Symmetry-protected creation of superposition states and entanglement using circulant Hamiltonians

We describe the use of a special interaction symmetry for the robust generation of the totally symmetric superposition state or entangled state of an $N$-state system. The required symmetry of the Hamiltonian is that of a circulant matrix. Such a matrix has the important property that its eigenstates are independent of the matrix elements as long as the circulant symmetry is maintained. One of the eigenvectors is the target superposition. By inducing a slow evolution of the Hamiltonian into the circulant form, adiabatic following will generate the desired superposition out of a convenient initial state such as a product state. The creation process is robust: it is insensitive to details of the interaction as long as the final Hamiltonian has the required symmetry. We illustrate the procedure with a simple example: a ring of quantum wells that permit interwell tunneling, into which a single atom is placed. By carrying out adiabatic evolution the state vector approaches an equal distribution of probability amplitudes in each well.

Figure 1

(Color online) Implementation of circulant Hamiltonian with ring of quantum wells (small circles) evenly spaced around a ring (large thin circle). Tunneling interactions appear as thick lines (strongest interactions, $V_1$), thin lines (weaker interactions, $V_2$) and dashed lines (weakest interactions, $V_3$). (a) Initially population is in one well, shown as dark. (b) After adiabatic passage to a smaller ring where tunneling becomes dominant, the population is evenly distributed.

Figure 2

(Color online) Top: Time variations $f\left(t\right)$ and $g\left(t\right)$. Bottom: Population evolution. After adiabatic passage to a smaller ring, the population is evenly distributed. The fields are given by Eqs. (16a, 16b) with $V_{1}T=10$, $V_{2}T=10∕\sqrt{3}$, $V_{3}T=5$, and $\Delta T=50$. The dashed line is the distance $D\left(t\right)=\mid \Psi \left(t\right)-{\Phi }_0\mid$ between the state vector $\Psi \left(t\right)$ and the target superposition state ${\Phi }_0$.