Using an SU(3)/U(2) Wigner Function to Represent Noisy Spin Ensembles

Abstract

The SU(2) Wigner function represents a quantum state of a spin-$J$ as a real-valued function on the surface of a 2-sphere. For an ensemble of $N$ spin-1/2 particles, this representation is useful when the dynamics is restricted to a single SU(2) irrep, e.g., the symmetric subspace with $J=N/2$. Physically relevant noise sources tend to be local, such as spontaneous emission, depolarizing, and incoherent optical pumping, all of which transfer the state outside of the initial irrep, and as such the SU(2) Wigner function is no longer a useful representation. In this work, we address this issue by encoding a noisy spin ensemble in an SU(3) irrep, and evaluating the SU(3) Wigner function for that irrep. We find that physical constraints enforced by the noise eliminate all but three real parameters from the input to the Wigner function, which can then be interpreted as a polar, azimuthal, and radial component. This interpretation leads us to refer to the resulting Wigner function as the solid spin Wigner function, visualized on a solid ball rather than a hollow sphere.

Figures (6)

• Figure 1: Diagram of the collective state basis, with half-integer SU(2) irreps included. The symmetric subspace, which is pure and nondegenerate, is indicated using a red circle. Two examples of collective states [Eq. (\ref{['eq:coll_state']})] are labeled. Also depicted are the directions of raising operators $\hat{J}_+, \hat{T}_{1/2,1/2}^{J+1/2,J}$ and $\hat{T}_{1/2,-1/2}^{J+1/2,J}$ in this diagram, which are given by the spherical tensor operators (more precisely, given by weighted sums across $J$ of spherical tensor operators). These operators and their hermitian conjugates arise naturally as the jump operators for atom loss and gain channels Forbes2024Zhang2018.
• Figure 2: Weight diagrams of SU(3) irreducible representations labeled by Dynkin indices $(\lambda,\mu)$. Each dot represents a state in the representation, labeled by the occupation number $\boldsymbol{\nu}=(\nu_1,\nu_2,\nu_3)$. States in darker red regions have more multiplicity. The directions of the raising operators are shown on the right side of the figure. A few specific states $\ket{(\lambda,\mu);\boldsymbol{\nu}I}$ have been labeled for clarity. The highest weight state of each irrep is indicated with a red circle. Note that two states correspond to the weight labeled in the $(3,3)$ irrep, due to that weight having multiplicity 2. The multiplicity is lifted by the multiplicity label $I$, which is 2 for one of the states, and 1 for the other.
• Figure 3: Diagram representing how the $(9,0)$ irrep (a) is used to encode the vectorized collective state space $\mathcal{V}$ (b). We highlight the highest weight state $\ket{\text{h.w.s.}}$ using a red circle, and note that in $\mathcal{V}$ it gets mapped to the $\ket{0,0}$ state. The symmetric subspace of $\mathcal{V}$, where $J=N/2$, is highlighted in red. The arrow corresponding to $\beta$ shows the direction that rotations by $\beta$ move the h.w.s. (in the $\{12\}$ subgroup), and similarly the arrow corresponding to $\theta$ shows how rotations by $\theta$ move the state (in the $\{23\}$ subgroup).
• Figure 4: Diagram representing the change of variables in Eq. (\ref{['eq:beta_to_r']}) from polar angle $\beta$ to radial component $r$.
• Figure 5: Slices of solid spin Wigner functions along $\phi=0,\pi$ for angular momentum eigenstates $\ketbra{J,M}$ for $N=8$. Each plot is normalized to its own minimum and maximum value, to ensure that the features of each plot are clearly visible. Also plotted are three examples of maximally mixed states. The $N=8$ and $N=16$ maximally mixed states are those defined by Eq. (\ref{['eq:maximally_mixed']}), and are the maximally mixed states of $N$ spin-1/2 particles. Finally, the SU(3) maximally mixed state, Eq. (\ref{['eq:su3_maximally_mixed']}), is plotted. This is the state corresponding to equal weight on all states in the $(N=8,0)$ irrep of SU(3), and its Wigner function is thus constant in its parameters.