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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 24 — Nov. 19, 2012
  • pp: 27198–27211
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Deterministic generation of an on-demand Fock state

Keyu Xia, Gavin K. Brennen, Demosthenes Ellinas, and Jason Twamley  »View Author Affiliations


Optics Express, Vol. 20, Issue 24, pp. 27198-27211 (2012)
http://dx.doi.org/10.1364/OE.20.027198


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Abstract

We theoretically study the deterministic generation of photon Fock states on-demand using a protocol based on a Jaynes Cummings quantum random walk which includes damping. We then show how each of the steps of this protocol can be implemented in a low temperature solid-state quantum system with a Nitrogen-Vacancy centre in a nanodiamond coupled to a nearby high-Q optical cavity. By controlling the coupling duration between the NV and the cavity via the application of a time dependent Stark shift, and by increasing the decay rate of the NV via stimulated emission depletion (STED) a Fock state with high photon number can be generated on-demand. Our setup can be integrated on a chip and can be accurately controlled.

© 2012 OSA

1. Introduction

The generation of on-demand photonic Fock states is at the heart of many photonic quantum technologies. Single-photon sources have been realized in a variety of quantum systems such as nitrogen-vacancy (NV) centres in diamond [1

1. J. T. Choy, B. J. M. Hausmann, T. M. Babinec, I. Bulu, M. Khan, P. Maletinsky, A. Yacoby, and M. Lončar, “Enhanced single photon emission from a diamond-silver aperture,” Nat. Photonics 5, 738–743 (2011). [CrossRef]

], or using quantum dots [2

2. K. Rivoire, S. Buckley, A. Majumdar, H. Kim, P. Petroff, and J. Vuckovic, “Fast quantum dot single photon source triggered at telecommunications wavelength,” Appl. Phys. Lett. 98, 083105 (2011). [CrossRef]

]. However the creation of photonic Fock states with a high photon number is an open challenge to date. In this paper we propose a novel Jaynes Cummings quantum random walk (QRW) protocol that drives the cavity to accumulate a photonic Fock state deterministically. We describe in detail how to implement the theoretical protocol using a Nitrogen-Vacancy defect in a nano-diamond evanescently coupled to circulating light modes in a high-Q toroidal resonator at moderately low temperatures (< 10 K). We show that even in the case where one has error in the timing of the control pulses there is very high probability to generate photonic Fock states up to n = 6.

Synthesising high-number Fock states has received much attention in the literature. A high-number Fock state has been conditionally produced with a probability Pn via the state collapse from a coherent or thermal state [3

3. C. Guerlin, J. Bernu, S. Deléglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J. Raimond, and S. Haroche, “Progressive field-state collapse and quantum non-demolition photon counting,” Nature 448, 889–893 (2007). [CrossRef] [PubMed]

5

5. M. Brune, S. Haroche, V. Lefevre, J. M. Raimond, and N. Zagury, “Quantum non-demolition measurement of small photon numbers by rydberg atom phase sensitive detection,” Phys. Rev. Lett. 65, 976–979 (1990). [CrossRef] [PubMed]

]. The probability Pn of success is equal to the initial overlap probability of the target Fock state with the initial state of light (P(n) = Tr[|n〉 〈n|ρinit]), and this probability can be quite low: for |n = 3〉, P3 ≈ 0.22 if starting from a pure coherent state |α=3. An arbitrary quantum state of a cavity field can also be engineered if the cavity-qubit coupling can be very accurately controlled and recent experiments using low-temperature superconducting circuit-QED have synthesised microwave cavity states up to nine photons [6

6. C. K. Law and J. H. Eberly, “Arbitrary control of a quantum electromagnetic field,” Phys. Rev. Lett. 76, 1055– 1058 (1996). [CrossRef] [PubMed]

8

8. M. Hofheinz, E. M. Weig, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, H. Wang, J. M. Martinis, and A. N. Cleland, “Generation of fock states in a superconducting quantum circuit,” Nature 454, 310–314 (2008). [CrossRef] [PubMed]

]. When an excited two level system interacts with a cavity mode via the Jaynes Cummings (JC) interaction of strength g, the emission probability Pemit(g, τ, n) of the excited atom depends on the duration of coupling τ and the choice of Fock state n. For certain values, terming trapping values, of (g, τ, n), Pemit vanishes and a Fock state can be trapped in the cavity [9

9. P. Filipowicz, J. Javanainen, and P. Meystre, “Quantum and semiclassical steady states of a kicked cavity mode,” J. Opt. Soc. Am. B 3, 906–910 (1986). [CrossRef]

11

11. B. T. H. Varcoe, S. Brattke, and H. Walther, “The creation and detection of arbitrary photon number states using cavity QED,” New J. Phys. 6, 97 (2004). [CrossRef]

]. In this way by sending a train of Rydberg atoms through a superconducting high-Q microwave cavity, Walther et al. trapped a microwave Fock state [10

10. S. Brattke, B. T. H. Varcoe, and H. Walther, “Generation of photon number states on demand via cavity quantum electrodynamics,” Phys. Rev. Lett. 86, 3534–3537 (2001). [CrossRef] [PubMed]

, 11

11. B. T. H. Varcoe, S. Brattke, and H. Walther, “The creation and detection of arbitrary photon number states using cavity QED,” New J. Phys. 6, 97 (2004). [CrossRef]

]. Indeed all the experimental demonstrations in high-Fock number state generation have been in the microwave regime [8

8. M. Hofheinz, E. M. Weig, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, H. Wang, J. M. Martinis, and A. N. Cleland, “Generation of fock states in a superconducting quantum circuit,” Nature 454, 310–314 (2008). [CrossRef] [PubMed]

, 10

10. S. Brattke, B. T. H. Varcoe, and H. Walther, “Generation of photon number states on demand via cavity quantum electrodynamics,” Phys. Rev. Lett. 86, 3534–3537 (2001). [CrossRef] [PubMed]

, 11

11. B. T. H. Varcoe, S. Brattke, and H. Walther, “The creation and detection of arbitrary photon number states using cavity QED,” New J. Phys. 6, 97 (2004). [CrossRef]

]. At optical frequencies, Brown et al. propose a system of N three-level atoms in a high-finesse cavity [12

12. K. R. Brown, K. M. Dani, D. M. Stamper-Kurn, and K. B. Whaley, “Deterministic optical Fock-state generation,” Phys. Rev. A 67, 043818 (2003). [CrossRef]

], for Fock state generation but this requires the preparation of complicated nonclassical states of the atoms. Until now only single photon sources at optical frequencies have been realized in solid-state quantum systems.

