppressing noise that would otherwise limit their performance. This tutorial aims to provide an intuitive interpretation of these protocols. Written from an experimentalist's perspective, we use the Bloch sphere picture to visualise quantum dynamics for a range of sequences. Here each dynamical decoupling operation can be viewed as an "effective field vector", whilst the phase and population of the two-level system is encoded into a "Bloch vector". The two-level dynamics are calculated by the cross product between these two vectors, such that the quantum state represented by the Bloch vector rotates around the effective field vector. Several prominent control schemes are simulated using this picture, starting with a Rabi oscillation, followed by the Ramsey sequence and a selection of pulsed and continuous dynamical decoupling techniques. These sequences are engineered to preserve control of the quantum system for as long as possible, with their efficacy often quantified by a coherence time. We clarify the definition of the longitudinal $T_1$, transverse $T_2$ and inhomogeneous $T_2^*$ coherence times, which are used across the literature. Each simulation has an accompanying animation to illustrate the protocol's two-level dynamics on the Bloch sphere. The custom simulation and Bloch sphere plotting scripts are also provided in an open access repository to allow the reader to reproduce, modify, and explore these results. Dynamical decoupling sequences have established themselves as an invaluable tool across the entire quantum technologies remit; by providing an open framework alongside this tutorial, we aim to make these protocols more accessible to new researchers discovering this thriving field.

quant-pharXiv:2608.10914Aug 12, 20260 citations

Bloch-vector model for dressed states of resonant Raman interaction

M. S. Shahriar, Philip Hemmer

It is well known that the resonant Raman interaction optically pumps atoms into a non-absorbing dressed-state.1 When this state is long-lived compared to the optical Rabi frequencies and the decay rate of the intermediate state, many interesting dynamics arise. These dynamics are significant in applications such as Raman clock AC–Stark shifts, microwave phase-sensitive optical absorption,2 and Raman cooling. By using appropriate approximations, we show that these dynamics can be well described with a Bloch-vector model, which involves only ground-state populations and coherence. This ground-state Bloch-vector model resembles the Bloch-vector model of a two-level system, but it has subtly different dynamics. Furthermore, we construct a novel dressed-state Bloch-vector that clearly shows the analogy between conventional optical pumping and coherent Raman trapping. In this model, one dressed-state is a nonabsorbing state, and the other is an absorbing state. We use the dressed-state Bloch vector to show how Raman AC–Stark shift arises and how it depends on experimental parameters. We also show how the microwave phase-sensitive optical absorption can be quantitatively calculated in a simple way.It is well known that the resonant Raman interaction optically pumps atoms into a non-absorbing dressed-state.1 When this state is long-lived compared to the optical Rabi frequencies and the decay rate of the intermediate state, many interesting dynamics arise. These dynamics are significant in applications such as Raman clock AC–Stark shifts, microwave phase-sensitive optical absorption,2 and Raman cooling. By using appropriate approximations, we show that these dynamics can be well described with a Bloch-vector model, which involves only ground-state populations and coherence. This ground-state Bloch-vector model resembles the Bloch-vector model of a two-level system, but it has subtly different dynamics. Furthermore, we construct a novel dressed-state Bloch-vector that clearly shows the analogy between conventional optical pumping and coherent Raman trapping. In this model, one dressed-state is a nonabsorbing state, and the other is an absorbing state. We use the dressed-state Bloch vector to show how Raman AC–Stark shift arises and how it depends on experimental parameters. We also show how the microwave phase-sensitive optical absorption can be quantitatively calculated in a simple way.

Annual Meeting Optical Society of AmericaJan 1, 19900 citations

\mathbb{S}^{7}

splits into two parts, a four-sphere

\mathbb{S}^{4}

and a two-sphere

\mathbb{S}^{2}

. 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.

T_1

, transverse

T_2

and inhomogeneous

T_2^*

coherence times, which are used across the literature. Each simulation has an accompanying animation to illustrate the protocol's two-level dynamics on the Bloch sphere. The custom simulation and Bloch sphere plotting scripts are also provided in an open access repository to allow the reader to reproduce, modify, and explore these results. Dynamical decoupling sequences have established themselves as an invaluable tool across the entire quantum technologies remit; by providing an open framework alongside this tutorial, we aim to make these protocols more accessible to new researchers discovering this thriving field.