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

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    [CrossRef] [PubMed]

2012 (3)

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]

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. Photonics6, 369–373 (2012).
[CrossRef]

V. N. Mochalin, O. Shenderova, D. Ho, and Y. Gogotsi, “The properties and applications of nanodiamonds,” Nat. Photonics7, 11–23 (2012).

2011 (6)

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. Photonics5, 738–743 (2011).
[CrossRef]

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]

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]

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,” Nature477, 574–578 (2011).
[CrossRef] [PubMed]

J. R. Maze, A. Gali, E. Togan, Y. Chu, A. Trifonov, E. Kaxiras, and M. D. Lukin, “Properties of nitrogen-vacancy centers in diamond: the group theoretic approach,” New J. Phys.13, 025025 (2011).
[CrossRef]

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,” Nature478, 221–224 (2011).
[CrossRef] [PubMed]

2010 (6)

V. M. Acosta, A. Jarmola, E. Bauch, and D. Budker, “Optical properties of the nitrogen-vacancy singlet levels in diamond,” Phys. Rev. B82, 201202(R) (2010).
[CrossRef]

Y. Ma, M. Rohlfing, and A. Gali, “Excited states of the negatively charged nitrogen-vacancy color center in diamond,” Phys. Rev. B81, 041204(R) (2010).
[CrossRef]

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]

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]

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. Photonics4, 46–49 (2010).
[CrossRef]

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]

2009 (6)

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. Express17, 8081–8097 (2009).
[CrossRef] [PubMed]

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]

M. Larsson, K. N. Dinyari, and H. Wang, “Composite optical microcavity of diamond nanopillar and silica microsphere,” Nano Lett.9, 1447–1450 (2009).
[CrossRef] [PubMed]

E. Rittweger, K. Y. Han, S. E. Irvine, C. Eggeling, and S. W. Hell, “STED microscopy reveals crystal colour centres with nanometric resolution,” Nat. Photonics3, 144–147 (2009).
[CrossRef]

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]

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]

2008 (3)

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]

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.” Science319, 1062–1065 (2008).
[CrossRef] [PubMed]

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,” Nature454, 310–314 (2008).
[CrossRef] [PubMed]

2007 (4)

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,” Nature448, 889–893 (2007).
[CrossRef] [PubMed]

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]

I. S. Grudinin, A. B. Matsko, and L. Maleki, “On the fundamental limits of Q factor of crystalline dielectric resonators,” Opt. Express15, 3390–3395 (2007).
[CrossRef] [PubMed]

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]

2006 (3)

I. S. Grudinin, V. S. Ilchenko, and L. Maleki, “Ultrahigh optical q factors of crystalline resonators in the linear regime,” Phys. Rev. A74, 063806 (2006).
[CrossRef]

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]

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]

2005 (1)

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]

2004 (4)

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]

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]

Y. Liu, L. F. Wei, and F. Nori, “Generation of non-classical photon states using a superconducting qubit in a quantum electrodynamic microcavity,” Europhys. Lett.67, 941–947 (2004).
[CrossRef]

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]

2003 (2)

K. J. Vahala, “Optical microcavities,” Nature424, 839–846 (2003).
[CrossRef] [PubMed]

K. R. Brown, K. M. Dani, D. M. Stamper-Kurn, and K. B. Whaley, “Deterministic optical Fock-state generation,” Phys. Rev. A67, 043818 (2003).
[CrossRef]

2001 (2)

M. F. Santos, E. Solano, and R. L. de Matos Filho, “Conditional large Fock state preparation and field state reconstruction in cavity QED,” Phys. Rev. Lett.87, 093601 (2001).
[CrossRef] [PubMed]

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]

1999 (1)

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

1998 (1)

1996 (2)

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

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

1990 (1)

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]

1987 (1)

G. J. Milburn, “Kicked quantized cavity mode - an open-systems- theory approach,” Phys. Rev. A36, 744–749 (1987).
[CrossRef] [PubMed]

1986 (1)

1985 (1)

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A31, 3761–3774 (1985).
[CrossRef] [PubMed]

1984 (1)

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A30, 1386–1391 (1984).
[CrossRef]

Acosta, V. M.

