## Remarks on the use of group theory in quantum optics

Optics Express, Vol. 8, Issue 2, pp. 76-85 (2001)

http://dx.doi.org/10.1364/OE.8.000076

Acrobat PDF (183 KB)

### Abstract

The relationships between certain important nonclassical states of the quantized field and the coherent states associated with the SU(2) and SU(1,1) Lie groups and associated Lie algebras is briefly reviewed. As an example of the utility of group theoretical methods in quantum optics, a method for generating maximally entangled photonic states is discussed. These states may be of great importance for achieving Heisenberg-limited interferometry and in beating the diffraction limit in lithography.

© Optical Society of America

## 1. Introduction

1. B. G. Wybourne, “The ‘Gruppen Pest’ yesterday, today, and tomorrow,” Intl. J. Quant. Chem. Symp. **7**, 35–43 (1973). [CrossRef]

^{th}birthday of Joseph Eberly, highlights the application of group theoretical methods to quantum optics. In fact, one of the earliest papers on the application of these methods to quantum optics is that of Wodkiewicz and Eberly [2

2. K. Wódkiewicz and J. H. Eberly, “Coherent states, squeezed fluctuations, and the SU(2) and SU(1,1) groups in quantum optics,” J. Opt. Soc. Am. B **2**, 458–466 (1985). [CrossRef]

## 2. Symmetry groups and dynamical groups

## 3. SU(2) and SU(1,1) in a nutshell

*z*

_{1}and

*z*

_{2}are complex numbers. In contrast, the group SU(1,1) consist of the set of all two-dimensional pseudo-unitary matrices (of determinant 1) preserving the quadratic form

*j,m*〉, satisfying the relations

*j*being 2

*j*+1.

*K*

_{0}is diagonal and has a discrete spectrum. The basis states of these representations we denote as |

*k,m*〉, where the number

*k*is known as the Bargmann index. These states satisfy the relations

*k*carries the fractional part of the spectrum of the operator

*K*

_{0}.

## 3. SU(1,1) in quantum optics

*realization*of the generators of the group in terms of the operators of the underlying physical system. As a first example, we consider a single mode field described by the usual annihilation (creation) operators

*a*(

*a*+) satisfying the bose algebra [

*a*,

*a*+]=1. A realization of the su(1,1) algebra in terms of these operators is

*C*

_{11}=-3/16 which in turn yields Bargmann indices

*k*=1/4,3/4. Note that these fall outside the list given above and thus the relevant representations are not, but can be considered as “continuations” of, the standard ones. The complete Hilbert space of the single mode field given in terms of the eigenstates of the number operator, the

*a*+

*a*Fock states |

*n*〉, becomes mapped onto two representations of SU(1,1) according to parity. The representation associated with the Bargmann index

*k*=1/4 consist of only the even numbered Fock states and that for

*k*=3/4 of only the odd numbered Fock states. This is easy to verify from Eqs. (8) and (9). To summarize, we have the correspondence

3. A. M. Perelomov, “Coherent states for an arbitrary Lie group,” Commun. Math. Phys. **26**, 222–236 (1972) [CrossRef]

*S*(

*z*)=exp(

*zK*

_{+}

*z**

*K*

_{-}) is an element of SU(1,1) often called the squeeze operator,

*z*=-(

*θ*/2)

*e*

^{-iϕ}, ξ=-tanh(

*θ*/2)

*e*

^{-iϕ}, and where

*θ*is a hyperbolic angle (0≤

*θ*<∞) and

*ϕ*is an azimuthal angl (0≤

*ϕ*≤2

*π*). Expanded in terms the SU(1,1) states we have

*k*=1 4 the squeeze operator acts on the vacuum state and the resulting coherent state is just the familiar squeezed vacuum state,

*θ*/2 is sometimes written as

*r*and is known as the squeeze parameter. For the case

*k*=3/4, the corresponding SU(1,1) Perelomov coherent state is just the squeezed one-photon state.

