Simple microcavity for single-photon generation - OSA Publishing

Simple microcavity for single-photon generation Taras Plakhotnik School of Physical Sciences, University of Queensland, St Lucia 4072, Australia [email protected]

Abstract: A new design of an optical resonator for generation of singlephoton pulses is proposed. The resonator is made of a cylindrical or spherical piece of a polymer squeezed between two flat dielectric mirrors. The mode characteristics of this resonator are calculated numerically. The numerical analysis is backed by a physical explanation. The decay time and the mode volume of the fundamental mode are sufficient for achieving more than 96% probability of generating a single-photon in a single-mode. The corresponding requirement for the reflectivity of the mirrors (~99.9%) and the losses in the polymer (100 dB/m) are quite modest. The resonator is suitable for single-photon generation based on optical pumping of a single quantum system such as an organic molecule, a diamond nanocrystal, or a semiconductor quantum dot if they are imbedded in the polymer. ©2005 Optical Society of America OCIS codes: (230.5750) Resonators; (230.3990) Microstructure devices; (270.5290) Photon statistics.

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D. Bouwmeester, A. Ekert, and A. Zeilinger eds., The physics of quantum information: quantum cryptography, quantum teleportation, quantum computation (Springer, Berlin , 2000). E. Knill, R. Laflamme and G. J. Milburn. “A scheme for efficient quantum computation with linear optics,” Nature 409 46 (2001). G. Brassard, N. Lütkenhaus, T. Mor, and B.C. Sanders. “Limitations on practical quantum cryptography,” Phys. Rev. Lett. 85, 1330-1333 (2000). C. Brunel, B. Lounis, P. Tamarat, and M. Orrit. “Triggered source of single photons based on controlled single molecule fluorescence,” Phys. Rev. Lett. 83, 2722-2725 (1999). B. Lounis, W.E. Moerner. “Single photons on demand from a single molecule at room temperature,” Nature 407, 491 (2000). P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, Lidong Zhang, E. Hu, A. Imamogùlu,“A Quantum Dot Single-Photon Turnstile Device,” Science 290, 2282-2285 (2000). M. Pelton, C. Santori, J. Vuckovic, B. Zhang, G.S. Solomon, J. Plant, and Y. Yamamoto . “Efficient Source of Single Photons: A single quantum dot in a microposts microcavity,” Phys. Rev. Lett. 89, 233602 (2002). A. J. Bennett , D. C. Unitt, P. Atkinson, D. A. Ritchie and A. J. Shields, “High performance single photon sources from photo lithographically defined pillar microcavities,” Opt. Express 13, 50-55 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-50. W.E. Moerner, “Single-photon sources based on single molecules in solids,” New J. Phys. 6, 27 (2004). A. Beveratos, S. Kühn, R. Brouri, T. Gacoin, J.-P. Poizat, and P. Grangier, “Room temperature stable single Photon source,” Eur. Phys. J. D 18, 191-196 (2002). 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 431, 1075-1078 (2004). Y. Ben, Z. Hao, C. Sun, F. Ren, N. Tan and Y. Luo, “Three-dimensional photonic-crystal cavity with an embedded quantum dot as a nonclassical light emitter,” Opt. Express 12, 5146-5152 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-21-5146. W.L. Barnes,a, G. Björk2, J.M. Gérard, P. Jonsson, J.A.E. Wasey, P.T. Worthing, and V. Zwiller, “Solidstate single photon sources: light collection strategies”, Eur. Phys. J. D 18, 197–210 (2002). For a review see K.J. Vahala, “Optical microcavities,” Nature 424, 840-846 (2003). J. Vuckovic, M. Pelton, A. Scherer, and Y. Yamamoto. “Optimization of three-dimensional microposts microcavities for cavity quantum electrodynamics,” Phys. Rev. A 66, 023808 (2002). M. Pelton, J. Vuckovic, G. S. Solomon, A. Scherer, and Y. Yamamoto. “Three-dimensionally confined modes in micropost microcavities: Quality Factors and Purcell Factors,” IEEE J. Quantum. Electron. 38, 170-177 (2002). J. McKeever, A. Boca, A. D. Boozer, R. Miller, J. R. Buck, A. Kuzmich, H. J. Kimble, “Deterministic generation of single photons from one atom trapped in a cavity,” Science 303, 1992-1994 (2004).

