## Amplitude damping of Laguerre-Gaussian modes |

Optics Express, Vol. 18, Issue 22, pp. 22789-22795 (2010)

http://dx.doi.org/10.1364/OE.18.022789

Acrobat PDF (946 KB)

### Abstract

We present an amplitude damping channel for Laguerre-Gaussian modes. Our channel is tested experimentally for a Laguerre-Gaussian mode, having an azimuthal index *l* = 1, illustrating that it decays to a Gaussian mode in good agreement with the theoretical model for amplitude damping. Since we are able to characterize the action of such a channel on orbital angular momentum states, we propose using it to investigate the dynamics of entanglement.

© 2010 OSA

## 1. Introduction

*p*and

*p*, respectively, illustrating that the amplitude of the excited state has been ‘damped’. This quantum operation is a fundamental tool in classifying the behaviour of many quantum systems that incur the loss of energy, from describing the evolution of an atom that spontaneously emits a photon to how the state of a photon evolves in an optical system due to scattering and attenuation. The amplitude damping channel is an elementary quantum operation which is used to model quantum noise in understanding the dynamics of open quantum systems [1].

*M*

_{0}and

*M*

_{1}are the Krauss operators defined asandThe initial state of the two-level system is written aswhere |0〉 and |1〉 represent the ground state and excited state of the system, respectively and

*α*and

*β*denote complex amplitudes with |α|

^{2}+ |β|

^{2}= 1.

*M*

_{0}and

*M*

_{1}, and coupling the system appropriately to an environment with two orthogonal states |

*K*= 0〉

*and |*

_{E}*K*= 1〉

*, the unitary time evolution operator,*

_{E}*U*, of the system,

_{SE}*S*, and the environment,

*E*, produces the following transformations where

*K*represents an environment observable, such as a path along which a photon propagates.

*l*= 0〉 and |

*l*= 1〉 are state vectors representing a single photon in the LG

_{0}mode (ground state) and LG

_{1}mode (excited state), respectively, and the upper indices

*A*and

*B*refer to the two states of the environment, the ground and excited states, respectively. Later it will be illustrated that the two environments are represented as two optical paths. Discarding the environment leaves the system in a statistical mixture of being excited with probability

*p*or de-excited with probability 1-

*p*, which is the characteristic trait of amplitude damping.

*) laser modes carry OAM of*

_{lp}*lħ*per photon [2

2. M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and the transfer of orbital angular momentum,” Opt. Commun. **96**(1-3), 123–132 (1993). [CrossRef]

3. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A **45**(11), 8185–8189 (1992). [CrossRef] [PubMed]

*ilφ*), of these modes, we neglect the radial index

*p*for the rest of the paper. The generation of LG

*beams (a variety of which are depicted in Fig. 1(a) - 1(d)) has advanced from cylindrical lens mode converters [2*

_{l}2. M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and the transfer of orbital angular momentum,” Opt. Commun. **96**(1-3), 123–132 (1993). [CrossRef]

4. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wave-front laser-beams produced with a spiral phaseplate,” Opt. Commun. **112**(5-6), 321–327 (1994). [CrossRef]

6. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. **207**(1-6), 169–175 (2002). [CrossRef]

7. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. **75**(5), 826–829 (1995). [CrossRef] [PubMed]

8. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature **412**(6844), 313–316 (2001). [CrossRef] [PubMed]

*et al*to sort and infer OAM states [8

8. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature **412**(6844), 313–316 (2001). [CrossRef] [PubMed]

*beams of odd and even orders of*

_{l}*l*could be sorted in a Mach-Zehnder interferometer incorporating Dove prisms in each arm [9

9. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. **88**(25), 257901 (2002). [CrossRef] [PubMed]

*beams.*

_{l}10. T. Konrad, F. de Melo, M. Tiersch, C. Kasztelan, A. Aragão, and A. Buchleitner, “Evolution equation for quantum entanglement,” Nat. Phys. **4**(4), 99–102 (2008). [CrossRef]

12. C. E. R. Souza and A. Z. Khoury, “A Michelson controlled-not gate with a single-lens astigmatic mode converter,” Opt. Express **18**(9), 9207–9212 (2010). [CrossRef] [PubMed]

## 2. Concept of the channel

*modes, which is depicted in Fig. 1(e), is based on a Mach-Zehnder interferometer with a Dove prism in each arm. Such an optical system has already been shown to be useful for the sorting of modes [9*

_{l}9. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. **88**(25), 257901 (2002). [CrossRef] [PubMed]

13. R. Zambrini and S. M. Barnett, “Quasi-intrinsic angular momentum and the measurement of its spectrum,” Phys. Rev. Lett. **96**(11), 113901 (2006). [CrossRef] [PubMed]

