Resonance fluorescence in photonic band gap waveguide architectures: Engineering the vacuum for all-optical switching

We describe the spectral characteristics of the radiation scattered by two-level atoms (quantum dots) driven by a strong external field, and coupled to a photonic crystal radiation reservoir. We show that in the presence of strong variations with the frequency of the photonic reservoir density of states, the atomic, Mollow, sideband components of the scattered intensity can be strongly modified. Consequently, a weak optical probe field experiences a substantial differential gain in response to slight variations in the intensity of an optical driving field. We suggest that these effects may be of relevance to all-optical transistor action in photonic crystals. Using a specific photonic crystal heterostructure, we suggest that an all-optical microtransistor based on photonic crystals may operate at less than $100\text{nW}$ switching threshold power. Collective $N$-atom effects substantially enhance this optical switching effect. Near the switching threshold intensity, collective effects are manifest in the $N^2$ scaling of the intensity spectrum (reminiscent of superradiance). Above and below this critical region, the gain spectrum widens (linearly with $N$). This correspondingly reduces the switching time scales of the atomic system in response to external fields. Furthermore, the quantum degree of second-order coherence exhibits unusual features. Scattered photons display a variable degree of antibunching as function of driving laser field intensity and the photonic density of states discontinuity. We analyze the effects of the inhomogeneous atomic line broadening on the amplification process. We show, using suitable photonic density of states engineering, that it is possible to select a narrow spectral range around the central frequency of the atomic frequency distribution over which amplification and switching occur. This is done either by spectral decoupling of the active elements from the electromagnetic field (through the introduction of band gaps at specified spectral locations) or through incoherent pumping to selectively saturate atoms outside the spectral region chosen for amplification.

Figure 1

Photonic band gap waveguide architecture for all-optical switching and transistor action. The microstructure consists in a waveguide channel in a two-dimensional photonic crystal, which is embedded in a 3D photonic crystal [11].

Figure 2

Dispersion relation of a photonic crystal heterostructure similar to one presented in Fig. 1 for propagation along the waveguide. The two-dimensional photonic crystal consists of Si cylinders with aspect ratio $r∕a=0.3$ (here $r$ is cylinder radius and $a$ dielectric lattice constant), and cylinder height of $h=0.4a$. The linear waveguide is generated by removing three rows of cylinders in the longitudinal direction. The three-dimensional photonic crystal is assumed to be an inverted square spiral structure (Si infiltrated) that presents a photonic band gap of $23.6\%$ [13]. For these parameters, the linear defect supports two waveguide modes, one of which experiences a sharp cutoff in the middle of the 3D PBG, same spectral range for which the second guided mode has a very large dispersion. Source: Ref. 12.

Figure 3

A schematic description of the local density of states (LDOS) generated by the structure presented in Figs. 1, 2, which illustrates the concept of band-edge quality factor. The spectral range used here roughly corresponds to the middle of the gap in Fig. 2, and the large dispersion mode (continuous curve in Fig. 2) generates a relatively small contribution to the LDOS, while the sharp frequency cutoff of the second guided mode (the dashed curve in Fig. 2) gives rise to the apparent jump in the LDOS. The band-edge quality factor is defined (analogously to the quality factor of a cavity) by the ratio between the central frequency ${\omega }_C$ and the frequency $\Delta {\omega }_C$ range over which the LDOS jumps to half of its peak, i.e., $Q_B={\omega }_C∕\Delta {\omega }_C$.

Figure 4

Relevant frequencies and frequency scales in the case of a steplike density of states.

Figure 5

Scattered radiation spectrum as a function of the detuning frequency, $\Delta =\omega -{\omega }_L$ (in units of ${\gamma }_{+}$), for different values of the Rabi frequency in the presence of dipolar dephasing. All Rabi frequencies used here are below the threshold value, which, for the specific choice of the parameters used in this figure, is ${\epsilon }_{thr}∕{\Delta }_{AL}=0.35$. The atomic system is negatively detuned from the laser field frequency, $\mid {\Delta }_{AL}\mid ∕{\gamma }_{+}=3$, and ${\gamma }_{-}∕{\gamma }_{+}=0.01$. The dipolar dephasing rate is ${\gamma }_p∕{\gamma }_{+}=0.1$.

Figure 6

Scattered radiation spectrum in the presence of dipolar dephasing, as a function of the detuning frequency, $\Delta =\omega -{\omega }_L$ (in units of ${\gamma }_{+}$), for different values of the Rabi frequency. All Rabi frequencies used here are above the threshold value, which, for the specific choice of the parameters used in this figure, is ${ϵ}_{thr}∕{\Delta }_{AL}=0.35$. The atomic system is negatively detuned from the laser field frequency $\mid {\Delta }_{AL}\mid ∕{\gamma }_{+}=3$, and ${\gamma }_{-}∕{\gamma }_{+}=0.01$. The dipolar dephasing rate is ${\gamma }_p∕{\gamma }_{+}=0.1$. The inset shows an expanded (on the $y$ axis) view of the sideband peak region.

