## Non-Markovian dynamics of a microcavity coupled to a waveguide in photonic crystals |

Optics Express, Vol. 18, Issue 17, pp. 18407-18418 (2010)

http://dx.doi.org/10.1364/OE.18.018407

Acrobat PDF (1306 KB)

### Abstract

In this paper, the non-Markovian dynamics of a microcavity coupled to a waveguide in photonic crystals is studied based on a semi-finite tight binding model. Using the exact master equation, we solve analytically and numerically the general and exact solution of the non-Markovain dynamics for the cavity coupled to the waveguide in different coupling regime. A critical transition is revealed when the coupling increases between the cavity and the waveguide. In particular, the cavity field becomes dissipationless when the coupling strength goes beyond a critical value, as a manifestation of strong non-Markovian memory effect. The result also indicates that the cavity can maintain in a coherent state with arbitrary small number of photons when it strongly couples to the waveguide at very low temperature. These properties can be measured experimentally through the photon current flowing over the waveguide in photonic crystals.

© 2010 Optical Society of America

## 1. Introduction

1. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature **425**, 944 (2003). [CrossRef] [PubMed]

2. A. R. Md Zain, N. P. Johnson, M. Sorel, and R. M. De La Rue, “Ultra high quality factor one dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI),” Opt. Express , **16**, 12084 (2008). [CrossRef] [PubMed]

3. S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photon. **1**, 449 (2007). [CrossRef]

4. P. Yao and S. Hughes, “Controlled cavity QED and single-photon emission using a photonic-crystal waveguide cavity system,” Phys. Rev. B **80**, 165128 (2009). [CrossRef]

5. M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, “Laser oscillation in a strongly coupled single-quantum-dot V nanocavity system,” Nat. Phys. **6**, 279 (2010). [CrossRef]

6. F. Bordas, C. Seassal, E. Dupuy, P. Regreny, M. Gendry, P. Viktorovich, M. J. Steel, and A. Rahmani, “Room temperature low-threshold InAs/InP quantum dot single mode photonic crystal microlasers at 1.5 gm using cavity-confined slow light,” Opt. Express **17**, 5439 (2009). [CrossRef] [PubMed]

7. M. Loncar and A. Scherer, “Photonic crystal laser sources for chemical detection,” Appl. Phys. Lett. **82**, 4648 (2003). [CrossRef]

8. M. Skorobogatiy and A. V. Kabashin, “Photon crystal waveguide-based surface plasmon resonance biosensor,” Appl. Phys. Lett. **89**, 143518 (2006). [CrossRef]

9. S. Mandal, X. Serey, and D. Erickson, “Nanomanipulation using silicon photonic crystal resonators,” Nano Lett. **10**, 99 (2010). [CrossRef]

10. T. Baba, “Slow light in photonic crystals,” Nat. Photon. **2**, 465 (2008). [CrossRef]

11. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. **87**, 253902 (2001). [CrossRef] [PubMed]

12. H. G. Park, C. J. Barrelet, Y. Wu, B. Tian, F. Qian, and C. M. Lieber, “A wavelength-selective photonic-crystal waveguide coupled to a nanowire light source,” Nat. Photon. **2**, 622 (2008). [CrossRef]

13. Y. Liu, Z. Wang, M. Han, S. Fan, and R. Dutton, “Mode-locking of monolithic laser diodes incorporating coupled-resonator optical waveguides,” Opt. Express **13**, 4539 (2005). [CrossRef] [PubMed]

14. P. Lambropoulos, G. Nikolopoulos, T. R. Nielsen, and S. Bay, “Fundamental quantum optics in structured reservoirs,” Rep. Prog. Phys. **63**, 455 (2000). [CrossRef]

15. S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: Photon bound states and dressed atoms,” Phys. Rev. Lett. **64**, 2418 (1990). [CrossRef] [PubMed]

16. S. John and T. Quang, “Spontaneous emission near the edge of a photonic band gap,” Phys. Rev. A **50**, 1764 (1994). [CrossRef] [PubMed]

18. A. G. Kofman, G. Kurizki, and B. Sherman, “Spontaneous and Induced Atomic Decay in Photonic Band Structures,” J. Mod. Opt. **41**, 353 (1994). [CrossRef]

19. D. Mogilevtsev, F. Moreira, S. B. Cavalcanti, and S. Kilin, “Field-emitter bound states in structured thermal reservoirs,” Phys. Rev. A **75**, 043802 (2007). [CrossRef]

