Abstract

We present a theoretical analysis of pulse propagation in a waveguide with a side-coupled Kerr nonlinear cavity. As a specific application, we demonstrate how the nonlinearity has a profound influence on transport properties of a photon-added coherent state. We find that the fields provided by the coherent states facilitate a tunable nonlinearity on the few-photon level, allowing for a robust gating of single photons.

© 2012 Optical Society of America

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  1. J. L. O’Brien, A. Furusawa, and J. Vuckovic, Nat. Photon. 3, 687 (2009).
    [CrossRef]
  2. O. Benson, Nature 480, 193 (2011).
    [CrossRef]
  3. M. J. Hartmann, F. G. S. L. Brandgāo, and M. B. Plenio, Nat. Phys. 2, 849 (2006).
    [CrossRef]
  4. M. Notomi, E. Kuramochi, and T. Tanabe, Nat. Photon. 2, 741 (2008).
    [CrossRef]
  5. J. Ward and O. Benson, Laser Photon. Rev. 5, 553(2011).
    [CrossRef]
  6. D. Jacksch and P. Zoller, Ann. Phys. 315, 52 (2005).
    [CrossRef]
  7. Z. R. Gong, H. Ian, Y.-X. Liu, C. P. Sun, and F. Nori, Phys. Rev. A 80, 065801 (2009).
    [CrossRef]
  8. A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, Phys. Rev. Lett. 79, 1467 (1997).
    [CrossRef]
  9. M. B. Plenio and P. L. Knight, Rev. Mod. Phys. 70, 101 (1998).
    [CrossRef]
  10. H. J. Carmichael, An Open Systems Approach to Quantum Optics (Lecture Notes in Physics) (Springer, 1993).
  11. G. S. Agarwal and K. Tara, Phys. Rev. A 43, 492 (1991).
    [CrossRef]
  12. P. Longo, P. Schmitteckert, and K. Busch, J. Opt. A 11, 114009 (2009).
    [CrossRef]
  13. P. Longo, P. Schmitteckert, and K. Busch, Phys. Rev. Lett. 104, 023602 (2010).
    [CrossRef]
  14. P. Longo, P. Schmitteckert, and K. Busch, Phys. Rev. A 83, 063828 (2011).
    [CrossRef]
  15. P. W. Nolte, D. Pergande, S. L. Schweizer, M. Geuss, R. Salzer, B. T. Makowski, M. Steinhart, P. Mack, D. Hermann, K. Busch, C. Weder, and R. B. Wehrspohn, Adv. Mater. 22, 4731 (2010).
    [CrossRef]

2011 (3)

O. Benson, Nature 480, 193 (2011).
[CrossRef]

J. Ward and O. Benson, Laser Photon. Rev. 5, 553(2011).
[CrossRef]

P. Longo, P. Schmitteckert, and K. Busch, Phys. Rev. A 83, 063828 (2011).
[CrossRef]

2010 (2)

P. W. Nolte, D. Pergande, S. L. Schweizer, M. Geuss, R. Salzer, B. T. Makowski, M. Steinhart, P. Mack, D. Hermann, K. Busch, C. Weder, and R. B. Wehrspohn, Adv. Mater. 22, 4731 (2010).
[CrossRef]

P. Longo, P. Schmitteckert, and K. Busch, Phys. Rev. Lett. 104, 023602 (2010).
[CrossRef]

2009 (3)

P. Longo, P. Schmitteckert, and K. Busch, J. Opt. A 11, 114009 (2009).
[CrossRef]

Z. R. Gong, H. Ian, Y.-X. Liu, C. P. Sun, and F. Nori, Phys. Rev. A 80, 065801 (2009).
[CrossRef]

J. L. O’Brien, A. Furusawa, and J. Vuckovic, Nat. Photon. 3, 687 (2009).
[CrossRef]

2008 (1)

M. Notomi, E. Kuramochi, and T. Tanabe, Nat. Photon. 2, 741 (2008).
[CrossRef]

2006 (1)

M. J. Hartmann, F. G. S. L. Brandgāo, and M. B. Plenio, Nat. Phys. 2, 849 (2006).
[CrossRef]

2005 (1)

D. Jacksch and P. Zoller, Ann. Phys. 315, 52 (2005).
[CrossRef]

1998 (1)

M. B. Plenio and P. L. Knight, Rev. Mod. Phys. 70, 101 (1998).
[CrossRef]

1997 (1)

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, Phys. Rev. Lett. 79, 1467 (1997).
[CrossRef]

1991 (1)

G. S. Agarwal and K. Tara, Phys. Rev. A 43, 492 (1991).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal and K. Tara, Phys. Rev. A 43, 492 (1991).
[CrossRef]

Benson, O.

