The symmetry properties of the N-particle multilevel system are studied. For indistinguishable particles, the total wave function must be either symmetric or antisymmetric in the exchange of any two particles. The internal states of the multilevel system are combined with suitable spatial wave functions to produce a totally symmetric or antisymmetric wave function. The problem of degeneracy and the method of combining a given permutational symmetry of the internal wave function with the same or the conjugate symmetry for the spatial part are described. It is found that Young’s diagram corresponding to the given irreducible representation of the permutation group plays an important role.
The application of transition operators ${\mathcal{P}}_{m,{m}^{\ensuremath{'}}}(t)=\mathrm{exp}(\mathrm{iHt})|m〉〈{m}^{\ensuremath{'}}|\mathrm{exp}(\ensuremath{-}iHt)$ is studied for the problem of an atomic system
S with eigenstates
m interacting with one or more damping reservoirs
R. The average value of these operators gives the reduced density matrix ${{\ensuremath{\rho}}_{{m}^{\ensuremath{'}},m}}^{(S)}(t)$ for
S. If
R consists of broad-band distributions of harmonic oscillators, (e.g., radiative damping), then damped equations of motion can be derived for all ${\mathcal{P}}_{m,{m}^{\ensuremath{'}}}(t)$, even if
S is a multilevel system. One need not specify the initial states of
R, nor restrict the treatment to second order in the S-R coupling. The formalism is illustrated for the case where
S consists of (i) a four-level atom in a resonant cavity (with broad-band modes also present), and (ii) a collection of atoms that can be treated as a multilevel spin system. Density-matrix equations are obtained for the case where no damping radiation is present initially. In (ii), the formalism is used to derive a two-time corrleation function without the aid of the fluctuation-regression theorem.
In dense and ordered atomic arrays, destructive interference in photon emission leads to the emergence of subradiant states, which are protected from radiative decay and are characterized by long lifetimes. Within linear optics, subradiant states of one-dimensional chains can be interpreted as guided modes. Collective interactions in arrays have attracted a lot of interest recently, with work showing that collective emission can be used to realize improved quantum memories [1, 2] as well as almost-perfect atomic mirrors [3, 4], among other exciting applications. While tantalizing, these ideas strongly rely on one assumption: that atoms are two-level systems and only have a unique electronic ground state. However, most atoms exhibit a rich and complex internal structure. In this talk, I will discuss the properties of subradiant states of two-level and multilevel atoms. In particular, I will present a new mechanism for subradiance and atomic wave-guiding in atoms with hyperfine structure, which crucially depends on many-body correlations, a radical departure from conventional subradiant states found in classical-dipole lattices or two-level-atom arrays [5].
We propose a new physical approach for encoding and processing of quantum information in ensembles of multilevel quantum systems, where the different bits are not carried by individual particles but associated with the collective population of different internal levels. One- and two-bit gates are implemented by collective internal state transitions taking place in the presence of an excitation blockade mechanism, which restricts the population of each internal state to the values zero and unity. Quantum computers with 10-20 bits can be built via this scheme in single trapped clouds of ground state atoms subject to the Rydberg excitation blockade mechanism, and the linear dependence between register size and the number of internal quantum states in atoms offers realistic means to reach larger registers.
Quantum systems of indistinguishable particles are commonly described using the formalism of second quantization, which relies on the assumption that any admissible quantum state must be either symmetric or antisymmetric under particle permutations. Coherence-induced many-body effects such as superradiance, however, can arise even in systems whose constituents are not fundamentally indistinguishable as long as all relevant dynamical observables are permutation-invariant. Such systems are not confined to symmetric or antisymmetric states and therefore require a different theoretical approach. Focusing on noninteracting systems, here we combine tools from representation theory and thermodynamically consistent master equations to develop such a framework. We characterize the structure and properties of the steady states emerging in permutation-invariant ensembles of arbitrary multilevel systems that are collectively weakly coupled to a thermal environment. As an application of our general theory, we further explore how these states can in principle be used to enhance the performance of quantum thermal machines. Our group-theoretical framework thereby makes it possible to analyze various limiting cases that would not be accessible otherwise. In addition, it allows us to show that the properties of multilevel ensembles differ qualitatively from those of spin ensembles, which have been investigated earlier using the standard Clebsch-Gordan theory. Our results have a large scope for future generalizations and pave the way for systematic investigations of collective effects arising from permutation invariance in quantum thermodynamics.
In describing the nonlinear interaction of radiation with matter it is well known that the perturbation expansion technique is used to calculate the atomic wave functions and the transitions between the states of a system. However the technique, though involving all states of the system, is valid only in the limit of weak interaction fields and non-resonant interactions. For strong fields and resonant interactions, density matrix or dressed atom formulisms are commonly used though these exact treatments are restricted to simple two level schemes or approximations of these; their application to general resonant-near resonant multilevel interactions being prohibitively complex.