20. S. Longhi, “Non-Markovian decay and lasing condition in an optical microcavity coupled to a structured reservoir,” Phys. Rev. A **74**, 063826 (2006). [CrossRef]

## 2. The microcavity dynamics coupled to a waveguide

### 2.1. Fano-type tight-binding model for a microcavity coupled to a waveguide

*h̄*= 1. The first term in Eq. (1) is the Hamiltonian of the microcavity in which

*a*† and

*a*are the creation and annihilation operators of the single mode cavity field, with the frequency

*ω*which can easily be tuned to any value within the band gap by changing the size or the shape of the defect. The second and third terms are the Hamiltonian of the waveguide where

_{c}*a*

^{†}

_{n}and

*a*are the photonic creation and annihilation operators of the resonator at site

_{n}*n*of the waveguide with an identical frequency

*ω*

_{0}, and

*ξ*

_{0}is the hopping rate between adjacent resonator modes. Both

*ω*

_{0}and

*ξ*

_{0}are experimentally turnable. In practical,

*ξ*

_{0}≪

*ω*

_{0}. Also, we consider the waveguide in this paper to be semi-infinite, namely

*N*→ ∞. The last term in Eq. (1) is the coupling between the microcavity and the waveguide with the coupling constant

*ξ*. While, the coupling between the cavity and the waveguide is also controllable by changing the geometrical parameters of the defect cavity and the distance between the cavity and the waveguide [13

13. Y. Liu, Z. Wang, M. Han, S. Fan, and R. Dutton, “Mode-locking of monolithic laser diodes incorporating coupled-resonator optical waveguides,” Opt. Express **13**, 4539 (2005). [CrossRef] [PubMed]

21. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. **24**, 711 (1999). [CrossRef]

22. J. K. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B **21**, 1665 (2004). [CrossRef]

23. U. Fano, “Effects of Configuration interaction on intensities and phase shifts,” Phys. Rev. **124**, 1866 (1961). [CrossRef]

24. S. Longhi, “Spectral singularities in a non-Hermitian Friedrichs-Fano-Anderson model,” Phys. Rev. B **80**, 165125 (2009). [CrossRef]

*k*≤

*π*,

*ω*and

_{k}*V*are given by:

_{k}*a*

^{†}

_{k},

*a*are the creation and annihilation operators of the corresponding Bloch modes of the waveguide, which is defined as follow:

_{k}### 2.2. Exact master equation

*ρ*(

*t*) ≡ tr

_{R}

*ρ*

_{tot}(

*t*), where the total density operator is governed by the quantum Liouville equation

25. A. J. Leggett, S. Chakravarty, A. T. Dorsey, M.P. Fisher, A. Garg, and W. Zwerger, “Dynamics of the dissipative two-state system,” Rev. Mod. Phys. **59**, 1 (1987). [CrossRef]

*ρ*

_{tot}(

*t*

_{0}) =

*ρ*(

*t*

_{0}) ⊗

*ρ*

_{E}(

*t*

_{0}), where

*ρ*(

*t*

_{0}) can be any arbitrary initial state of the cavity, and the reservoir is initially in a thermal equilibrium state:

26. R. P. Feynman and F. L. Vernon, “The theory of a general quantum system interacting with a linear dissipative system,” Ann. Phys. **24**, 118 (1963). [CrossRef]

27. W. M. Zhang, D. H. Feng, and R. Gilmore, “Coherent states: theory and some applications,” Rev. Mod. Phys. **62**, 867 (1990). [CrossRef]

28. M. W. Y. Tu and W. M. Zhang, “Non-Markovian decoherence theory for a double-dot charge qubit,” Phys. Rev. B **78**, 235311 (2008). [CrossRef]

29. M. W. Y. Tu, M. T. Lee, and W. M. Zhang, “Exact master equation and non-markovian decoherence for quantum dot quantum computing,” Quantum Inf. Process **8**, 631 (2009). [CrossRef]

30. J. H. Au and W. M. Zhang, “Non-Markovian entanglement dynamics of noisy continuous-variable quantum channels,” Phys. Rev. A , **76**, 042127 (2007). [CrossRef]

*ω*ʹ

_{c}(

*t*) is the renormalized frequency of the cavity which contains the frequency shift due to the coupling with the reservoir, while

*κ*(

*t*) and

*(*κ ˜

*t*) describe the dissipation and noise to the cavity field induced by the reservoir. These coefficients are non-perturbatively determined by the following relations:

*u*(

*t*) and

*v*(

*t*) satisfy the integrodifferential equations of motion:

*u*(

*t*

_{0}) = 1 while

*ū*(

*τ*) ≡

*u*(

*t*+

*t*

_{0}−

*τ*).

