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Physical Review A

Optical ring cavities as tailored four-level systems: An application of the group U(2,2)

R. J. C. Spreeuw, Marco W. Beijersbergen, J. P. Woerdman

Published Jan 1, 1992• 19 citations
Publisher

Abstract

We report on the experimental realization and theoretical analysis of four-level systems produced in passive optical ring cavities by introducing various intracavity elements. Birefringent elements couple waves with different polarizations and thus lift the polarization degeneracy of the cavity modes. Reflecting elements couple waves propagating in different directions and lift the propagation degeneracy. A combination of these elements leads, in general, to a splitting of every longitudinal cavity mode into a four-mode system. Under the restriction that the optical components are loss-free, the description of these ring cavities in a transmission-matrix formalism leads in a natural way to the study of the Lie group U(2,2) and its Lie algebra u(2,2). We associate each of the 16 generators of this algebra with a specific type of optical element, some of which are standard components, others not. From the commutation relations of the generators, we derive a recipe for the construction of the nonstandard components as a sequence of standard ones. It follows that the entire U(2,2) group can actually be realized. The number of independent parameters---16 for a general U(2,2) element---is shown to be reduced substantially if the optical components are selected out of a restricted number of types, provided that the corresponding generators define a subalgebra of u(2,2). In such cases, a subgroup of U(2,2) is realized and the number of independent parameters of the optical system is given by the number of generators of the subalgebra. A connection is established between the subalgebras and symmetry properties of the optical components in the cavity. We consider the influence of symmetries on the mode structure and consider our experimental results from this viewpoint. In particular, we discuss time-reversal invariance, leading to the symplectic group USp(2,2), and we identify antiunitary symmetries leading to Kramers's degeneracy in the mode spectrum. We propose the application of the group U(2,2) in optics as a tool yielding direct practical results.