O. Benson, Nature 480, 193 (2011).
[CrossRef]

J. Ward and O. Benson, Laser Photon. Rev. 5, 553(2011).
[CrossRef]

Brandgao, F. G. S. L.

M. J. Hartmann, F. G. S. L. Brandgāo, and M. B. Plenio, Nat. Phys. 2, 849 (2006).
[CrossRef]

Busch, K.

P. Longo, P. Schmitteckert, and K. Busch, Phys. Rev. A 83, 063828 (2011).
[CrossRef]

P. Longo, P. Schmitteckert, and K. Busch, Phys. Rev. Lett. 104, 023602 (2010).
[CrossRef]

P. W. Nolte, D. Pergande, S. L. Schweizer, M. Geuss, R. Salzer, B. T. Makowski, M. Steinhart, P. Mack, D. Hermann, K. Busch, C. Weder, and R. B. Wehrspohn, Adv. Mater. 22, 4731 (2010).
[CrossRef]

P. Longo, P. Schmitteckert, and K. Busch, J. Opt. A 11, 114009 (2009).
[CrossRef]

Carmichael, H. J.

H. J. Carmichael, An Open Systems Approach to Quantum Optics (Lecture Notes in Physics) (Springer, 1993).

Deutsch, M.

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, Phys. Rev. Lett. 79, 1467 (1997).
[CrossRef]

Furusawa, A.

J. L. O’Brien, A. Furusawa, and J. Vuckovic, Nat. Photon. 3, 687 (2009).
[CrossRef]

Geuss, M.

P. W. Nolte, D. Pergande, S. L. Schweizer, M. Geuss, R. Salzer, B. T. Makowski, M. Steinhart, P. Mack, D. Hermann, K. Busch, C. Weder, and R. B. Wehrspohn, Adv. Mater. 22, 4731 (2010).
[CrossRef]

Gong, Z. R.

Z. R. Gong, H. Ian, Y.-X. Liu, C. P. Sun, and F. Nori, Phys. Rev. A 80, 065801 (2009).
[CrossRef]

Hartmann, M. J.

M. J. Hartmann, F. G. S. L. Brandgāo, and M. B. Plenio, Nat. Phys. 2, 849 (2006).
[CrossRef]

Hermann, D.

P. W. Nolte, D. Pergande, S. L. Schweizer, M. Geuss, R. Salzer, B. T. Makowski, M. Steinhart, P. Mack, D. Hermann, K. Busch, C. Weder, and R. B. Wehrspohn, Adv. Mater. 22, 4731 (2010).
[CrossRef]

Ian, H.

Z. R. Gong, H. Ian, Y.-X. Liu, C. P. Sun, and F. Nori, Phys. Rev. A 80, 065801 (2009).
[CrossRef]

Imamoglu, A.

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, Phys. Rev. Lett. 79, 1467 (1997).
[CrossRef]

Jacksch, D.

D. Jacksch and P. Zoller, Ann. Phys. 315, 52 (2005).
[CrossRef]

Knight, P. L.

M. B. Plenio and P. L. Knight, Rev. Mod. Phys. 70, 101 (1998).
[CrossRef]

Kuramochi, E.

M. Notomi, E. Kuramochi, and T. Tanabe, Nat. Photon. 2, 741 (2008).
[CrossRef]

Liu, Y.-X.

Z. R. Gong, H. Ian, Y.-X. Liu, C. P. Sun, and F. Nori, Phys. Rev. A 80, 065801 (2009).
[CrossRef]

Longo, P.

P. Longo, P. Schmitteckert, and K. Busch, Phys. Rev. A 83, 063828 (2011).
[CrossRef]

P. Longo, P. Schmitteckert, and K. Busch, Phys. Rev. Lett. 104, 023602 (2010).
[CrossRef]

P. Longo, P. Schmitteckert, and K. Busch, J. Opt. A 11, 114009 (2009).
[CrossRef]

Mack, P.

P. W. Nolte, D. Pergande, S. L. Schweizer, M. Geuss, R. Salzer, B. T. Makowski, M. Steinhart, P. Mack, D. Hermann, K. Busch, C. Weder, and R. B. Wehrspohn, Adv. Mater. 22, 4731 (2010).
[CrossRef]

Makowski, B. T.