*g*(

*τ*−

*τ*ʹ) and

*g͂*(

*τ*−

*τ*ʹ), in the above equations are the time correlation functions of the waveguide. These two time-correlation functions characterize all the non-Markovian memory structures between the cavity and the waveguide. By defining the spectral density of the waveguide:

*J*(

*ω*) = 2

*π*Σ

_{k}|

*V*|

_{k}^{2}

*δ*(

*ω*−

*ω*), the time-correlation functions are explicitly given by

_{k}*t*

_{0}. With the spectrum of the photonic crystal, Eq. (3), the spectral density becomes

*J*(

*ω*) = 2

*πg*(

*ω*)|

*V*(

*ω*)|

^{2}, and

*g*(

*ω*) is the density of state:

*ω*

_{0}− 2

*ξ*

_{0}<

*ω*<

*ω*

_{0}+ 2

*ξ*

_{0}. Then the spectral density can be explicitly written as

*ξ*

_{0}≪

*ω*

_{0}, namely the waveguide has a very narrow band.

*ω*ʹ

_{0}(

*t*),

*κ*(

*t*) and

*(*κ ˜

*t*), in Eq. (5) through the integrodifferential equations of motion, Eqs. (9) and (10). Thus, the non-Markovian memory structure is non-perturbatively built into the integral kernels in these equations. The equations (9) and (10) show that

*u*(

*t*) is just the propagating function of the cavity field (the retarded Green function in nonequilibrium Green function theory [32]), and

*v*(

*t*) is the corresponding correlation (Green) function, as we will see next. Therefore, the exact master equation, Eq. (5), depicts the full nonequilibrium dynamics of the cavity system as well as the waveguide.

### 2.3. Exact solutions of the microcavity dynamics

*a*(

*t*)〉 = tr[

*aρ*(

*t*)]. From Eq. (5), it is easy to find that 〈

*a*(

*t*)〉 obeys the equation of motion

*u*(

*t*), which indicates that

*u*(

*t*) is the exact propagating function characterizing the cavity field evolution.

*n*(

*t*) =tr[

*a*

^{†}

*aρ*(

*t*)]. From the exact master equation, it is also easy to find that

*κ*(

*t*) = [

*u̇*/

*u*(

*t*) + H.c.]. Combing these equations together, we obtain the exact solution for the time-dependent photon number in terms of

*u*(

*t*) and

*v*(

*t*):

28. M. W. Y. Tu and W. M. Zhang, “Non-Markovian decoherence theory for a double-dot charge qubit,” Phys. Rev. B **78**, 235311 (2008). [CrossRef]

30. J. H. Au and W. M. Zhang, “Non-Markovian entanglement dynamics of noisy continuous-variable quantum channels,” Phys. Rev. A , **76**, 042127 (2007). [CrossRef]

*t*becomes

*α*(

*t*) =

*u*(

*t*)

*α*

_{0}. It is interest to see that Eq. (21) is indeed a mixed state of generalized coherent states

*n*(

*t*) = |

*u*(

*t*)

*α*

_{0}|

^{2}+

*v*(

*t*), as we expected. An initial state other than the coherent state will result in different reduced density operator, as we have also shown explicitly in [34].

*u*(

*t*) decays to zero due to the dissipation induced by the coupling to the waveguide. The corresponding reduced density operator asymptotically becomes a thermally state with the asymptotic photon number

*n*(

*t*) =

*v*(

*t*→ ∞) ~

*n̄*(

*ω*,

_{c}*T*). This solution shows precisely how the cavity field loses its coherence (i.e. decoherence) due to the coupling to the waveguide. This decoherence arises from the decay of the cavity field amplitude

*α*(

*t*) =

*u*(

*t*)

*α*

_{0}as well as the thermal-fluctuation-induced noise effect manifested through the correlation function

*v*(

*t*), as shown in Eq. (21). The later describes a process of randomly losing or gaining the thermal energy from the reservoir (here is the waveguide), upon the initial temperature of the waveguide.

