Effective Two-State Model and NOON States for Double-Well Bose-Einstein Condensates in Strong-Interaction Regime

William J. Mullin, Franck Laloë

Xiaoxue Yang, Ying Wu

The model of double-well Bose–Einstein condensates in the strong-interaction regime is shown to reduce adiabatically to an effective two-state model describing the Rabi oscillations between the two atomic Fock states |N, 0〉 and |0, N〉, and the NOON states of arbitrary ultracold atoms can therefore be generated periodically from the initial state of either one of the Fock states.

Chen, Ai-Xi, Shoulong Deng +1 more

Tania J. Haigh

This thesis considers measurements that could be used to verify the existence of a coherent superposition, using the specific example of a superposition of a Bose-Einstein condensate in a double-well. There is significant interest in superposition states in atomic condensates, due to the possibility of large numbers of atoms being involved in the superposition, and while various experimental proposals exist for generating such states none have yet been successfully verified in this system. We use a number-state description of the condensate in a double well, which is a number-conserving approach with basis states specifying the number of atoms in each well. This allows us to consider 'NOON`-type states, i.e. a superposition of the 'N,0' state of all atoms localized in the left well and the '0,N' state where all N atoms are localized in the right hand well. MATLAB is used to solve for the ground state, dynamics, and measurement expectation values for the double-well condensate in this basis. We consider measurements that would distinguish this coherent superposition from a statistical mixture of the same states. Such a suitable measurement is the parity of the atoms in one well, after a Ramsey-type interference experiment has been performed. We have found that unambiguously distinguishing even a perfect 'NOON' state from a statistical mixture using standard detection methods will present experimental difficulties. We find that interactions between condensate atoms degrade the visibility of the parity oscillations, and we also consider the effects of detector inefficiency on these parity measurements. We suggest that a 'NOON'-type superposition state in the double-well system could be generated in a reasonable time frame by making use of a Feshbach resonance to tune the inter-atomic interactions from repulsive to negative. We find that even with imperfect superpositions, parity measurements retain their frequency dependence, which is the signature of NOON-type coherence.

Bogdan Opanchuk, L. Rosales-Zárate, Run Yan Teh +1 more

We examine how to signify and quantify the mesoscopic quantum coherence of approximate two-mode NOON states and spin-squeezed two-mode Bose-Einstein condensates (BEC). We identify two criteria that verify a nonzero quantum coherence between states with quantum number different by $n$. These criteria negate certain mixtures of quantum states, thereby signifying a generalized $n$-scopic Schr\"odinger cat-type paradox. The first criterion is the correlation $\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{\ifmmode\dagger\else\textdagger\fi{}n}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}^{n}\ensuremath{\rangle}\ensuremath{\ne}0$ (here $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}$ are the boson operators for each mode). The correlation manifests as interference fringes in $n$-particle detection probabilities and is also measurable via quadrature phase amplitude and spin-squeezing measurements. Measurement of $\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{\ifmmode\dagger\else\textdagger\fi{}n}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}^{n}\ensuremath{\rangle}$ enables a quantification of the overall $n\text{th}$ order quantum coherence, thus providing an avenue for high efficiency verification of high-fidelity photonic NOON states. The second criterion is based on a quantification of the measurable spin-squeezing parameter ${\ensuremath{\xi}}_{N}$. We apply the criteria to theoretical models of NOON states in lossy interferometers and double-well trapped BECs. By analyzing existing BEC experiments, we demonstrate generalized atomic ``kitten'' states and atomic quantum coherence with $n⪆10$ atoms.

Weizhu Bao, Yongyong Cai, Xinran Ruan