Quantum quenches of ion Coulomb crystals across structural instabilities. II. Thermal effects

We theoretically analyze the efficiency of a protocol for creating mesoscopic superpositions of ion chains, described in Baltrusch et al. [Phys. Rev. A 84, 063821 (2011)], as a function of the temperature of the crystal. The protocol makes use of state-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. Ion Coulomb crystals are well isolated from the external environment and should therefore experience a coherent, unitary evolution, which follows the quench and generates structural Schrödinger-cat-like states. The temperature of the chain, however, introduces a statistical uncertainty in the final state. We characterize the quantum state of the crystal by means of the visibility of Ramsey interferometry performed on one ion of the chain and determine its decay as a function of the crystal's initial temperature. This analysis allows one to determine the conditions on the chain's initial state in order to efficiently perform the protocol.

Figure 1

Ramsey interferometry with an ion chain whose vibrations are at temperature $T$. A quench across the linear-zigzag instability is performed by exciting the central ion with a laser pulse in the presence of state-dependent forces. In (a) the collective motion is initially in a thermal 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) The ion in state $|e\rangle$ experiences a tighter state-dependent potential. The corresponding conditional dynamics entangle the ions' internal and external degrees of freedom. A subsequent laser pulse performs a $-\pi /2$ rotation on the ion's internal transition. The final occupation of the ground state $|g\rangle$ as a function of the time $t$ elapsed between the two pulses allows one to extract information on quantum coherence and entanglement created by these dynamics.

Figure 2

Visibility signal as a function of the elapsed time $t$ between the two Ramsey pulses for three ${}^9{\mathrm{Be}}^{+}$ ions. The signal is evaluated for temperatures $0\mu \mathrm{K}$ (black line), $5\mu \mathrm{K}$ (brown/dark gray line), $10\mu \mathrm{K}$ (green/medium gray line), $50\mu \mathrm{K}$ (blue/medium light gray line), and $100\mu \mathrm{K}$ (light pink/light gray line). The parameters are $\Delta =0.025$ and (a) $g=0.02$, (b) $g=-0.005$, (c) $g=-0.1$. The insets display magnifications of (b) the first double-peak and (c) the third peak.

Figure 3

Spectra for the logarithmic visibility, Eq. (66), for the visibility curves in Fig. 2 (using same color/grayscale). The vertical dash-dotted (green) lines show the zigzag eigenfrequency ${\omega }_1^e$ [in (a) the line is at $2{\omega }_1^e$]. The dashed (orange) line shows the location of frequency ${\omega }_{\mathrm{beat}}=|{\omega }_1^e-{\omega }_1^g|$. The insets display a magnification of the low-frequency part of the corresponding spectrum, highlighting the peak structure of the spectrum at multiples of ${\omega }_{\mathrm{beat}}$.

Figure 4

Visibility as a function of the time $t$ elapsed between the two Ramsey pulses at $T=100\mu \mathrm{K}$ and $\Delta =0.005$ (light pink), $\Delta =0.010$ (blue), $\Delta =0.015$ (green), $\Delta =0.020$ (brown), and $\Delta =0.025$ (black line). In a scale of gray, as $\Delta$ increases the line becomes darker. The quench is performed for (a) $g=0.02$ and (b) $g=-0.1$, such that the initial and quenched equilibrium structures are either both linear or zigzag chains.

Figure 5

Visibility as a function of the elapsed time $t$ for $100\mu \mathrm{K}$, $\Delta =0.005$, and (a) $g=0.02$ and (b) $g=-0.1$. The vertical lines indicate the location of ${\omega }_{\mathrm{beat}}$ (dashed orange line) and of the beating frequency between the transverse center-of-mass frequencies (dash-dotted green line). The inset in (b) shows a magnification for the interval centered about ${\omega }_{\mathrm{beat}}$.

Figure 6

Spectrum of the logarithm of the visibility, Eq. (66), for $\Delta =0.005$ and (a) $g=0.02$ (linear to linear) and (b) $g=-0.1$ (zigzag to zigzag). The curves correspond to the temperatures $T=0$ (black line), $T=5\mu \mathrm{K}$ (brown), $T=10\mu \mathrm{K}$ (green), $T=50\mu \mathrm{K}$ (blue), and $T=100\mu \mathrm{K}$ (light pink), corresponding to a scale of gray from dark to light. The inset is a magnification in the low-frequency part. The vertical dash-dotted (green) line shows the location of the zigzag eigenfrequency in panel (b) and of double the zigzag frequency in panel (a). The vertical dashed (orange) line shows the location of ${\omega }_{\mathrm{beat}}$.

Figure 7

Visibility as a function of the elapsed time for $\Delta =0.025$ and (a) $g=0.02$, (b) $g=-0.005$, and (c) $g=-0.1$. The pink and black lines correspond to the crystal initially at $T=10$ and $100\mu \mathrm{K}$. The green (dark gray) line is for the zigzag mode at $100\mu \mathrm{K}$ and all other modes cooled at $10\mu \mathrm{K}$, the blue (medium gray) line is for the zigzag mode cooled at $10\mu \mathrm{K}$ and all other modes at $100\mu \mathrm{K}$.

Figure 8

Visibility as a function of the elapsed time $t$ between the Ramsey pulses at $T=0\mathrm{K}$. The parameters are $\Delta =0.025$ and (a) $g=0.02$, (b) $g=-0.005$, and (c) $g=-0.1$. A Raman transition of the central ion is driven by two laser beams at 313 nm which are copropagating (black line), orthogonal (green/dark gray), or counterpropagating (pink line/light gray). In panels (b) and (c), the curves differ moderately from each other.