*u*(

*t*) may not decay to zero, as we shall show explicitly in the numerical calculation in the next section. Then the reduced density operator remains as a mixed coherent state. On the other hand, at zero-temperature limit

*T*= 0, we have

*n̄*(

*ω*,

*T*) = 0 so that

*g̃*(

*τ*−

*τ*ʹ) = 0. As a result, we obtain

*v*(

*t*) = 0. The reduced density operator at zero temperature limit is given by

*u*(

*t*) may not decay to zero in the strong coupling regime and

*v*(

*t*) = 0 at

*T*= 0] indicate that enhancing the coupling between the cavity and the waveguide and meantime lowing the initial temperature of the waveguide can significantly reduce the cavity’s decoherence effect in photonic crystals.

## 3. Numerical analysis of the exact non-Markovian dynamics

35. M. Bayindir, B. Temelkuran, and E. Ozbay, “Tight-Binding Description of the coupled defect modes in three-dimensional photonic crystals,” Phys. Rev. Lett. **84**, 2140 (2000). [CrossRef] [PubMed]

*ω*

_{0}= 12.15 GHz = 50.25

*µ*eV (in the unit

*h̄*= 1), and the coupling between the adjacent resonators to be

*ξ*

_{0}= 1.24

*µ*eV. The initial temperature of the waveguide is set at

*T*= 5

*K*so that

*k*= 430.75

_{B}T*µ*eV ≈ 8.57

*ω*

_{0}. The frequency of the single mode cavity

*ω*and the coupling

_{c}*ξ*between the cavity and the waveguide are tunable parameters by changing the geometry of the cavity and the distance between the cavity and the waveguide [13

13. Y. Liu, Z. Wang, M. Han, S. Fan, and R. Dutton, “Mode-locking of monolithic laser diodes incorporating coupled-resonator optical waveguides,” Opt. Express **13**, 4539 (2005). [CrossRef] [PubMed]

36. A. Faraon, E. Waks, D. Englund, I. Fushman, and J. Vuckovic, “Efficient photonic crystal cavity-waveguide couplers,” Appl. Phys. Lett. **90**, 073102 (2007). [CrossRef]

37. S. Hughes and H. Kamada, “Single-quantum-dot strong coupling in a semiconductor photonic crystal nanocavity side coupled to a waveguide,” Phys. Rev. B **70**, 195313 (2004). [CrossRef]

*ω*=

_{c}*ω*

_{0}). ii), the cavity coupled to the waveguide near the upper band edge (

*ω*

_{0}<

*ω*<

_{c}*ω*

_{0}+ 2

*ξ*

_{0}). iii), the cavity coupled apart from the band of the waveguide (

*ω*>

_{c}*ω*

_{0}). Detailed numerical results are plotted in Figs. 2 through 5.

*a*(

*t*)〉/〈

*a*(

*t*

_{0})〉| = |

*u*(

*t*)| (see Eq. (16)), in different coupled configuration from the weak coupling to strong coupling regime. For

*ω*lies outside the band of the waveguide, in both the weak and strong coupling regimes, |

_{c}*u*(

*t*)| remains unchanged beside a short time very small oscillation at the beginning, see Fig. 2(a). This result indicates that when its frequency lies outside the band of the waveguide, the cavity effectively decouples from the waveguide. However, when

*ω*lies inside the band of the waveguide, the time evolution behavior of the field amplitude is totally different in different coupling regime, see Fig. 2(b) and 2(c). In weak coupling regime (in terms of a dimensionless coupling rate

_{c}*u*(

*t*)| decays to zero monotonically, as a typical Markov process. However, increasing the coupling such that

*η*> 1.0, after a short time decay at the beginning, the field amplitude begins to revives, and more than that, it keeps oscillating below an nonzero value. This behavior shows that the cavity field no longer decays monotonically in strong coupling regime, as a significant non-Markovian memory effect.

*ω*=

_{c}*ω*

_{0}.

