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arXiv:0706.3133 · Physical Review A

Tunable photonic band gaps with coherently driven atoms in optical lattices

David Petrosyan

Published Nov 16, 2007• 40 citations
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Abstract

An optical lattice loaded with cold atoms can exhibit a tunable photonic band gap for a weak probe field under the conditions of electromagnetically induced transparency. This system possesses a number of advantageous properties, including reduced relaxation of Raman coherence and the associated probe absorption, and simultaneous enhancement of the index modulation and the resulting reflectivity of the medium. This flexible system has the potential to serve as a test bed for various designs of linear and nonlinear photonic band gap materials at very low light level and can be employed to realize deterministic entanglement between weak quantum fields.

DOI
10.1103/physreva.76.053823

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An optical lattice loaded with cold atoms can exhibit a tunable photonic band gap for a weak probe field under the conditions of electromagnetically induced transparency. This system possesses a number of advantageous properties, including reduced relaxation of Raman coherence and the associated probe absorption, and simultaneous enhancement of the index modulation and the resulting reflectivity of the medium. This flexible system has the potential to serve as a test bed for various designs of linear and nonlinear photonic band gap materials at very low light level and can be employed to realize deterministic entanglement between weak quantum fields.
We illustrate a mechanism based on coherent optical nonlinearities to realize optically tunable photonic crystals built from media supporting electromagnetically induced transparency. These exhibit specific periodic patterns where a light probe can experience a fully developed photonic band-gap with negligible absorption. An analytical method based on a two-mode approximation is developed to study the optical response of such a periodically modulated medium driven into a regime of standing-wave electromagnetically induced transparency. A comparison with a transfer matrix approach and with techniques based on the coupled Maxwell–Liouville equations shows that our method is very accurate to describe the optical properties of such photonic metamaterials in the frequency region of interest. Nearly perfect reflectivities may be attained for ultracold 87Rb atoms samples which are seen to reflect with little loss and deformation light pulses whose frequency components are contained within the gap. When the same mechanism is implemented for inhomogeneously broadened optical transitions of nitrogen-vacancy centers in diamond, which is more amenable to device applications, fully developed photonic band-gap structures with negligible absorption, reflectivities very close to unity and sufficiently large bandwidths may also be attained. We further examine more flexible schemes based on four level systems under double driving conditions showing more accurate and efficient coherent optical control of the photonic band-gap. The remarkable experimental simplicity of the schemes that we presented is set to ease quantum nonlinear optics applications.
The photonic band gap is a spectral range which cannot propagate in a periodic optical nanostructure, that is, the structure itself has a “forbidden band”. It has been successfully applied to the filters, amplifiers, mixers, etc. As is well known, dynamically tunable photonic band gaps in cold atomic lattices are of great importance in various research fields. However, the photonic band gaps of a traditional photonic crystal are non-tunable because the periodic structure is determined once the photonic crystal is grown. On the other hand, a majority of previous researches focused on improving the reflectivity of photonic band gap, which can only keep approaching to 1. Due to the action of the vacuum of the radiation field, near-degenerate lower level has an additional coherence term, the spontaneously generated coherence term. In this paper, we consider a three-level ∧-type atomic system driven by a strong coherent field, a weak coherent field and an incoherent pump, in which the two ground states are of hyperfine structure. The one-dimensional photonic band gaps are formed by cold atoms trapped in a one-dimensional-ordered optical lattice and this system may create two photonic band gaps (PBGs). The trapped cold atoms have a Gaussian density distribution in each period as determined by the optical potential depth and the average atomic temperature. We investigate in detail how the reflectivities of the two PBGs are influenced by the coherent effect of spontaneously generated coherence. Then, we find that the reflectivities of the two band gaps can be significantly improved by the spontaneously generated coherence. The reflectivities of such two band gaps can be dynamically manipulated by varying the intensity of incoherent driving field and the relative phase between the probe field and the coupling field, which cannot be realized in a conventional ∧-type atomic system. Besides, by adjusting the parameters appropriately, the reflectivities of these two band gaps can be higher than 1, which is because probe field gain stems from the spontaneously generated coherence. In the future, photonic transport properties can be investigated in the three-dimensional atomic lattices and this work is meaningful for the optical routing, photodiode and transistor.
Using the technique of electromagnetically induced transparency, three photonic bandgaps can be established and manipulated at any time due to the refraction modulated periodically by the one-dimensional optical lattice in a tripod atomic system which is trapped in a one-dimensional optical lattice with a Gaussian density distribution. Using the density-matrix equations to describe the interaction between laser and atoms and the transfer-matrix equation to describe the scattering of light waves in periodic media, we can obtain the steady reflection and transmission spectra. It can be found that the position and width as well as the reflectivity of the photonic band-gap could be tuned by changing the detunings and intensities of the coupling fields and the geometric Bragg detuning.
We study a five-level $\text{quasi}\ensuremath{-}\ensuremath{\Lambda}$ system of cold atoms for achieving high-order photonic band gaps (PBGs) probed by an ultraviolet field in two cases where either an infrared or a visible control field works in the standing-wave configuration to induce one atomic grating. Transfer-matrix calculations for appropriate rubidium levels demonstrate that three fifth-order band gaps or two second-order band gaps can be generated near the probe resonance when the standing-wave control field is modulated to satisfy relevant Bragg conditions in the regime of electromagnetically induced transparency. These high-order PBGs, as characterized by homogeneous high reflectivities, can be dynamically tuned in positions and widths on demand and may be extended to develop efficient devices (like diodes and reflectors) for manipulating weak high-frequency light with strong low-frequency light.
We experimentally study the photonic properties of a cold-atom sample trapped in a one-dimensional optical lattice under the conditions of electromagnetically induced transparency. We show that such a medium has two photonic band gaps. One of them is in the transparency window and gives rise to a Bragg mirror, which is spectrally very narrow and dynamically tunable. We discuss the advantages and the limitations of this system. As an illustration of a possible application we demonstrate a two-port all-optical switch.
We study an ensemble of cold atoms driven to the four-level double-$\ensuremath{\Lambda}$ configuration and trapped in the one-dimensional optical lattices for achieving dynamically controlled photonic band gaps (PBGs) via the balanced four-wave mixing interaction. Numerical results show that a pair of PBGs characterized by high platforms of probe reflectivities and reduced density of photonic states are established, simultaneously, on both probe transitions in one symmetric driving scheme. Such two-color PBGs can be easily manipulated in position and width by modulating, e.g., the balanced Rabi frequencies of two coupling fields and the Gaussian widths of atomic density distributions, to change the equal probe detunings fulfilling both Bragg conditions. Each two-color PBG as predicted in the coherent atomic lattices may be explored to attain robust light entanglement, even for single-probe photons, via an efficient frequency conversion during the nonlinearly correlated reflection.