We propose a simple physical implementation of the quantum Householder reflection ͑QHR͒ M͑v͒ = I −2͉v͗͘v͉ in a quantum system of N degenerate states ͑forming a qunit͒ coupled simultaneously to an ancillary ͑excited͒ state by N resonant or nearly resonant pulsed external fields. We also introduce the generalized QHR M͑v ; ͒ = I + ͑e i −1͉͒v͗͘v͉, which can be produced in the same N-pod system when the fields are appropriately detuned from resonance with the excited state. We use these two operators as building blocks in constructing arbitrary preselected unitary transformations. We show that the most general U͑N͒ transformation can be factorized ͑and thereby produced͒ by either N − 1 standard QHRs and an N-dimensional phase gate, or N −1 generalized QHRs and a one-dimensional phase gate. Viewed mathematically, these QHR factorizations provide parametrizations of the U͑N͒ group. As an example, we propose a recipe for constructing the quantum Fourier transform ͑QFT͒ by at most N interaction steps. For example, the QFT requires a single QHR for N = 2, and only two QHRs for N = 3 and 4.

We show that the solution of a multistate system composed of N degenerate lower (ground) states and one upper (excited) state can be reduced by using the Morris-Shore transformation to the solution of a two-state system involving only the excited state and a (bright) superposition of ground states. In addition, there are N − 1 dark states composed of ground states. We use this decomposition to derive analytical solutions for degenerate extensions of the most popular exactly soluble models: the resonance solution, the Rabi, Landau-Zener, Rosen-Zener, Allen-Eberly and Demkov-Kunike models. We suggest various applications of the multistate solutions, for example, as tools for creating multistate coherent superpositions by generalized resonant π-pulses. We show that such generalized π-pulses can occur even when the upper state is far off resonance, at specific detunings, which makes it possible to operate in the degenerate ground-state manifold without populating the (possibly lossy) upper state, even transiently.

In this work we show that stationary light-matter excitations generated inside a hollow onedimensional waveguide filled with atoms, can be made to generate a photonic two-component Lieb Liniger model. We explain how to prepare and drive the atomic system to a strongly interacting regime where spin-charge separation could be possible. We then proceed by explaining how to measure the corresponding effective spin and charge densities and velocities through standard optical methods based in measuring dynamically the emitted photon intensities or by analyzing the photon spectrum. The relevant interactions exhibit the necessary tunability both to generate and efficiently observe spin charge separation with current technology. PACS numbers:One of the most counterintuitive characteristics of one dimensional electron gases is spin-charge separation. In this case the electrons cease to behave as single particles comprised of spin and charge. Instead collective excitations appear carrying only charge (and no spin) or only spin (and no charge) which can propagate through the system with different velocities [1]. Spin charge separation was predicted in tunneling experiments in metallic chains [2], organic conductors [3], carbon nanotubes [4] and more recently in quantum wires [5]. However due to the complexity of the structures used, measuring the spectral function and observing distinct spinon and holon branches has yet to be conclusive. In parallel with these works, artificially engineered many-body systems that could simulate condensed matter effects in well controllable environments have been developed the last decade. Cold atoms and ions traps are the most famous example [6][7][8], and strongly interacting photons (SIP) the most recent developments. Proposals to observe spin charge separation have also been in place for some time in cold atoms, including both bosonic and fermionic species [15][16][17][18][19][20]. However the lack of necessary individual accessibility and measurement, and the challenges in trapping and cooling especially fermionic gases make current results inconclusive so far SIPs on the other hand, as hybrid light-matter quantum simulators promise to provide the necessary extra manipulation and measurement lacking so far from other proposals. The efficient quantum simulation of photonic and polaritonic Mott transitions and the crystallization of photons was shown to be possible using both arrays of coupled resonators or stationary polaritons in atomic gases [9][10][11][12][13][14]. We show here for the first time that spincharge separation could be efficiently observed in a such strongly correlated quantum optical system. * Electronic address: dimitris.angelakis@gmail.orgIn parallel with the seminal works in Luttinger liquid and in cold atoms physics, significant progress has also been made in a different field. Quantum nonlinear optics has shown that strong interactions between light pulses, even at the single photon level is possible with numerous applications in photon switching and quantum memorie...