Figure 7

The peak height $H_{max}^0$ of the central component of the scattered spectrum (in arbitrary units) as a function of the Rabi frequency $ϵ∕\mid {\Delta }_{AL}\mid$ for $\mid {\Delta }_{AL}\mid ∕{\gamma }_{+}=3$, $\text{sgn}\left({\Delta }_{AL}\right)=-1$, and the magnitude of the jump in the photonic DOS, ${\gamma }_{-}∕{\gamma }_{+}$, in the absence of the dipolar dephasing, ${\gamma }_p∕{\gamma }_{+}=0$. The atomic resonant frequency is detuned negatively from the laser field frequency ${\Delta }_{AL}={\omega }_A-{\omega }_L<0$.

Figure 8

Second-order correlation function $g^{\left(2\right)}\left(\tau \right)$ as a function of the scaled time $\tau =\gamma t$ and the Rabi laser field frequency, $ϵ∕\mid {\Delta }_{AL}\mid$, for ordinary vacuum case ${\gamma }_{-}∕{\gamma }_{+}={\gamma }_0$, and in the absence of the dipolar dephasing, ${\gamma }_p∕{\gamma }_{+}=0$. The atomic resonant frequency is detuned negatively from the laser field frequency, ${\Delta }_{AL}={\omega }_A-{\omega }_L<0$.

Figure 9

Second-order correlation function $g^{\left(2\right)}\left(\tau \right)$ as the function of the scaled time $\tau ={\gamma }_{+}t$ and the Rabi laser field frequency, $ϵ∕\mid {\Delta }_{AL}\mid$, for a photonic crystal characterized by ${\gamma }_{-}∕{\gamma }_{+}=0.001$, ${\gamma }_0∕{\gamma }_{+}=1$, and in the absence of the dipolar dephasing, ${\gamma }_p∕{\gamma }_{+}=0$. The atomic resonant frequency is detuned negatively from the laser field frequency, ${\Delta }_{AL}={\omega }_A-{\omega }_L<0$.

Figure 10

Second-order correlation function $g^{\left(2\right)}\left(0\right)$ as a function of the Rabi laser field frequency, $ϵ∕\mid {\Delta }_{AL}\mid$, in the absence of the dipolar dephasing, ${\gamma }_p∕{\gamma }_{+}=0$, for different magnitudes of the jump in the photonic DOS. The atomic resonant frequency is detuned negatively from the laser field frequency, ${\Delta }_{AL}={\omega }_A-{\omega }_L<0$.

Figure 11

Second-order correlation function $g^{\left(2\right)}\left(0\right)$ as a function of the Rabi laser field frequency, $ϵ∕\mid {\Delta }_{AL}\mid$, in the presence of the dipolar dephasing, ${\gamma }_p∕{\gamma }_{+}=0.2$, for different magnitudes of the jump in the photonic DOS. The atomic resonant frequency is detuned negatively from the laser field frequency, ${\Delta }_{AL}={\omega }_A-{\omega }_L<0$.

Figure 12

Bare atom and dressed atom states diagram.

Figure 13

The imaginary part of the linear susceptibility of the probe beam (absorption spectrum) as a function of the detuning of the probe beam frequency from the driving laser field frequency $\omega -{\omega }_L$, for ${\gamma }_{-}∕{\gamma }_{+}=0.001$, ${\gamma }_0∕{\gamma }_{+}=1$, ${\gamma }_p∕{\gamma }_{+}=0.2$, $\mid {\Delta }_{AL}\mid ∕{\gamma }_{+}=10$. The two curves correspond to different Rabi frequencies, $ϵ∕\mid {\Delta }_{AL}\mid$, one of them (continuous curve) below the inversion threshold $\left(ϵ∕\mid {\Delta }_{AL}\mid =0.176\right)$, and the other one (dashed curve) above the threshold intensity $\left(ϵ∕\mid {\Delta }_{AL}\mid =0.190\right)$. The atomic resonant frequency is detuned negatively from the laser field frequency, $\text{sgn}\left({\Delta }_{AL}\right)=-1$, and, for this choice of parameters, the inversion threshold is ${ϵ}_{thr}∕\mid {\Delta }_{AL}\mid =0.183$.

Figure 14

The peak height to width ratio [Eq. (4.6)] of the components of the absorption spectrum of the probe field as a function of the Rabi frequency of the driving field $ϵ∕\mid {\Delta }_{AL}\mid$, for ${\gamma }_{-}∕{\gamma }_{+}=0.001$, ${\gamma }_0∕{\gamma }_{+}=0.001$, and in the absence of the dipolar dephasing, ${\gamma }_p∕{\gamma }_{+}=0$. The atomic resonant frequency is detuned negatively from the laser field frequency, ${\Delta }_{AL}={\omega }_A-{\omega }_L<0$.

