## Abstract

We present theoretical results of a low-loss all-optical switch based on electromagnetically induced transparency and the quantum Zeno effect in a microdisk resonator. We show that a control beam can modify the atomic absorption of the evanescent field which suppresses the cavity field buildup and alters the path of a weak signal beam. We predict more than 35 dB of switching contrast with less than 0.1 dB loss using just 2 *μ*W of control-beam power for signal beams with less than single photon intensities inside the cavity.

© 2013 OSA

## 1. Introduction

Over the past few decades transistors and other computing components have dropped in size while simultaneously increasing performance. However, power dissipation is increasingly becoming a fundamental limitation to performance [1]. All-optical switches and transistors seek to address this issue, while simultaneously pushing forward the technology needed to create all-optical quantum computing devices [2–6]. One of the fundamental issues limiting this technology is the strength of the nonlinear effects that couple the signal and control fields. Electromagnetically induced transparency (EIT) [7, 8] has been investigated as a resource for optical switches [9–11] and quantum memories [12–15] due to its large nonlinearity, which is enhanced by coherent effects. We demonstrate how it can be used along with a microdisk resonator to create a high-speed low-loss all-optical switch.

The quantum Zeno effect (QZE) [16] is a process whereby frequent measurements of a quantum system can inhibit transitions. It has been demonstrated that the QZE can be used to create an all-optical switch using two-photon absorption (TPA) in a resonant optical cavity, by using rapid measurements to suppress cavity field buildup [17, 18]. The switch design consists of a four-port resonator evanescently coupled to Rubidium (Rb) vapor. The presence of two input beams results in sufficiently strong TPA to suppress the cavity field build-up thereby altering the path of the beams due to inteference effects. Similar all-optical switches have also been proposed using other nonlinearities such as inverse Raman induced loss [19, 20].

Here we report theoretical results and performance estimates showing that EIT and Autler–Townes splitting [21] can also be used to implement an all-optical switch in a microdisk cavity. We make use of single photon absorption (SPA) to suppress the resonant field buildup in a cavity, and use a control beam to modulate the absorption by inducing EIT. The benefit of this approach is that on-resonant SPA has a higher absorption cross-section than known nonlinear processes, potentially enabling better switching results as long as SPA can be sufficiently reduced by EIT. In addition this approach allows the use of very weak signal beams, making it a good candidate to switch single-photon intensities with low loss.

To understand how the switch operates, consider the operation of a four-port resonator, as shown in Fig. 1, in the absence of any atomic interaction (see e.g. [22] and [23] for detailed discussions of four-port resonators and nomenclature). On resonance, a weakly-coupled signal *E _{in}* will be almost completely transferred to the drop-port as a result of destructive interference between the resonant build-up of the cavity field and the light in the input waveguide. Next, consider how this system would change with the addition of a strong loss mechanism in the cavity, such as shown in Fig. 1(a). The resonant field buildup of the signal beam will be suppressed which dramatically reduces the destructive interference in the input waveguide and results in nearly all of the incident light exiting the through-port.

This is directly analogous to the suppression of probability amplitudes via measurement in the quantum Zeno effect. This is made more clear when considering the case of single photons in the signal beam. In that case, a null absorption (measurement) event by the atoms surrounding the cavity suppresses the probability buildup of the photon in the cavity, causing it to bypass the resonator. The stronger this potential absorption process, the more strongly that the photon probability in the cavity is suppressed. Counter-intuitively, the presence of the strong loss mechanism will not dramatically increase the loss of the system but will instead alter the coupling condition of the cavity which changes the output path of the light. More information on this can be found in Refs. [17, 24].

Now we consider a loss mechanism based on a cascade EIT scheme using the 5*S*_{1/2}, 5*P*_{3/2} and 5*D*_{5/2} states of Rb. We take the signal beam to be resonant with the 5*S*_{1/2} → 5*P*_{3/2} transition near 780 nm and the control beam to be resonant with the 5*P*_{3/2} → 5*D*_{5/2} transition near 776 nm. When the 776 nm EIT control beam is present, a transmission window is created in the 780 nm single-photon absorption line. This allows the signal beam to build in the resonator and exit through the drop port, as shown schematically in Fig. 1(b).

In this paper, we theoretically analyze the performance of such a device using high-fidelity numerical models and present detailed estimates of switching performance. Our results indicate that this scheme enables high-contrast, low-loss switching at timescales on the order of the total cavity relaxation time which is roughly ∼ 100 picoseconds for the devices under consideration in this paper.