Jaynes Cummings Damped Quantum Random Walk:- A coined quantum random walk involves a coin, which we take as a qubit with Hilbert space ℋc = span{|e〉, |g〉}, together with a walk on the discretised non-negative real line ℋw = span{|n〉; n = 0, 1,···}. The normal coined quantum random walk on the full real line (−∞ ≤ n ≤∞), is an iteration of a basic step involving a conditional displacement of the walker on the line depending on the internal state of the walker Ûd ≡ |e〉〈e|⊗|n+1〉〈n|+|g〉〈g|⊗|n−1〉〈n|, followed by a “scrambling” of the internal state of the walker by the action of a Hadamard operation on the internal states. This coined version of the QRW where the walker moves on the discretized real line 𝕑 has been studied intensively over the past decade. In the following we will examine the case when the space upon which the walker walks is the Fock ladder, n ∈ 𝕑*, i.e. the non-negative integers. It is no longer possible to have a unitary operator that implements a conditional displacement with constant displacement independent of the position of the walker. To achieve a unitary operation for the conditional displacement the walker can execute a step up/down the half-line with “step sizes” that depend on n. Our QRW step will consist of a period of Jaynes Cummings evolution between the internal states of the coin and conditional displacements up/down the Fock ladder, followed by a manipulation of the internal states of the walker. Rather than a complete scrambling of the internal states we will just consider a flip where |g ↔|e〉, are swapped. We have found that such a unitary QRW on the half line using the JC walk step exhibits complex temporal dynamics but simplifies greatly when we allow for periodic damping of the internal state of the coin.

We now consider analytically the above QRW on the half-line and derive a formulae for the resulting map on the reduced Fock space of the walker. We will see that, if starting at |n = 0〉, the walker will, on average, step to greater values of n and will hit a ceiling value of n that depends on the chosen value for the JC interaction strength/time. Let the JC Hamiltonian in the RWA be given by HJC = g(|e〉 〈g| ⊗ â + |g〉 〈e| ⊗ â), then the resulting unitary evolution operator ÛJC(τ) = eiHJCτ, can be expressed as
U^JC(τ)=(cosgτN+1isin(gτN+1)N+1aiasin(gτN+1)N+1cosgτN),
(1)
where N = aa is the photon number operator. Considering the initial product state for the density matrix of the coin and the walker to be ρCρW, then following evolution by the JC Hamiltonian we obtain ρCρWU^JC(ρCρW)U^JC. Subsequently we allow spontaneous emission (amplitude damping channel) to operate on the atomic system i.e.
ρCρWU^JC(ρCρW)U^JC^SEid^W[U^JC(ρCρW)U^JC],
(2)
where id^W stands for the identity map in walker’s space. Here ℰ̂SE = AdŜ0 + AdŜ1 is the spontaneous emission channel with non-unitary Kraus generators
S^0=|gg||ee|η,S^1=|ge|1η,
(3)
where η(t) = et/T a positive parameter quantifying the degree by which the atomic system is reset by the channel, with t the nominal time over which the channel operates and T a constant characterising how rapid the reset process it. We have also used the notation of the adjoint action Ad(Â) of an operator  on some other operator X̂ as follows: X̂ → Ad(Â)X̂ =ÂX̂Â, noticing the property Ad(ÂB̂)X̂ = Ad(Â)Ad(B̂)X̂. For a general pure input state of the coin:
ρC=(α|g+β|e)(α*g|+β*e|)=(|β|2α*βαβ*|α|2),
(4)
the channel ℰ̂SE outputs
^SE[ρC](t)=(η|β|2α*βηαβ*η1η|β|2).
(5)
In view of the limit limt→∞ η(t) = 0, and normalization relation |α|2 +|β|2 = 1, the last expression leads to the reset state limt^SE[ρat](t)=|gg|=(0001).

Fig. 1 Jaynes Cummings quantum random walk: Plots showing how the walker evolves starting in the vacuum, i.e. Tr{|n〉〈n|ℰ̂m[|e〉 〈e|⊗|0〉〈0|]}, as a function of the number of steps m and Fock number n. (a) for a completely unitary evolution with ℰ̂ = AdĤAdÛJC, where one executes a Hadamard on the coin space (b) completely unitary evolution with ℰ̂ = AdX̂ ○ AdÛJC, where one executes a π flip instead of the Hadamard and (c) including spontaneous damping channel ℰ̂ = AdX̂ ○ Adℰ̂SEAdÛJC, and the Jaynes Cumming coupling strength and duration chosen so that |n = 16〉 is a trapping state. We see that the latter evolution clearly leads to accumulation of the walker at the target trapping state.

Now we theoretically discuss the maximum Fock number that can be reached with high fidelity. We consider a case where the noise in the timing of the JC interactions is vanishingly small. However the decay of cavity and the effect of population in the ground state of the qubit must be taken into account. We denote the effective cavity loss of the photons with the rate γc. The higher the Fock state is, the larger this effective decay rate. For the target Fock state nT, the effective decay rate increases to nTγc. Another factor limiting whether one can achieve the target state relates to the downward transfer of population with probability PD from the target state |nT〉 to the lower Fock state |nT − 1〉 due to the net population Pg of the ground state. In the stationary state, the pumping probability PU from the state |nT − 1〉 must balance the loss from the target Fock state |nT〉. A formula describing this balance takes the form
PnT(1enTγct)+PgPD=PnT1PU,
(12)
where PnT(PnT −1) is the population in Fock state |nT〉 (|nT −1〉). Because η can not be practically zero after waiting for a time t, the net population in the excited state of qubit is η = eq with the effective decay rate of qubit γq which can be modified using STED beam in our setup. After the state flipping, this population is transferred to the ground state. If the time t is measured as t=Mγq1, Pg = eM. We are interested in the case of high fidelity F of achieving the target state. We observe that the population is a good approximation of fidelity, PnTF. The population of the lower Fock state is PnT −1 = α(1 − F) with constant 0 ≤ α ≤ 1. For our Hermitian system, we have PU=PD=sin2(πnTnT+1). There we have
F(1eNTγcMγq1)+eMsin2(πnTnT+1)=α(1F)sin2(πnTnT+1).
(13)
This formula shows the relation between the decay rates, target state Fock number and the achievable fidelity. Assuming that NTγcMγq11 and nTnT+11, the fidelity as a function of the decay rates and the photon number nT takes the form
F=π2(αeM)π2α+4MnT3γc/γq.
(14)

