Dynamical Engineering of Interactions in Qudit Ensembles

We propose and analyze a method to engineer effective interactions in an ensemble of $d$-level systems (qudits) driven by global control fields. In particular, we present (i) a necessary and sufficient condition under which a given interaction can be decoupled, (ii) the existence of a universal sequence that decouples any (cancelable) interaction, and (iii) an efficient algorithm to engineer a target Hamiltonian from an initial Hamiltonian (if possible). We illustrate the potential of this method with two examples. Specifically, we present a 6-pulse sequence that decouples effective spin-1 dipolar interactions and demonstrate that a spin-1 Ising chain can be engineered to study transitions among three distinct symmetry protected topological phases. Our work enables new approaches for the realization of both many-body quantum memories and programmable analog quantum simulators using existing experimental platforms.

Figure 1

Schematic diagram of interaction engineering. Black solid, red dotted, and blue dashed lines indicate full interactions, isotropic components, and anisotropic components, respectively. Dotted arrows represent applications of the dynamical decoupling sequence. (a) When both source and target interactions are purely anisotropic ($s_0=s_f=0$), one directly maps interactions. (b) For interactions with both isotropic and anisotropic components, one engineers only the anisotropic component and matches the relative strength by canceling some fraction.

Figure 2

(a) Level diagram for an anharmonic three level system. (b) Decoupling sequence for spin-1 dipolar interactions. Pulse durations are indicated by rotation angles, and phase choices are color coded. (c) Numerical simulations of decoupling dipolar interactions among $N=6$ spin-1 particles. Black solid line indicates $\mathcal{F}(t)$ in the absence of the pulse sequence. Blue, red, and yellow solid lines correspond to $\mathcal{F}(t)$ under a decoupling sequence with $1/JT=3,5,10$, respectively. Dashed lines are for symmetrized sequences. (d) Two generators $\{a,x\}$ of the symmetry group $A_4$. (e) Phase diagram. Three SPT phases (I, II, and III) are distinguished by the transformation of ground state wave functions under the action of $a\in A_4$. The colored area indicates the domain of $(p,q)$ that can be engineered from Ising interactions. Blue dot indicates the AKLT point $(p,q)=(1/3,0)$.