## 2. Theoretical model

We model the transmission characteristics of the four-port microdisk shown in Fig. 1 evanescently coupled to Rubidium vapor. The device specifications have been chosen to be consistent with current fabrication capabilities. The design consists of an unclad, suspended Si_{3}N_{4} disk with a free spectral range equal to the 4 nm difference between the 5*S*_{1/2} → 5*P*_{3/2}*D*_{2} line at 780 nm and the 5*P*_{3/2} → 5*D*_{5/2} line at 776 nm, allowing for simultaneous resonance at 776 nm and 780 nm (for an example of a similar device operating at a higher wavelength see Ref. [18]). The thickness is chosen such that roughly 30% of the electric field energy of the fundamental mode is outside the resonator, allowing considerable interaction with the Rubidium vapor surrounding the cavity.

We estimate the field profile of the fundamental mode in the cavity using a fully vectorial 2-D axially symmetric weighted residual formulation of Maxwell’s Eqs. implemented in Comsol Multiphysics software (as shown in Fig. 2). The field profile is used to estimate the evanescent interaction of signal and control beams with an ensemble of three-level atoms by calculating the effective absorption coefficient. This calculated absorption coefficient is combined with a classical model of a resonator to predict switching performance. The intent is to show that the presence of the control beam will induce EIT, modifying the absorption coefficient of the signal beam in the resonator. This causes a change in the overall coupling between the waveguide and the resonator resulting in switching.

#### 2.1. Atomic Model

The device under consideration is designed to support simultaneous cavity resonances at the 780 nm and 776 nm spectral lines of Rubdium. To model this interaction between the cavity fields and the Rubidium vapor, we approximate the Rubidium atom as the four-level atomic system shown in Fig. 3. The fourth level, indicated on the right side of the figure, is the decay channel from the excited state level |3〉 to the ground state level |1〉 through the 6*P*_{3/2} level. In the steady-state approximation, the population of that level can be neglected, when no fields are near resonance with it, and replaced by just the decay rate from the excited state to the intermediate state. Therefore it is included in our atomic model only as a decay term. The Hamiltonian of this system is the standard three-level cascade model given by

*d⃗*·

*ε*̂〉

*ℰ*/

*h*̄, where

*d⃗*is the dipole moment,

*ℰ*is the slowly varying electric field amplitude,

*ε*̂ is the polarization vector, and Δ is the detuning of the field from its associated transition. The angular brackets denote an averaging over the random orientation of the dipole moment with respect to the electric field polarization that reduces the Rabi frequency due to random atomic orientations [25]. This Hamiltonian, along with the addition of phenomenological decay terms, gives the following density matrix Eqs.:

*is the homogeneous decay term associated with the*

_{mn}*m*→

*n*transition, and

*γ*is the transverse decay term associated with the off-diagonal elements. We take

_{mn}*γ*

_{12}= Γ

_{12}/2,

*γ*

_{13}= (Γ

_{13}+ Γ

_{23})/2, and

*γ*

_{23}= (Γ

_{12}+ Γ

_{13}+ Γ

_{23})/2. These off-diagonal decay rates assume that all decoherence is due to population decay, which is suitable for atomic vapors. The values were chosen to be physically consistent with a positive density matrix, which does not always hold for arbitrary decoherence terms [26, 27]. The branching ratio between the Γ

_{23}and Γ

_{13}decay channels was set to 0.65 : 0.35 as given in Ref. [28].

Many analytic solutions to Eqs. (2) and similar models have been analyzed [8, 29, 30]. However, to ensure accurate device performance estimates, we use numerical solutions to the density matrix Eqs. above. This allows us to solve Eqs. (2), with only a steady-state approximation, which is consistent with our experiments using CW lasers. However, we do not make any perturbation approximations since we assume a high-Q resonator, which can develop intense fields. The steady-state approximation can easily be lifted to model pulsed phenomena, however we include it here in order to reduce the run-time of our numerical simulations.

We include Doppler broadening in our simulations to account for thermal motion of the atoms. We do so by averaging the density matrix elements over the Doppler profile given by

*=*

_{s,c}*ω*−

_{s,c}*ω*

_{12,23}±

*k*

_{v}*z*is the laser field detuning of the signal and control field respectively due to the thermal velocity of the atom,

*k*is the wave-vector due to the Doppler shift,

_{v}*z*is taken to be the dimension of propagation inside the resonator,

*σ*is the Doppler linewidth, given by the Maxwell-Boltzmann velocity distribution, and

_{D}*h*̄

*ω*

_{12,23}is the energy difference between the |1〉 → |2〉 and |2〉 → |3〉 transitions respectively. Because the signal and control fields are counter-propagating in the resonator, the signal and control experience opposite Doppler shifts as denoted by the opposite signs within the detuning coefficient.

