Ramsey interferometry with a spin embedded in a Coulomb chain

We show that the statistical properties of a Coulomb crystal can be measured by means of a standard interferometric procedure performed on the spin of one ion in the chain. The ion spin, constituted by two internal levels of the ion, couples to the crystal modes via spatial displacement induced by photon absorption. The loss of contrast in the interferometric signal allows one to measure the autocorrelation function of the crystal observables. Close to the critical point, where the chain undergoes a second-order phase transition to a zigzag structure, the signal gives the behavior of the correlation function at the critical point.

Figure 1

(Color online) Ramsey interferometry with a chain. (I) A transverse laser pulse prepares one ion of the chain in a superposition of the internal states $\mid g⟩$ and $\mid e⟩$. The mechanical effect, associated with the absorption of a laser photon and conditioned to the ion being in state $\mid e⟩$, displaces the ion transversally and excites the modes of the chain. (II) The chain is let freely evolve for a time $t$. (III) A second laser pulse, addressing the same ion of the chain, is sent in the opposite direction of the first pulse. The probability that the ion is in state $\mid g⟩$ is then measured as a function of $t$.

Figure 2

(a) Visibility $\mathcal{V}$ as a function of the time elapsed between two Ramsey pulses, in units of $1∕{\omega }_0$, for $N=100$ ions in a linear chain. The parameters are ${\nu }_t={\nu }_t^{\left(c\right)}+\Delta$, with $\Delta ={10}^{-1}{\omega }_0$, and ${\eta }^{\left(c\right)}=0.25$. (b) Fourier spectrum $\mathcal{F}\left(\omega \right)$ of the visibility $\mathcal{V}\left(t\right)$, obtained by computing numerically the discrete Fourier transform of $\mathcal{V}\left(t\right)$ in a finite time interval $t∊[-T_F∕2;T_F∕2]$ with $T_F={10}^4∕{\omega }_0$. The interval is sampled with a time slicing $dt=T_F∕n_s$ with $n_s={10}^5$. We checked convergence of the spectrum amplitude $\mathcal{F}$ by increasing $T_F$ and $n_s$. The spectrum is normalized such that $\mathcal{F}(\omega =0)=1$. The dashed lines indicate the minimum and maximum frequencies of the transverse spectrum (${\omega }_{\min }=\sqrt{2\Delta {\nu }_t+{\Delta }^2}$ and ${\omega }_{\max }={\nu }_t$).

Figure 3

Same as in Fig. 2 except for ${\nu }_t={\nu }_t^{\left(c\right)}+\Delta$ with $\Delta ={10}^{-4}{\omega }_0$.

Figure 4

(Color online) Function $A\left(t\right)$ as a function of the time elapsed between two Ramsey pulses (in units of $1∕{\omega }_0$). The solid line corresponds to Eq. (37) and the symbols correspond to the short-time expansion in Eq. (40). The curve has been evaluated for ${\eta }^{\left(c\right)}=0.05$ in a linear chain of $N={10}^3$ ions.

Figure 5

(Color online) Coefficient $\Gamma$ (in units of ${\omega }_0^2$) as a function of $\Delta ={\nu }_t-{\nu }_t^{\left(c\right)}$ (in units of ${\omega }_0$). The solid line corresponds to the coefficient $\Gamma$, given in Eq. (41), and the symbols correspond to the value ${\Gamma }_{\text{fit}}$, obtained by fitting the function $A\left(t\right)$ in Eq. (37) with the function ${\Gamma }_{\text{fit}}{t}^2$. The parameters are the same as in Fig. 4. Inset: Derivative of $\Gamma$ with respect to $\Delta$ (in units of ${\omega }_0$) as a function of $\Delta$ (in units of ${\omega }_0$ and logarithmic scale) for $\Delta >0$ (symbols). An analytical analysis of the asymptotic behavior, as well as a numerical fit of the curve give a logarithmic dependence of the derivative on $\Delta$ for $\Delta \to 0^{+}$ (line).

Figure 6

(Color online) Behavior of the visibility signal for long times. The exact result for the visibility $\mathcal{V}$ (solid) obtained using Eqs. (36, 37) and the approximate function $\exp [-{\tilde{A}}_{\infty }-\tilde{B}\left(t\right)]$ (symbols) [see Eqs. (47, 48)] as a function of time (in units of $1∕{\omega }_0$) for $\Delta ={10}^{-3}{\omega }_0$ (linear chain) with $N={10}^3$ and ${\eta }^{\left(c\right)}=0.25$. The vertical dashed line corresponds to an estimate of the revival time according to the formula in Eq. (43), which gives $t^{*}=1229∕{\omega }_0$. Inset: The same plot in a smaller region for a closer comparison between $\mathcal{V}$ and the approximated expression $\exp [-{\tilde{A}}_{\infty }-\tilde{B}\left(t\right)]$.

Figure 7

The asymptotic value of the correlation function, as given by $A_{\infty }$ in Eq. (45), as a function of $\Delta$ when the linear chain is close to the critical point, where it makes a transition to the zigzag structure. The symbols correspond to the curve given by Eq. (45), and the solid line is the curve obtained from the analytical prediction in Eq. (47). The parameters are $N={10}^3$ and ${\eta }^{\left(c\right)}\simeq 0.05$.