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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 9 — Apr. 28, 2008
  • pp: 6240–6250
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Towards a picosecond transform-limited nitrogen-vacancy based single photon source

Chun-Hsu Su, Andrew D. Greentree, and Lloyd C. L. Hollenberg  »View Author Affiliations


Optics Express, Vol. 16, Issue 9, pp. 6240-6250 (2008)
http://dx.doi.org/10.1364/OE.16.006240


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Abstract

We analyze a nitrogen-vacancy (NV-) colour centre based single photon source based on cavity Purcell enhancement of the zero phonon line and suppression of other transitions. Optimal performance conditions of the cavity-centre system are analyzed using Master equation and quantum trajectory methods. By coupling the centre strongly to a high-finesse optical cavity [Q~𝒪(104-105), V3] and using sub-picosecond optical excitation the system has striking performance, including effective lifetime of 70 ps, linewidth of 0.01 nm, near unit single photon emission probability and small [𝒪(10-5)] multi-photon probability.

© 2008 Optical Society of America

1. Introduction

A source that produces single photons on demand is an invaluable tool for precision optical measurement [1

1. E. S. Polzik, J. Carri, and H. J. Kimble, “Spectroscopy with squeezed light,” Phys. Rev. Lett. 68, 3020–3023 (1992). [CrossRef] [PubMed]

, 2

2. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: Beating the standard quantum limit,” Science 306, 1330–1336 (2004). [CrossRef] [PubMed]

] and is a crucial building block for many quantum computing and communication applications. For an attenuated laser, the number of photons per pulse follows a Poisson distribution and the multi-photon probability becomes negligible only in the limit of small mean photon number at the expense of the single-photon probability. In quantum computing, using single photons to store and transport quantum information is natural as information can be easily encoded and manipulated over photonic degrees of freedom e.g. polarization. In linear optical quantum computing (LOQC) [3

3. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature (London) 409, 46–52 (2001). [CrossRef] [PubMed]

, 4

4. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing,” Rev. Mod. Phys. 79, 135–174 (2007). [CrossRef]

], it is straightforward to perform single qubit operations on photons with elementary optical components and projective measurements generate photon-photon interactions. In quantum communication, single photon sources can be used for unconditionally secure quantum key distribution (QKD) protocols [5

5. E. Waks, K. Inoue, C. Santori, D. Fattal, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, “Quantum cryptography with a photon turnstile,” Nature (London) 420, 762 (2002). [CrossRef] [PubMed]

, 6

6. E. Waks, C. Santori, and Y. Yamamoto, “Security aspects of quantum key distribution with sub-Poisson light,” Phys. Rev. A 66, 042315 (2002). [CrossRef]

]. To date, single photon generation has been demonstrated with a variety of single quantum emitters such as atoms [7–10

7. A. Kuhn, M. Hennrich, and G. Rempe, “Deterministic single-photon source for distributed quantum networking,” Phys. Rev. Lett. 89, 067901 (2002). [CrossRef] [PubMed]

], ions [11

11. M. Keller, B. Lange, K. Hayasaka, W. Lange, and H. Walther, “Continuous generation of single photons with controlled waveform in an ion-trap cavity system,” Nature (London) 431, 1075–1078 (2004). [CrossRef] [PubMed]

], molecules [12

12. C. Brunel, B. Lounis, Ph. Tamarat, and M. Orrit, “Triggered source of single photons based on controlled single molecule fluorescence,” Phys. Rev. Lett. 83, 2722–2725 (1999). [CrossRef]

,13

13. B. Lounis and W. E. Moerner, “Single photons on demand from a single molecule at room temperature,” Nature (London) 407, 491–493 (2000). [CrossRef] [PubMed]

], diamond colour centres [14–17

14. C. Kurtsiefer, S. Mayer, P. Zarda, and H. Weinfurter, “Stable solid-state source of single-photons,” Phys. Rev. Lett. 85, 290–293 (2000). [CrossRef] [PubMed]

] and semiconductor quantum dots [18–21

18. D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vučković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904 (2005). [CrossRef] [PubMed]

].