#### 2.2. Cavity – Waveguide Coupling Model

We model the coupling of the waveguides to the cavity using the coupled-mode Eqs. of Haus [22] (see also [23]). This method is related to the E&M approach taken in [17], and we get nearly identical results using either technique. We will label the ports as done in Fig. 1. The input port will be called “in” and corresponds to the upper left waveguide in the figures, and the two output ports will be labelled the “drop” port and “through” port for the lower left and upper right waveguides respectively.

The Eq. governing the field amplitude inside the resonator for a single input field is is given by

*a*is the field amplitude inside the resonator, Δ = (

*ω*−

*ω*

_{0}) is the detuning of the field with frequency

*ω*from the cavity resonance of frequency

*ω*

_{0},

*κ*

_{0}is the intrinsic cavity linewidth,

*κ*is related to the absorption rate of the atoms,

_{e}*κ*

_{1}and

*κ*

_{2}are the cavity – waveguide coupling rates for waveguides 1 and 2 respectively, and |

*s*|

_{in}^{2}is the input power.

The steady state solution to Eq. (4) is

*T*= |

*s*|

_{t}^{2}/|

*s*|

_{in}^{2}and

*D*= |

*s*|

_{d}^{2}/|

*s*|

_{in}^{2}which yield

The intrinsic quality factor is defined as *Q*_{0} = *ω*_{0}/*κ*_{0} and the external quality factor is *Q _{ext}* =

*ω*

_{0}/(

*κ*+

_{e}*κ*

_{1}+

*κ*

_{2}). By controlling the atomic absorption rate

*κ*we can modify the external

_{e}*Q*of the atom-resonator system causing the coupling conditions to change, resulting in the ability to switch between the drop port and through port.

#### 2.3. Cavity Field – Atomic Interaction Model

The interaction of the evanescent fields to the Rubidium surrounding the cavity is the source of the added loss *κ _{e}* in the waveguide–resonator coupling model above. To give performance estimates for the switch, we calculate the average absorption coefficient of the signal beam in the resonator.

To calculate *κ _{e}* we first solve the atomic Eqs., including Doppler broadening, given in (2) for a particular value of the signal and control beam fields. The base absorption coefficient from standard quantum-optics theory is given by:

*χ*

_{12}(Ω

*, Ω*

_{s}*) = Im[*

_{c}*ρ*

_{12}(Ω

*, Ω*

_{s}*)]/Ω*

_{c}*is related to the linear susceptibility and is dependent upon: signal and control beam Rabi frequencies, the density*

_{s}*N*of Rubidium atoms, the dipole moment

*d*and the angular frequency

*ω*of the 780 nm transition. Here we explicitly note the functional dependence of all terms that depend upon the Rabi frequencies (field amplitudes) for clarity.

The small mode volume and high Q of the microdisk produce large intra-cavity intensities for even modest input power levels. This can induce large splitting of the intermediate level. However, the field distribution of the control beam outside the resonator falls off sharply as shown in Fig. 2. To capture this, we compute the density matrix elements for each Rabi frequency corresponding to the different field values, and then average over the field distributions of both the signal and control fields as shown in Eq. (10) below. These solutions are then averaged and weighted by the normalized signal beam intensity, to give the average absorption coefficient for the signal beam. This average absorption coefficient *α*̄ is given by

*V*, and the weighting function

_{e}The normalization integral in the denominator of the weighting function extends over the entire volume of integration *V _{t}*, including the region interior to the resonator, as opposed to the integral in Eq. (10) which is only over the region exterior to the resonator. This accounts for the fact that the field inside the resonator does not interact with the Rubidium thus reducing the overall strength of the interaction. This weighting also accounts for the fact that the scattering cross section is larger near regions of high signal beam intensity than regions of low intensity. We assume that both beams are contained in the same cavity mode for this calculation.

We can relate the average absorption coefficient, *α*̄ to the external cavity loss rate *κ _{e}* through the simple relation

*κ*=

_{e}*c*

*α*̄, where

*c*is the speed of light in the cavity. Thus, by changing the atomic absorption coefficient

*α*̄ through EIT, we can modify the total cavity Q, resulting in changing coupling conditions and switching.