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Received 17 February 2005; revised 4 April 2005; accepted 5 April 2005

18 April 2005 / Vol. 13, No. 8 / OPTICS EXPRESS 3049

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V.B. Braginsky, M.L. Gorodetsky, and V.S. Ilchenko. “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393 (1989). L. Pang, Y. Shen, K. Tetz and Y. Fainman, “PMMA quantum dots composites fabricated via use of prepolymerization,” Opt. Express 13, 44-49 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-44 F. De Martini, G. Di Giuseppe, and M. Marrocco. “Single-mode generation of quantum photon states by excited single molecules in a microcavity trap,” Phys. Rev. Lett. 76, 900-903 (1996). B. Gayral, J. M. Gérard, A. Lemaître, C. Dupuis, L. Manin, J. L. Pelouard, “ High-Q wet-etched GaAs microdisks containing InAs quantum boxes,” Appl. Phys.Lett. 75, 1908–1910 (1999). T. Plakhotnik, E.A. Donley, U.P. Wild, “Single molecule spectroscopy”, Annu. Rev. Phys. Chem. 48, 181-212 (1997). X. S. Xie, J.K. Trautman, “Optical studies of single molecules at room temperatures”, Annu. Rev. Phys. Chem. 49, 441-480 (1998). M. Ehrl, F.W. Deeg, C. Bräuchle, O. Franke, A. Sobbi, G. Schulz-Ekloff, D. Wöhrle, “High-temperature non-photochemical hole-burning phthalocyanine-Zinc derivatives embedded in hydrated AlPO4-5 molecular sieve“, J. Phys. Chem. 98, 47-52 (1994). L. Eldada, L. W. Shacklette, “Advances in Polymer Integrated Optics,” IEEE J. Selected Topics in Quantum Electron. 6, 54-68 (2000). .

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1. Introduction Generation of optical pulses consisting of exactly one photon is a hot topic of modern quantum optics. Application of such pulses to quantum cryptography substantially improves robustness of quantum communication channels against enemy attacks [1]. I has been suggested that single-photon states can also be used for quantum computation [2]. Ordinary weak laser pulses have Poisson probability distribution function for the photon numbers. Accordingly, the probability to have more than one photon per pulse for such pulses is 1 − (1 + n ) exp ( −n ) , where n is the average photon number, and is quite high unless n 2000 × λ 3 ) mode volume and therefore a very long mode decay-time is required. Practically, such cavity can operate only in vacuum and requires expensive mirrors with losses below 10−6 . A micropost type of cavity [6 - 8] exploits microchip production technology and is hardly usable for organic molecules and diamond nanocrystals. High quality factors of microsphere resonances [18] can also be exploited but this type of microcavity has a mode volume similar to a bulk Fabry-Perot cavity. 2. Cavity design and numerical simulations We propose a cavity which can be built using moderate quality, relatively cheap and easily available mirrors with losses of about 10−3 . The active region of the cavity is filled with a polymer. This polymer can be used to host a large variety of single-photon emitters including organic molecules [4, 5, 9], diamond nanocrystals [10], and semiconductor quantum dots [6, 19]. The design of the cavity is shown in Fig. 1. It is made of two flat highly reflecting dielectric mirrors with a small cylindrically shaped spacer made of a polymer. The spacer is placed and squeezed between the two mirrors. If squeezing is very gentle, the spacer stays cylindrical. If a substantial force if applied, the side surface of the spacer is curved. Such a cavity also can be fabricated, for example, by spin coating on one cavity mirror a very dilute solution of polymer microspheres doped with organic dye molecules or other single-photon emitters. After the spheres are distributed, they are squeeze with the second mirror. Spin coating will ensure that the micro cavities are spread over the mirror surface without touching each other. It is assumed that the concentration of the SQSs is sufficiently low so that only one SQS per cavity is in resonance with a cavity mode. The numerical simulation shows (see details below) that the exact form of the spacer side shape is of little importance even in an extreme case when the radius of the curvature is equal to the spacer thickness s. (see Table 1, superscript a) and therefore most simulations are done for a cavity with cylindrical spacer (Table 1, no superscript or superscripts b, c or d ). This cavity resembles a Fabry-Perot microcavity used for single photon generation by De Martini et al [20] but has a substantial difference. The space between the two mirrors forming the Fabry-Perot resonator is not filled with a homogeneous dielectric. This inhomogeneity leads to strong localization of the electromagnetic field if the field oscillation frequency is close to the cavity resonances. The cavity has been analyzed using 3-dimentional finite-difference time-domain (FDTD) simulation technique with perfectly matching boundary conditions at the boundary of the simulation region (a commercially available package Lumerical FDTD Solution has been exploited). Distributions of the electric field amplitude in the cavity modes were obtained by simulating propagation of a short electromagnetic pulse (initial polarization is parallel to the #6618 - $15.00 US

