Quantum superpositions of crystalline structures

A procedure is discussed for creating coherent superpositions of motional states of ion strings. The motional states are across the structural transition linear-zigzag, and their coherent superposition is achieved by means of spin-dependent forces, such that a coherent superposition of the electronic states of one ion evolves into an entangled state between the chain's internal and external degrees of freedom. It is shown that the creation of such an entangled state can be revealed by performing Ramsey interferometry with one ion of the chain.

Figure 1

Scheme for the creation of coherent superpositions of different motional states. (a) The ions are initially in the internal state $|g\rangle$ and the collective motion is in the ground state of a zigzag structure. A resonant laser pulse (block arrow) prepares the central ion in the superposition $\frac{1}{\sqrt{2}}\left(\right|g\rangle +|e\rangle )$. (b) A state-dependent potential, acting only when the ion is in state $|e\rangle$, induces a conditional dynamics such that the ions internal and external degrees of freedom get entangled. When the state-dependent potential is sufficiently tight, the excited component will start oscillating around the equilibrium positions of the linear chain.

Figure 2

Normal mode frequencies for three ions confined in a harmonic potential, in units of the trap frequency ${\nu }_x$, as a function of the aspect ratio $\alpha ={\nu }_y/{\nu }_x$. The dashed vertical line indicates the critical value ${\alpha }_c\approx 1.5492$, separating the zigzag ($\alpha <{\alpha }_c$) from the linear array ($\alpha >{\alpha }_c$).

Figure 3

Stable configurations for a three-ion chain as a function of the aspect ratio $\alpha$ and variation $\delta \alpha$ (here due to a state-dependent potential acting on the central ion). The labels “LIN X,” “ZZ X,” and “ZZ Y” correspond, respectively, to a linear chain along the $x$ axis, a zigzag chain along the $x$ axis, and a zigzag along the $y$ axis. The solid lines are the boundaries given in Eqs. (26) and (32). The other boundary was determined numerically (the wiggled boundary is here due to the chosen numerical resolution). The variation of the state-dependent potential corresponds to a vertical displacement in the plot.

Figure 4

Overlap signal $\left|I\right(t\left)\right|$ as a function of the elapsed time $t$ (in $\mu$s) for a Ramsey setup in which the central ion of a three-ion chain is excited. The curves are evaluated for ${}^9{\mathrm{Be}}^{+}$ ions for the parameters ${\nu }_x=2\pi \times 500$ kHz, ${\nu }_y=\alpha {\nu }_x$, and ${\nu }_y^{\prime }={\nu }_y+2\pi \times 10$ kHz (corresponding to $\delta \alpha =0.02$). The values of $\alpha$ (and thus ${\nu }_y$) are given in the boxes. The equilibrium structures for ground and excited states are (a) zigzag arrays with different transverse displacements (here ${\nu }_y=2\pi \times \{725,745,765\}$ kHz); (b) a zigzag and a linear array, respectively (${\nu }_y=2\pi \times \{772.5,773.0,773.5\}$ kHz); (c) linear arrays with different transverse confinements (${\nu }_y=2\pi \times \{775.0,776.0,780.0\}$ kHz).

Figure 5

Fourier transform of the overlap signal $\left|I\right(t\left)\right|$ in Fig. 4. Here, (a) corresponds to the curve at $\alpha =1.49$, (b) to $\alpha =1.547$, and (c) to $\alpha =1.55$.

Figure 6

Same as Fig. 4 but for $\alpha =1.5470$ (${\nu }_y=2\pi \times 773.5$ kHz) and $\delta \alpha =\{0.01,0.04,0.10\}$ (corresponding to $\delta {\nu }_y=2\pi \times \{5,20,50\}$ kHz). In this case the equilibrium structures for ground and excited states are a zigzag and a linear array, respectively.

Figure 7

Same as Fig. 6 but for $\delta \alpha =0.02$ and different axial trap frequencies ${\nu }_x$ (the values are reported in the box). The value for $\alpha$ corresponds to ${\nu }_y=2\pi \times \{154.7,309.4,773.5,1547\}$ kHz, the frequency shift due to the optical potential is $\delta {\nu }_y=2\pi \times \{2,4,10,20\}$ kHz (ordering corresponds to the values of ${\nu }_x$ reported in the legends in the box, from bottom to top). The different abscissae indicate the time (in $\mu$s) for the corresponding curves.

Figure 8

Same as Fig. 6 but for $\delta \alpha =0.02$ but for different ion species.

Figure 9

Same as Fig. 3 when one outer ion experiences the state-dependent potential. The labels “LIN X*” and “TRIA*” correspond, respectively, to a linear chain along the $x$ axis, and a triangular structure. The solid lines correspond to the stability boundaries in Eqs. (32) and (A1); the others were calculated numerically. The white regions are such that there is no stable structure with one outer ion excited.