Spatial correlation functions for the collective degrees of freedom of many trapped ions

Abstract

Spatial correlation functions provide a glimpse into the quantum correlations within a quantum system. Ions in a linear trap collectively form a nonuniform, discretized background on which a scalar field of phonons propagates. Trapped ions have the experimental advantage of each having their own "built-in" motional detector: electronic states that can be coupled, via an external laser, to the ion's vibrational motion. The post-interaction electronic state can be read out with high efficiency, giving a stochastic measurement record whose classical correlations reflect the quantum correlations of the ions' collective vibrational state. Here we calculate this general result, then we discuss the long detection-time limit and specialize to Gaussian states, and finally we compare the results for thermal versus squeezed states.

Simulating quantum effects of cosmological expansion using a static ion trap

“…This is because all terms in the Dyson series are integrals involving the interaction Hamiltonian, and such integrals can always be recast from cosmic time to conformal time, resulting in a new interpretation as the response of Minkowski-spacetime detectors with timedependent frequencies, as long as the window function is also appropriately modified. This leads us to suggest the possibility of studying the two-point correlations of scalar-field modes in de Sitter spacetime by measuring the correlation function for simultaneous electronic excitations in multiple ions [28] under the detector modulation prescription proposed here. Such correlations are believed, in inflationary cosmology, to be responsible for structure formation in the early universe [13,14].…”

“…Regardless of the experimental methods needed to achieve it, the new interaction Hamiltonian remains as in Eq. (28).…”

“…This is the probability that an ion will get flipped to its excited electronic state by interaction with the field. The interaction is assumed to be weak enough to only excite one transition from |g to |e during the detection, thus allowing the use of time-dependent perturbation theory [28]. We have extended the limits to ±∞, with finite limits able to be enforced using the window function F (ζ), as required.…”

“…Because of the restriction to conformally invariant settings in the current proposal, our goals are comparatively modest. By measuring correlation functions in the detection probability for excitations in multiple ions [28] under a detector modulation corresponding to various scale factors, we can analyze the spectrum of fluctuations that should result from the corresponding cosmological predictions [13,14].…”