## Deterministic generation of an on-demand Fock state |

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

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]

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]

*n*= 6.

*P*via the state collapse from a coherent or thermal state [3

_{n}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. 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]

*P*of success is equal to the initial overlap probability of the target Fock state with the initial state of light (

_{n}*P*(

*n*) = Tr[|

*n*〉 〈

*n*|

*ρ*]), and this probability can be quite low: for |

_{init}*n*= 3〉,

*P*

_{3}≈ 0.22 if starting from a pure coherent state

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. 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]

*g*, the emission probability

*P*(

_{emit}*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*),

*P*vanishes and a Fock state can be trapped in the cavity [9

_{emit}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. 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]

*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. 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]

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. 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. 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]

*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]

*Jaynes Cummings Damped Quantum Random Walk:-*A coined quantum random walk involves a coin, which we take as a qubit with Hilbert space ℋ

*=*

_{c}*span*{|

*e*〉, |

*g*〉}, together with a walk on the discretised non-negative real line ℋ

*=*

_{w}*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

*Û*≡ |

_{d}*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.

*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

*H*=

_{JC}*g*(|

*e*〉 〈

*g*| ⊗

*â*+ |

*g*〉 〈

*e*| ⊗

*â*

^{†}), then the resulting unitary evolution operator

*Û*(

_{JC}*τ*) =

*e*

^{−iHJCτ}, can be expressed as where

*N*=

*a*

^{†}

*a*is the photon number operator. Considering the initial product state for the density matrix of the coin and the walker to be

*ρ*⊗

_{C}*ρ*, then following evolution by the JC Hamiltonian we obtain

_{W}*=*

_{SE}*AdŜ*

_{0}+

*AdŜ*

_{1}is the spontaneous emission channel with non-unitary Kraus generators where

*η*(

*t*) =

*e*

^{−t/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

*Â*on some other operator

*X*̂ as follows:

*X*̂ →

*Ad*(

*Â*)

*X*̂ =

*ÂX*̂

*Â*

^{†}, noticing the property

*Ad*(

*ÂB*̂)

*X*̂ =

*Ad*(

*Â*)

*Ad*(

*B*̂)

*X*̂. For a general pure input state of the coin: the channel ℰ̂

*outputs In view of the limit lim*

_{SE}_{t→∞}

*η*(

*t*) = 0, and normalization relation |

*α*|

^{2}+|

*β*|

^{2}= 1, the last expression leads to the reset state

*X*̂ ≡ exp(−

*iπ*/2

*σ*) and denote the entire process by ℰ̂: where ℰ̂ acts in total coin-walker (atom-mode) density matrix. Choosing the action ℰ̂(|

_{x}*e*〉〈

*e*| ⊗

*ρ*), gives [13

_{W}13. D. Ellinas and I. Smyrnakis, “Asymptotics of a quantum random walk driven by an optical cavity,” J. Opt. B: Quant. Semiclass. Opt. **7**, S152–S157 (2005). [CrossRef]

14. A. J. Bracken, D. Ellinas, and I. Tsohantji, “Pseudo memory effects, majorization and entropy in quantum random walks,” J. Phys. A: Math. Gen. **37**, L91–L97 (2004). [CrossRef]

*m*> 1 steps the reduced state of the walker is Consider the case of number state input for the walker

*ρ*= |

_{W}*n*〉 〈

*n*|. Then and repeated action of ℰ̂ in the

*η*→ 0 limit leads to a progressive increase in Fock number

*n*. From the form of ℰ̂

*in this limit we now observe that we can halt this upwards motion to accumulate at the trapping value*

_{W}*n*=

*n*〉, if we choose the Jaynes Cummings coupling strength and duration

_{T}*τ*to satisfy the trapping condition:

*k*∈ .

*This is the main result of this section: by executing a sequence of operations: Jaynes Cummings for a period, followed by spontaneous decay and then a complete flip of the coin space, one can, with unit probability, arrange for the walker to reach a steady state at the position n*=

*n*. The behaviour of the quantum random walk on the half line with and without damping can be observed in Fig. 1. The dramatically different behaviour from the position independent step operator used in the conventional quantum walk is apparent. Next we show how to implement this map for an optical cavity-QED setup consisting of a Nitrogen-Vacancy centre in a nano diamond coupled to a high-Q optical cavity to produce multi-photon optical Fock states. We perform more detailed numerical simulations taking into account cavity and atomic decay to determine a figure of merit to synthesis Fock states of light.

