Turning off the $J$-Coupling

The coupling between the nuclear spins in a molecule cannot be physically turned off. But for QIP, we need to be able to maintain a state in memory and to couple qubits selectively. Fortunately, NMR spectroscopists solved this problem well before the development of modern quantum information concepts. The idea is to use the control of single spins to cancel the interaction's effect over a given period. This technique is called refocusing and requires applying a $180^\circ$ pulse to one of two coupled spins at the midpoint of the desired period. To understand how refocusing works, consider again the visualization of Fig. 5. A general state is in a superposition of the four logical states of the two spins. By linearity, it suffices to consider the evolution with spin $\mathsf {1}$ being in one of its two logical states, up or down, along the $z$-axis. Suppose we wish to remove the effects of the coupling over a period of $2{\mathchoice{\mbox{ms}}{\mbox{ms}}{\mbox{\small ms}}{\mbox{\tiny ms}}}\,$. To do so, wait $1{\mathchoice{\mbox{ms}}{\mbox{ms}}{\mbox{\small ms}}{\mbox{\tiny ms}}}\,$. In a sequence of pulses, this waiting period is called a $1{\mathchoice{\mbox{ms}}{\mbox{ms}}{\mbox{\small ms}}{\mbox{\tiny ms}}}\,$ ``delay''. The effect on spin $\mathsf {2}$ in its rotating frame is to precess counterclockwise if spin $\mathsf {1}$ is up, and clockwise for the same angle if spin $\mathsf {1}$ is down. Now, apply a pulse that rotates spin $\mathsf {1}$ by $180^\circ$ around the $x$-axis. This is called an ``inversion'', or in the current context, a ``refocusing'' pulse. It exchanges the up and down states. For the next $1{\mathchoice{\mbox{ms}}{\mbox{ms}}{\mbox{\small ms}}{\mbox{\tiny ms}}}\,$, the effect of the coupling on spin $\mathsf {2}$ is to undo the earlier rotation. At the end of the second $1{\mathchoice{\mbox{ms}}{\mbox{ms}}{\mbox{\small ms}}{\mbox{\tiny ms}}}\,$ delay, one can apply another $180^\circ$ pulse to reverse the inversion and recover the initial state. The pulse sequence is depicted in Fig. 6.

FIG. 6: Pulse sequence for refocusing the coupling. The sequence of events is shown with time running from left to right. The two spins' lifelines are shown in blue, and the RF power targeted at each spin is indicated by the black line above. Pulses are applied to spin $\mathsf {1}$ only, as indicated by the rectangular rises in RF power at $1{\mathchoice{\mbox{ms}}{\mbox{ms}}{\mbox{\small ms}}{\mbox{\tiny ms}}}\,$ and $2{\mathchoice{\mbox{ms}}{\mbox{ms}}{\mbox{\small ms}}{\mbox{\tiny ms}}}\,$. The axis for each pulse is given with the pulse. The angle is determined by the area under the pulse and is also given explicitly. Ideally for pulses of this type, the pulse times (the widths of the rectangles) should be zero. In practice, for hard pulses, they can be as small as $\approx .01{\mathchoice{\mbox{ms}}{\mbox{ms}}{\mbox{\small ms}}{\mbox{\tiny ms}}}\,$. Any ${\sigma_z}^{({\mathsf {1}})}{\sigma_z}^{({\mathsf {2}})}$ coupling's effect is refocused by the sequence shown, so that the final state of the two spins is the same as the initial state. The axis for the pair of refocusing pulses can be changed to any other axis in the plane.

Turning off couplings between more than two nuclear spins can be quite complicated unless one takes advantage of the fact that non-adjacent nuclear spins tend to be relatively weakly coupled. Methods that scale polynomially with the number of nuclear spins and that can be used to selectively couple pairs of nuclear spins can be found in [18,19]. These techniques can be used in other physical systems where couplings exist that are difficult to turn off directly. An example is qubits represented by the state of one or more electrons in tightly packed quantum dots.