Diamond defects hold promise as a platform for solid-state quantum optical and quantum computing applications [22

22. A. D. Greentree, P. Olivero, M. Draginski, E. Trajkov, J. R. Rabeau, P. Reichart, B. C. Gibson, S. Rubanov, S. T. Huntington, D. N. Jamieson, and S. Prawer, “Critical components for diamond-based quantum coherent devices,” J. Phys.: Cond. Matt. 18, S825–S842 (2006). [CrossRef]

], and for the study of condensed-matter analogues [23–25

23. M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nature Physics 2, 849–855 (2006). [CrossRef]

]. One of the most well-studied systems, garnering much attention lately, is the optically active negatively-charged nitrogen-vacancy defect (NV- centre). The centre consists of a substitutional nitrogen atom and an adjacent vacancy in the carbon lattice. It forms naturally or may be engineered within a diamond matrix using techniques such as single ion implantation [26–28

26. J. Meijer, B. Burchard, M. Domhan, C. Wittmann, T. Gaebel, I. Popa, F. Jelezko, and J. Wrachtrup, “Generation of single color centers by focused nitrogen implantation,” Appl. Phys. Lett. 87, 261909 (2005). [CrossRef]

] or chemical vapour deposition [29

29. J. R. Rabeau, A. Stacey, A. Rabeau, F. Jelezko, I. Mirza, J. Wrachtrup, and S. Prawer, “Single nitrogen vacancy centers in chemical vapor deposited diamond nanocrystals,” Nano Letters 7, 3433–3437 (2007). [CrossRef] [PubMed]

]. It has a combination of remarkable properties that render it a suitable single photon source candidate. These include robustness against photobleaching, structural stability at room temperature and demonstrated antibunching, which is the hallmark of a single photon source [14

14. C. Kurtsiefer, S. Mayer, P. Zarda, and H. Weinfurter, “Stable solid-state source of single-photons,” Phys. Rev. Lett. 85, 290–293 (2000). [CrossRef] [PubMed]

]. The centre has also been used to realize Wheeler’s delayed-choice experiment [30

30. V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Experimental realization of Wheeler’s delayed-choice gedanken experiment,” Science 315, 966–968 (2007). [CrossRef] [PubMed]

]. However, the centre has a relatively long photoluminescence lifetime of 11.6 ns and broad spectral width of 150 nm [31

31. N. B. Manson, J. P. Harrison, and M. J. Sellars, “Nitrogen-vacancy center in diamond: Model of the electronic structure and associated dynamics,” Phys. Rev. B 74, 104303 (2006). [CrossRef]

], which is not optimal for daylight or optical fibre operation in QKD [32

32. A. Beveratos, R. Brouri, T. Gacoin, A. Villing, J.-P. Poizat, and P. Grangier, “Single photon quantum cryptography,” Phys. Rev. Lett. 89, 187901 (2002). [CrossRef] [PubMed]

, 33

33. R. Alléaume, F. Treussart, G. Messin, Y. Dumeige, J.-F. Roch, A. Beveratos, R. Brouri-Tualle, J.-P. Poizat, and P. Grangier, “Experimental open-air quantum key distribution with a single-photon source,” New J. Phys. 6, 92 (2004). [CrossRef]

]. Furthermore, the photons are not time-bandwidth limited (or indistinguishable) for the purpose of LOQC where photon indistinuishability is crucial for Hong-Ou-Mandel interference [34

34. C. K. Hong, Z. Y Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987). [CrossRef] [PubMed]

] and hence quantum gates [35

35. P. P. Rohde, T. C. Ralph, and M. A. Nielsen, “Optimal photons for quantum-information processing,” Phys. Rev. A 72, 052332 (2005). [CrossRef]

].

Preparing the centre in a high-finesse quantum cavity offers a solution to these problems. Cavity quantum electrodynamics has been shown to induce a single-photon Kerr nonlinearity [36

36. Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710–4713 (1995). [CrossRef] [PubMed]

] and assist quantum gate operation [37

37. L.-M. Duan and H. J. Kimble, “Scalable photonic quantum computation through cavity-assisted interactions,” Phys. Rev. Lett. 92, 127902 (2004). [CrossRef] [PubMed]

, 38

38. L.-M. Duan, B. Wang, and H. J. Kimble, “Robust quantum gates on neutral atoms with cavity-assisted photon scattering,” Phys. Rev. A 72, 032333 (2005). [CrossRef]

]. Under strong photonic confinement, the quantum emitter (the centre in this case) in the cavity interacts coherently with photon states with the effect of modifying photon-emission dynamics. As a result, we show that the spectral properties of the centre can be improved to fulfill the stringent criteria for quantum information applications. Additionally, the emission can be directed into an application or experiment as desired. A suitable cavity is the planar photonic-band-gap (PBG) cavity that defines an excellent cavity with small mode volume (of order one cubic wavelength) and low loss that provides strong centre-photon coupling [39–41

39. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature (London) 425, 944–947 (2003). [CrossRef] [PubMed]

] suitable for our purposes.