*A*or

*B*, where

*A*and

*B*represent the two states of the environment, the ground and excited states, respectively. By placing a Dove prism, which flips the transverse cross section of a transmitted beam [14], in each arm of the interferometer, a relative phase shift between the two arms is introduced. More specifically, this relative phase shift, Δ

*ϕ*, is proportional to both the helicity,

*l,*of the incoming LG

*beam and the relative angle,*

_{l}*θ*, between the two Dove prisms: Δ

*ϕ*= 2

*lθ*. In varying the relative angle,

*θ*, between the two Dove prisms, the LG

*beam will either exit solely in path*

_{l}*A*or path

*B*or in a weighted superposition of both paths

*A*and

*B*. One is able to control how much of the initial LG

*mode will exit into the two environments,*

_{l}*A*(the ground state of the environment) and

*B*(the excited state of the environment). A phase mask, which decreases the azimuthal mode index by 1, is placed in path

*B*resulting in the ‘excited’ LG

*mode decaying to an ‘unexcited’ LG*

_{l}*mode, thus performing the action of amplitude damping.*

_{l-1}*mode through arms*

_{l}*A*1 and

*A*2 of the interferometer to output path

*A*, the additional phase picked up by the beam from each of the components is given byandrespectively.

*d*1 and

*d*2 are the path lengths of the two arms,

*A*1 and

*A*2, respectively and

*t*is the optical path length of the beam-splitter. The phase difference in output path

*A*is consequently given bysince the interferometer is constructed such that the path lengths of the two arms are equal.

*B*iswhere the additional phase shift of

*π*is due to the reflection in the first beam-splitter. The amplitude of the field for a LG

*mode emerging after the second beam-splitter in path*

_{l}*A*and path

*B*is described asandrespectively.

*U*denotes the amplitude of the field and the negative sign in Eq. (13) for the field emerging in path

*B*from arm

*A*2 is due to an additional phase shift of

*π*, evident in Eq. (11).

_{1}mode through the interferometer there is a non-vanishing relative phase difference (Δ

*ϕ*= 2

*θ*) between the two arms of the interferometer, leading to partially constructive and destructive interference of the LG

_{1}mode in both paths

*A*and

*B*

_{1}mode exiting in the two paths,

*A*and

*B*, is proportional to cos

^{2}

*θ*and sin

^{2}

*θ*, respectively and consequently the probabilities of the LG

_{1}mode existing in the two paths follow the same trend. The incoming LG

_{1}mode exits in a superposition of paths

*A*and

*B*with probabilities cos

^{2}

*θ*and sin

^{2}

*θ*, respectively and a phase mask, in path

*B*, decreases the azimuthal mode index,

*l*, by 1, converting the ‘exited’ LG

_{1}mode to an ‘unexcited’ Gaussian mode

_{0}mode enters the device there is no azimuthal phase dependence in such a mode, the field is unaffected by the Dove prisms and no phase difference occurs between the two arms; the result is that the Gaussian mode exits in path

*A*. The transformation of the Gaussian mode through the interferometer is denoted asCombining Eqs. (15) and (16), the general equation for the amplitude damping of OAM, given in Eq. (7), is obtained, where

*p*, it is sufficient to experimentally investigate each case individually in order to verify the action of our amplitude damping channel.

## 3. Experimental methodology and results

*π*phase shift at λ = 632.8 nm). The mode of the field was prepared before it entered the channel by programming an appropriate phase pattern. The two exiting paths of the interferometer were monitored for various angles,

*θ*, between the two Dove prisms using a CCD camera (Spiricon LW130).

9. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. **88**(25), 257901 (2002). [CrossRef] [PubMed]

*θ*=

*π*/2, the interferometer sorted the incoming field into even LG

*modes (path*

_{l}*A*) and odd LG

*modes (path*

_{l}*B*). We can confirm this result for incoming LG

*modes having indices*

_{l}*l*= 0 to 4, as shown in Fig. 2(a) . In the special case that

*l*= 0 (Gaussian mode) the interferometer only produces an output in path

*A*(Fig. 2(b)). This experiment is instructive when considering the errors in the system due to imperfect alignment and environmental fluctuations. The variance in the data set of Fig. 2(b) for path

*A*(where we should expect 100% transmission) and for path

*B*(where we should expect no transmission) is given as 0.88 ± 0.06 and 0.15 ± 0.08, respectively. In performing the measurements depicted in Fig. 3 , for each occurrence for which the Dove prism was rotated, constructive and destructive interference was first achieved in paths

*A*and

*B*, respectively, for the case of an incoming Gaussian beam. This ensured that the interferometer was correctly aligned before the LG

_{1}mode entered the channel.