_{T}*γ*. The higher the Fock state is, the larger this effective decay rate. For the target Fock state

_{c}*n*, the effective decay rate increases to

_{T}*n*. Another factor limiting whether one can achieve the target state relates to the downward transfer of population with probability

_{T}γ_{c}*P*from the target state |

_{D}*n*〉 to the lower Fock state |

_{T}*n*− 1〉 due to the net population

_{T}*P*of the ground state. In the stationary state, the pumping probability

_{g}*P*from the state |

_{U}*n*− 1〉 must balance the loss from the target Fock state |

_{T}*n*〉. A formula describing this balance takes the form where

_{T}*P*

_{nT}(

*P*

_{nT −1}) is the population in Fock state |

*n*〉 (|

_{T}*n*−1〉). Because

_{T}*η*can not be practically zero after waiting for a time

*t*, the net population in the excited state of qubit is

*η*=

*e*

^{−tγq}with the effective decay rate of qubit

*γ*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

_{q}*t*is measured as

*P*=

_{g}*e*

^{−}

*. We are interested in the case of high fidelity*

^{M}*F*of achieving the target state. We observe that the population is a good approximation of fidelity,

*P*

_{nT}∼

*F*. The population of the lower Fock state is

*P*

_{nT −1}=

*α*(1 −

*F*) with constant 0 ≤

*α*≤ 1. For our Hermitian system, we have

*n*takes the form

_{T}*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

*n*is the number of photons in the cavity,

*g*is the JC coupling strength, and

*τ*is the interaction time. Choosing

*τ*=

*τ*such that the

_{T}*n*photon is trapped in the cavity we have

_{T}*P*(

_{emit}*g*,

*τ*,

_{T}*n*) = 0. Using a fixed

_{T}*τ*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

_{T}*n*=

*n*. 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

_{T}*τ*, 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

_{T}*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

*P*into the cavity. (ii) After a time

_{g}*τ*, the JC coupling is turned off by bringing the NV’s optical transition out of resonance with the cavity via electrical Stark control [15

_{T}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]

*π*pulse. Repeating these operations, the field in the cavity can be trapped in a selected target Fock state.

*The System:-*Our setup for creation of photonic Fock state is shown in Fig. 2. An optical toroidal cavity with high quality factor

*Q*and resonance frequency

*ω*couples to the optical ZPL transition in an NV

_{c}^{−}center in a type IIa nanodiamond with

*C*

_{3}

*symmetry, which is positioned on or nearby the toroid so that it has a large evanescent overlap with the whispering gallery optical modes of the toroidal resonator. The nanodiamond is oriented so that the [111] axis of NV center is fabricated to be along the*

_{v}*z*direction (see Fig. 2). This setup can be realized using current technology [16

16. W. L. Yang, Z. Q. Yin, Z. Y. Xu, M. Feng, and J. F. Du, “One-step implementation of multiqubit conditional phase gating with nitrogen-vacancy centers coupled to a high-Q silica microsphere cavity,” Appl. Phys. Lett. **96**, 241113 (2010). [CrossRef]

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]

*b*̂ and counterclockwise (CCW) mode

*â*, which propagate around the cavity along two opposite directions and form a standing wave if both modes were excited [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]

*h*from the NV center and rough surface. Depending on it’s position the coupling of NV

^{−}centre to one of these two normal cavity modes

*Â*or

_{s}*Â*may occur predominantly or even exclusively with respect to the other mode [19

_{a}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]

^{−}’s position to be at the antinode of mode

*Â*(node of the antisymmetric mode

_{s}*Â*). Thus the NV center couples dominantly to the mode

_{a}*Â*. We neglect the small coupling to mode

_{s}*Â*. The mode

_{a}*A*is set to be red detuned to the ZPL transition of the NV center in the absence of any applied Stark shift [15

_{s}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]

*κ*via the input-output relations [20

_{ext}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. 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]

*â*,

_{in}*b*̂

*,*

_{in}*â*,

_{out}*b*̂

*}, respectively.*

_{out}*â*and

_{out}*b*̂

*are mixed by a 50 : 50 directional coupler [22*

_{out}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]

*â*〉 = 〈

_{in}*b*̂

*〉 = 0, and thus the outputs*

_{in}*â*and

_{out}*b*̂

*are proportional to*

_{out}*â*and

*b*̂, respectively. Thus the outputs of the directional coupler yields modes

*Â*and

_{s}*Â*[23

_{a}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]

*Â*. Assuming the intrinsic loss of the cavity is denoted by the decay rate

_{s}*γ*, if we take into account the scattering

_{c}*h*between two modes

*â*and

*b*̂, the critical coupling condition is given by

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]

*μ*m 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]

*(*

_{SE}*ρ*̂), 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 (

*λ*= 775 nm), to dynamically create a fast decay channel from the excited state to the ground state of the NV center [25

_{STED}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]

*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. 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. 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]

*π*−pulse. This is the final

*X*̂ portion of the map Eq. (6).