## 3. Results

Using the methodology above, the average absorption coefficient of the signal beam is calculated and plotted in Fig. 4, with the control beam on and off (blue and red curves respectively). All relevant atomic parameters for this simulation are given in Table 1. The spectral shape of the absorption coefficient is modified from the typical double-Lorentzian peaks associated with Autler–Townes splitting because of Doppler broadening and the transverse variation of control beam intensity in the evanescent field. This causes the signal beam to see a distribution of EIT windows rather than a single split line as would be the case in a uniform field. Very large Autler–Townes splitting on the order of GHz is possible, because the high-Q and small mode volume of the resonator can generate high-intensity fields with relatively modest input powers.

Upon calculating the average absorption coefficient of the signal beam in the resonator, we estimate the transmission of the through-port (Fig. 5(a)) and drop-port (Fig. 5(b)) with and without the control beam, using the formulas in Eq. (8). We take the waveguides to be strongly over-coupled to the cavity with *κ*_{1} and *κ*_{2} roughly two orders of magnitude larger than *κ*_{0}. The coupling rates were chosen to equalize the bandwidth on the through-port and drop-port. With this constraint, we estimate the on-resonant switching contrast to be 50 dB in the through-port and 25 dB in the drop-port, along with only 0.5 dB and 0.02 dB of loss in each respectively. We define the bandwidth as the point at which the switching contrast reaches 20 dB, on Figs. 5(a) and 5(b). This gives a bandwidth of approximately 516 MHz for each output port. These results are tabulated in Table 2.

Switching performance trade-offs can be analyzed by varying the coupling rates between the waveguides and cavity. In the previous example, we chose to equalize the bandwidth in the through-port and drop-port. This required strongly over-coupling the waveguides to the cavity. We can instead choose to equalize the contrast, and by extension the loss in each port. To do this, we lower the fixed waveguide-cavity coupling rates which could be done by increasing the distance between the waveguides and the cavity for example. These results are shown in Figs. 5(c) and 5(d), with the corresponding performance metrics given in Table 3. On cavity resonance we estimate 38 dB switching contrast with only 0.1 dB loss in each port. The tradeoff for equalized contrast, and lower loss is a reduced bandwidth in the through-port of 330 MHz, however the drop-port bandwidth is increased a comparable amount due to conservation of energy constraints.

## 4. Conclusions

We have presented numerical simulations demonstrating the effectiveness of using EIT to create an all-optical switch in a micro-resonator. All of the parameters used for the microdisk and atomic system were chosen to be consistent with physical values, or to be within experimentally achievable design constraints. The model does not take into account adsorption of rubidium onto the waveguides and resonator structure that is known to be a considerable hindrance to experimental demonstration of devices such as these. We note that recent experiments have shown how one might mitigate the problem of rubidium adsorption justifying its neglect in this model, however until this problem is solved the switching contrast and loss rates predicted here would be affected [31].

Assuming the effects of rubidium adsorption can be mitigated, we show that the large resonant build-up within micro-resonators allows one to create very high intensity control fields that can create excellent switching contrast with low loss, even with modest input powers. In addition, this method is compatible with a very low input power signal. These properties suggest that such a device could be suitable for switching single-photon intensities, ideal for some quantum information applications.

## Acknowledgments

Funding was provided in part by IRAD and the DARPA ZOE program (Contract No. W31P4Q-09-C-0566). We acknowledge thought provoking discussions with Jim Franson and Todd Pittman.

## References and links

**1. **N. Kim, T. Austin, D. Baauw, T. Mudge, K. Flautner, J. Hu, M. Irwin, M. Kandemir, and V. Narayanan, “Leakage current: Moore’s law meets static power,” Computer **36**, 68 – 75 (2003) [CrossRef] .

**2. **A. M. C. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” Science **308**, 672–674 (2005) [CrossRef] [PubMed] .

**3. **X. Hu, P. Jiang, C. Ding, H. Yang, and Q. Gong, “Picosecond and low-power all-optical switching based on an organic photonic-bandgap microcavity,” Nat. Photonics **2**, 185–189 (2008) [CrossRef] .

**4. **M. Waldow, T. Plötzing, M. Gottheil, M. Först, J. Bolten, T. Wahlbrink, and H. Kurz, “25ps all-optical switching in oxygen implanted silicon-on-insulator microring resonator,” Opt. Express **16**, 7693–7702 (2008) [CrossRef] [PubMed] .

**5. **D. Miller, “Are optical transistors the logical next step?” Nat. Photonics **4**, 3–5 (2010) [CrossRef] .

**6. **M. Albert, A. Dantan, and M. Drewsen, “Cavity electromagnetically induced transparency and all-optical switching using ion coulomb crystals,” Nat. Photonics **5**, 633–636 (2011) [CrossRef] .

