Quantum quenches of ion Coulomb crystals across structural instabilities

Quenches in an ion chain can create coherent superpositions of motional states across the linear-zigzag structural transition. The procedure has been described in Baltrusch et al. [Phys. Rev. A 84, 063821 (2011)] and makes use of spin-dependent forces, so 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. The properties of the crystalline state so generated are theoretically studied by means of Ramsey interferometry on one ion of the chain. An analytical expression for the visibility of the interferometric measurement is obtained for a chain of an arbitrary number of ions and as a function of the time elapsed after the quench. Sufficiently close to the linear-zigzag instability the visibility decays very fast, but exhibits revivals at the period of oscillation of the mode that drives the structural instability. These revivals have a periodicity that is independent of the crystal size, and they signal the creation of entanglement by the quantum quench.

Figure 1

A quench across the linear-zigzag instability is performed by exciting the central ion with a laser pulse in the presence of spin-dependent forces. In (a) the collective motion is initially in the ground state of a zigzag structure, and the central ion is in the internal state $|g\rangle$. A $\pi /2$ laser pulse prepares it in the superposition $\left(\right|g\rangle +|e\rangle )/\sqrt{2}$. (b) A tighter state-dependent potential, acting only when the ion is in state $|e\rangle$, induces 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. (c) A laser pulse performs a $-\pi /2$ rotation on the central ion. The final occupation of the ground state $|g\rangle$ as a function of the time between the two pulses contains information on the properties of the chain across the linear-zigzag instability.

Figure 2

Visibility signal as a function of the time $t$ elapsed between the Ramsey pulses for three ${}^9{\mathrm{Be}}^{+}$ ions with an axial trap frequency of ${\nu }_x=2\pi \times 1\mathrm{MHz}$ and the following transverse frequencies: (a) ${\nu }_y=2\pi \times 1.565\mathrm{MHz}$, (b) ${\nu }_y=2\pi \times 1.545\mathrm{MHz}$, (c) ${\nu }_y=2\pi \times 1.470\mathrm{MHz}$. The critical value is ${\nu }_c=2\pi \times 1.549\mathrm{MHz}$. The frequency of the transverse potential for the central ion when it is excited is $\sqrt{{\nu }_y^2+{\nu }_{\mathrm{dip},y}^2}$, where ${\nu }_{\mathrm{dip},y}=2\pi \times 245\mathrm{kHz}$.

Figure 3

Phase diagram for three ions as a function of the dimensionless parameters $g$ and $\Delta$. The solid red line separates the parameter regions where the ions form a zigzag (bottom left) or a linear chain (top right). The crystalline structure corresponding to state $|g\rangle$ is at $\Delta =0$. Three regimes are considered, depending on where the state before and after the pulse are located in the diagram: When they are both in the left region, $g<g_c$, the chain is in a zigzag structure, when they are both in the right region, $g>0$, the chain is linear, while when they are across the line, $g_c<g<0$, the initial ground state is a zigzag and the final is a linear chain.

Figure 4

Curvature $\eta$ of the visibility signal at short time [Eq. (3.48)] as a function of $g$ and $\Delta$. Subplot (b) shows the corresponding contour plot. The curve has been evaluated for three ${}^9{\mathrm{Be}}^{+}$ ions with axial trap frequency ${\nu }_x=2\pi \times 1\mathrm{MHz}$.

Figure 5

Density plot of the visibility signal as a function of $g$ and $t$ (in $\mu$s) for $\Delta =0.025$. The other parameters are the same as in Fig. 4. The inset enlarges the region about $g_c$ for short elapsed times.

Figure 6

(a) Times (in $\mu$s) corresponding to the first peak of the visibility as a function of $g$ for different values of $\Delta =0.025$ (symbol: $▵$), 0.015 ($\square$), 0.005 ($◯$). The dashed vertical lines denote the different values of $g_c$. (Inset) The same curves are plotted as a function of $g-g_c\left(\Delta \right)$. The other parameters are the same as in Fig. 4. (b) Same as (a) but for a number of ions $N$ = 3 ($◯$), 5 ($◊$), 7 ($▵$), 9 ($\square$), 11 ($\times$) with $\Delta =0.025$. In the inset the curves are plotted as a function of $g-g_c\left(N\right)$ with the time in units of $1/{\nu }_c\left(N\right)$.

Figure 7

The spectrum of the logarithm of the visibility, ${\mathcal{F}}_L\left({\omega }_n\right)$ in Eq. (4.2), corresponding to the plots in Fig. 2. The vertical solid red lines indicate the normal-mode frequencies of the crystal when the central ion is excited, the dashed green lines give the doubled frequencies. The dash-dotted magenta lines show sums of frequencies. The insets show the corresponding spectrum of the visibility, $\mathcal{F}\left({\omega }_n\right)$ in Eq. (4.1). The parameter $g$ is, from left to right, $0.02,-0.005$, and $-0.1$, while the critical value is $g_c=-0.0165$ and $\Delta =0.025$.

Figure 8

Density plot of the logarithmic spectrum ${\mathcal{F}}_L\left(\omega \right)$ of the visibility for the parameters of Fig. 4, $\Delta =0.025$, and $g<0$. The dashed line shows the critical value $g_c=-0.0165$.