## 2. Detailed model of experimental protocol

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]

^{3}

**A**

_{2}and an orbital excited state

^{3}

**E**. Both the ground and excited states are

*S*= 1 spin triplets. The ground state has

^{3}

*A*

_{2}symmetry and is split into an

*S*,

_{x}*S*doublet 2.87GHz above an

_{y}*S*singlet due to the zero-field splitting [32

_{z}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]

^{3}

**E**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]

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]

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]

**E**

*,*

_{x}*S*〉 is taken to be negligible. Such intersystem crossing decay contributes to an effective decay to the singlet ground state

_{z}*S*.

_{z}^{3}

**E**to split into an orbital upper branch

**E**

*and an orbital lower branch*

_{x}**E**

*(see Fig. 3(b)). Each branch is a spin triplet formed by three spin states*

_{y}*S*,

_{x}*S*and

_{y}*S*. The sublevel |

_{z}**E**

*,*

_{x}*S*〉 is well isolated from the other five sublevels by several GHz. Actually the state |

_{z}**E**

*,*

_{y}*S*〉 can be isolated from |

_{z}**E**

*,*

_{x}*S*〉 because these two sublevels are associated to orthogonal transition dipoles [26

_{z}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. 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]

^{3}

**A**

_{2},

*S*〉 ↔|

_{z}**E**

*,*

_{x}*S*〉, can be excited resonantly at low strain [26

_{z}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]

**102**, 195506 (2009). [CrossRef] [PubMed]

**103**, 256404 (2009). [CrossRef]

**102**, 195506 (2009). [CrossRef] [PubMed]

^{−}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

**103**, 256404 (2009). [CrossRef]

**102**, 195506 (2009). [CrossRef] [PubMed]

*g*〉 ↔|

*e*〉, (|

^{3}

**A**

_{2},

*S*〉 ↔|

_{z}**E**

*,*

_{x}*S*〉) and excludes other sublevels [27

_{z}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]

*Ĥ*in the rotating wave approximation (RWA) where

*σ*̂

_{−}= |

*g*〉〈

*e*|. Δ

*=*

_{g}*ω*−

_{zpl}*ω*is the detuning of the mode

_{s}*Â*and the ZPL transition between states |

_{s}*g*〉 and |

*e*〉 (with frequency

*ω*), in the absence of any Stark shift.

_{zpl}*ω*=

_{s}*ω*+

_{c}*h*is the resonant frequency of mode

*Â*shifted by the scattering

_{s}*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

*Â*. This is reasonable because this coupling can be predominately to mode

_{s}*Â*by specialized positioning of the nanodiamond [19

_{s}**319**, 1062–1065 (2008). [CrossRef] [PubMed]

*A*and the ZPL of NV center. The Stark shift Δ

_{a}*(*

_{s}*t*) is used to dynamically control the creation of cavity photons by the NV center due to the JC coupling [15

**108**, 206401 (2012). [CrossRef] [PubMed]

*(*

_{s}*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. Δ

*(*

_{s}*t*) = −Δ

*. According to our numerical simulation, a fast relaxation from |*

_{g}*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]

*γ*becomes to

_{STED}*I*/

_{STED}γ*I*, with

_{s}*I*/

_{STED}*I*denoting the ratio of the STED pulse intensity

_{s}*I*and the saturation intensity

_{STED}*I*. For a lifetime 11.6 ns,

_{s}*I*is ∼ 1.85 MW cm

_{s}^{−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]

^{−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.

*g*〉 after a short 532 nm laser pulse optically prepares the defect into the

*m*= 0(|

_{s}^{3}

*A*

_{2},

*S*〉) state. A

_{z}*π*laser pulse Ω

*is used to resonantly pump it to the excited state |*

_{x}*e*〉. Then the ZPL is tuned on-resonance with the mode

*Â*to enable the JC coupling

_{s}*g*using a SEF. After time

*τ*, we turn off the JC coupling but use the STED laser beam to create a fast decay channel to the ground state |

_{T}*g*〉. Waiting for time

*τ*, almost all population decays to the ground state |

_{γ}*g*〉 from |

*e*〉. Then a further optical

*π*pulse generated by Ω

*is applied to resonantly excite the NV center to |*

_{x}*e*〉 again. We repeat these operations until the target state is trapped.