Advances in fabrication techniques are nearing the stage where preparing a single crystal diamond with PBG structures using lithography and lift-off [42–46

42. P. Olivero, S. Rubanov, P. Reichart, B. C. Gibson, S. T. Huntington, J. R. Rabeau, A. D. Greentree, J. Salzman, D. Moore, D. N. Jamieson, and S. Prawer, “Ion-beam-assisted lift-off techniques for three-dimensional micromachining of freestanding single-crystal diamond,” Advanced Materals (Weinheim, Ger.) 17, 2427–2430 (2005). [CrossRef]

] and placing an individual NV centre in the centre of the cavity using ion implantation techniques may be possible. The latter technique permits locating the centre to achieve full emission enhancement. We note that cavities have been used to enhance the emission of quantum-dots [18

18. D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vučković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904 (2005). [CrossRef] [PubMed]

, 20

20. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoǧlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature (London) 445, 896–899 (2007). [CrossRef] [PubMed]

] and atoms [10

10. M. Hijlkema, B. Weber, H. P. Specht, S. C. Webster, A. Kuhn, and G. Rempe, “A single-photon server with just one atom,” Nature Physics 3, 253–255 (2007). [CrossRef]

] for photon generation. Our study is made in a similar spirit, but a diamond-based device has the advantages of robustness against the environmental noise combined with simplicity of its setup. Here, without loss of generality, we theoretically study the effects of placing the centre within a high-finesse single-moded diamond PBG cavity. We establish the operation criteria (cavity specification and excitation scheme) for an efficient cavity-centre based single photon source.

2. Theory

A model of an NV- centre has been proposed as a vibronic system with ground (g) and excited (e) electronic states, each given by a series of vibrational sublevels |gi〉 and |ej〉 respectively, where i and j label the vibrational states [47

47. G. Davies and M. F. Hamer, “Optical studies of the 1.945eV vibronic band in diamond,” Proc. R. Soc. Lond. A: Math. and Phys. Sci. 348, 285–298 (1976). [CrossRef]

]. Its emission spectra consists of several phonon lines iPL, corresponding to the transition |e0〉-|gi〉. For simplicity, |e0〉 is denoted as |e〉 from now on. Phononic relaxation of the excited state is much faster than the radiative relaxation to the ground state and the photonic transition probabilities were calculated by Davies and Hamer [47

47. G. Davies and M. F. Hamer, “Optical studies of the 1.945eV vibronic band in diamond,” Proc. R. Soc. Lond. A: Math. and Phys. Sci. 348, 285–298 (1976). [CrossRef]

] under the WKB approximation. Because of the rapid phononic transitions, we treat the centre as an atomic system with a single excited state |e〉 and a ground state with ten sublevels {|gi〉} (Fig. 1). The fluorescence from the centre corresponds to a transition from the excited spin triplet state (3 E) to an electron spin triplet ground state (3 A) and the dynamics are influenced by the presence of a possible metastable state (1 A). We have explicitly ignored the metastable state as its role in de-excitation process remains unclear [31

31. N. B. Manson, J. P. Harrison, and M. J. Sellars, “Nitrogen-vacancy center in diamond: Model of the electronic structure and associated dynamics,” Phys. Rev. B 74, 104303 (2006). [CrossRef]

, 48

48. F. Jelezko, T. Gaebel, I. Popa, A. Gruber, and J Wrachtrup, “Observation of coherent oscillations in a single electron spin,” Phys. Rev. Lett. 92, 076401 (2004). [CrossRef] [PubMed]

, 49

49. Ph. Tamarat, N. B. Manson, R. L. McMurtie, A. Nitsovtsev, C. Santori, P. Neumann, T. Gaebel, F. Jelezko, P. Hemmer, and J. Wrachtrup, “The excited state structure of the nitrogen-vacancy center in diamond,” http://arxiv.org/abs/cond-mat/0610357 (2006).