Implementation:- To implement the above protocol we propose to use a single nitrogen-vacancy (NV) center in a nanodiamond coupled to a high-finesse toroidal optical cavity at low temperature, while the latter is also connected to an optical interferometer and where the NV’s optical transition is initialised via optical pumping, brought in/out of resonance with the cavity via Stark shift tuning resulting from an electric field, and undergoes periodic optical π flips via resonant optical laser pulses. In more detail: when the cavity interacts on-resonance with the zero-phonon line (ZPL) of the single NV center, the de-excitation probability of the NV (treated as a two level system (TLS)), [and consequently excitation probability of the cavity], is given by Pemit(g,τ,n)=sin2(n+1gτ), where n is the number of photons in the cavity, g is the JC coupling strength, and τ is the interaction time. Choosing τ = τT such that the nT photon is trapped in the cavity we have Pemit(g, τT, nT) = 0. Using a fixed τT as the time step in the damped JCQRW above and starting the cavity in the vacuum state leads to the cavity field undergoing a deterministic ratchet-like increase in Fock number until it accumulates at n = nT. The trapping-state condition means that the field in the cavity reaches an upper bound and is prevented from being excited to a higher photonic number state. Thus via a precise control of the Jaynes Cummings coupling τT, an on-demand Fock state can be deterministically trapped in the cavity starting from the vacuum state. To do this we start the following process (Eq. (6)) from the excited state of the NV center, which is resonantly prepared by a π laser pulse: (i) we first switch on the JC coupling by tuning off the electric field. During this stage, the NV center emits a photon with the probability Pg into the cavity. (ii) After a time τT, the JC coupling is turned off by bringing the NV’s optical transition out of resonance with the cavity via electrical Stark control [15

15. V. M. Acosta, C. Santori, A. Faraon, Z. Huang, K.-M. C. Fu, A. Stacey, D. A. Simpson, K. Ganesan, S. Tomljenovic-Hanic, A. D. Greentree, S. Prawer, and R. G. Beausoleil, “Dynamic stabilization of the optical resonances of single nitrogen-vacancy centers in diamond,” Phys. Rev. Lett. 108, 206401 (2012). [CrossRef] [PubMed]

] and the NV center is allowed to completely decay to its ground state (GS). (iii) Then we resonantly pumping the NV center to its excited state again via a π pulse. Repeating these operations, the field in the cavity can be trapped in a selected target Fock state.

Fig. 2 Solid state setup for deterministic generation of an on-demand Fock state of photons. A two level system (nitrogen vacancy in a nanodiamond - here shown as yellow at top of the toroidal resonator), interacts with counter propagating optical modes â and b̂ in a high-Q toroidal resonator with intrinsic decay rate γc and which is coupled to a nearby waveguide interferometer at a coupling rate κext. Shown are the input and output modes â/b̂in/out and the resulting anti/symmetric modes Âa/s modes from the interferometer with each associated photon detector Da/s. Also shown is the incident (red arrow), laser pulse resonant on the NV zero phonon line required to implement an optical π−pulse and the Stark shift electrode used to bring the NV’s optical transition in/out of resonance with the cavity. Not shown is the initialising green laser and stimulated depletion laser.

An optical waveguide precisely positioned close to the cavity couples light in/out to/from the cavity with external coupling rate κext via the input-output relations [20

20. M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984). [CrossRef]

, 21

21. C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985). [CrossRef] [PubMed]

]
a^out=a^in+2κexta^,
(15)
b^out=b^in+2κextb^,
(16)
where the input and output fields of the waveguide are denoted by {âin, b̂in, âout, b̂out}, respectively. [a^in(t),a^in(t)]=[a^out(t),a^out]=δ(tt) and similarly [b^in(t),b^in(t)]=[b^out(t),b^out(t)]=δ(tt). The output fields âout and b̂out are mixed by a 50 : 50 directional coupler [22

22. L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled state measurement on a chip,” Phys. Rev. Lett. 105, 200503 (2010). [CrossRef]

]. We take the inputs to the cavity to be vacuum states, i.e. 〈âin〉 = 〈b̂in〉 = 0, and thus the outputs âout and b̂out are proportional to â and b̂, respectively. Thus the outputs of the directional coupler yields modes Âs and Âa [23

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

], leading to the detectors. Here we aim to create Fock state of the symmetric mode Âs. Assuming the intrinsic loss of the cavity is denoted by the decay rate γc, if we take into account the scattering h between two modes â and b̂, the critical coupling condition is given by κext=h2+γc2 [19

19. B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom.” Science 319, 1062–1065 (2008). [CrossRef] [PubMed]

].

To switch the interaction between the NV center and the cavity, an electric field perpendicular to the axis of NV center is applied to induce a Stark shift. This static electric field (SEF) can be created by two electrodes positioned 10 μm above the setup [24

24. L. C. Bassett, F. J. Heremans, C. G. Yale, B. B. Buckley, and D. D. Awschalom, “Electrical tuning of single nitrogen-vacancy center optical transitions enhanced by photoinduced fields,” Phys. Rev. Lett. 107, 266403 (2011). [CrossRef]

]. This distance is much larger than the wavelength of the field in the cavity and the extent of the evanescent field and thus results in negligible scattering loss of the cavity modes. During the excitation of cavity, this SEF is applied to shift the NV center in/out of resonance with the cavity, thus executing the JC step in the map Eq. (6).