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]

V. M. Acosta, A. Jarmola, E. Bauch, and D. Budker, “Optical properties of the nitrogen-vacancy singlet levels in diamond,” Phys. Rev. B82, 201202(R) (2010).
[CrossRef]

Alkemade, P. F. A.

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,” Nature477, 574–578 (2011).
[CrossRef] [PubMed]

Ansmann, M.

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,” Nature454, 310–314 (2008).
[CrossRef] [PubMed]

Aoki, T.

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.” Science319, 1062–1065 (2008).
[CrossRef] [PubMed]

Awschalom, D. D.

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]

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Figures (5)

Fig. 1
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.

Fig. 2
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.

Fig. 3
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.

Fig. 4
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.

Fig. 5
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.

Equations (19)

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U ^ J C ( τ ) = ( cos g τ N + 1 i sin ( g τ N + 1 ) N + 1 a i a sin ( g τ N + 1 ) N + 1 cos g τ N ) ,
ρ C ρ W U ^ J C ( ρ C ρ W ) U ^ J C ^ S E i d ^ W [ U ^ J C ( ρ C ρ W ) U ^ J C ] ,
S ^ 0 = | g g | | e e | η , S ^ 1 = | g e | 1 η ,
ρ C = ( α | g + β | e ) ( α * g | + β * e | ) = ( | β | 2 α * β α β * | α | 2 ) ,
^ S E [ ρ C ] ( t ) = ( η | β | 2 α * β η α β * η 1 η | β | 2 ) .
^ A d X ^ ( ^ S E i d ^ W ) A d U ^ J C ,
^ ( | e e | ρ W ) = ( ^ W ( ρ W ) cos ( g τ N + 1 ) ρ W cos ( g τ N + 1 ) η i a sin ( g τ N + 1 ) N + 1 ρ W cos ( g τ N + 1 ) η i cos ( g τ N + 1 ) ρ W sin ( g τ N + 1 ) N + 1 a η cos ( g τ N + 1 ) ρ W cos ( g τ N + 1 η ) ) ,
^ W ( ρ W ) = cos ( g τ N + 1 ) ρ W cos ( g τ N + 1 ) + a sin ( g τ N + 1 ) N + 1 ρ W sin ( g τ N + 1 ) N + 1 a ,
Tr ^ W ( ρ W ) = Tr { [ cos ( g τ N + 1 ) 2 + sin ( g τ N + 1 ) 2 ] ρ W } + Tr ρ W = 1.
ρ W ( m ) = Tr c [ ^ m ( | e e | ρ W ) ] = ^ W m ( ρ W ) + 𝒪 ( η 3 2 ) .
^ W ( | n n | ) = cos ( g τ n + 1 ) 2 | n n | + sin ( g τ n + 1 ) 2 | n + 1 n + 1 | ,
P n T ( 1 e n T γ c t ) + P g P D = P n T 1 P U ,
F ( 1 e N T γ c M γ q 1 ) + e M sin 2 ( π n T n T + 1 ) = α ( 1 F ) sin 2 ( π n T n T + 1 ) .
F = π 2 ( α e M ) π 2 α + 4 M n T 3 γ c / γ q .
a ^ out = a ^ in + 2 κ ext a ^ ,
b ^ out = b ^ in + 2 κ ext b ^ ,
H ^ = H ^ s + H ^ a + H ^ x , H ^ s = h ¯ ( Δ g + Δ s ( t ) ) A ^ s A ^ s + h ¯ g [ A ^ s σ + H . c . ] , H ^ x = h ¯ Ω x ( t ) [ σ ^ + H . c . ] ,
[ ρ ^ ] = e ( L ^ ^ i H ^ ^ s / h ¯ ) τ γ ρ ^
ρ ^ l + 1 = X ^ S E [ U ^ J C ρ ^ l U ^ J C ] X ^ .

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