Visualizing operators of coupled spin systems

The state of quantum systems, their energetics, and their time evolution is modeled by abstract operators. How can one visualize such operators for coupled spin systems? A general approach is presented that consists of several shapes representing linear combinations of spherical harmonics. It is applicable to an arbitrary number of spins and can be interpreted as a generalization of Wigner functions. The corresponding visualization transforms naturally under nonselective spin rotations as well as spin permutations. Examples and applications are illustrated for the case of three spins 1/2.

Figure 1

Visualization of an arbitrary $8\times 8$ matrix $A$ whose complex matrix elements have been randomly chosen. The matrix $A$ corresponds to an operator acting on a system of three spins $1/2$. In this particular example, $A$ is not Hermitian. Each droplet is associated with a specific linear combination $f_A^{\left(\ell \right)}$ of spherical harmonics [see Eq. (4b)]. The labels $\ell \in L$ are defined in Table 1.

Figure 2

Examples of characteristic droplets for Cartesian product operators [3]. The red (dark gray) and cyan (light gray) colors refer, respectively, to positive and negative values of the spherical functions. The droplets for most of the remaining Cartesian product operators can be obtained by rotating the displayed ones. The droplets for $4I_{1y}{I}_{2x}{I}_{3x}$ differ from the ones for $4I_{1x}{I}_{2y}{I}_{3x}$ only in an inversion of color for the ${\tau }_3^{\left[3\right]}$ component [74].

Figure 3

Nontrivial droplets for coupling Hamiltonians: Ising-ZZ (or Heisenberg-Ising) model ${\mathcal{H}}_{\mathrm{long}}$, Heisenberg-XXX model ${\mathcal{H}}_{\text{iso}}$, Heisenberg-XX model ${\mathcal{H}}_{\mathrm{plan}}$, and dipolar coupling ${\mathcal{H}}_{\mathrm{dip}}$ as detailed in the text [74].

Figure 4

Example of an NMR pulse sequence creating triple-quantum coherences, which consists of a $\pi /2$ pulse (with phase $x$) followed by a delay and a second $\pi /2$ pulse (with phase $y$). The upper left panel shows (the traceless part of) the initial density operator $\rho \left(t_0\right)=I_{1z}+I_{2z}+I_{3z}$. The final density operator $\rho \left(t_3\right)=4I_{1y}{I}_{2x}{I}_{3x}+4I_{1x}{I}_{2y}{I}_{3x}+4I_{1x}{I}_{2x}{I}_{3y}$ is given in the upper right panel. The effective Hamiltonian ${\mathcal{H}}_{\mathrm{eff}}$ [3] and the effective propagator $U_{\mathrm{eff}}$ for the pulse sequence are shown in the lower left and lower right panels, respectively. Since $\rho \left(t_0\right), \rho \left(t_3\right)$, and ${\mathcal{H}}_{\mathrm{eff}}$ are Hermitian, the corresponding droplets feature only the colors red (dark gray) and cyan (light gray). Note that $U_{\mathrm{eff}}$ is not Hermitian and hence further colors appear in its DROPS representation.

Figure 5

Visualizations of density matrices for separable and entangled pure states. The upper panels show the product state $|00\rangle$ (as a separable state) and the maximally entangled Bell state $|{\Phi }^{+}\rangle$ for two spins. In the lower panels, the $W$ state $|W\rangle$ and the Greenberger-Horne-Zeilinger state $|\text{GHZ}\rangle$ are shown as entangled quantum states for three spins [74].

Figure 6

DROPS visualization based on multipole tensor operators of the density matrices for the pure states of Fig. 5 [74].

Figure 7

Different choices $(a_0,a_1,a_2)$ for the signs of the bilinear tensors $T_0^{\{k,l\}}, T_1^{\{k,l\}}$, and $T_2^{\{k,l\}}$ with rank 0, 1, and 2 result in different visualizations for the operators (a) $2I_{kz}{I}_{lz}$, (b) $2I_{kx}{I}_{lx}+2I_{ky}{I}_{ly}$, and (c) $2I_{kx}{I}_{ly}$. The red (dark gray) and the cyan (light gray) color [74] refers to the positive and the negative value of the droplet function $f^{\{k,l\}}\left(A\right)$, respectively. The signs for the LISA basis are given in the last column, which is highlighted by a dashed rectangle.

Figure 8

DROPS visualization in the LISA basis of characteristic multiple-quantum coherences for three coupled spins. The operators are classified according to their linearity and their coherence order $p\in \mathbb{Z}$. For visualization purposes, the droplets that are always empty are not displayed. The four trilinear droplets are ordered from left to right according to the symmetry of each droplet, i.e., from ${\tau }_1^{\left[3\right]}$ to ${\tau }_4^{\left[3\right]}$ [see Eq. (B1)]. The above pictures correspond to the DROPS visualization of the tensors after normalization, where the droplets for the trilinear operators are depicted in a smaller size. For definitions of the zero-, double-, and triple-quantum operators ${\left({\mathit{ZQ}}_{\eta }\right)}_{kl}, {\left({\mathit{DQ}}_{\eta }\right)}_{kl}$, and ${\left({\mathit{TQ}}_{\eta }\right)}_{kl}$ with $\eta \in \{x,y\}$ refer to [3, 145].

Figure 9

Bilinear Cartesian product operator $2I_{kx}{I}_{ly}$ represented as a linear combination of double- and zero-quantum operators ${\left({\mathit{DQ}}_y\right)}_{kl}$ and ${\left({\mathit{ZQ}}_y\right)}_{kl}$ [74].

Figure 10

Experimental NMR pulse sequence to create triple-quantum coherences starting from the thermal equilibrium density operator in the high-temperature limit [3]. The pulse sequence consists of a ${90}^{\circ }$ pulse (with phase $x$) followed by a delay ($t_2-t_1$) and a second ${90}^{\circ }$ pulse (with phase $y$). The density operators $\rho \left(t_i\right)$ are depicted in the middle row. The Hamiltonians $\mathcal{H}(t_i,t_{i+1})$ (which are rescaled for better visibility) are given in the second row and the effective Hamiltonian [3] of the experiment is shown at the top. In the fourth row, DROPS representations of the propagators associated with the individual time steps are displayed. The effective propagator is visualized at the bottom.

Figure 11

LISA representation of the density matrix for characteristic entangled and separable pure states, as detailed in the text [74]. Compare to Fig. 12.

Figure 12

DROPS visualization in the multipole tensor basis for the examples of Fig. 11 [74].