A critical step in the process Eq. (6) is the rapid decay of the two level system ℰ̂SE(ρ̂), the spontaneous emission decay of the two level system (coin). This decay must be executed with a rate much higher than the cavity decay rate. The natural excited state lifetime of the NV ZPL is ∼ 11ns and this is too long to permit many repetitions of our process Eq. (6) even with high-Q cavities. To shorten this, after the JC coupling is switched off by applying the SEF we use a stimulated emission depletion (STED) laser beam (λSTED = 775 nm), to dynamically create a fast decay channel from the excited state to the ground state of the NV center [25

25. E. Rittweger, K. Y. Han, S. E. Irvine, C. Eggeling, and S. W. Hell, “STED microscopy reveals crystal colour centres with nanometric resolution,” Nat. Photonics 3, 144–147 (2009). [CrossRef]

], and this can increase the effective decay rate of the NV by almost four orders of magnitude. After almost all of the population has decayed to the ground state |g〉, another laser beam on-resonant with the ZPL repumps the NV center from |g〉 to |e〉 [15

15. V. M. Acosta, C. Santori, A. Faraon, Z. Huang, K.-M. C. Fu, A. Stacey, D. A. Simpson, K. Ganesan, S. Tomljenovic-Hanic, A. D. Greentree, S. Prawer, and R. G. Beausoleil, “Dynamic stabilization of the optical resonances of single nitrogen-vacancy centers in diamond,” Phys. Rev. Lett. 108, 206401 (2012). [CrossRef] [PubMed]

,26

26. K.-M. C. Fu, C. Santori, P. E. Barclay, L. J. Rogers, N. B. Manson, and R. G. Beausoleil, “Observation of the dynamic jahn-teller effect in the excited states of nitrogen-vacancy centers in diamond,” Phys. Rev. Lett. 103, 256404 (2009). [CrossRef]

28

28. A. Batalov, V. Jacques, F. Kaiser, P. Siyushev, P. Neumann, L. J. Rogers, R. L. McMurtrie, N. B. Manson, F. Jelezko, and J. Wrachtrup, “Low temperature studies of the excited-state structure of negatively charged nitrogen-vacancy color centers in diamond,” Phys. Rev. Lett. 102, 195506 (2009). [CrossRef] [PubMed]

], i.e. an optical π−pulse. This is the final X̂ portion of the map Eq. (6).

2. Detailed model of experimental protocol

We propose to implement the JC Damped QRW Fock state synthesis using a NV cavity-QED setup. The relevant energy level scheme for the NV center is shown in Fig. 3(a). We neglect both the hyperfine electron-nuclear spin coupling and the weak electronic spin-spin interaction [33

33. X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nature 478, 221–224 (2011). [CrossRef] [PubMed]

]. The center has an optically allowed transition between an orbital ground state 3A2 and an orbital excited state 3E. Both the ground and excited states are S = 1 spin triplets. The ground state has 3A2 symmetry and is split into an Sx, Sy doublet 2.87GHz above an Sz singlet due to the zero-field splitting [32

32. P. Tamarat, N. B. Manson, J. P. Harrison, R. L. McMurtrie, A. Nizovtsev, C. Santori, R. G. Beausoleil, P. Neumann, T. Gaebel, F. Jelezko, P. Hemmer, and J. Wrachtrup, “Spin-flip and spin-conserving optical transitions of the nitrogen-vacancy centre in diamond,” New J. Phys. 10, 045004 (2008). [CrossRef]

]. The lifetime of excited state 3E is about 11.6 ns, corresponding to a decay rate γ = 14 MHz [34

34. P. 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 centers in diamond,” Phys. Rev. Lett. 97, 083002 (2006). [CrossRef] [PubMed]

]. Low temperature transform-limited single photon emission spectra from individual NV defects [34

34. P. 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 centers in diamond,” Phys. Rev. Lett. 97, 083002 (2006). [CrossRef] [PubMed]

] indicates that dephasing is negligible at these low temperatures. Further low temperature experiments demonstrate that the optical excited states of the NV can be isolated from the effects of the nearby phonon sidebands [28

28. A. Batalov, V. Jacques, F. Kaiser, P. Siyushev, P. Neumann, L. J. Rogers, R. L. McMurtrie, N. B. Manson, F. Jelezko, and J. Wrachtrup, “Low temperature studies of the excited-state structure of negatively charged nitrogen-vacancy color centers in diamond,” Phys. Rev. Lett. 102, 195506 (2009). [CrossRef] [PubMed]

] at low temperatures. Throughout our operation below the non-radiative decay to the intersystem state from |Ex,Sz〉 is taken to be negligible. Such intersystem crossing decay contributes to an effective decay to the singlet ground state Sz.

Fig. 3 (a) Level diagram of a NV center showing spin-triplet ground and excited states, as well as the singlet system involved in intersystem crossing [2931]. Another triplet excited state Ey is not shown here. Also shown is the JC coupling (blue) g, decay rate from |e〉 to |g〉, the STED illumination (red), and the laser transition for the π flip generated by Ωx (orange). (b) Eigenvalues of the excited state triplet as a function of applied SEF [32]. The vertical dashed line at V = 0 marks the splitting due to the strain. At this position the NV center can be excited resonantly by a π laser pulse Ωx. An electric field is applied to bring the NV center into resonance with the symmetric mode (resonant frequency ωs). (c) Time sequence for generation of photonic Fock state showing the initialisation, JC coupling, Decay and X flip.

At low temperature (T< 10 K), strain in the nanodiamond causes the excited state 3E to split into an orbital upper branch Ex and an orbital lower branch Ey (see Fig. 3(b)). Each branch is a spin triplet formed by three spin states Sx, Sy and Sz. The sublevel |Ex,Sz〉 is well isolated from the other five sublevels by several GHz. Actually the state |Ey,Sz〉 can be isolated from |Ex,Sz〉 because these two sublevels are associated to orthogonal transition dipoles [26

26. K.-M. C. Fu, C. Santori, P. E. Barclay, L. J. Rogers, N. B. Manson, and R. G. Beausoleil, “Observation of the dynamic jahn-teller effect in the excited states of nitrogen-vacancy centers in diamond,” Phys. Rev. Lett. 103, 256404 (2009). [CrossRef]