DOI
10.1103/physreva.45.1213

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We report on the experimental realization and theoretical analysis of four-level systems produced in passive optical ring cavities by introducing various intracavity elements. Birefringent elements couple waves with different polarizations and thus lift the polarization degeneracy of the cavity modes. Reflecting elements couple waves propagating in different directions and lift the propagation degeneracy. A combination of these elements leads, in general, to a splitting of every longitudinal cavity mode into a four-mode system. Under the restriction that the optical components are loss-free, the description of these ring cavities in a transmission-matrix formalism leads in a natural way to the study of the Lie group U(2,2) and its Lie algebra u(2,2). We associate each of the 16 generators of this algebra with a specific type of optical element, some of which are standard components, others not. From the commutation relations of the generators, we derive a recipe for the construction of the nonstandard components as a sequence of standard ones. It follows that the entire U(2,2) group can actually be realized. The number of independent parameters---16 for a general U(2,2) element---is shown to be reduced substantially if the optical components are selected out of a restricted number of types, provided that the corresponding generators define a subalgebra of u(2,2). In such cases, a subgroup of U(2,2) is realized and the number of independent parameters of the optical system is given by the number of generators of the subalgebra. A connection is established between the subalgebras and symmetry properties of the optical components in the cavity. We consider the influence of symmetries on the mode structure and consider our experimental results from this viewpoint. In particular, we discuss time-reversal invariance, leading to the symplectic group USp(2,2), and we identify antiunitary symmetries leading to Kramers's degeneracy in the mode spectrum. We propose the application of the group U(2,2) in optics as a tool yielding direct practical results.
This thesis mainly focuses on the experimental investigations of electromagnetically induced transparency (EIT) related phenomena in various systems involving multilevel atoms inside an optical ring cavity. Semiclassical methods, e.g. density-matrix equations, are used through out this thesis to simulate the experimental results. First, the cavity transmission spectrum can be significantly modified when multilevel atoms are placed inside an optical ring cavity. Such coupled atom-cavity systems are well explained by the intracavity dispersion/absorption properties. Specifically, three-level lambda-type, four-level N-type and double-lambda-type atoms inside an optical ring cavity are investigated by examining their cavity transmission spectra. Second, optical multistability (OM) has been demonstrated with EIT atoms inside an optical ring cavity. Such OM has been utilized to realize a controllable optical multistate switch, which can be modeled as a triple-well system. Third, self-Kerr nonlinearities of multilevel atoms are measured in an optical ring cavity by scanning the cavity length. Fourth, bright Stokes and anti-Stokes fields are generated simultaneously in a doubly-resonant atomic optical parametric oscillator (AOPO). In addition, vacuum-induced absorption and noise correlations are studied in the AOPO system. Last, a theoretical model is proposed to realize parity-time (PT) symmetry in a four-level N-type atomic system by spatially modifying the complex refractive index in free space. Moreover, the experimental progress is made to observe discrete diffraction pattern in an optically induced lattice by interfering with plane waves in a coherent atomic medium.
We consider the dynamical behavior of light in a cavity that contains birefringent elements with time-dependent settings. The evolution of the two-dimensional polarization vector can be described by an equation of the Schr\"odinger type if the intracavity elements change little on the time scale of the round-trip time. A coupled-mode analysis is given, which models the multimode dynamics of the optical system. We show that a description in terms of isolated two-level systems is justified if the optical band gap is small.
Starting from the symplectic Lie groups Sp (2 n , ) of an n -mode system in classical and quantum mechanics we discuss, in particular, Sp (2, )~ SO (2,1) for a single mode and Sp (4, )~ SO (3,2) with some of its important subgroups such as SU (1,1)~ Sp (2, ) and SU (2). We then look at their applications in quantum optics (squeezing, phase states, beamsplitter and polarization). We explicitly construct the root scheme of the Lie algebra sp (4, ) in two different bases as it is an important means for the investigation of the group properties and the construction of the irreducible representations. Apart from the well known realizations of Sp (2, ) by single-mode and two-mode systems, we discuss some other realizations of this group by fractal combinations of boson operators.
Questions at the intersection of the AdS/CFT correspondence and quantum information theory motivate the study of projectors in sequences of subalgebras of finite-dimensional commutative associative semisimple algebras A\mathcal{A}A, obtained by incrementally adjoining one generator at each step to produce a non-linear generating set for A\mathcal{A}A. We define degeneracy graphs, which are finite layered tree graphs whose nodes represent projectors in the successive subalgebras. Using combinatorial properties of the degeneracy graph, we give a simple formula for constructing a linear basis of A\mathcal{A}A in terms of monomials in the generators. The nodes can be labelled by formal variables corresponding to the eigenvalues of the generators added at each layer. We prove that the construction is compatible with the required counting of projectors in A\mathcal{A}A, and give explicit constructions of the projectors in terms of the monomials, in the cases of one- and two-layer degeneracy graphs with arbitrary numbers of nodes. More generally, we provide extensive computational evidence for the invertibility of the matrix relating the proposed monomial basis to the projector basis, by evaluating its determinant. In the 1-layer case, this is a Vandermonde determinant. A simple formula for the non-vanishing determinant in the general layer case is conjectured and supported by the computational data. The construction is illustrated with examples including centres of symmetric group algebras and maximally commuting subalgebras generated by JucysMurphy elements. We outline applications of the monomial basis to algorithms for constructing matrix units in non-commutative semisimple algebras, with relevance to orthogonal bases of multi-matrix gauge-invariant operators and to quantum information theory.
Systems of coupled cavities have the potential to provide bright quantum optical states of light in a highly versatile manner. Micro-ring resonators for instance are highly scalable candidates for photon sources thanks to CMOS fabrication techniques and their small footprint, however, surface roughness of the wave-guides, and defects in the coupler geometry routinely induce splitting of the cavity modes due to back-scattering and back-coupling. The parasitic back-propagating mode in the micro-ring leads to hybridisation of the modes, altering the linear and nonlinear properties of this coupled cavity system, and ultimately constraining the fidelity of quantum light sources that can be produced. In this paper, we derive a comprehensive general model for Gaussian non-linear processes in coupled cavity systems, based on an effective field Hamiltonian and a dispersive input-output model. The resulting dynamics of the equations of motion are evaluated in a Gaussian process formalism via the symplectic transformations on the optical modes. We numerically model and explore the problem of back-scattering in micro-ring resonators in physically relevant parameter regimes, involving the splitting of various resonances, we calculate the consequent impurity and heralding efficiency of various heralded photon schemes, we explore a perturbative explanation of the observations and assess the correspondence between spontaneous and stimulated processes in these systems.
We study the set of nnn-qubit pure quantum states. Specifically, we consider their incarnation as density matrices in Bloch representation. Building on prior work, we define a family of quadratic functions in Pauli expectation values, and fully characterise all instances that result in state-independent purity constraints. The resulting family of pure-state invariants is surprisingly rich in structure, with strong ties to the geometry of symplectic polar spaces. After charting the general landscape, we define a sub-family of pure-state invariants that satisfies the dynamical constraint of being stable under state update. We prove that it corresponds with (different embeddings of) symplectic Lie algebras sp(2n−1)⊂su(2n)\mathfrak{sp}(2^{n-1})\subset\mathfrak{su}(2^n)sp(2n−1)⊂su(2n), which admit a generating set of Pauli operators. We carefully distinguish the symplectic case from its orthogonal counterpart, using the Arf invariant of quadratic refinements of the symplectic form on the projective Pauli group, which we further identify with quaternionic structures on C2n\mathbb{C}^{2^n}C2n that are both generated by and restricted to Pauli operators. As a consequence of this analysis, we reveal previously largely overlooked structure in the nnn-qubit Pauli group. We highlight this in a number of examples. We characterise all state-independent uncertainty relations in variance for Pauli observables, as well as all Pauli channels for which the output fidelity and output purity are state-independent. We further show that the orbits of the projective stabiliser groups of pure-state invariants, which for n≥2n\geq2n≥2 are proper subgroups of the Clifford group, form complex projective state 222-designs, while their unitary lifts form symplectic 222-designs.