]. Also, in the absence of strain and magnetic field, the transitions from the ms=0 spin sublevel of the ground state are found to be spin-conserving [31

31. N. B. Manson, J. P. Harrison, and M. J. Sellars, “Nitrogen-vacancy center in diamond: Model of the electronic structure and associated dynamics,” Phys. Rev. B 74, 104303 (2006). [CrossRef]

], hence we assume this in our treatment.

Fig. 1. Theoretical model of a cavity-centre system for single photon generation: the NV centre is modelled as a multi-level atom with a single excited state |e〉 and a ground state with vibrational sublevels {|gj〉}. The centre is pumped with an external classical field r(t) (white arrow) acting as the trigger pulse, the transition |e〉-|gi〉 is coupled to a lossy single-modal cavity with coupling strength Ωi (grey) and cavity decay rate κ (black). γgie (dashed) is the atomic decay rate for radiative transitions |e〉-|gi〉 while γgmgn (dotted) are that for the non-radiative phononic transitions |gn〉-|gm〉.

To study the transient behaviour of the cavity-centre system, we extend the basic Jaynes-Cummings (JC) model [50

50. E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theory with application to the beam maser,” Proc. IEEE 51, 89–109 (1963). [CrossRef]

, 51

51. B. W. Shore and P. L. Knight, “The Jaynes-Cummings model,” J. Mod. Opt. 40, 1195–1238 (1993). [CrossRef]

], which treats the interaction between a single-mode electromagnetic field or cavity of resonance frequency ωC and a two-level atom, to consider our more complex atomic system. In the rotating-wave approximation (RWA), the JC Hamiltonian of our cavity-centre system is expressed in terms of the atomic projection operators σ^ αβ=|α〉〈β| and the annihilation (creation) operators â (â ) for single cavity mode as, (=1)

𝓗̂JC=i=0NA2ωgiσ̂gigi+ωeσ̂ee+ωCââ+12i=0NA2(Ωiâσ̂gie+H.c),
(1)

where NA=11 is the total number of atomic states, ωα is the energy of the atomic level |α〉 and Ωi is the cavity-centre coupling constant between atomic transition |e〉-|gi〉. In the dipole approximation, the coupling is Ωi=di[ωC/(2h̄ε 0 V)]1/2 where di is the dipole moment of the respective transition. The evolution of the cavity-centre system obeys the Liouville equation of motion for the density matrix ρ,

dρdt=i[𝓗̂JC,ρ]+j=0NA2γgje𝓛[σ̂gje,ρ]+r(t)𝓛[σ̂eg0,ρ]+i=0NA3γgigi+1𝓛[σ̂gigi+1,ρ]+κ𝓛[b̂â,ρ],
(2)

with the Lindbladian terms for some operator Ô,

𝓛[Ô,ρ]=ÔρÔ12(ÔÔρ+ρÔÔ).
(3)

The spontaneous transition (|e〉-|gj〉) couples to any non-cavity field modes at the characteristic rates γgje and non-radiative phononic decays from |g i+1〉 to |gi〉 with rates γgigi+1 . Here represents the creation operator for electromagnetic (waveguide) mode outside the cavity, which the cavity couples to via decay rate κ. The rate κ=ωC/(2Q) is parameterized by the quality factor of the cavity Q. Incoherent excitation, acting as the trigger for photon emission, is represented by the phenomenological term with a pump absorption rate r(t). In this model, we have explicitly ignored thermal broadening by assuming a zero temperature operating environment. Broadening can be introduced phenomenologically for a more realistic estimate of the linewidth of the emitted wave packet.

Efficient single photon generation requires minimizing loss and fast outcoupling of the excitation via the cavity channel. The appropriate regime to optimise the output is the strong Purcell regime [52

52. C. K. Law and H. J. Kimble, “Deterministic generation of a bit-stream of single-photon pulses,” J. Mod. Opt. 44, 2067–2074 (1997).