, 28

28. A. Batalov, V. Jacques, F. Kaiser, P. Siyushev, P. Neumann, L. J. Rogers, R. L. McMurtrie, N. B. Manson, F. Jelezko, and J. Wrachtrup, “Low temperature studies of the excited-state structure of negatively charged nitrogen-vacancy color centers in diamond,” Phys. Rev. Lett. 102, 195506 (2009). [CrossRef] [PubMed]

]. Therefore the spin-conserving transition |3A2,Sz〉 ↔|Ex,Sz〉, can be excited resonantly at low strain [26

26. K.-M. C. Fu, C. Santori, P. E. Barclay, L. J. Rogers, N. B. Manson, and R. G. Beausoleil, “Observation of the dynamic jahn-teller effect in the excited states of nitrogen-vacancy centers in diamond,” Phys. Rev. Lett. 103, 256404 (2009). [CrossRef]

28

28. A. Batalov, V. Jacques, F. Kaiser, P. Siyushev, P. Neumann, L. J. Rogers, R. L. McMurtrie, N. B. Manson, F. Jelezko, and J. Wrachtrup, “Low temperature studies of the excited-state structure of negatively charged nitrogen-vacancy color centers in diamond,” Phys. Rev. Lett. 102, 195506 (2009). [CrossRef] [PubMed]

]. To suppress further any small spin mixing and phonon-induced transitions within these two excited states [26

26. K.-M. C. Fu, C. Santori, P. E. Barclay, L. J. Rogers, N. B. Manson, and R. G. Beausoleil, “Observation of the dynamic jahn-teller effect in the excited states of nitrogen-vacancy centers in diamond,” Phys. Rev. Lett. 103, 256404 (2009). [CrossRef]

,28

28. A. Batalov, V. Jacques, F. Kaiser, P. Siyushev, P. Neumann, L. J. Rogers, R. L. McMurtrie, N. B. Manson, F. Jelezko, and J. Wrachtrup, “Low temperature studies of the excited-state structure of negatively charged nitrogen-vacancy color centers in diamond,” Phys. Rev. Lett. 102, 195506 (2009). [CrossRef] [PubMed]

], our setup works with low-strain NV centers at cryogenic temperatures. Moreover, the JC coupling g is assumed to be much larger than the decay rate γ and the thermal orbital coupling and relaxation rates. In our protocol the duration when the NV center is excited into state |e〉 state is small and thus the spin mixing can be neglected [26

26. K.-M. C. Fu, C. Santori, P. E. Barclay, L. J. Rogers, N. B. Manson, and R. G. Beausoleil, “Observation of the dynamic jahn-teller effect in the excited states of nitrogen-vacancy centers in diamond,” Phys. Rev. Lett. 103, 256404 (2009). [CrossRef]

, 28

28. A. Batalov, V. Jacques, F. Kaiser, P. Siyushev, P. Neumann, L. J. Rogers, R. L. McMurtrie, N. B. Manson, F. Jelezko, and J. Wrachtrup, “Low temperature studies of the excited-state structure of negatively charged nitrogen-vacancy color centers in diamond,” Phys. Rev. Lett. 102, 195506 (2009). [CrossRef] [PubMed]

]. Our protocol primarily involves the transition between |g〉 ↔|e〉, (|3A2, Sz〉 ↔|Ex, Sz〉) and excludes other sublevels [27

27. L. Robledo, L. Childress, H. Bernien, B. Hensen, P. F. A. Alkemade, and R. Hanson, “High-fidelity projective read-out of a solid-state spin quantum register,” Nature 477, 574–578 (2011). [CrossRef] [PubMed]

]. Thus we are able to treat the NV as a two level system with a transition in the optical ∼ 637nm.

The dynamics of our system is given by the Hamiltonian Ĥ in the rotating wave approximation (RWA)
H^=H^s+H^a+H^x,H^s=h¯(Δg+Δs(t))A^sA^s+h¯g[A^sσ+H.c.],H^x=h¯Ωx(t)[σ^+H.c.],
(17)
where σ̂ = |g〉〈e|. Δg = ωzplωs is the detuning of the mode Âs and the ZPL transition between states |g〉 and |e〉 (with frequency ωzpl), in the absence of any Stark shift. ωs = ωc + h is the resonant frequency of mode Âs shifted by the scattering h. g is the JC coupling strength between a single NV center and a single photon in the cavity. In our scheme, the NV center only couples to the symmetric mode Âs. This is reasonable because this coupling can be predominately to mode Âs by specialized positioning of the nanodiamond [19

19. B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom.” Science 319, 1062–1065 (2008). [CrossRef] [PubMed]

], and the scattering can also introduce a large detuning between the unwanted mode Aa and the ZPL of NV center. The Stark shift Δs(t) is used to dynamically control the creation of cavity photons by the NV center due to the JC coupling [15

15. V. M. Acosta, C. Santori, A. Faraon, Z. Huang, K.-M. C. Fu, A. Stacey, D. A. Simpson, K. Ganesan, S. Tomljenovic-Hanic, A. D. Greentree, S. Prawer, and R. G. Beausoleil, “Dynamic stabilization of the optical resonances of single nitrogen-vacancy centers in diamond,” Phys. Rev. Lett. 108, 206401 (2012). [CrossRef] [PubMed]

]. For Δs(t) = 0, the cavity decouples from the NV center because of the large detuning. This process can be considered as “the JC coupling off”. The excitation of cavity is turned on if “the JC coupling on”, i.e. Δs(t) = −Δg. According to our numerical simulation, a fast relaxation from |e〉 to the ground state |g〉 is required following the JC coupling phase for the preparation of a high-number Fock state with a high fidelity. Here we make use of the concept of “Stimulated Emission Depletion” (STED) to dynamically enhance the relaxation process [25

25. E. Rittweger, K. Y. Han, S. E. Irvine, C. Eggeling, and S. W. Hell, “STED microscopy reveals crystal colour centres with nanometric resolution,” Nat. Photonics 3, 144–147 (2009). [CrossRef]

] during the “decay phase” of the map (see Eq. (6) and Fig. 3(c)). When the STED beam is applied, the stimulated emission rate γSTED becomes to ISTEDγ/Is, with ISTED/Is denoting the ratio of the STED pulse intensity ISTED and the saturation intensity Is. For a lifetime 11.6 ns, Is is ∼ 1.85 MW cm−2 [25

25. E. Rittweger, K. Y. Han, S. E. Irvine, C. Eggeling, and S. W. Hell, “STED microscopy reveals crystal colour centres with nanometric resolution,” Nat. Photonics 3, 144–147 (2009). [CrossRef]

] if a continuous wave (cw) STED beam is applied. A cw STED beam of 20 GW cm−2 can enhance the decay rate by four orders of magnitude. During the initial “JC coupling on” phase (see Fig. 3(c)), we turn off the STED beam and the nominal decay rate of NV center remains γ ∼ 14 MHz.