*u*(

*t*)| for

*ω*=

_{c}*ω*

_{o}by varying the coupling rate

*η*and the time

*t*, where the critical transition from the Markov to non-Markovian dynamics is manifested with the critical coupling

*η*≃ 0.7 ~ 1.0. To understand the underlying mechanism of this critical transition, we also plot in Fig. 3 the decay coefficient in Eq. (5),

_{c}*κ*(

*t*) = −Re[

*u̇*(

*t*)/

*u*(

*t*)], for different coupling configurations. The decay coefficient

*κ*(

*t*) dominates the dissipation behavior of the cavity field, roughly given by the damping factor

*κ*(

*t*) approaches to a stationary positive value, see Fig. 3(c). This leads to a monotonic decay for the cavity field, i.e. a dissipation process. However, in strong coupling regime, the behavior of

*κ*(

*t*) is totally different, it keeps oscillation in all the time between an equal positive and negative bound value without approaching to zero, see Fig. 3(b). This oscillation process means that the cavity dissipates energy to the waveguide and then fully regains it back from the waveguide repeatedly. The overall effect of this reviving process is that no energy dissipates (or photon losses) into the waveguide. In other words, the cavity dynamics becomes dissipationless in the strong coupling regime. Thus, the critical transition from weak to strong coupling regime reveals the transition from dissipation into dissipationless processes for the cavity dynamics, as a manifestation of the non-Markovian memory effect.

15. S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: Photon bound states and dressed atoms,” Phys. Rev. Lett. **64**, 2418 (1990). [CrossRef] [PubMed]

16. S. John and T. Quang, “Spontaneous emission near the edge of a photonic band gap,” Phys. Rev. A **50**, 1764 (1994). [CrossRef] [PubMed]

18. A. G. Kofman, G. Kurizki, and B. Sherman, “Spontaneous and Induced Atomic Decay in Photonic Band Structures,” J. Mod. Opt. **41**, 353 (1994). [CrossRef]

20. S. Longhi, “Non-Markovian decay and lasing condition in an optical microcavity coupled to a structured reservoir,” Phys. Rev. A **74**, 063826 (2006). [CrossRef]

15. S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: Photon bound states and dressed atoms,” Phys. Rev. Lett. **64**, 2418 (1990). [CrossRef] [PubMed]

*u*(

*t*). This is not surprising since Eq. (16) shows that the Green function

*u*(

*t*) determines the photon propagation in cavity states containing arbitrary number of photons. Therefore it must cover the special case involving only one single photon. The non-Markovian dynamics found in [15

**64**, 2418 (1990). [CrossRef] [PubMed]

16. S. John and T. Quang, “Spontaneous emission near the edge of a photonic band gap,” Phys. Rev. A **50**, 1764 (1994). [CrossRef] [PubMed]

18. A. G. Kofman, G. Kurizki, and B. Sherman, “Spontaneous and Induced Atomic Decay in Photonic Band Structures,” J. Mod. Opt. **41**, 353 (1994). [CrossRef]

20. S. Longhi, “Non-Markovian decay and lasing condition in an optical microcavity coupled to a structured reservoir,” Phys. Rev. A **74**, 063826 (2006). [CrossRef]

*v*(

*t*) given by Eq. (4). Physically, Eq. (19) shows that if the cavity is initially empty, then

*v*(

*t*) is the average photon number inside the cavity, induced by the thermal fluctuation of the waveguide. In Fig. 4(b), we plot

*v*(

*t*) with a few different coupling strength. As we see in the weak coupling regime (

*η*< 0.7), the exact

*v*(

*t*) increases monotonically and approaches to

*n̄*(

*ω*,

_{c}*T*) gradually. However, in strong coupling regime (

*η*> 1.0), the behavior of

*v*(

*t*) is qualitatively different from that in weak coupling case. It increases much faster within a very short time in the beginning, then keeps oscillation in a long time, in response to the corresponding dissipationless oscillation of the cavity amplitude

*u*(

*t*).

*v*(

*t*) in Fig. 4(a) with a very low temperature,

*T*= 5 mK. The value of

*v*(

*t*) is reduced dramatically [< 10

^{−8}as shown in Fig. 4(a)]. This clearly shows that

*v*(

*t*) characterizes the noise effect of the thermal fluctuation from the waveguide. Lowing the initial temperature of the waveguide can efficiently suppress the thermal noise effect. Based on the analytical solution of the reduced density matrix in the last section, if the cavity is initially in a coherent state, and if the initial temperature of the waveguide is low enough such that

*v*(

*t*) → 0, the cavity state is given by Eq. (22) where

*α*(

*t*) =

*u*(

*t*)

*α*(0). As a result, we can maintain well the cavity’s coherence by enhancing the coupling between the cavity and the waveguide such that the dissipation can also be suppressed.