], κ>Ωiγgje , where the rate of coherent coupling between the centre and cavity mode, Ω2 i/κ, dominates the rates γgje (∀j) of the incoherent coupling to the non-cavity modes. The cavity mode is chosen to match the transition |e〉-|gi〉. The cavity loss rate κ sets the time scale for photon outcoupling and when greater than Ωi, suppresses the vacuum Rabi oscillations, which otherwise lead to unwanted spectral features on the output photon. For a two-level atom in a cavity, altogether embedded in a medium of refractive index n, the enhancement of emission into the cavity is parameterized by the Purcell factor, the ratio of the emission rate to the cavity mode to the unmodified rate into the non-cavity modes [53–55

53. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

],

Fp=3(λCn)34π2QV,
(4)

where V is the cavity mode volume and λC is the cavity wavelength.

To determine the fraction of pulse cycles that lead to useful output, we must consider the fraction of emitted photons from the atom that enter the desired cavity mode. This fraction is set by the spontaneous coupling factor β=Fp/(1+Fp). We demand β to be near unity, which implies a high-Q/V cavity to maximize efficiency, but a small Q for fast cavity loss. These considerations lead to an upper bound for Q, or equivalently, a lower bound on κ given by κ≥2Ωi. Beyond this, we enter the strong cavity regime [54

54. S. Haroche and J. M. Raimond, “Radiative properties of Rydberg states in resonant cavities,” in Advances in Atomic and Molecular Physics Vol. XX, D. Bates and B. Bederson, eds. (Academic, 1985), pp. 350–411.

]. In this regime, the vacuum Rabi oscillations are not sufficiently suppressed by cavity decay leading to an effective timing jitter or equivalently temporal/spectral features which will degrade overall device performance. Alternatively, if κ<γgje for large Q, the excitation will be outcoupled as atomic decoherence, which leads to loss of photons through non-guided modes, and manifests as an increased zero photon probability. A rigorous treatment of cavity-assisted emission was made in Ref. [56

56. M. Khanbekyan, D.-G. Welsh, C. Di Fidio, and W. Vogel, “Cavity-assisted spontaneous emission as a single-photon source: Pulse shape and efficiency of one-photon Fock state preparation,” http://arxiv.org/abs/0709.2998 (2007).

].

3. Analysis and results

To investigate dynamical processes of photon generation from the cavity-centre system described by Eq. 2, we use direct numerical integration to study the specifications on the trigger pulse that ensures high-fidelity single photon emission and the effect of the cavity on its characteristics. Additionally, we use the quantum trajectory approach to simulate photodetection experiments [57

57. L. Tian and H. J. Carmichael, “Quantum trajectory simulations of the two-state behavior of an optical cavity containing one atom,” Phys. Rev. A 46, R6801–R6804 (1992). [CrossRef] [PubMed]

, 58

58. H. J. Carmichael, An Open System Approach to Quantum Optics (Springer, 1993).

]. We show that it yields results that agree with the former approach and demonstrate bit-stream photon generation and, most importantly, antibunching with the Hanbury-Brown-Twiss (HBT) experiment.

3.1. Determination of the pulsed excitation parameters

The specification of the pulsed excitation scheme for the cavity-centre system is crucial to ensure efficient and high-fidelity single photon emission. The complete treatment of Eq. 2 for this study is computationally demanding as it requires adopting a considerably large state space given by {|α〉}atomic⊗{|0C〉,|1C〉,…, |NC〉}cavity⊗{|0W〉,|1W〉,…,|NW〉}waveguide, where the state evolution is taken to span over NW (or equivalently NC) excitation states. ‘C’ denotes the field mode in the cavity and ‘W’ labels the external waveguide mode. However, for the purposes of understanding just the photon emission, we may reduce the system state space by setting NC=2, NW=0 and enforcing a choice of parameters for the pulsed excitation that limits system population to the ≤2-quantum states throughout the cycle.

Fig. 2. a. Probability of the cavity-centre system (ωC=ωZPL, V=λ 3 ZPL) to emit a single photon per top-hat excitation pulse as a function of pulse width T and absorption rate r 0. Dash-dotted line denotes the pulse width parameter used in Ref. [59] where the illumination irreversibly transforms the centre into a different centre and is therefore a practical cutoff. Circle labels the parameters [yield P 1=0.996 and P ≥2~𝩄(10-5)] used for the demonstration of single-photon generation with the cavity-centre system. b. Zero (dotted), single (solid) and multi- (dashed) photon probability as a function of pulse width with constant absorption rate r 0=1013 Hz.