We now describe the detailed steps in synthesising the process described in Eq. (6). We assume that the optical modes in both the cavity and waveguide are initially in the vacuum state yielding initially zero photon number in the cavity. The time sequence for creating a photonic Fock state is shown in Fig. 3(c): Initially the NV center is in its ground state |g〉 after a short 532 nm laser pulse optically prepares the defect into the ms = 0(|3A2, Sz〉) state. A π laser pulse Ωx is used to resonantly pump it to the excited state |e〉. Then the ZPL is tuned on-resonance with the mode Âs to enable the JC coupling g using a SEF. After time τT, we turn off the JC coupling but use the STED laser beam to create a fast decay channel to the ground state |g〉. Waiting for time τγ, almost all population decays to the ground state |g〉 from |e〉. Then a further optical π pulse generated by Ωx is applied to resonantly excite the NV center to |e〉 again. We repeat these operations until the target state is trapped.

When Δgs = 0, the NV center resonantly couples to the cavity mode Âs. The dynamics of the system can then be described by a unitary time evolution operator and we further now assume that the time duration of this unitary may not be precisely controlled, i.e. we assume some noise in the target JC coupling time τT. More precisely we take UJC = eT(1+δτ)Hs/, where δτ is a normally distributed additional noise in timing with a standard deviation given by a parameter σn. The excitation probability of the cavity to state |n + 1〉 when the NV is in the excited state is thus now given by Pg=sin2[gτTn+1(1+δτ)]. Once the SEF is turned off, the STED laser pulse is switched on. The NV center is decoupled from the cavity and relaxes to its GS Sz. The timing noise during the decay is not considered in τγ because this damping process is insensitive to the timing error. The population in the excited state |3〉 decays at an effective rate γSTED to the ground state |g〉. Such process can be described by a supperoperator ε as [35

35. S. M. Tan, “A computational toolbox for quantum and atomic optics,” J. Opt. B: Quantum Semiclass. Opt. 1, 424–432 (1999). [CrossRef]

]
[ρ^]=e(L^^iH^^s/h¯)τγρ^
(18)
where the superoperators are defined as H^^sρ^=[H^s,ρ^], L^^ρ^=γc/2(2a^ρ^a^a^a^ρ^ρ^a^a^)+γSTED/2(2σ^ρ^σ^+σ^zρ^ρ^σ^z) with σ̂z = |e〉 〈e|−|g〉 〈g| and the density matrix ρ̂. After a time τγ=5γSTED1, the GS Sz is polarized more than 99% again. The flip π laser pulse generated by Ωx (λx ≈ 637 nm) turns on successively to flip the NV center to the ES Sz. We define a flip operator X = e(1+δx)σx/2 with σx = σ+ + σ to model this flip process as X̂ρ̂X̂. δx is a noise having the same statistic property but independent of δτ. Then the density matrix after l + 1 steps is determined by a recurrence relation
ρ^l+1=X^SE[U^JCρ^lU^JC]X^.
(19)
The system is initialized in the state ρ0=|ee|ρvacc, where ρvacc is the density matrix of vacuum state of cavity mode.

3. Results

To control the JC coupling, we consider a setup shown in Fig. 2, in which the cavity is designed to be off-resonance with the transition |g〉↔|e〉 such that |Δg| ≫ |g|. A detuning of Δg = 10g is large enough to decouple the cavity from the NV center. To switch on the JC coupling, the transition |g〉↔|e〉 is tuned to be on resonance with the cavity, i.e. Δg + Δs = 0, by the Stark shift Δs induced by the SEF [15

15. V. M. Acosta, C. Santori, A. Faraon, Z. Huang, K.-M. C. Fu, A. Stacey, D. A. Simpson, K. Ganesan, S. Tomljenovic-Hanic, A. D. Greentree, S. Prawer, and R. G. Beausoleil, “Dynamic stabilization of the optical resonances of single nitrogen-vacancy centers in diamond,” Phys. Rev. Lett. 108, 206401 (2012). [CrossRef] [PubMed]

]. However during the JC step the NV center will decay to its ground state |g〉 at a rate given by γ and this decay during the JC step decreases the fidelity of the target Fock state. We assume a static, larger JC coupling g = 30γ to suppress this detrimental process. By assuming a good optical cavity with γc γ, in combination with the large JC coupling strength further improves the ultimate fidelity of the trapped photon state.

In the ideal case of no cavity decay, a complete switching on/off JC coupling and no timing error, one can stably trap a Fock state |nT〉 with unit fidelity, see the black solid line (i) in Fig. 4(a) for instance.

Fig. 4 (a) Time evolution of fidelities of target Fock state |n = 6〉. (i) σn = 0, γc = 0; (ii) σn = 0, γc = 0.1γ; (iii) σn = 1%, γc = 0.1γ; (iv) σn = 2%, γc = 0. (b) Probabilities of photon number states at step 73. Red bar for only cavity decay σn = 0, γc = 0.1γ; blue bar for σn = 1%, γc = 0.1γ. Other parameters are g = 30γ, Δg = 300γ, γSTED = 104γ.

If only the cavity decay is included (red dashed-dotted line (ii) in Fig. 4(a)), the target state is stable once it is prepared after about 100 steps, and then the fidelity F is very high, about 0.97. The loss of cavity photon cancel the small probability of pumping from |n = 5〉 to |n = 6〉 if F is large, and consequently leads to the reduce of fidelity. Thus the excitation of state |n = 5〉 is considerable, see red bar in Fig. 4(b).