*z*|=2|

*λ*|

*t*and

*ϕ*=2arg(

*λ*).

4. A. O. Barut and L. Girardello, “New ‘coherent’ states associated with non-compact groups,” Commun. Math. Phys. **21**, 41–55 (1971). [CrossRef]

*η*is an arbitrary complex number, in obvious analogy to the annihilation operator eigenstate definition of the ordinary coherent states. (But unlike those states, SU(1,1) coherent states are inequivalent under different definitions.) The solution to Eq. (16) is

*I*

_{2k-1}being a modified Bessel function. However, the solutions may economically be written as superpositions of the ordinary coherent states |

*α*〉 and |-

*α*〉 as

*η*=

*α*

^{2}/2 and where

5. C. C. Gerry and E. E. Hach, “Generation of even and odd coherent states in a competitive two-photon process,” Phys. Lett. A **117**, 185–189 (1993). [CrossRef]

*κ*is proportional to the cross section for two photon absorption. The steady-state long-time solutions, for which

*∂ρ*/

*∂t*=0, must satisfy (

*K*

_{-}-

*λ*/

*κ*)

*ρ*=0. The particular steady-state solution depends highly on the initial state. Because parity is conserved by the interactions, if the initial state of the field is a pure state containing only even number states, typically this might be just the vacuum, the steady state solution is

*ρ*(∞)=|

*η*,1/4〉〈

*η*,1/4| and if only odd number are in the initial state we shall have

*ρ*(∞)=

*η*,3/4〉〈

*η*,3/4 where in both cases

*η*=

*λ*/

*κ*. Many other schemes have been discussed for producing these states and we refer the reader to relevant reviews [6].

*a*and

*b*, is given by

*q*=0 we just have the familiar two-mode squeezed vacuum state

7. G. S. Agarwal, “Nonclassical statistics of fields in pair coherent states,” J. Opt. Soc. Am. B **5**, 1940–1947 (1988). [CrossRef]

*q*,0〉 results in the pure pair coherent state of Eq. (28).

## 4. SU(2) in quantum optics for two-mode fields

*J*

_{0}commutes with all the others and where

*C*

_{2}=

*J*

_{0}(

*J*

_{0}+1). The two-mode number states map onto angular momentum states according to the rule

*J*

_{0}|

*j,m*〉=

*j*|

*j,m*〉. It is well known that certain passive optical devices such as beam splitters can well be described as “rotations” typically represented by the unitary operator of the form [8

8. R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A. **40**, 1371–1348 (1989). [CrossRef] [PubMed]

*J*

_{1}=(

*J*

_{+}+

*J*

_{-})/2 and

*J*

_{2}=(

*J*

_{+}-

*J*

_{-})/

*2i*. Also, parametric frequency converters can be represented by Hamiltonians of the form

*N*〉=|

*j*=

*N*/2,

*m*=-

*N*/2〉 then the output state is just the SU(2) coherent state defined as [9

9. J. M. Radcliffe, “Some properties of spin coherent states,” J. Phys. A **4**, 313–323 (1971). [CrossRef]

*N*is the total photon number,

*ζ*=tan(|

*β*|/2)exp(-

*iψ*), and

*ψ*=arg(

*β*). The

*N*photons are binomially distributed over the two modes.

## 5. Nonlinear four-wave mixer and generation of maximally entangled photonic states

10. B. Yurke and D. Stoler, “Quantum behavior of a four-way mixer operated in nonlinear regime,” Phys. Rev. A **35**, 4846–4849 (1987). [CrossRef] [PubMed]

*′*=1/

*n*̄

_{sv}where

*n*̄

_{sv}=sinh

^{2}(

*θ*/2) is the average photon number in the squeezed vacuum state. A similar numerical relationship holds for the even coherent state but such a state is much harder to generate.