For the excitation, we assume a top-hat form of the trigger pulse r(t)=r 0 for t∈[0,T], where r 0 is the absorption rate and T the pulse width which must be much shorter than the photoluminescence lifetime. In the strong Purcell regime the centre may be approximated as a two-level system in resonance with the cavity, and for very short times we ignore the effects of spontaneous emission and cavity outcoupling. Under these approximations the lower bound for zero photon emission probability (P 0) and the upper bound for one (P 1) are

P0=er0T,
(5)
P1=2er0T{er0T2[16Ωi2+r02cosh(ηT2)]η21},
(6)

and upper bound for multi-photon probabilities is P ≥2=1-(P 0+P 1), where η=(r 2 0-16Ω2 i)1/2. In the limit r 0≫Ωi, Eq. 6 reduces to P 1=1-exp(-r 0 T), but the second order term reveals that the single photon probability decreases with increasing pulse width according to exp(-4Ω2 i T/r 0)-P0, where P0=[2-exp(-4Ω2 i T/r 0)]exp(-r 0 T). There is therefore a clear trade-off between the requirement to produce a photon on demand and that of having no more than one photon per pulse. Fig. 2 illustrates the single photon emission probability for an NV centre in a cavity resonant with the zero-phonon line (ZPL) transition (|e〉-|g 0〉), as a function of excitation parameters. In contrast to an attenuated laser where small P ≥2 is achieved in expense of P 1, the cavity-centre system offers P 1~𝒪(1) while keeping P ɥ2𝒪(10-2). The mean photon number per pulse is =∑i iPiP 1+2P ≥2 if Pi is negligible for i≥3, for these parameter choices.

The photon source must operate in the limit of large r 0 and maintain short T to ensure near unit single photon probability and efficient operation. However, Dumeige et al. showed that under intense femtosecond illumination, the centre becomes photo-ionized, resulting in blinking [59

59. Y. Dumeige, F. Treussart, R. Alléaume, T. Gacoin, J.-F. Roch, and P. Grangier, “Photo-induced creaton of nitrogen-related color centers in diamond nanocrystals under femtosecond illumination,” J. Lumin. 109, 61–67 (2004). [CrossRef]

], thus this sets the lower bound for the pulse width.

Fig. 3. Evolution of a cavity-centre system (ωC=ωZPL, V3 ZPL, κ=2.5Ω0) in response to a top-hat excitation pulse (r 0=1013 Hz, T=0.56 ps). a. Population in |e,0C,0W〉 (the excited centre, black dash-dotted) and |g 0,0C,1W〉 or ρWW (the outcoupled waveguide mode, black/red solid) as a function of time. b. Time derivatives of ρWW, proportional to the output intensity, with an integrated area of 0.99 (solid) and of 1.01 (dashed red). Simulation is performed by direct integration of Eq. 2 (black solid/dash-dotted) and by quantum trajectory approach as a direct photodetection experiment (red dashed).

The optimal excitation parameters for the operation of cavity-centre system depend on the desired application for the photon source. The source can be tailored by varying pulse parameters T and r 0 to optimize properties P 1 and for a specific application. We adopted technically feasible values T=0.56 ps and r 0=1013 Hz that yield P 1=0.996 and P ≥2=𝒪(10-5).

3.2. Simulation of single photon generation with the cavity-centre system

The analysis in Sec. 3.1 allows a determination of the parameter range for single photon emission, and this is obviously the regime in which we want our device to work. In this section, therefore, we work in the single photon emission regime and assume single photon output. This affords a considerable saving in computational complexity and we thus truncate the photonic state space to one excitation by setting NC=NW=1. We may then solve Eq. 2 numerically to simulate the response of the cavity-centre system to a pulsed excitation, with the result given in Fig. 3. The cavity of volume V3 ZPL is chosen to maximize coupling and is in resonance with the ZPL. We choose Q of 36500 so that κ=2.5Ω0. A photon is being issued at a mean time 70 ps from the pump with excitation outcoupling into the external waveguide mode illustrated by increasing population in |g 0,0C,1W〉 (or ρWW≡〈g 0,0C,1W|ρ|g 0,0C,1W〉). The integral of the derivative ρ˙ WW is near unity at 0.99, as required of a single photon pulse. Fourier transform of the temporal profile yields an emission spectrum centered at λZPL with effective linewidth of 0.01 nm.