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x-axis) inside the resonator. Simulations were repeated with different space and time resolution to verify their stability and reproducibility. An example of such distribution taken at a frequency close to the lowest EM11 cavity resonance is shown in Fig.1. The numerical values of the refraction indexes, the thicknesses of the dielectric layers, and the geometry of the polymer spacer are given in the figure caption. The cavity characteristics were systematically investigated as a function of the diameter D of the polymer spacer. The results are given in Table 1. Note that the best performance of the cavity can be achieved if the SQS is placed at the highest electric field intensity with SQS transition dipole moment parallel to the electric field vector. This can be achieved approximately by selecting a best performing micro cavity from a number of cavities with randomly distributed SQSs. 3. Discussion If the mirrors were perfectly reflecting plains, the solution of the problem would be a single period of a standing wave in an infinitely long cylindrical fiber. It is expected that for large D the volume of the lowest axial mode is proportional to the volume of the polymer spacer or to D 2 , if the spacer length stays constant. As follows from the Table 1, this dependence holds quite well already when D > 2λ . When D gets smaller, the electromagnetic field extends substantially outside the spacer and the proportionality to D 2 violates. Note, that for the

(

cavity with D = 0.6 µm the mode volume is just 0.83 ⋅ λ / n p

)

3

and is about 2 times smaller

than a minimum mode volume of a micropost cavity reported in [15]. If the mirrors were perfectly reflecting plains, there would be no losses except for the 1

z

1

0.9 0.8

0.8

0.6 0.4

t2

0.7 0.2

0.6

s t1

x

0

−1

0

1

0

2

x (µm)

0.5 1

0.4

D 0.3

0.8

0.6

0.4

0.2 0.2

0.1

0

−2

z (µm) Fig.1. Distribution the electric field ( |Ex | ) oscillating at the resonance frequency of the microcavity is shown as a false color image. The color bar on the right shows the color code for the relative amplitude of the electric field. The relative values of Ex2 n 2 are shown for z = 0 (upper-right panel)

and r = 0 (lower-right panel). Thirty pairs of the dielectric layers form two mirrors (with 15 pairs for each mirror). The alternating refraction indexes of the layers are 2.4 (TiO2) and 1.46 (SiO2). Their thicknesses are t1 = 0.06 µm and t2 = 0.1 µm (for the layers with the smaller refraction index). The polymer spacer has a refraction index of n p = 1.45 and a thickness of s = 0.24 µm .

Two possible shapes of the polymer spacer (cylindrical and elliptical) are shown. The influence of the spacer shape on the cavity characteristics was marginal. The cavity characteristics have been studied as a function of the diameter of the polymer spacer (see Table 1). For the simulation results presented . #6618 - $15.00 USin this figure D = 0.8 µm Received 17 February 2005; revised 4 April 2005; accepted 5 April 2005

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losses due to absorption and/or scattering in the material of the fiber (polymer spacer). Because the mirrors are distributed Bragg reflectors (DBR) the electromagnetic field can penetrate into the mirrors and can propagate sidewise inside the layers. To get a simple physical picture for the losses due to this sidewise propagation we first consider a ray of light incident on the mirror at a small angle θ with the z-axis (see Fig. 1). The sidewise displacement δ of this ray during a unit time will be proportional to sin (θ ) . The energy transmitted by all rays propagating at the angle θ irrespective of their polar angle ϕ in the x,y-plane is proportional to sin (θ ) dθ , where we take into consideration that the problem is cylindrically symmetrical. Therefore the sidewise energy flow transmitted by these rays is proportional to δ ⋅ sin (θ ) dθ ∝ θ 2 dθ , where we have substituted θ for sin (θ ) . The total energy flow out of the cavity is proportional to

θ max

∫0

3 θ 2 dθ ∝ θ max ∝ ( λ / D ) , where we take 3

into account that for a confined electromagnetic field θ max ∝ λ / D due to diffraction. Thus we expect the decay time of the cavity mode to be proportional to D 3 . Such scaling together with Vm ∝ D 2 means that F / D ≈ constant . This holds surprisingly well (see Tab. 1) despite of quite a rough model. Table 1. Characteristics of the Micro Resonator as Calculated by 3D-FDTD Method

D/µ m

τ ⊥ / ps

0.5 0.6 0.6 a 0.8 0.8 a

0.09 0.14 0.13 a 0.27 0.27 a

0.8 b c

0.8

Vm / µ m 3

ν/THz

F

F/D · µm

115 224 209 a 422 424 a

0.056 0.057 0.056 a 0.092 0.088 a

518 509 511 a 498 500 a

13 21 20 a 27 26 a

26 35

0.24 b

376 b

0.092 b

499 b

22

c

c

c

c

18 c

0.19

Q

298

0.092

499

34

0.8 d

0.15 d

235 d

0.092 d

498 d

14 d

1.2 1.6 1.8

0.82 1.4 1.85

1260 2130 2810

0.14 0.21 0.26

489 485 484

50 54 57

42 34 32

2.0 2.4

2.9 4.2

4400 6360

0.33 0.44

483 482

72 78

32 32

2.8

7.23

11000

0.58

481

102

37

3.2 3.2 a

11.6 12.4 a

17500 18700 a

0.74 0.71 a

481 481 a

127 141 a

40

4.8

31.4

47500

1.6

480

158

33

a

with polymer inset having a curved side surface. with both mirrors having reflectivity of ~ 0.998 (8 pairs of layers). c with both mirrors having reflectivity of ~ 0.996 (7 pairs of layers). d with both mirrors having reflectivity of ~ 0.989 (6 pairs of layers). b