We notice that the timing error in the JC coupling causes a leakage of the population to higher photon number states. This leakage results in a reduction of the fidelity as the operation continues (iii and iv). To provide a limit for the fidelity of a Fock state we can prepare with nT ≤ 6, we perform the simulation including both kinds of imperfection: the timing error (σn = 1%) and the cavity decay γc = 0.1γ. In this case, the probability of |n = 6〉 is about 0.9 from step 64 to 94 (blue dashed line (iii) in Fig. 4(a)). Obviously, the prepared Fock state is stable within a wide operation step range. This is an advantage of the trapping state [10

10. S. Brattke, B. T. H. Varcoe, and H. Walther, “Generation of photon number states on demand via cavity quantum electrodynamics,” Phys. Rev. Lett. 86, 3534–3537 (2001). [CrossRef] [PubMed]

]. However about 6% population leaks to the higher photon number state (blue bar in Fig. 4(b)). The fidelity gradually reduces as the Q factor of cavity decreases. Hailin Wang’s group has demonstrated a microspherical cavity with γc ∼ 0.4γ coupling to a nanodiamond [18

18. Y. S. Park, A. K. Cook, and H. Wang, “Cavity QED with diamond nanocrystals and silica microspheres,” Nano Lett. 6, 2075–2079 (2006). [CrossRef] [PubMed]

]. Using this number, our simulation shows that the fidelity can still be 0.88 if σn = 1%.

A timing error of 2% substantially destroys the trapping condition (iv) and causes a considerable excitation of higher state. As a result, a considerable part, about 15%, of population leaks to higher photon number states. The fidelity of the target state |n = 6〉 decreases fast from a maximum F = 0.81 after 60 steps. However if the operation stops at step 60, one still can obtain the Fock state |n = 6〉 with high probability. A large error in the interaction time is the crucial reason why a train of atoms successively entering a cavity can not trap a Fock state with high probability [11

11. B. T. H. Varcoe, S. Brattke, and H. Walther, “The creation and detection of arbitrary photon number states using cavity QED,” New J. Phys. 6, 97 (2004). [CrossRef]

, 37

37. G. J. Milburn, “Kicked quantized cavity mode - an open-systems- theory approach,” Phys. Rev. A 36, 744–749 (1987). [CrossRef] [PubMed]

]. Thanks to a solid-state setup, this timing error can be much smaller during our operation. In contrast, our setup can generate a higher Fock state with a higher fidelity.

Even if the timing of JC interaction can be controlled perfectly, the decay of the cavity also limit the available number of photon of Fock state for a set fidelity F. Equation (14) provides a good estimation for the maximum of nT if F is set. The constant α (about 0.5) is numerically evaluated for σn = 0, γSTED/γc = 105. Using this value in Eq. (14), the fidelity for a certain target state |nT〉 is shown in Fig. 5. Clearly, the estimation given by Eq. (14) is consistent with the numerical results.

Fig. 5 Numerical proof of Relation Eq. (14). Blue solid line shows the available fidelity F evaluated by Eq. (14) as a function of target state |nT〉 for α = 0.5 and M = 5, γq/γc = 105. Blue triangle marks the numerical results for nT = 2, 4, 6, 8, 10, 12, 14, 20, 30 and σn = 0, γc = 0.1γ. Here γq is equal to γSTED. Numerical evaluation of α is marked as red stars. Other parameters are g = 30γ, Δg = 300γ.

To generate a Fock state |n = 6〉, we need γSTED > 104γc. In the presence of noise, the decay rate γSTED need be larger. To perform these simulations we used a cavity with Q ∼ 3 × 108 [38

38. H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012). [CrossRef]

43

43. I. S. Grudinin, A. B. Matsko, and L. Maleki, “On the fundamental limits of Q factor of crystalline dielectric resonators,” Opt. Express 15, 3390–3395 (2007). [CrossRef] [PubMed]

], corresponding to a decay rate of γc ∼ 2π × 1.4 MHz. The nanodiamond embedded in the cavity contributes an extra loss channel to the cavity and subsequently reduce the Q factor. However this induced loss is proportional to r6, where r is the radius of particle [44

44. A. Mazzei, S. Gẗzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating Whispering-Gallery modes by a single Rayleigh scattering: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99, 173603 (2007). [CrossRef] [PubMed]

, 45

45. J. Zhu, S. K. Ozdemir, Y. Xiao, L. Li, L. He, D. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010). [CrossRef]

]. This contribution of loss is negligible if r < 10 nm and nanodiamonds containing nitrogen vacancy centres in such small nano diamonds have been made [46

46. V. N. Mochalin, O. Shenderova, D. Ho, and Y. Gogotsi, “The properties and applications of nanodiamonds,” Nat. Photonics 7, 11–23 (2012).

, 47

47. B. R. Smith, D. W. Inglis, B. Sandnes, J. R. Rabeau, A. V. Zvyagin, D. Gruber, C. J. Noble, R. Vogel, E. Ōsawa, and T. Plakhotnik, “Five-Nanometer Diamond with Luminescent Nitrogen-Vacancy Defect Centers,” Small. 5, 1649–1653 (2009). [CrossRef] [PubMed]

]. Experiments have demonstrated that the Q factor of a cavity embedding a nanoparticle, such as a nanodimaond [42

42. I. S. Grudinin, V. S. Ilchenko, and L. Maleki, “Ultrahigh optical q factors of crystalline resonators in the linear regime,” Phys. Rev. A 74, 063806 (2006). [CrossRef]

], or potassium chloride nanoparticle [45

45. J. Zhu, S. K. Ozdemir, Y. Xiao, L. Li, L. He, D. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010). [CrossRef]

], can be larger than 3 × 108. The nanodiamond also causes scattering in the cavity and leads to a doubling of the linewidth of the cavity or mode splitting. This scattering rate decreases quickly (∝ r3) as the size of particle decreases [44

44. A. Mazzei, S. Gẗzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating Whispering-Gallery modes by a single Rayleigh scattering: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99, 173603 (2007). [CrossRef] [PubMed]

, 45

45. J. Zhu, S. K. Ozdemir, Y. Xiao, L. Li, L. He, D. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010). [CrossRef]

]. On the other hand we use the nanodiamond to selectively excite the symmetric mode. Therefore the effect of scattering on the generation of the target state can be neglected for r < 10 nm. We use the typical value γ/2π = 14 MHz for the decay rate of the excited state |3〉 of a single NV center [34