P. W. Nolte, D. Pergande, S. L. Schweizer, M. Geuss, R. Salzer, B. T. Makowski, M. Steinhart, P. Mack, D. Hermann, K. Busch, C. Weder, and R. B. Wehrspohn, Adv. Mater. 22, 4731 (2010).
[CrossRef]

Nolte, P. W.

P. W. Nolte, D. Pergande, S. L. Schweizer, M. Geuss, R. Salzer, B. T. Makowski, M. Steinhart, P. Mack, D. Hermann, K. Busch, C. Weder, and R. B. Wehrspohn, Adv. Mater. 22, 4731 (2010).
[CrossRef]

Nori, F.

Z. R. Gong, H. Ian, Y.-X. Liu, C. P. Sun, and F. Nori, Phys. Rev. A 80, 065801 (2009).
[CrossRef]

Notomi, M.

M. Notomi, E. Kuramochi, and T. Tanabe, Nat. Photon. 2, 741 (2008).
[CrossRef]

O’Brien, J. L.

J. L. O’Brien, A. Furusawa, and J. Vuckovic, Nat. Photon. 3, 687 (2009).
[CrossRef]

Pergande, D.

P. W. Nolte, D. Pergande, S. L. Schweizer, M. Geuss, R. Salzer, B. T. Makowski, M. Steinhart, P. Mack, D. Hermann, K. Busch, C. Weder, and R. B. Wehrspohn, Adv. Mater. 22, 4731 (2010).
[CrossRef]

Plenio, M. B.

M. J. Hartmann, F. G. S. L. Brandgāo, and M. B. Plenio, Nat. Phys. 2, 849 (2006).
[CrossRef]

M. B. Plenio and P. L. Knight, Rev. Mod. Phys. 70, 101 (1998).
[CrossRef]

Salzer, R.

P. W. Nolte, D. Pergande, S. L. Schweizer, M. Geuss, R. Salzer, B. T. Makowski, M. Steinhart, P. Mack, D. Hermann, K. Busch, C. Weder, and R. B. Wehrspohn, Adv. Mater. 22, 4731 (2010).
[CrossRef]

Schmidt, H.

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, Phys. Rev. Lett. 79, 1467 (1997).
[CrossRef]

Schmitteckert, P.

P. Longo, P. Schmitteckert, and K. Busch, Phys. Rev. A 83, 063828 (2011).
[CrossRef]

P. Longo, P. Schmitteckert, and K. Busch, Phys. Rev. Lett. 104, 023602 (2010).
[CrossRef]

P. Longo, P. Schmitteckert, and K. Busch, J. Opt. A 11, 114009 (2009).
[CrossRef]

Schweizer, S. L.

P. W. Nolte, D. Pergande, S. L. Schweizer, M. Geuss, R. Salzer, B. T. Makowski, M. Steinhart, P. Mack, D. Hermann, K. Busch, C. Weder, and R. B. Wehrspohn, Adv. Mater. 22, 4731 (2010).
[CrossRef]

Steinhart, M.

P. W. Nolte, D. Pergande, S. L. Schweizer, M. Geuss, R. Salzer, B. T. Makowski, M. Steinhart, P. Mack, D. Hermann, K. Busch, C. Weder, and R. B. Wehrspohn, Adv. Mater. 22, 4731 (2010).
[CrossRef]

Sun, C. P.

Z. R. Gong, H. Ian, Y.-X. Liu, C. P. Sun, and F. Nori, Phys. Rev. A 80, 065801 (2009).
[CrossRef]

Tanabe, T.

M. Notomi, E. Kuramochi, and T. Tanabe, Nat. Photon. 2, 741 (2008).
[CrossRef]

Tara, K.

G. S. Agarwal and K. Tara, Phys. Rev. A 43, 492 (1991).
[CrossRef]

Vuckovic, J.

J. L. O’Brien, A. Furusawa, and J. Vuckovic, Nat. Photon. 3, 687 (2009).
[CrossRef]

Ward, J.

J. Ward and O. Benson, Laser Photon. Rev. 5, 553(2011).
[CrossRef]

Weder, C.

P. W. Nolte, D. Pergande, S. L. Schweizer, M. Geuss, R. Salzer, B. T. Makowski, M. Steinhart, P. Mack, D. Hermann, K. Busch, C. Weder, and R. B. Wehrspohn, Adv. Mater. 22, 4731 (2010).
[CrossRef]

Wehrspohn, R. B.

P. W. Nolte, D. Pergande, S. L. Schweizer, M. Geuss, R. Salzer, B. T. Makowski, M. Steinhart, P. Mack, D. Hermann, K. Busch, C. Weder, and R. B. Wehrspohn, Adv. Mater. 22, 4731 (2010).
[CrossRef]

Woods, G.