The decay time or the quality factor Q ≡ πτ ⊥ν of proposed cavity (see Table 1) is comparable to the quality factor of micropost [15] and microdisk [21] cavities (at equal mode volumes). The Purcell factor of this cavity is from 2 to 16 times (depending on the spacer diameter) larger than a theoretically estimated value reported in [8]. The comparison with very high-Q cavities [14] is more difficult because these cavities have much larger mode #6618 - $15.00 US

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volumes than presented in Table 1. Theoretically, by increasing D one can increase Q (according to the scaling derived above one gets Q ≈ 106 when Vm ≈ 103 µm 3 ) and in this way even a strong coupling regime [14] can be achieved but such a high Q will be very sensitive to any imperfections and losses in the cavity (see below about losses in the polymer spacer and DBR penetration). This may compromise the main idea of this cavity – its low cost and simplicity. It follows from Eq. (3) and the Purcell factor obtained in the simulations, that the probability of spontaneous emission in the fundamental cavity mode is about 96% for D = 0.6 µm and even higher for larger D. The Purcell factor marginally depends on the shape of the spacer (compare the results labeled with a superscript a to those without a subscript). The applicability of the theory outlined in the introduction is limited. As an example we consider a dye molecule with a typical Γ 0 = 50 MHz [22]. At room temperatures, a characteristic homogeneous linewidth of such a molecule is around 15 THz [23] but this number drops below 0.3 THz at 80 K [24] and can be as small as 50 MHz at 2K [22]. The cavity bandwidth γ cav ≡ 1/(πτ ⊥ ) ≈ 2.2 THz (at D = 0.6 µm ) and g 0 / π ≈ 23 GHz (at Γ 0 = 50 MHz and D = 0.6 µm ) are practically temperature independent. Therefore the validity of Eq. (3) depends on the temperature and the cavity diameter. For example, using scaling g 0 ∝ D −1 and γ cav ∝ D −3 one can prove that the strong coupling conditions are fulfilled at liquid He temperatures if D > 8 µm . To estimate the DBR penetration effect on the decay rate of the cavity field, we performed simulations with just 8 pairs of dielectric layers on each side of the spacer (see simulation results labeled in Table 1 with a superscript b). The reflectivity R of such mirrors is only 0.998, but if D < 0.8 µm additional losses introduced by mirrors are negligible. Therefore relatively cheap mirrors can be used to achieve a Purcell factor up to 30. τ m−1 ≈ ν (1 − R) . For even smaller reflectivity (0.996 and 0.989, see Table 1), the effect of DBR penetration becomes significant but not totally destructive (the corresponding Purcell factors are still larger than 10). The field decay times τ ⊥ were calculated assuming no absorption or scattering in the polymer. The effect of the losses in the polymer can be estimated as follows. A round trip inside the resonator tikes time equal to one period of oscillations of the electromagnetic field of the lowest longitudinal cavity mode. If the energy loss per one passage through the polymer insert (including both absorption and scattering) is L, the related field decay rate τ −p 1 can be estimated as τ p−1 ≈ ν L . Optical loses in PMMA fibers are less than α = 1 dB/m [25] but as a conservative estimate we take 100 dB/m and obtain L = 1 − 10− sα /10 ≈ 6 × 10−6 for a polymer length s = 0.24 µm . The total decay rate of the cavity field is given by τ ⊥−1 + τ p−1 . The probability that a photon is absorbed or scattered by the polymer is given by the ratio of absorption and/or scattering rate to the total rate τ p−1 /(τ ⊥−1 + τ p−1 ) . This is smaller than 1% if

τ ⊥ < 5 ps . Therefore losses in the polymer can be neglected unless D > 2.5 µm (see Table 1). 4. Conclusions The proposed micro cavity can be used for substantial improvement of the fidelity of singlephoton sources based on spontaneous emission of a single quantum system. The achievable theoretical efficiency of this cavity is comparable to that of already known cavities but the design of this cavity is suitable for a larger variety of quantum emitters (including organic molecules, diamond nanocrystals, and others) which can be used for efficient single-photon generation.

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