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  • Editor: Alan E. Willner
  • Vol. 38, Iss. 18 — Sep. 15, 2013
  • pp: 3693–3695
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Correlated photons in one-dimensional waveguides

Matthias Moeferdt, Peter Schmitteckert, and Kurt Busch  »View Author Affiliations


Optics Letters, Vol. 38, Issue 18, pp. 3693-3695 (2013)
http://dx.doi.org/10.1364/OL.38.003693


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Abstract

We study two-photon transport in a one-dimensional waveguide with a side-coupled two-level system. Depending on the momentum of the incoming photons, we find that the nature of the scattering process changes considerably. We further show that bunching behavior can be found in the scattered light. As a result, we find that the waveguide dispersion has a strong influence on the photon correlations. By modifying the momentum of the pulse, the nature of the correlations can therefore be altered or optimized.

© 2013 Optical Society of America

In the endeavor of controlling quantum states in order to realize quantum networks, light–matter interaction has been scientists’ focus for many years [1

1. H. J. Kimble, Nature 453, 1023 (2008). [CrossRef]

].

Waveguiding structures—photonic crystals tailored to posses the desired optical properties—are coupled to quantum dots and form a waveguide QED setup. In this way, all-optical switching [2

2. P. Bermel, A. Rodriguez, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, Phys. Rev. A 74, 043818 (2006). [CrossRef]

] and single photon sources [3

3. T. Lund-Hansen, S. Stobbe, B. Julsgaard, H. Thyrrestrup, T. Sünner, M. Kamp, A. Forchel, and P. Lodahl, Phys. Rev. Lett. 101, 113903 (2008). [CrossRef]

] are realized, to name only a few applications. Furthermore, photon transport and scattering have been studied in these systems. In the few-photon regime, photon scattering can be controlled [4

4. L. Zhou, Z. R. Gong, Y.-X. Liu, C. P. Sun, and F. Nori, Phys. Rev. Lett. 101, 100501 (2008). [CrossRef]

] and interesting effects, such as interaction-induced radiation trapping and a photon–atom bound state can be observed [5

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

7

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

]. It has also been shown that strongly correlated photons are generated upon scattering off two-level systems [8

8. J. T. Shen and S. Fan, Phys. Rev. Lett. 98, 153003 (2007). [CrossRef]

] or off three- or four-level systems [9

9. H. Zheng, D. J. Gauthier, and H. U. Baranger, Phys. Rev. A 85, 043832 (2012). [CrossRef]

].

In this Letter, we investigate two-photon transport in a one-dimensional waveguide with a cosine-shaped dispersion relation. It is equipped with a two-level system (the atom), as is the case in [5

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

7

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

]. For our numerical calculations we employ the rotating wave approximation. We show that not only is a photon–atom bound state (PABS) formed, but the photon statistics is also influenced by the scattering process. We find that strongly correlated photons can be generated, as was shown for a setup with a linear dispersion relation (without a PABS) in [8

8. J. T. Shen and S. Fan, Phys. Rev. Lett. 98, 153003 (2007). [CrossRef]

], and that these correlations depend on the center frequency of the pulse relative to the band edges.

The Hamiltonian has the form of a multimode Jaynes–Cummings Hamiltonian. The cosine-shaped dispersion relation corresponds to nearest-neighbor hopping in real space. The lattice constant is set to a=1. The Hamiltonian then reads (=1)
H=Jx(ax+1ax+axax+1)+Ω2σz+V(σ+ax0+σax0),
(1)
where ax,ax denote the photon creation and annihilation operators at site x, respectively, and σz,σ± represent the Pauli operators that describe the excitation and de-excitation of the atom, respectively. Furthermore, Ω is the atomic transition energy and V is the coupling strength between the atom and the waveguide. The atom is side-coupled to one particular site x0 and the hopping element J describes the coupling between neighboring sites.