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, Phys. Rev. Lett. 79, 1467 (1997).
[CrossRef]

Zoller, P.

D. Jacksch and P. Zoller, Ann. Phys. 315, 52 (2005).
[CrossRef]

Adv. Mater. (1)

P. W. Nolte, D. Pergande, S. L. Schweizer, M. Geuss, R. Salzer, B. T. Makowski, M. Steinhart, P. Mack, D. Hermann, K. Busch, C. Weder, and R. B. Wehrspohn, Adv. Mater. 22, 4731 (2010).
[CrossRef]

Ann. Phys. (1)

D. Jacksch and P. Zoller, Ann. Phys. 315, 52 (2005).
[CrossRef]

J. Opt. A (1)

P. Longo, P. Schmitteckert, and K. Busch, J. Opt. A 11, 114009 (2009).
[CrossRef]

Laser Photon. Rev. (1)

J. Ward and O. Benson, Laser Photon. Rev. 5, 553(2011).
[CrossRef]

Nat. Photon. (2)

J. L. O’Brien, A. Furusawa, and J. Vuckovic, Nat. Photon. 3, 687 (2009).
[CrossRef]

M. Notomi, E. Kuramochi, and T. Tanabe, Nat. Photon. 2, 741 (2008).
[CrossRef]

Nat. Phys. (1)

M. J. Hartmann, F. G. S. L. Brandgāo, and M. B. Plenio, Nat. Phys. 2, 849 (2006).
[CrossRef]

Nature (1)

O. Benson, Nature 480, 193 (2011).
[CrossRef]

Phys. Rev. A (3)

Z. R. Gong, H. Ian, Y.-X. Liu, C. P. Sun, and F. Nori, Phys. Rev. A 80, 065801 (2009).
[CrossRef]

G. S. Agarwal and K. Tara, Phys. Rev. A 43, 492 (1991).
[CrossRef]

P. Longo, P. Schmitteckert, and K. Busch, Phys. Rev. A 83, 063828 (2011).
[CrossRef]

Phys. Rev. Lett. (2)

P. Longo, P. Schmitteckert, and K. Busch, Phys. Rev. Lett. 104, 023602 (2010).
[CrossRef]

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, Phys. Rev. Lett. 79, 1467 (1997).
[CrossRef]

Rev. Mod. Phys. (1)

M. B. Plenio and P. L. Knight, Rev. Mod. Phys. 70, 101 (1998).
[CrossRef]

Other (1)

H. J. Carmichael, An Open Systems Approach to Quantum Optics (Lecture Notes in Physics) (Springer, 1993).

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Figures (3)

Fig. 1.
Fig. 1.

Time evolution of the normalized intensity for the coherent state’s part of the total state (“mean field”) and the “added” single-photon wave function (“fluctuations”) for an initial field strength of the coherent state of α(0)=1. The system parameters are Δ=0.05J, J=J, and Γ=0 (lossless case) for a coupled-resonator waveguide with 299 cavities. The pulses’ centers and widths are xc=100a and s=7a, respectively. The corresponding wavenumbers are k0MF=0.6π/a and k0FL=0.5π/a. For further details on the numerical scheme, the parameters, and the hard-wall boundary conditions employed, we refer to [12].

Fig. 2.
Fig. 2.

Time evolution of the normalized intensities analogous to Fig. 1 except for an initial field strength of the coherent state of α(0)=7.

Fig. 3.
Fig. 3.

Fidelities as defined in Eq. (7) as a function of the decay rate Γ of the side-coupled cavity. As a reference, the field distributions after scattering are used for the two lossless cases shown in Figs. 1 and 2. Dashed curves correspond to fluctuations, while solid curves represent mean fields. Black curves represent the case α(0)=1 (cf. Fig. 1), and blue curves denote the case α(0)=7 (cf. Fig. 2). The dashed green line indicates the value of the nonlinearity Δ=0.05J.

Equations (7)

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H=Ji(ai+1ai+aiai+1)J(ax0c+ax0c)12Γcc+Δcccc.
itαi=J(αi+1+αi1)Jδix0γ,
itai=J(ai+1+ai1)Jδix0c,
itγ=Jαx0+2Δ|γ|2iΓ/2γ,
itc=Jαx0+4Δ|γ|2ciΓ/2c.
Heff=Ji(ai+1ai+aiai+1)J(ax0c+ax0c)+4Δ|γ(t)|2cciΓ/2cc.
F(ρ)=|xρx*ρx(ref)|NρNρ(ref)

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