Figure 15

The widths of the components of the absorption spectrum of the probe field in the presence of a photonic crystal, ${\Gamma }_{\mathit{coh}}^{PC}$, and in free space ${\Gamma }_{\mathit{coh}}^{FS}$, as a function of the Rabi frequency of the driving laser field $ϵ∕\mid {\Delta }_{AL}\mid$, for ${\gamma }_{-}∕{\gamma }_{+}=0.001$, ${\gamma }_0∕{\gamma }_{+}=0.001$, and in the absence of the dipolar dephasing, ${\gamma }_p∕{\gamma }_{+}=0$. The atomic resonant frequency is detuned negatively from the laser field frequency, ${\Delta }_{AL}={\omega }_A-{\omega }_L<0$.

Figure 16

Weakly inhomogeneously broadened Mollow spectrum near a steplike discontinuity in the electromagnetic density of states. We assume that the density of states is constant over spectral ranges larger than the inhomogeneous linewidth $\Delta {\omega }_I$, surrounding the Mollow triplet frequencies ${\omega }_L$, ${\omega }_L\pm 2\Omega$.

Figure 17

The imaginary part of the linear susceptibility of the probe beam (absorption spectrum) as a function of the detuning of the probe beam frequency $\omega$ with respect to the driving laser field frequency ${\omega }_L$, for ${\gamma }_{-}∕{\gamma }_{+}=0.001$, ${\gamma }_0∕{\gamma }_{+}=1$, $ϵ∕\mid {\Delta }_{AL}\mid =0.178$ and in the absence of the phonon mediated dephasing $\left({\gamma }_p∕{\gamma }_{+}=0\right)$. The width of the inhomogeneous line distribution varies from $\Delta {\omega }_I∕\mid {\Delta }_{AL}\mid =0$ (no inhomogeneous broadening) for the solid curve to $\Delta {\omega }_I∕\mid {\Delta }_{AL}\mid =0.9$ for the double-dot-dashed curve. The center of the atomic frequency distribution is detuned negatively from the laser field frequency, ${\Delta }_{AL}={\omega }_A-{\omega }_L<0$.

Figure 18

The imaginary part of the linear susceptibility of the probe beam (absorption spectrum) as a function of the detuning of the probe-pump detuning frequency, $\omega -{\omega }_L$ and the driving field Rabi frequency, $ϵ∕\mid {\Delta }_{AL}\mid$, for ${\gamma }_{-}∕{\gamma }_{+}=0.001$, ${\gamma }_0∕{\gamma }_{+}=1$, $ϵ∕\mid {\Delta }_{AL}\mid =0.178$, and ${\gamma }_p∕{\gamma }_{+}=0$. The inhomogeneous frequency distribution has a width $\Delta {\omega }_I∕\mid {\Delta }_{AL}\mid =0.1$ and is centered below the laser field frequency, ${\Delta }_{AL}={\omega }_A-{\omega }_L<0$.

Figure 19

The engineered photonic DOS proposed to quench the unfavorable members of the inhomogeneous quantum dot distribution. The upper panel shows the steplike density of states (the staircase curve) used in the absence of inhomogeneous broadening. The dot-arrow curves depict the evolution of the Mollow sideband peaks for a particular quantum dot as its deviation $\Delta {\omega }_A$ from the center of the inhomogeneous distribution is increased. The lower panel depicts a simple engineered photonic DOS, which quenches dots detuned by $\mid \Delta {\omega }_A\mid >{\Omega }_{MIN}$. This quenching prevents the smearing out of the switching effect.

Figure 20

The evolution of the atomic frequency detuning with respect to the laser field frequency (the thick continuous curve) and the spectral features (the left, central, and right components of the Mollow spectrum) (the thick dashed lines) as a function of the position of the atomic frequency with respect to the central frequency of the inhomogeneous distribution. To simplify the interpretation, the plot is rotated by $90°$. The numerical values correspond to the parameters used above (with a negative detuning of the central frequency of the inhomogeneous broadening with respect to the laser field frequency). The driving field Rabi frequency is $ϵ={ϵ}_{thr}$ and ${\overline{\Delta }}_{AL}=5{ϵ}_{thr}$.

Figure 21

Same as Fig. 20, but now we introduce a colored incoherent pump that saturates some members of the atomic distribution. The frequency profile of the pump rate is represented by the criss-crossed region (centered on the right sideband component of the Mollow spectrum), and its presence has as consequence the elimination of the upper permitted region in Fig. 20.

Figure 22

The full recovery of the amplification effect in the presence of inhomogeneous broadening by band gap engineering and incoherent pump saturation mechanisms. The dashed curve represents the signal in the absence of inhomogeneous broadening. The continuous curve is obtained using the photonic DOS and the incoherent pump presented in Fig. 21. The width of the allowed passbands of the DOS is $\Delta {\omega }_{PB}\approx {ϵ}_{thr}=0.183\mid {\Delta }_{AL}\mid$, and their magnitude is given by ${\gamma }_{-}∕{\gamma }_{+}=0.001$, ${\gamma }_{-}∕{\gamma }_{+}=1$. The incoherent pump is characterized by $\Delta {\omega }_P=\Delta {\omega }_{PB}\approx {ϵ}_{thr}$, ${\rho }_{+}∕{\gamma }_{+}=50$, ${\rho }_{-}∕{\gamma }_{+}=0$.