With the Hamiltonian at hand, we can numerically compute the time evolution of arbitrary initial states |ψ(t=t0) by solving the equation
|ψ(t)=eiH(tt0)|ψ(t=t0).
(2)
We compute the time evolution for two-photon states using a Krylov subspace method [10

10. Y. Saad, SIAM J. Numer. Anal. 29, 209 (1992). [CrossRef]

,11

11. R. B. Sidje, ACM Trans. Math. Softw. 24, 130 (1998). [CrossRef]

]. A two-photon state for a system with M sites can be written as
|ψ(t)=i=1Mj=1Mugij(t)axiaxj|0,g+i=1Muei(t)axi|0,e.
(3)
We start out with the initial pulse
|ψ(t0)=1Ni=1Mj=1Meikx1ieikx2j×e(x1ix¯)2/2σ2e(x2jx¯)2/2σ2ax1iax2j|0,g.
(4)
This is a Gaussian pulse containing two indistinguishable photons, both centered around x¯ and with spatial extent σ. Both photons travel toward x0 with the same central momentum k. The factor 1/N is for normalization.

As was described in [5

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

7

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

], trapping of radiation occurs because the PABS that lies outside of the cosine-shaped band is excited. The PABS is a polaritonic eigenstate of mixed photonic and atomic excitation. It can be shown [7

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

] that the Hamiltonian [Eq. (1)] exhibits a nonlinear term that makes it possible to excite the PABS in the case of two or more photons. In [6

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

], it was also determined that in order to efficiently excite the PABS, the resonance condition needs to be fulfilled:
Ω=2Jcos(k).
(5)
In Fig. 1, we display the occupation of the waveguide and the two-level system after the resonant scattering of the two photons for the cases k=(3π/4) and k=(π/2), with Eq. (5) fulfilled. The waveguide consists of M=199 lattice sites, and the atom is coupled to site x=x0=100. The width of the initial pulses is σ2=300, and the simulations are terminated before the pulses reach the ends of the waveguide. In both cases we find the characteristic exponentially decaying occupation of the waveguide sites near the site x0=100—the trapped radiation—and a finite occupation of the PABS in the long time limit.

Fig. 1. Final states after scattering for (a) k=(3π/4) at time t=70/J and (b) k=(π/2) at t=55/J. In the first case, the occupation of the atom after scattering is much higher, and less light is transmitted into the right lead of the waveguide.

We would now like to further investigate the nature of the photonic excitations after the scattering process. In order to do this, we introduce the following matrix representation of the coefficients ug and ue of the wave function:
(ugM1ugM2ugMMug21ug22ug11),(ue1ue2ueM).
(6)
Because of the bosonic commutation relations, we have ugij=ugji, and thus we require only one half of the matrix, plus the diagonal. This representation displays all the information contained in the wavefunction in a physically transparent way. Instead of giving us the occupation of only one site, as is the case in Fig. 1, we obtain information about the relative location of the two photons. The horizontal axis represents the location of the first photon and the vertical axis the location of the second one. The diagonal, therefore, represents those contributions to the wave function that correspond to both photons being at the same site. As we move away from the diagonal, the distance between the two photons increases. The coefficients uei denote the position of the photon that is paired up with an atomic excitation. In Fig. 2 we depict the matrix representation for the same cases (band edge case k=(3π/4) and linear dispersion case k=(π/2)) as in Fig. 1. It can be readily seen that the transport characteristics differ considerably, depending on the central momentum of the incoming pulse. For k=(3π/4), besides having a large reflection (lower left corner), we have two main contributions to the wave function; both involve one photon being trapped in the vicinity of the site x=100 and the other one traveling forward or backward in the waveguide. The contributions to the transmitted light of the two photons that are paired up are small, and virtually no contribution to the trapped radiation stems from a pair of photons being trapped at x=100.

Fig. 2. (a) and (b) Final state after scattering in the two-dimensional representation of the wave function and the position of the second photon while the first has been absorbed to excite the atom in the case k=(3π/4) at time t=70/J. (c) and (d) The same for the case k=(π/2) at t=55/J. In the first case one photon is trapped at the site x=100 while the other photon propagates, whereas in the second case both photons are trapped in the same position or propagate alongside one another [enlarged in the inset in (c)].

The case k=(π/2) is the exact opposite: Here, the trapped radiation comes from pairs of photons being located at x=100, which we see from the strong excitations in the center of the plot. The transmitted light now consists of photons propagating alongside one another in pairs (along the diagonal), which we would expect to correspond to photon bunching. The counter-diagonal represents photon pairs of one photon propagating to the right and the other one propagating to the left. It is interesting to note that the diagonal in the reflected regime is almost empty (lower left), and no photon pairs propagate in the negative direction.

