Quantum simulations of a freely rotating ring of ultracold and identical bosonic ions

We report on two types of quantum calculations pertaining to a ring of ultracold and identical bosonic ions in a static, external magnetic field. The purpose of these calculations is to give examples of what could be expected in certain types of measurements of the rotation of this ring. The first type of calculation explores how well the ring reaches the rotational ground state when it starts in the ground state of a trapping potential that is then adiabatically switched off; the trapping potential is designed to frustrate the rotation and raise the energy scale to the point where standard cooling techniques could reach the ground state. The second type of calculation shows what kind of rotation signal might be measured using two types of measuring schemes. The focus is on how the signal changes with the temperature of the ring and with how the measurement is performed.

Figure 1

Schematic of how a standing light wave can pin the ion ring. The lines represent the maxima of the light intensity for the standing wave. The ions are attracted to the light maxima. See text for discussion of the qualitative difference between even and odd numbers of ions.

Figure 2

Calculation of the $-V$ from Eq. (6) for 90 and 91 ions. The solid line is for $k=0.52N/R$. The dotted line is for $k=1.014N/R$ for $N=90$ and $1.024N/R$ for $N=91$. These values for $k$ were chosen to maximize the well depths. The potential energy is typically proportional to $-V$ for red detuned light.

Figure 3

Lowest seven eigenvalues (in units of kHz) of the Hamiltonian in Eq. (7) for different values of $\epsilon$ (in units of $\mu \mathrm{K}$). The calculations are performed for $a_0=1/4$, $R=60\mu \mathrm{m}$, and a mass of 9 amu.

Figure 4

Population in the ground (+), first excited (*), second excited (diamonds), and third excited (triangles) states as a function of the turn-off time.

Figure 5

Angular distribution of the marked ion as a function of angle for times of 0 s (dotted), 1/2 s (dashed), 1 s (dash-dot), 3/2 s (dash-dot-dot-dot), and 2 s (long dash) after the marking occurs. All of the $P$'s have been scaled by the same factor so that the maximum at $t=0$ is approximately 1. The calculations are performed for $a_0=1/4$, $R=60\mu \mathrm{m}$, and a mass of 9 amu. Using the approximation from Eq. (23) gives results indistinguishable from those plotted here.

Figure 6

Angular distribution of the marked ion as a function of angle at the time 1 s for different amounts of contamination from the first and second excited states. The population in the first excited state was $f$ times that in the ground state and the population in the second excited state was $f^2$ times the ground state. The calculations are $f=0$ (dotted), $f=0.1$ (dashed), $f=0.2$ (dot-dash), $f=0.3$ (dash-dot-dot-dot), $f=0.4$ (long dash), and $f=0.5$ (solid).