*v*(

*t*), dominates the photon number in the cavity, where

*v*(

*t*→ ∞) ~

*n̄*(

*ω*,

_{c}*T*) which is about a few tens [~ 50 for

*T*= 5 K, as shown in Fig. 4(b)] when

*ω*is in the microwave region. However, in a very low temperature,

_{c}*v*(

*t*) approaches to zero [

*v*(

*t*) < 10

^{−8}at

*T*= 5 mK, as shown in Fig. 4(a)]. Then

*n*(

*t*) is fully dominated by the evolution of the initial photon number in the cavity, i.e. the first term |

*u*(

*t*)|

^{2}

*n*(

*t*

_{0}) in Eq. (19).

*η*< 0.7),

*n*(

*t*) approaches gradually and monotonically to

*n̄*(

*ω*,

_{c}*T*). However, in the strong coupling regime (

*η*> 1.0),

*n*(

*t*) quickly reaches to

*n̄*(

*ω*,

_{c}*T*) and then oscillates around

*n̄*(

*ω*,

_{c}*T*) due to the dissipationless oscillation of

*u*(

*t*). In fact, the dissipationless oscillation of

*u*(

*t*) in strong coupling regime indicates that the cavity field can produce subsequent pulses with a small number of photons in the low temperature region. From Fig. 5(a), we can see that the cavity can generate indeed single photon pulses at

*T*= 5 mK when the coupling rate

*η*=

*ξ*/

*ξ*

_{0}= 2, namely the coupling between the cavity and the waveguide is twice of the coupling between the adjacent resonators in the waveguide, which is experimentally feasible. Fig. 5(c) and 5(d) also plot the photon current in the waveguide, which shows the corresponding oscillation associated with the amplitude oscillation of the cavity field. Physically this result indicates that the photons tunnel between the cavity and the waveguide repeatedly without loss of the coherence in the strong coupling regime. Experimentally, one can directly measure the photon current flowing over the waveguide to demonstrate these properties. These properties may provide new applications for the microcavity in photonic crystals.

*ω*lies outside the band of the waveguide, the cavity dynamics effectively decouples from the waveguide. However, when

*ω*locates inside the band of the waveguide, the non-Markovian memory effect can qualitatively change the dissipation as well as the noise dynamics of the cavity field. In particular, the coupling between the microcavity and the waveguide can be used to manipulate well the dissipation behavior of the cavity dynamics. Meanwhile, in the very low temperature limit, one can also control efficiently the cavity coherence as well as the photon number. Otherwise, the thermal fluctuation may induce non-negligible noise effect.

## 4. Conclusion

**64**, 2418 (1990). [CrossRef] [PubMed]

**50**, 1764 (1994). [CrossRef] [PubMed]

**41**, 353 (1994). [CrossRef]

**74**, 063826 (2006). [CrossRef]

## Acknowledgements

## References and links

1. | Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature |

2. | A. R. Md Zain, N. P. Johnson, M. Sorel, and R. M. De La Rue, “Ultra high quality factor one dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI),” Opt. Express , |

3. | S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photon. |

4. | P. Yao and S. Hughes, “Controlled cavity QED and single-photon emission using a photonic-crystal waveguide cavity system,” Phys. Rev. B |

5. | M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, “Laser oscillation in a strongly coupled single-quantum-dot V nanocavity system,” Nat. Phys. |

6. | F. Bordas, C. Seassal, E. Dupuy, P. Regreny, M. Gendry, P. Viktorovich, M. J. Steel, and A. Rahmani, “Room temperature low-threshold InAs/InP quantum dot single mode photonic crystal microlasers at 1.5 gm using cavity-confined slow light,” Opt. Express |

7. | M. Loncar and A. Scherer, “Photonic crystal laser sources for chemical detection,” Appl. Phys. Lett. |

8. | M. Skorobogatiy and A. V. Kabashin, “Photon crystal waveguide-based surface plasmon resonance biosensor,” Appl. Phys. Lett. |

9. | S. Mandal, X. Serey, and D. Erickson, “Nanomanipulation using silicon photonic crystal resonators,” Nano Lett. |

10. | T. Baba, “Slow light in photonic crystals,” Nat. Photon. |

11. | M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. |

12. | H. G. Park, C. J. Barrelet, Y. Wu, B. Tian, F. Qian, and C. M. Lieber, “A wavelength-selective photonic-crystal waveguide coupled to a nanowire light source,” Nat. Photon. |

13. | Y. Liu, Z. Wang, M. Han, S. Fan, and R. Dutton, “Mode-locking of monolithic laser diodes incorporating coupled-resonator optical waveguides,” Opt. Express |