*+Δ*

_{g}*= 0, the NV center resonantly couples to the cavity mode*

_{s}*Â*. 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

_{s}*τ*. More precisely we take

_{T}*U*=

_{JC}*e*

^{−iτT(1+δτ)Hs/h̄}, where

*δτ*is a normally distributed additional noise in timing with a standard deviation given by a parameter

*σ*. The excitation probability of the cavity to state |

_{n}*n*+ 1〉 when the NV is in the excited state is thus now given by

*S*. The timing noise during the decay is not considered in

_{z}*τ*because this damping process is insensitive to the timing error. The population in the excited state |3〉 decays at an effective rate

_{γ}*γ*to the ground state |

_{STED}*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]

*σ*̂

*= |*

_{z}*e*〉 〈

*e*|−|

*g*〉 〈

*g*| and the density matrix

*ρ*̂. After a time

*S*is polarized more than 99% again. The flip

_{z}*π*laser pulse generated by Ω

*(*

_{x}*λ*≈ 637 nm) turns on successively to flip the NV center to the ES

_{x}*S*. We define a flip operator

_{z}*X*=

*e*

^{−iπ(1+δx)σx/2}with

*σ*=

_{x}*σ*

_{+}+

*σ*

_{−}to model this flip process as

*X*̂

*ρ*̂

*X*̂

^{†}.

*δ*is a noise having the same statistic property but independent of

_{x}*δ*. Then the density matrix after

_{τ}*l*+ 1 steps is determined by a recurrence relation The system is initialized in the state

## 3. Results

*Â*. This requisite can be satisfied by positioning the NV center at the antinode of

_{a}*Â*[19

_{s}**319**, 1062–1065 (2008). [CrossRef] [PubMed]

*g*〉↔|

*e*〉 such that |Δ

*| ≫ |*

_{g}*g*|. A detuning of Δ

*= 10*

_{g}*g*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. Δ

*+ Δ*

_{g}*= 0, by the Stark shift Δ*

_{s}*induced by the SEF [15*

_{s}**108**, 206401 (2012). [CrossRef] [PubMed]

*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*= 30

*γ*to suppress this detrimental process. By assuming a good optical cavity with

*γ*

_{c}*≪*

*γ*, in combination with the large JC coupling strength further improves the ultimate fidelity of the trapped photon state.

*n*〉 with unit fidelity, see the black solid line (i) in Fig. 4(a) for instance.

_{T}*n*= 6. Before the fifth step, only the Fock states with |

_{T}*n*< 6〉 are excited, because in each step the prepared photon number state can only excite the next one. As the operation continues, the target state |

*n*= 6〉 is essentially populated. It can be clearly seen from Fig. 4(a) that the fidelity

_{T}*F*= T

*r*[

*ρ*̂

*ρ*̂

*] with |*

_{T}*n*= 6〉 [36

_{T}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]

*P*(

*g*,

*τ*,

_{T}*n*= 6) is large when the population in |

_{T}*n*= 6〉 is small. As more population transfers to |

*n*= 6〉, the probability to excite this state decreases. Overall we have found the generation of Focks states is fairly robust with the number of repetitions of the process Eq. (6), providing the timing error is not too large (ii and iii). The cavity mode becomes stable after ∼ 73 steps.

*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).

*n*≤ 6, we perform the simulation including both kinds of imperfection: the timing error (

_{T}*σ*= 1%) and the cavity decay

_{n}*γ*= 0.1

_{c}*γ*. 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]

*γ*∼ 0.4

_{c}*γ*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]

*σ*= 1%.

_{n}*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

**6**, 97 (2004). [CrossRef]

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

*F*. Equation (14) provides a good estimation for the maximum of

*n*if

_{T}*F*is set. The constant

*α*(about 0.5) is numerically evaluated for

*σ*= 0,

_{n}*γ*/

_{STED}*γ*= 10

_{c}^{5}. Using this value in Eq. (14), the fidelity for a certain target state |

*n*〉 is shown in Fig. 5. Clearly, the estimation given by Eq. (14) is consistent with the numerical results.

_{T}*n*= 6〉, we need

*γ*> 10

_{STED}^{4}

*γ*. In the presence of noise, the decay rate

_{c}*γ*need be larger. To perform these simulations we used a cavity with

_{STED}*Q*∼ 3 × 10

^{8}[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. 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]

*γ*∼ 2

_{c}*π*× 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

*r*

^{6}, 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. 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]

*r*< 10 nm and nanodiamonds containing nitrogen vacancy centres in such small nano diamonds have been made [46, 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]

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]

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]

^{8}. 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 (∝

*r*

^{3}) 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. 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]

*r*< 10 nm. We use the typical value

*γ*/2

*π*= 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. 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]

^{−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*= 30

*γ*∼ 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]

*g*allows for shorter

*τ*and on these times scales the mixing between states

_{T}**E**

*and*

_{x}**E**

*is negligible. For this coupling strength, a Stark shift of Δ*

_{y}*= 10*

_{s}*g*∼ 2

*π*× 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

*μ*m and positioned 10

*μ*m 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

^{−}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

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**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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### References

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