With the above, we already have made statements on photon correlations. Traditionally, as a measure for photon correlations, the second-order coherence function is employed:
g(2)(xi,ti;xj,tj)=axi(ti)axj(tj)axj(tj)axi(ti)axi(ti)axi(ti)axj(tj)axj(tj).
(7)
Since we have full knowledge of the wave function, we can compute the normalized second-order correlation function between two sites at equal times, g(2)(xi,t;xj,t), via the photon number operators nxk,k=i,j:
g(2)(xi,t;xj,t)=nxinxjnxiδijnxinxj.
(8)
However, we have to keep in mind that for our nonstationary wave packets, the function g(2)(xi,t;xj,t) depends on the simulation time t.

As a first test, we computed g(2)(xi,t;xj,t) for freely propagating pulses with n=2 and n=3 photons (three-photon calculations were performed using a density matrix renormalization group (DMRG) algorithm [12

12. S. R. White, Phys. Rev. Lett. 69, 2863 (1992). [CrossRef]

14

14. S. T. Carr, D. A. Bagrets, and P. Schmitteckert, Phys. Rev. Lett. 107, 206801 (2011). [CrossRef]

]). We found the expected results:
g(2)(xi,t,xj,t)=11n,n=2,3,
(9)
for all times t after the pulse has arrived at the measurement sites xi,xj and independent of the distance xixj. This corresponds to the solution for a single-mode field [15

15. H. Römer, Theoretical Optics: An Introduction (Wiley-VCH, 2005).

], thus proving the validity of our approach. In Fig. 3, we depict the second-order correlation functions for the full system (waveguide plus two-level system) and for xi=150 and xj ranging from 100 to 199. For k=(π/2) we find bunching behavior, as expected from Fig. 2(c), as g(2)(xi,t,xj,t) decreases with increasing |xixj| [16

16. X. T. Zou and L. Mandel, Phys. Rev. A 41, 475 (1990). [CrossRef]

]. For distances |xixj|>10, g(2)(xi,t,xj,t) is essentially zero. At this point, we would like to clarify that the bunching of the photons does not imply the existence of a photon–photon bound state in the sense of a paired state with finite binding energy. The energies of the propagating wave packets equal the energy of independent single photon pulses.

Fig. 3. Second-order correlation function for (a) k=(3π/4) and (b) k=(π/2) at different times. In (b) we clearly observe photon bunching.

The case k=(3π/4) does not allow for such a straightforward interpretation. At first, for small distances |xixj|, g(2)(xi,t,xj,t) also decreases, but not in the same orderly fashion as for k=(π/2). We also notice that the correlations toward the right (the end of the waveguide) differ from the correlations toward the left (where the trapped radiation is located). To the right we find a steady decrease of g(2), whereas to the left, after a number of dents, g(2) increases again as we approach the site x=100. This behavior does not fit into either of the two categories, bunching or antibunching. While strong correlations between the site x=100 and the site x=150 were to be expected from Fig. 2(a), it is surprising that the bunching remains strong. Note that although the resonance condition is fulfilled, we still have a finite transmission window for pulses of finite width. This effect can be optimized by pulse engineering.

In conclusion, we have presented a method that allows us to both analyze and visualize photon correlations in one-dimensional waveguides. For the special case of a cosine-shaped dispersion relation, we find for frequencies in the center of the band (where the dispersion is essentially linear) a similar bunching behavior as in a waveguide with strictly linear dispersion without a PABS (cf. [8

8. J. T. Shen and S. Fan, Phys. Rev. Lett. 98, 153003 (2007). [CrossRef]

]). For frequencies near the band edge, the nonlinear scattering mechanisms that involve the PABS become relevant, and we obtain a considerable contribution of a single photon traveling forward with the other photon being trapped near the atom. We thus find that the dispersion itself has an influence on the photon correlations, and by adjusting the position of the pulses’ center frequencies relative to the band edge, we can alter the correlation properties of the photons.

Using the DMRG, the above method can be extended to handle three or more photons. For a representation equivalent to Figs. 2(a) and 2(c), the reduced density matrix can be used.

We thank P. Longo for fruitful discussions.

References

1.

H. J. Kimble, Nature 453, 1023 (2008). [CrossRef]

2.

P. Bermel, A. Rodriguez, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, Phys. Rev. A 74, 043818 (2006). [CrossRef]

3.