34. P. 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 centers in diamond,” Phys. Rev. Lett. 97, 083002 (2006). [CrossRef] [PubMed]

, 48

48. L. Robledo, H. Bernien, I. van Weperen, and R. Hanson, “Control and coherence of the optical transition of single nitrogen vacancy centers in diamond,” Phys. Rev. Lett. 105, 177403 (2010). [CrossRef]

]. This decay rate can be enhanced by four orders in magnitude if a 20 GW cm−2 cw STED beam is applied. To suppress the decay of population from the state |3〉 during the JC coupling on, we need a JC coupling strength g = 30γ ∼ 400 MHz, which can be reached in the current experiments [49

49. P. E. Barclay, C. Santori, K. Fu, R. G. Beausoleil, and O. Painter, “Coherent interference effects in a nano-assembled diamond NV center cavity-QED system,” Opt. Express 17, 8081–8097 (2009). [CrossRef] [PubMed]

]. This large value of g allows for shorter τT and on these times scales the mixing between states Ex and Ey is negligible. For this coupling strength, a Stark shift of Δs = 10g ∼ 2π × 4 GHz is large enough to switch on/off the excitation of cavity. Such Stark shift can be created using two electrodes separated by 10 μm and positioned 10 μm above the NV center [24

24. L. C. Bassett, F. J. Heremans, C. G. Yale, B. B. Buckley, and D. D. Awschalom, “Electrical tuning of single nitrogen-vacancy center optical transitions enhanced by photoinduced fields,” Phys. Rev. Lett. 107, 266403 (2011). [CrossRef]

].

4. Conclusion

In conclusion, we have proposed a solid-state setup consisting of a single NV center and a high-Q toroidal cavity for the generation of a multi-photon optical Fock state through the iteration of a damped Jaynes Cummings quantum random walk. By iterating this walk step we found a method to trap an on-demand photonic Fock state with high fidelity within the cavity.

Acknowledgments

One of us (D.E.) is grateful to the Macquarie University Research Centre for Quantum Science and Technology, for hospitality during a sabbatical stay during which this work was initiated. We also acknowledge support from the ARC Centre of Excellence in Engineered Quantum Systems and EU Project Quantip.

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P. Tamarat, N. B. Manson, J. P. Harrison, R. L. McMurtrie, A. Nizovtsev, C. Santori, R. G. Beausoleil, P. Neumann, T. Gaebel, F. Jelezko, P. Hemmer, and J. Wrachtrup, “Spin-flip and spin-conserving optical transitions of the nitrogen-vacancy centre in diamond,” New J. Phys. 10, 045004 (2008). [CrossRef]

33.

X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nature 478, 221–224 (2011). [CrossRef] [PubMed]

34.

P. 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 centers in diamond,” Phys. Rev. Lett. 97, 083002 (2006). [CrossRef] [PubMed]

35.

S. M. Tan, “A computational toolbox for quantum and atomic optics,” J. Opt. B: Quantum Semiclass. Opt. 1, 424–432 (1999). [CrossRef]

36.

F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillation of a single nuclear spin and realization of a two-qubit conditional quantum gate,” Phys. Rev. Lett. 93, 130501 (2004). [CrossRef] [PubMed]

37.

G. J. Milburn, “Kicked quantized cavity mode - an open-systems- theory approach,” Phys. Rev. A 36, 744–749 (1987). [CrossRef] [PubMed]

38.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012). [CrossRef]

39.

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003). [CrossRef] [PubMed]

40.

D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High-Q measurements of fused-silica microspheres in the near infrared,” Opt. Lett. 23, 247–249 (1998). [CrossRef]

41.

M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996). [CrossRef] [PubMed]

42.

I. S. Grudinin, V. S. Ilchenko, and L. Maleki, “Ultrahigh optical q factors of crystalline resonators in the linear regime,” Phys. Rev. A 74, 063806 (2006). [CrossRef]

43.

I. S. Grudinin, A. B. Matsko, and L. Maleki, “On the fundamental limits of Q factor of crystalline dielectric resonators,” Opt. Express 15, 3390–3395 (2007). [CrossRef] [PubMed]

44.

A. Mazzei, S. Gẗzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating Whispering-Gallery modes by a single Rayleigh scattering: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99, 173603 (2007). [CrossRef] [PubMed]

45.

J. Zhu, S. K. Ozdemir, Y. Xiao, L. Li, L. He, D. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010). [CrossRef]

46.

V. N. Mochalin, O. Shenderova, D. Ho, and Y. Gogotsi, “The properties and applications of nanodiamonds,” Nat. Photonics 7, 11–23 (2012).

47.

B. R. Smith, D. W. Inglis, B. Sandnes, J. R. Rabeau, A. V. Zvyagin, D. Gruber, C. J. Noble, R. Vogel, E. Ōsawa, and T. Plakhotnik, “Five-Nanometer Diamond with Luminescent Nitrogen-Vacancy Defect Centers,” Small. 5, 1649–1653 (2009). [CrossRef] [PubMed]

48.

L. Robledo, H. Bernien, I. van Weperen, and R. Hanson, “Control and coherence of the optical transition of single nitrogen vacancy centers in diamond,” Phys. Rev. Lett. 105, 177403 (2010). [CrossRef]

49.

P. E. Barclay, C. Santori, K. Fu, R. G. Beausoleil, and O. Painter, “Coherent interference effects in a nano-assembled diamond NV center cavity-QED system,” Opt. Express 17, 8081–8097 (2009). [CrossRef] [PubMed]

OCIS Codes
(020.1670) Atomic and molecular physics : Coherent optical effects
(270.5290) Quantum optics : Photon statistics
(270.5580) Quantum optics : Quantum electrodynamics
(140.3948) Lasers and laser optics : Microcavity devices
(270.5585) Quantum optics : Quantum information and processing

ToC Category:
Quantum Optics

History
Original Manuscript: September 5, 2012
Revised Manuscript: October 26, 2012
Manuscript Accepted: October 27, 2012
Published: November 16, 2012

Citation
Keyu Xia, Gavin K. Brennen, Demosthenes Ellinas, and Jason Twamley, "Deterministic generation of an on-demand Fock state," Opt. Express 20, 27198-27211 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-27198


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