**7. **S. E. Harris, “Electromagnetically induced transparency,” Phys. Today **50**, 36–42 (1997) [CrossRef] .

**8. **M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. **77**, 633–673 (2005) [CrossRef] .

**9. **J. Zhang, G. Hernandez, and Y. Zhu, “All-optical switching at ultralow light levels,” Opt. Lett. **32**, 1317–1319 (2007) [CrossRef] [PubMed] .

**10. **M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. **102**, 203902 (2009) [CrossRef] [PubMed] .

**11. **M. Fleischhauer, “Switching light by vacuum,” Science **333**, 1228–1229 (2011) [CrossRef] [PubMed] .

**12. **D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. **86**, 783–786 (2001) [CrossRef] [PubMed] .

**13. **A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. **88**, 023602 (2001) [CrossRef] .

**14. **A. Mair, J. Hager, D. F. Phillips, R. L. Walsworth, and M. D. Lukin, “Phase coherence and control of stored photonic information,” Phys. Rev. A **65**, 031802 (2002) [CrossRef] .

**15. **M. D. Lukin, “*Colloquium* : Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. **75**, 457–472 (2003) [CrossRef] .

**16. **B. Misra and E. Sudarshan, “The Zeno’s paradox in quantum theory,” J. of Math. Phys. **18**, 756 – 763 (1977) [CrossRef] .

**17. **B. C. Jacobs and J. D. Franson, “All-optical switching using the quantum zeno effect and two-photon absorption,” Phys. Rev. A **79**, 063830 (2009) [CrossRef] .

**18. **S. M. Hendrickson, C. N. Weiler, R. M. Camacho, P. T. Rakich, A. I. Young, M. J. Shaw, T. B. Pittman, J. D. Franson, and B. C. Jacobs, “All-optical-switching demonstration using two-photon absorption and the zeno effect,” Phys. Rev. A **87**, 023808 (2013) [CrossRef] .

**19. **Y. H. Wen, O. Kuzucu, T. Hou, M. Lipson, and A. L. Gaeta, “All-optical switching of a single resonance in silicon ring resonators,” Opt. Lett. **36**, 1413–1415 (2011) [CrossRef] [PubMed] .

**20. **K. Kieu, L. Schneebeli, E. Merzlyak, J. M. Hales, A. DeSimone, J. W. Perry, R. A. Norwood, and N. Peyghambarian, “All-optical switching based on inverse raman scattering in liquid-core optical fibers,” Opt. Lett. **37**, 942–944 (2012) [CrossRef] [PubMed] .

**21. **S. H. Autler and C. H. Townes, “Stark effect in rapidly varying fields,” Phys. Rev. **100**, 703–722 (1955) [CrossRef] .

**22. **H. A. Haus, *Wave and Fields in Optoelectronics* (Prentice-Hall, 1984).

**23. **T. J. Kippenberg, S. M. Spillane, D. K. Armani, B. Min, L. Yang, and K. J. Vahala, *Fabrication, Coupling and Nonlinear Optics of Ultra-High-Q Microcavities* (World Scientific Publishing, 2004, vol. 5, Chap. 5, pp. 177–238) [CrossRef] .

**24. **J. D. Franson, B. C. Jacobs, and T. B. Pittman, “Quantum computing using single photons and the zeno effect,” Phys. Rev. A **70**, 062302 (2004) [CrossRef] .

**25. **D. A. Steck, “Rubidium 87 d line data,” http://steck.us/alkalidata/ (2009).

**26. **S. G. Schirmer and A. I. Solomon, “Constraints on relaxation rates for *n*-level quantum systems,” Phys. Rev. A **70**, 022107 (2004) [CrossRef] .

**27. **P. R. Berman and R. C. O’Connell, “Constraints on dephasing widths and shifts in three-level quantum systems,” Phys. Rev. A **71**, 022501 (2005) [CrossRef] .

**28. **O. S. Heavens, “Radiative transition probabilities of the lower excited states of the alkali metals,” J. Opt. Soc. Am. **51**, 1058–1061 (1961) [CrossRef] .

**29. **R. G. Brewer and E. L. Hahn, “Coherent two-photon processes: Transient and steady-state cases,” Phys. Rev. A **11**, 1641–1649 (1975) [CrossRef] .

**30. **J. Gea-Banacloche, Y.-q. Li, S.-z. Jin, and M. Xiao, “Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: Theory and experiment,” Phys. Rev. A **51**, 576–584 (1995) [CrossRef] [PubMed] .

**31. **L. Stern, B. Desiatov, I. Goykhman, and U. Levy, “Evanescent light-matter interactions in atomic cladding wave guides,” arXiv:1204.0393 (2012).