%******************************************************************************* %*********************************** Seventh Chapter ***************************** %******************************************************************************* \chapter{Summary and Conclusions} %Title of the Seventh Chapter \ifpdf \graphicspath{{Chapter7/Figs/Raster/}{Chapter7/Figs/PDF/}{Chapter7/Figs/}} \else \graphicspath{{Chapter7/Figs/Vector/}{Chapter7/Figs/}} \fi Quantum optics of quantum gases explores the ultimate quantum regime of light-matter interactions where both the optical and matter fields are fully quantised. It provides a very rich system in which new phenomena can be observed, engineered, and controlled beyond what would be possible in condensed matter. Combined with rapid and promising experimental progress in this field the theoretical proposals have the potential of directing the research in the foreseeable future \cite{baumann2010, wolke2012, schmidt2014, klinder2015, landig2016}. In this thesis we focused on the coupling between global quantised optical fields and an ultracold bosonic quantum gas. By considering global fields as opposed to localised light-matter interactions we were able to introduce several nonlocal properties to the Hamiltonian in a controllable manner which would otherwise be impossible to implement. We showed how this can be useful in the context of nondestructive probing by showing that it can easily distinguish between a highly delocalised quantum state such as a superfluid and insulating states such as the Mott insulator and the Bose glass phases which is currently a challenge \cite{derrico2014}. Furthermore, we have seen how the correlation length, which would be inaccessible in localised measurements, was immediately visible in our scheme and lead to an angular scattering pattern that was far richer than it was for the classical case. This is best highlighted by the fact that it would be visible even when classically no light would scatter coherently at all. More interestingly, the global nature of the measurement was also capable of creating such long-range correlations itself when we considered measurement backaction. This was most visible when we saw how weak measurement was capable of driving global macroscopic multimode oscillations between different spatial modes, such as odd and even sites, across the whole lattice which could be used for quantum information and metrology. Such dynamical states show spatial density-density correlations with nontrivial periods and long-range coherence, thus having supersolid properties, but as an essentially dynamical version. Furthermore, the tunability of the optical arrangement meant that we had extreme flexibility in choosing our observables, effectively tailoring the long-range entanglement and correlations in the system. We have also shown how global measurement when combined with both atomic tunnelling and interactions leads to highly nontrivial dynamics in which backaction can either compete or cooperate with on-site repulsion in squeezing the atomic variables. In the limit of strong measurement when quantum Zeno dynamics occurs we showed that these nonlocal spatial modes created by the global measurement lead to long-range correlated tunnelling events whilst suppressing any other dynamics between different spatial modes of the measurement. Such globally paired tunneling due to a fundamentally novel phenomenon can enrich physics of long-range correlated systems beyond relatively shortrange interactions expected from standard dipole-dipole interactions \cite{sowinski2012, omjyoti2015}. These nonlocal high-order processes entangle regions of the optical lattice that are disconnected by the measurement. Using different detection schemes, we showed how to tailor density-density correlations between distant lattice sites. Quantum optical engineering of nonlocal coupling to environment, combined with quantum measurement, can allow the design of nontrivial system-bath interactions, enabling new links to quantum simulations \cite{stannigel2013} and thermodynamics \cite{erez2008}. Interestingly, these dynamics also provide a link to non-Hermitian quantum mechanics as this regime of measurement can be accurately described with a non-Hermitian Hamiltonian. Furthermore, we show that this allows for a rather novel type of competition between measurement and tunnelling where both processes actually cooperate to produce a steady state in which tunnelling is suppressed by destructive matter-wave interference. A unique feature of our global measurement scheme meant that we could couple directly to the phase observables of the system by coupling to the interference between the lattice sites, which represents the shortest meaningful distance in an optical lattice, rather than their on-site density. This defines most processes in optical lattices. For example, matter-field phase changes may happen not only due to external gradients, but also due to intriguing effects such quantum jumps leading to phase flips at neighbouring sites and sudden cancellation of tunneling \cite{vukics2007}, which should be accessible by this method. Furthremore, in mean-field one can measure the matter-field amplitude (which is also the order parameter), quadratures and their squeezing. This can link atom optics to areas where quantum optics has already made progress, e.g., quantum imaging \cite{golubev2010, kolobov1999}, using an optical lattice as an array of multimode nonclassical matter- field sources with a high degree of entanglement for quantum information processing. We have also shown how this scheme of coupling to phase observables can be used in the context of quantum measurement backaction to achieve a new degree of control. We used this result to show a generalisation of weak measurement on dynamical systems by showing that there is now a new class of projections available even when the measurement is not a compatible observable of the Hamiltonian. This an interesting result as the projections themselves are unlike those postulated by the Copenhagen interpretation, those present in quantum Zeno dynamic, or even those possible to engineer using dissipative methods. In this thesis we have covered significant areas of the broad field that is quantum optics of quantum gases, but there is much more that has been left untouched. Here, we have only considered spinless bosons, but the theory can also been extended to fermions \cite{atoms2015, mazzucchi2016, mazzucchi2016af} and molecules \cite{LP2013} and potentially even photonic circuits \cite{mazzucchi2016njp}. Furthermore, the question of quantum measurement and its properties has been a subject of heated debate since the very origins of quantum theory yet it is still as mysterious as it was at the beginning of the $20^\mathrm{th}$ century. However, this work has hopefully demonstrated that coupling quantised light fields to many-body systems provides a very rich playground for exploring new quantum mechanical phenomena beyond what would otherwise be possible in other fields.