## 6. Conclusion

## Acknowledgements

## References and links

1. | B. G. Wybourne, “The ‘Gruppen Pest’ yesterday, today, and tomorrow,” Intl. J. Quant. Chem. Symp. |

2. | K. Wódkiewicz and J. H. Eberly, “Coherent states, squeezed fluctuations, and the SU(2) and SU(1,1) groups in quantum optics,” J. Opt. Soc. Am. B |

3. | A. M. Perelomov, “Coherent states for an arbitrary Lie group,” Commun. Math. Phys. |

4. | A. O. Barut and L. Girardello, “New ‘coherent’ states associated with non-compact groups,” Commun. Math. Phys. |

5. | C. C. Gerry and E. E. Hach, “Generation of even and odd coherent states in a competitive two-photon process,” Phys. Lett. A |

6. | V. Buzek and P. L. Knight, “Quantum interference, superposition states of light, and nonclassical effects,” in |

7. | G. S. Agarwal, “Nonclassical statistics of fields in pair coherent states,” J. Opt. Soc. Am. B |

8. | R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A. |

9. | J. M. Radcliffe, “Some properties of spin coherent states,” J. Phys. A |

10. | B. Yurke and D. Stoler, “Quantum behavior of a four-way mixer operated in nonlinear regime,” Phys. Rev. A |

11. | C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A |

12. | K. Mølmer and A. Sørensen, “Multiparticle entanglement of hot trapped ions,” Phys. Rev. Lett. |

13. | A. Boto et al., “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(270.6570) Quantum optics : Squeezed states

**ToC Category:**

Focus Issue: Quantum control of photons and matter

**History**

Original Manuscript: November 7, 2000

Published: January 15, 2001

**Citation**

Christopher Gerry, "Remarks on the use of group theory in quantum optics," Opt. Express **8**, 76-85 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-2-76

Sort: Journal | Reset

### References

- B. G. Wybourne, "The 'Gruppen Pest' yesterday, today, and tomorrow," Intl. J. Quant. Chem. Symp. 7, 35-43 (1973). [CrossRef]
- K. W�dkiewicz and J. H. Eberly, "Coherent states, squeezed fluctuations, and the SU(2) and SU(1,1) groups in quantum optics," J. Opt. Soc. Am. B 2, 458-466 (1985). [CrossRef]
- A. M. Perelomov, "Coherent states for an arbitrary Lie group," Commun. Math. Phys. 26, 222-236 (1972) [CrossRef]
- A. O. Barut and L. Girardello, "New 'coherent' states associated with non-compact groups," Commun. Math. Phys. 21, 41-55 (1971). [CrossRef]
- C. C. Gerry and E. E. Hach, "Generation of even and odd coherent states in a competitive two-photon process," Phys. Lett. A 117, 185-189 (1993). [CrossRef]
- V. Buzek and P. L. Knight, "Quantum interference, superposition states of light, and nonclassical effects," in Progress in Optics XXXIV, E. Wolf, ed. (Elesevier, Amsterdam, 1995).
- G. S. Agarwal, "Nonclassical statistics of fields in pair coherent states," J. Opt. Soc. Am. B 5, 1940-1947 (1988). [CrossRef]
- R. A. Campos, B. E. A. Saleh, and M. C. Teich, "Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics," Phys. Rev. A. 40, 1371-1348 (1989). [CrossRef] [PubMed]
- J. M. Radcliffe, "Some properties of spin coherent states," J. Phys. A 4, 313-323 (1971). [CrossRef]
- B. Yurke and D. Stoler, "Quantum behavior of a four-way mixer operated in nonlinear regime," Phys. Rev. A 35, 4846-4849 (1987). [CrossRef] [PubMed]
- C. C. Gerry, "Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime," Phys.Rev.A 61, 043811-1-043811-7 (2000). [CrossRef]
- K. M�lmer and A. S�rensen, "Multiparticle entanglement of hot trapped ions," Phys. Rev. Lett. 82, 1835-1838 (1999). [CrossRef]
- A. Boto et al., "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 85, 2733-2736 (2000). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.