T. Lund-Hansen, S. Stobbe, B. Julsgaard, H. Thyrrestrup, T. Sünner, M. Kamp, A. Forchel, and P. Lodahl, Phys. Rev. Lett. 101, 113903 (2008). [CrossRef]

4.

L. Zhou, Z. R. Gong, Y.-X. Liu, C. P. Sun, and F. Nori, Phys. Rev. Lett. 101, 100501 (2008). [CrossRef]

5.

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

6.

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

7.

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

8.

J. T. Shen and S. Fan, Phys. Rev. Lett. 98, 153003 (2007). [CrossRef]

9.

H. Zheng, D. J. Gauthier, and H. U. Baranger, Phys. Rev. A 85, 043832 (2012). [CrossRef]

10.

Y. Saad, SIAM J. Numer. Anal. 29, 209 (1992). [CrossRef]

11.

R. B. Sidje, ACM Trans. Math. Softw. 24, 130 (1998). [CrossRef]

12.

S. R. White, Phys. Rev. Lett. 69, 2863 (1992). [CrossRef]

13.

P. Schmitteckert, Phys. Rev. B 70, 121302 (2004). [CrossRef]

14.

S. T. Carr, D. A. Bagrets, and P. Schmitteckert, Phys. Rev. Lett. 107, 206801 (2011). [CrossRef]

15.

H. Römer, Theoretical Optics: An Introduction (Wiley-VCH, 2005).

16.

X. T. Zou and L. Mandel, Phys. Rev. A 41, 475 (1990). [CrossRef]

OCIS Codes
(270.0270) Quantum optics : Quantum optics
(270.5290) Quantum optics : Photon statistics
(290.0290) Scattering : Scattering

ToC Category:
Quantum Optics

History
Original Manuscript: April 16, 2013
Revised Manuscript: July 26, 2013
Manuscript Accepted: August 28, 2013
Published: September 13, 2013

Citation
Matthias Moeferdt, Peter Schmitteckert, and Kurt Busch, "Correlated photons in one-dimensional waveguides," Opt. Lett. 38, 3693-3695 (2013)
http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-38-18-3693


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References

  1. H. J. Kimble, Nature 453, 1023 (2008). [CrossRef]
  2. P. Bermel, A. Rodriguez, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, Phys. Rev. A 74, 043818 (2006). [CrossRef]
  3. T. Lund-Hansen, S. Stobbe, B. Julsgaard, H. Thyrrestrup, T. Sünner, M. Kamp, A. Forchel, and P. Lodahl, Phys. Rev. Lett. 101, 113903 (2008). [CrossRef]
  4. L. Zhou, Z. R. Gong, Y.-X. Liu, C. P. Sun, and F. Nori, Phys. Rev. Lett. 101, 100501 (2008). [CrossRef]
  5. P. Longo, P. Schmitteckert, and K. Busch, J. Opt. A 11, 114009 (2009). [CrossRef]
  6. P. Longo, P. Schmitteckert, and K. Busch, Phys. Rev. Lett. 104, 023602 (2010). [CrossRef]
  7. P. Longo, P. Schmitteckert, and K. Busch, Phys. Rev. A 83, 063828 (2011). [CrossRef]
  8. J. T. Shen and S. Fan, Phys. Rev. Lett. 98, 153003 (2007). [CrossRef]
  9. H. Zheng, D. J. Gauthier, and H. U. Baranger, Phys. Rev. A 85, 043832 (2012). [CrossRef]
  10. Y. Saad, SIAM J. Numer. Anal. 29, 209 (1992). [CrossRef]
  11. R. B. Sidje, ACM Trans. Math. Softw. 24, 130 (1998). [CrossRef]
  12. S. R. White, Phys. Rev. Lett. 69, 2863 (1992). [CrossRef]
  13. P. Schmitteckert, Phys. Rev. B 70, 121302 (2004). [CrossRef]
  14. S. T. Carr, D. A. Bagrets, and P. Schmitteckert, Phys. Rev. Lett. 107, 206801 (2011). [CrossRef]
  15. H. Römer, Theoretical Optics: An Introduction (Wiley-VCH, 2005).
  16. X. T. Zou and L. Mandel, Phys. Rev. A 41, 475 (1990). [CrossRef]

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