Adiabatic Quantum Search Scheme With Atoms In a Cavity Driven by Lasers

We propose an implementation of the quantum search algorithm of a marked item in an unsorted list of $N$ items by adiabatic passage in a cavity-laser-atom system. We use an ensemble of $N$ identical three-level atoms trapped in a single-mode cavity and driven by two lasers. In each atom, the same level represents a database entry. One of the atoms is marked by having an energy gap between its two ground states. Appropriate time delays between the two laser pulses allow one to populate the marked state starting from an initial entangled state within a decoherence-free adiabatic subspace. The time to achieve such a process is shown to exhibit the $\sqrt{N}$ Grover speedup.

Figure 1

Linkage pattern for the individual atoms. The unmarked atoms have two degenerate ground states $|g⟩$ and $|g^{\prime }⟩$. One atom is marked with the state $|g^{\prime }⟩$ shifted in energy. The laser of Rabi frequency ${\Omega }^{\prime }$ ($\Omega$) is resonant with the $g^{\prime }\mathrm{\text{−}}e$ transition for the marked (unmarked) atom(s). The cavity of Rabi frequency $G$ is resonant with the $g\mathrm{\text{−}}e$ transition.

Figure 2

(a) Coupling scheme in the cavity. The cavity $G$, laser $\Omega$, and laser ${\Omega }^{\prime }$ Rabi frequencies are featured by thick, thin, and dashed arrows, respectively. (b) Equivalent scheme where the states $|g_i^{\prime },0⟩$ ($|e_i,0⟩$), $i=1$, $N-1$, of frame (a) form the collective unmarked ground state $|g_u^{\prime },0⟩$ (excited state $|e_u,0⟩$). The effective cavity Rabi frequency to the collective unmarked excited state is $\sqrt{N-1}G$.

Figure 3

Dynamics of the effective Hamiltonian for $N=8$, $\epsilon =0.05$ and a Rabi frequency with a Gaussian profile $\Omega (t)={\Omega }_{\mathrm{peak}}{e}^{-(t/T{)}^2}$, with ${\Omega }_{\mathrm{peak}}T=\sqrt{N-1}/(\epsilon \sqrt{\pi })$. ${\Omega }^{\prime }(t)$ is determined from (15). Upper panel: Rabi frequencies $\Omega (t)$ and ${\Omega }^{\prime }(t)$. Lower panel: Populations $P_u(t)\equiv |⟨g_u^{\prime },0|\varphi ⟩(t){|}^2$ of the collective unmarked state, $P_N(t)\equiv |⟨g_N^{\prime },0|\varphi ⟩(t){|}^2$ of the marked state, and $P_0(t)\equiv |⟨{\gamma }_0,0|\varphi ⟩(t){|}^2$ of the superposition of excited states, where $|\varphi ⟩(t)$ is the dynamical state vector.