Relaxation-optimized transfer of spin order in Ising spin chains

In this paper, we present relaxation optimized methods for the transfer of bilinear spin correlations along Ising spin chains. These relaxation optimized methods can be used as a building block for the transfer of polarization between distant spins on a spin chain, a problem that is ubiquitous in multidimensional nuclear magnetic resonance spectroscopy of proteins. Compared to standard techniques, significant reduction in relaxation losses is achieved by these optimized methods when transverse relaxation rates are much larger than the longitudinal relaxation rates and comparable to couplings between spins. We derive an upper bound on the efficiency of the transfer of the spin order along a chain of spins in the presence of relaxation and show that this bound can be approached by the relaxation optimized pulse sequences presented in the paper.

Figure 1

The system that we study in this paper is a linear chain of $n$ weakly interacting spins $\frac{1}{2}$ with Ising coupling between the next neighbors. The coupling constant is the same for all pairs of connected spins.

Figure 2

Energy level diagram for the spin order transfer $2I_{1z}{I}_{2z}\to 2I_{2z}{I}_{3z}$. The relative vertical position of the states represents their relative energy. The dark circles represent population excess. State $\alpha$ corresponds to spin up and state $\beta$ to spin down. The configuration depicted here corresponds to initial density matrix $2I_{1z}{I}_{2z}$. After the performance of the selective population inversions shown by the arrows, the configuration achieved corresponds to final density matrix $2I_{2z}{I}_{3z}$.

Figure 3

Auxiliary variables $r_i$.

Figure 4

Optimal pulse (dashed line) calculated using a numerical optimization method based on a steepest descent algorithm, for various values of the normalized relaxation parameter $\xi =k∕\left(J\sqrt{2}\right)$. Time $t$ is measured in units $1∕\left(\pi J\sqrt{2}\right)$ and rf field ${\Omega }_y$ in units $\pi J\sqrt{2}$. The Gaussian pulse (solid line) approximates very well the optimal pulse shape and gives a similar efficiency. This suggests that instead of using the initial numerical optimization method, we can use Gaussian pulses of the form (16), optimized with respect to $A$ and $\sigma$ for each value of $\xi$.

Figure 5

Efficiency for the conventional method (CINEPT), Eq. (9), and for our method (SPORTS ROPE), for the values of the normalized relaxation parameter $\xi =k∕\left(J\sqrt{2}\right)$ shown in Table . The upper bound (12) for the efficiency is also shown.

Figure 6

Gray-scale topographic plot of the efficiency as a function of the amplitude $A$ (units $\pi J\sqrt{2}$) and the standard deviation $\sigma$ [units $1∕\left(\pi J\sqrt{2}\right)$] of the Gaussian pulse for $\xi =1$. The maximum value can be found from Table and is 0.2510. The white region corresponds to values $⩾0.24$, while the black region to values $<{\eta }_{\mathit{CI}}(\xi =1)=0.1727$. The intermediate gray regions correspond to the intervals [0.23, 0.24), [0.22, 0.23), [0.21, 0.22), [0.20, 0.21), and [0.1727, 0.20) (from white to black).

Figure 7

(a) Time evolution of the various transfer functions (expectation values of operators) participating in the transfer $z_1\to z_3$, when the optimal Gaussian pulse for $\xi =1$, shown in Fig. 4(c), is applied to system (8). Time is measured in units $1∕\left(\pi J\sqrt{2}\right)$. (b) The angle ${\theta }_3={\tan }^{-1}(z_3∕x_3)$ of the vector ${\mathit{r}}_3$ with the $x$ axis, expressed in radians, as a function of time. Observe that initially ${\mathit{r}}_3$ is parallel to the $x$ axis $({\theta }_3=0)$, but under the action of the Gaussian pulse is rotated gradually to the $z$ axis $({\theta }_3=\pi ∕2)$.