14. | P. Lambropoulos, G. Nikolopoulos, T. R. Nielsen, and S. Bay, “Fundamental quantum optics in structured reservoirs,” Rep. Prog. Phys. |

15. | S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: Photon bound states and dressed atoms,” Phys. Rev. Lett. |

16. | S. John and T. Quang, “Spontaneous emission near the edge of a photonic band gap,” Phys. Rev. A |

17. | S. Kilin and D. Mogilevtsev, ““Freezing” of decay of a quantum system with a dip in a spectrum of the heat bath-coupling constants,” Laser Phys. |

18. | A. G. Kofman, G. Kurizki, and B. Sherman, “Spontaneous and Induced Atomic Decay in Photonic Band Structures,” J. Mod. Opt. |

19. | D. Mogilevtsev, F. Moreira, S. B. Cavalcanti, and S. Kilin, “Field-emitter bound states in structured thermal reservoirs,” Phys. Rev. A |

20. | S. Longhi, “Non-Markovian decay and lasing condition in an optical microcavity coupled to a structured reservoir,” Phys. Rev. A |

21. | A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. |

22. | J. K. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B |

23. | U. Fano, “Effects of Configuration interaction on intensities and phase shifts,” Phys. Rev. |

24. | S. Longhi, “Spectral singularities in a non-Hermitian Friedrichs-Fano-Anderson model,” Phys. Rev. B |

25. | A. J. Leggett, S. Chakravarty, A. T. Dorsey, M.P. Fisher, A. Garg, and W. Zwerger, “Dynamics of the dissipative two-state system,” Rev. Mod. Phys. |

26. | R. P. Feynman and F. L. Vernon, “The theory of a general quantum system interacting with a linear dissipative system,” Ann. Phys. |

27. | W. M. Zhang, D. H. Feng, and R. Gilmore, “Coherent states: theory and some applications,” Rev. Mod. Phys. |

28. | M. W. Y. Tu and W. M. Zhang, “Non-Markovian decoherence theory for a double-dot charge qubit,” Phys. Rev. B |

29. | M. W. Y. Tu, M. T. Lee, and W. M. Zhang, “Exact master equation and non-markovian decoherence for quantum dot quantum computing,” Quantum Inf. Process |

30. | J. H. Au and W. M. Zhang, “Non-Markovian entanglement dynamics of noisy continuous-variable quantum channels,” Phys. Rev. A , |

31. | J. H. Au, M. Feng, and W. M. Zhang, “Non-Markovian decoherence dynamics of entangled coherent states,” Quant. Info. Comput. |

32. | L. P. Kadanoff and G. Baym, |

33. | J. S. Jin, M. W. Y. Tu, W. M. Zhang, and Y. J. Yan, “A nonequilibrium theory for transient transport dynamics in nanostructures via the Feynman-Vernon influence functional approach,” arXiv:0910.1675 (to appear in N. J. Phys., 2010). |

34. | H. N. Xiong, W. M. Zhang, X. G. Wang, and M. H. Wu, “Exact non-Markovian cavity dynamics strongly coupled to a reservoir,” arXiv:1005.0904 (to appear in Phys. Rev. A, 2010). |

35. | M. Bayindir, B. Temelkuran, and E. Ozbay, “Tight-Binding Description of the coupled defect modes in three-dimensional photonic crystals,” Phys. Rev. Lett. |

36. | A. Faraon, E. Waks, D. Englund, I. Fushman, and J. Vuckovic, “Efficient photonic crystal cavity-waveguide couplers,” Appl. Phys. Lett. |

37. | S. Hughes and H. Kamada, “Single-quantum-dot strong coupling in a semiconductor photonic crystal nanocavity side coupled to a waveguide,” Phys. Rev. B |

38. | D. Mogilevtsev, S. Kilin, F. Moreira, and S. B. Cavalcanti, “Markovian and non-Markovian decay in pseudo-gaps,” Photon Nanostruct.: Fundam Appl. |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(130.5296) Integrated optics : Photonic crystal waveguides

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: June 2, 2010

Revised Manuscript: June 22, 2010

Manuscript Accepted: July 14, 2010

Published: August 12, 2010

**Citation**

Wei-Min Zhang, Meng-Hsiu Wu, Chan U Lei, and Heng-Na Xiong, "Non-Markovian dynamics of a microcavity coupled to a waveguide in photonic crystals," Opt. Express **18**, 18407-18418 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-17-18407

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