We investigate the generation, characterization and measurement of non-classical correlations and entanglement in ultracold atomic gases. Specifically, we propose new tests to demonstrate non-classical correlations, Einstein-Podolsky-Rosen (EPR) entanglement and Bell inequality violations, in systems involving dilute gas Bose-Einstein condensates (BECs). We focus on the challenges of generating and preserving these correlations in atom-optics schemes with massive particles and define appropriate operational measurements to demonstrate them experimentally. Further, we characterize how measures of EPR entanglement and violations of a Bell inequality evolve with time and scale with system size.
To begin, we detail a theoretical proposal to demonstrate the well-known optical HongOu-Mandel (HOM) effect of destructive quantum interference with massive particles. Our proposed matter-wave experiment, realized recently in a related experimental setup [Lopes et. al., Nature 520, 66 (2015)], utilizes pair-correlated atoms produced via spontaneous fourwave mixing in colliding BECs. The atom pairs are then subjected to Bragg pulses – the atom-optics equivalent of optical mirrors and beam-splitters – to realize a HOM atom interferometer. By taking advantage of the multimode nature of the four-wave mixing process we formulate a measurement protocol which, unlike the optical case, does not require repeated measurements for different beam-splitter settings. We perform numerical simulations of a realistic experimental system and predict a HOM ‘dip’ visibility of ∼ 69%, indicating the correlations between atom pairs are stronger than classically allowed.
In Chapter 5 we outline a theoretical proposal to demonstrate a violation of a motionalstate Bell inequality with massive particles. Identically to Chapter 4, the proposal uses pairs of momentum-entangled atoms produced via spontaneous four-wave mixing in colliding BECs. However, this scheme requires two pairs of atoms which are used as the input state of an atom-optics analog of the Rarity-Tapster interferometer, constructed via a sequence of Bragg pulses. We formulate a measurable form of the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality taking into account experimental limitations, and perform numerical simulations of a realistic experimental system for a range of parameters. We predict values of the CHSH-Bell parameter up to S ≃ 2.5, demonstrating a violation of the CHSH-Bell inequality which is bounded by S ≤ 2 for local hidden-variable theories.
In Chapter 6 we investigate the prospect of demonstrating EPR entanglement between massive particles via spin-changing collisions in a spinor BEC. This investigation is motivated by recent experimental work of Gross et al. [Nature 480, 219 (2011)] who reported inconclusive results in an attempt to measure EPR entanglement. In the experiment, spinchanging collisions between atoms in the (F, mF ) = (2, 0) state lead to pairs of strongly correlated atoms being created in opposing (F, mF ) = (2, ±1) states. For mF = ±1 states initially prepared as vaccuum, our calculations predict strong EPR entanglement. However,
we consider the possibility that the inconclusive experimental result was due to the spin-changing collisions being initiated by a small thermally populated occupation in the mF = ±1 modes. For experimentally realistic condensates of 150 − 200 atoms, we find that a thermal population as low as ¯nth ≃ 1 (currently experimentally undetectable) initially present in the mF = ±1 states is sufficient to destroy EPR entanglement.
Next, in Chapter 7 we demonstrate a practical application of the non-classical correlations studied in Chapter 6, by investigating how one can use phase-sensitive correlations generated in spin-changing collisions to realize an ‘active’ atomic interferometer. Instead of probing these correlations with passive elements, such as beam-splitters, we use the inherent nonlinearity of inter-atomic collisions in the spinor condensate to implement an active nonlinear beam-splitter. Such a scheme, known as an SU(1,1) interferometer, has been shown to have interferometric sensitivity at the Heisenberg limit. Our investigation is focused on recent experimental work in M.K. Oberthaler’s group in Heidelberg [D. Linnemann (private communication)]; in particular, we outline how novel features unique to the atomic realization, such as experimental control of the spin-changing collisions, must be well characterized to experimentally demonstrate an atomic SU(1,1) interferometer.
Lastly, in Chapter 8 we build on the phase-space techniques used in the thesis and examine the interpretation of individual stochastic trajectories in the truncated Wigner approximation as corresponding to possible outcomes of single experimental trials. Specifically, we investigate the relation between the true (measured) single-mode number distribution Pn and that obtained by discretely binning the individual stochastic realisations of squared mode amplitudes |α| 2 of the sampled Wigner distribution W(α), denoted via P˜ n. We find there is a close quantitative correspondence between Pn and P˜ n for a range of states, justifying the broadly accepted view that, for highly occupied modes, individual stochastic realisations of Wigner trajectories should approximately correspond to outcomes of single experiments. However, we also find counterexamples for which high mode occupation may not be a sufficient criterion; we find instead that a more relevant and sufficient requirement is the relative smoothness and broadness of W(α).