Due to the difference between cavity-centre coupling Ω0 and cavity outcoupling κ, the resultant photon pulse does not take the form of a Gaussian function which is optimal for LOQC [35

35. P. P. Rohde, T. C. Ralph, and M. A. Nielsen, “Optimal photons for quantum-information processing,” Phys. Rev. A 72, 052332 (2005). [CrossRef]

]. In principle, Stark tuning [60

60. Ph. Tamarat, T. Gaebel, J. R. Rabeau, M. Khan, A. D. Greentree, H. Wilson, L. C. L. Hollenberg, S. Prawer, P. Hemmer, F. Jelezko, and J. Wrachtrup, “Stark shift control of single optical centres in diamond,” Phys. Rev. Lett. 97, 083002 (2006). [CrossRef] [PubMed]

] can be used to optimize the atom-cavity coupling to reshape the temporal pulse profile [61

61. A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A 73, 013818 (2006). [CrossRef]

, 62

62. M. J. Fernée, H. Rubinsztein-Dunlop, and G. J. Milburn, “Improving single-photon sources with Stark tuning,” Phys. Rev. A 75, 043815 (2007). [CrossRef]

] and suppress timing jitter. Note that there is a gentle hump at 300 ps, representing the Rabi remnant, that can be eliminated with greater cavity damping.

To simulate photodetection experiments via the quantum trajectory approach, we again adopt a two-level model of the centre, but truncate NC=4 to allow for the possibility of multi-photon occupation and subsequent emission. There is a good agreement between this result and that from direct integration, as shown in Fig. 3.

Fig. 4. Comparison of the probability of the cavity-centre system (ωC=ωZPL,V=λ 3 ZPL) to emit a photon of ωZPL via the cavity channel and to emit iPL photon via atom decoherence as a function of cavity quality factor Q.

It is instructive to observe how the probability of the system to emit a photon via the cavity and via atomic decoherence varies under the influence of the cavity. We recalculate the emission (Fig. 4), but this time treat (NA-1) vibrational ground states, and only consider one photonic excitation. In the weak Purcell regime, the excitation is outcoupled via the atomic decoherence channel. The relative transition probabilities of the respective phonon lines jPL approach the unmodified atomic branching ratios γgje in the limit of small Q. However, in the strong Purcell regime with Q~𝒪(104-105), the ZPL transition is enhanced by the cavity while other transitions are suppressed accordingly. The excitation is predominantly outcoupled via the cavity relaxation channel, representing the optimal regime for the single-photon source. We note that such Q’s are technically achievable and have been demonstrated in Ref. [39

39. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature (London) 425, 944–947 (2003). [CrossRef] [PubMed]

, 40

40. B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Materials 4, 207–210 (2005). [CrossRef]

] in silicon, and are feasible in diamond [43

43. S. Tomljenovic-Hanic, M. J. Steel, C. Martijn de Sterke, and J. Salzman, “Diamond based photonic crystal microcavities,” Opt. Express 14, 3556–3562 (2006). [CrossRef] [PubMed]

]. Finally, in the strong cavity regime, the Purcell enhancement diminishes as the time scale for cavity relaxation becomes much longer with κ<γgje .

We have also considered the possibility of enhancing the higher order phonon side bands. Such an enhancement is attractive from the point of view of shifting the emission further into the IR, and taking advantage of the increased dipole moments. The presence of the vibrational sublevels and phononic decays introduces an additional complication to the dynamics of the basic JC model. By explicitly considering a reduced centre model with |e〉 and |gi〉 (i=0,1,2 only) that couples the cavity that is in resonance with the 1PL, we have obtained an analytical expression to the modified decay rate by solving the master equation for its eigenfrequencies and assuming Ω0(κ+γgmgn) . Single excitation relaxes at an overall damping rate,

γoverall=γg0g1+2Ω02ωC(2Q)+γgmgn2γg0g1+𝒪(Ω04).
(7)

Fig. 5. Photon correlation histogram of emission from the cavity-centre system under pulsed excitation of top-hat functional form obtained using a HBT setup with quantum trajectory approach. The simulations involve a. excitation pulse of temporal width T=0.56 ps and constant absorption rate r 0=1013 Hz at a repetition rate of 1 GHz for a trajectory time of 5 µs, and b. excitation pulse of varying temporal widths and constant r 0=1013 Hz at repetition rate of 0.5 GHz for total time 1.5 ms.