_{1}and the angle between the two Dove prisms varied from 0° through to 360°. The results are shown graphically in Fig. 3. In Fig. 3(a) it is evident that when the relative angle between the two Dove prisms is an even multiple of 90°, maximum transmission of the LG

_{1}mode occurs, which is in agreement with the ‘LG

*sorting’ nature of this device. At an angle of an even multiple of 90°, the reverse occurs in path*

_{l}*B*(shown in Fig. 3(b)), where minimum transmission of the LG

_{1}mode occurs. When the relative angle between the two Dove prisms is an odd multiple of 90°, minimum transmission of the LG

_{1}mode occurs in path

*A*and maximum transmission in path

*B*. For angles varying between 0° and 90° (and multiples of these angles) the LG

_{1}mode exits in both paths

*A*and

*B*. As the angle increases from 0° to 90° the transmission of the LG

_{1}mode in path

*A*decreases, but consequently increases in path

*B*. In our theoretical model, the probability that the LG

_{1}mode exits in paths

*A*and

*B*is given by cos

^{2}

*θ*and sin

^{2}

*θ*, respectively (depicted in Eq. (17)) and it is evident from Figs. 3(a) and 3(b) that our measured data are in very good agreement with the predicted model. The errors given in the measurements in Fig. 3 are the standard deviations of the measurements in Fig. 2(b), for paths

*A*(0.06) and

*B*(0.08).

## 4. Conclusion

_{0}and LG

_{1}laser modes. We have shown excellent agreement between theory and experiment, and believe this is the first time such a concept has been outlined in the literature. With this idea, one is able to mimic the well known quantum operation where an excitation (or in this case OAM) is lost to the environment, a key testing bed for the interaction of entangled states with an environment. The advantage of this particular concept – amplitude damping – is that the entanglement decay can be predicted analytically, while the concurrence of the entanglement may easily be measured both before and after the channel with standard state tomography techniques, thus allowing for a quantitative comparison of theory and experiment in the decoherence of entanglement due to interactions with a noisy environment.

## References and links

1. | M. A. Nielsen and I. L. Chaung, |

2. | M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and the transfer of orbital angular momentum,” Opt. Commun. |

3. | L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A |

4. | M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wave-front laser-beams produced with a spiral phaseplate,” Opt. Commun. |

5. | V. Yu Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. |

6. | J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. |

7. | H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. |

8. | A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature |

9. | J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. |

10. | T. Konrad, F. de Melo, M. Tiersch, C. Kasztelan, A. Aragão, and A. Buchleitner, “Evolution equation for quantum entanglement,” Nat. Phys. |

11. | B. Pors, C. H. Monken, E. R. Eliel, and J. P. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” Quant-ph, 1–5 (2009). |

12. | C. E. R. Souza and A. Z. Khoury, “A Michelson controlled-not gate with a single-lens astigmatic mode converter,” Opt. Express |

13. | R. Zambrini and S. M. Barnett, “Quasi-intrinsic angular momentum and the measurement of its spectrum,” Phys. Rev. Lett. |

14. | M. Born and E. Wolf, |

**OCIS Codes**

(050.0050) Diffraction and gratings : Diffraction and gratings

(270.0270) Quantum optics : Quantum optics

(050.4865) Diffraction and gratings : Optical vortices

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: August 10, 2010

Revised Manuscript: October 4, 2010

Manuscript Accepted: October 5, 2010

Published: October 13, 2010

**Citation**

Angela Dudley, Michael Nock, Thomas Konrad, Filippus S. Roux, and Andrew Forbes, "Amplitude damping of Laguerre-Gaussian modes," Opt. Express **18**, 22789-22795 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-22-22789

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### References

- M. A. Nielsen and I. L. Chaung, Quantum computation and quantum information, Cambridge University Press, 380, Cambridge (2000).
- M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and the transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993). [CrossRef]
- L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]
- M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wave-front laser-beams produced with a spiral phaseplate,” Opt. Commun. 112(5-6), 321–327 (1994). [CrossRef]
- V. Yu Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431 (1990).
- J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1-6), 169–175 (2002). [CrossRef]
- H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995). [CrossRef] [PubMed]
- A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). [CrossRef] [PubMed]
- J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002). [CrossRef] [PubMed]
- T. Konrad, F. de Melo, M. Tiersch, C. Kasztelan, A. Aragão, and A. Buchleitner, “Evolution equation for quantum entanglement,” Nat. Phys. 4(4), 99–102 (2008). [CrossRef]
- B. Pors, C. H. Monken, E. R. Eliel, and J. P. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” Quant-ph, 1–5 (2009).
- C. E. R. Souza and A. Z. Khoury, “A Michelson controlled-not gate with a single-lens astigmatic mode converter,” Opt. Express 18(9), 9207–9212 (2010). [CrossRef] [PubMed]
- R. Zambrini and S. M. Barnett, “Quasi-intrinsic angular momentum and the measurement of its spectrum,” Phys. Rev. Lett. 96(11), 113901 (2006). [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1980), 6th ed.

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