3.3. Antibunching with a HBT setup

Photon correlation obtained from a HBT experiment is a test for single-photon emission. The HBT setup is simulated with a pulsed excitation of rate 1 GHz for a simulation time of 5 µs and the correlation histogram of emission from the cavity-centre system is shown in Fig. 5(a). The experiment assumes the use of a detector with a time bin size of 10 ps. The result shows a series of peaks separated by the clock period of the pulse. The suppression of the coincidences observed at zero delay signifies antibunching. More importantly, the suppression is observed during the period of a single excitation cycle for the short excitation pulse width T=0.56 ps. The single photon probability per excitation trigger, estimated from the ratio of the number of single photon events to total number of pulses simulated is 0.99, while multi-photon probability is zero. With an effective lifetime of 70 ps, the system is capable of operating at an excitation rate of 10 GHz. However, to ensure all photon pulses are well-localized within their respective time bins, a bit-stream rate of 1 GHz is preferable. Finally, in Fig. 5(b), we show an increase in multi-photon probability with increasing excitation pulse width. In agreement with the result from Fig. 2(b), a considerably long simulation is needed to observe an appreciable multi-photon probability of 𝒪(1)% for T~103 ps, the simulation was performed over a trajectory period of 1.5 ms and resolution set to 10 ps.

4. Conclusion

We have studied the effect of a cavity on an NV- defect centre in enhancing its spectral properties for the purpose of single photon generation for quantum computing and communication. Assuming an atomic-vibronic NV model in single-mode cavity, we have shown that by coupling the centre strongly to a high-Q/V [Q~𝒪(104-105), V3] cavity that is resonant with the ZPL and with excitation using a sub-picosecond pump, the cavity-centre system is capable of issuing a photon of wavelength 638 nm with high spectral purity. We predict that the cavity-enhanced NV centre can have an effective lifetime of 70 ps and linewidth of 0.01 nm, in contrast with an unmodified centre’s photoluminescence lifetime of 11.6ns and spectral width of 150 nm. Photons are emitted with near unit single photon probability of 0.99 while maintaining small multi-photon probability 𝒪(10-5), thus making it a relatively efficient triggered photon source compared to a bare NV centre or an attenuated laser. Finally, the device can potentially operate at a repetition rate of 1 GHz, considerably greater than demonstrated NV systems for QKD applications [32

32. A. Beveratos, R. Brouri, T. Gacoin, A. Villing, J.-P. Poizat, and P. Grangier, “Single photon quantum cryptography,” Phys. Rev. Lett. 89, 187901 (2002). [CrossRef] [PubMed]

, 33

33. R. Alléaume, F. Treussart, G. Messin, Y. Dumeige, J.-F. Roch, A. Beveratos, R. Brouri-Tualle, J.-P. Poizat, and P. Grangier, “Experimental open-air quantum key distribution with a single-photon source,” New J. Phys. 6, 92 (2004). [CrossRef]

].

Acknowledgments

We thank J. H. Cole and A. M. Stephens for helpful discussions. This project was supported by Quantum Communications Victoria (QCV), which is funded by the Victorian Government’s Science, Technology and Innovation (STI) initiative, and by the Australian Research Council (ARC), the Department of Education Science and Technology under the International Science Linkages scheme and the EU 6th Framework under the EQUIND collaboration. ADG is the recipient of an Australian Research Council Queen Elizabeth II Fellowship (project number DP0880466), LCLH is the recipient of an Australian Research Council Australian Professorial Fellowship (project number DP0770715).

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OCIS Codes
(160.2220) Materials : Defect-center materials
(160.4760) Materials : Optical properties
(230.0230) Optical devices : Optical devices
(230.6080) Optical devices : Sources

ToC Category:
Optical Devices

History
Original Manuscript: November 27, 2007
Revised Manuscript: January 30, 2008
Manuscript Accepted: March 22, 2008
Published: April 18, 2008

Citation
Chun-Hsu Su, Andrew D. Greentree, and Lloyd C. L. Hollenberg, "Towards a picosecond transform-limited nitrogen-vacancy based single photon source," Opt. Express 16, 6240-6250 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-9-6240


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References

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