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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 5 — Feb. 28, 2011
  • pp: 4405–4410
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Direct generation of a multi-transverse mode non-classical state of light

Benoît Chalopin, Francesco Scazza, Claude Fabre, and Nicolas Treps  »View Author Affiliations


Optics Express, Vol. 19, Issue 5, pp. 4405-4410 (2011)
http://dx.doi.org/10.1364/OE.19.004405


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Abstract

Quantum computation and communication protocols require quantum resources which are in the continuous variable regime squeezed and/or quadrature entangled optical modes. To perform more and more complex and robust protocols, one needs sources that can produce in a controlled way highly multimode quantum states of light. One possibility is to mix different single mode quantum resources. Another is to directly use a multimode device, either in the spatial or in the frequency domain. We present here the first experimental demonstration of a device capable of producing simultanuously several squeezed transverse modes of the same frequency and which is potentially scalable. We show that this device, which is an Optical Parametric Oscillator using a self-imaging cavity, produces a multimode quantum resource made of three squeezed transverse modes.

© 2011 Optical Society of America

1. Introduction

As the complexity of quantum communication and computation protocols increases, the need for quantum systems of high dimensionality increases accordingly. As far as light is concerned, different modes can be used as different quantum channels or variables in such a protocol. The different modes can be either frequency modes [3

3. A. S. Coelho, F. A. S. Barbosa, K. N. Cassemiro, A. S. Villar, M. Martinelli, and P. Nussenzveig, “Three-color entanglement,” Science 326, 823–826 (2009). [CrossRef] [PubMed]

5

5. A. Eckstein and C. Silberhorn, “Broadband frequency mode entanglement in waveguided parametric downconversion,” Opt. Lett. 33, 1825–1827 (2008). [CrossRef] [PubMed]

] and/or spatial modes [6

6. O. Jedrkiewicz, Y-K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. A. Lugiato, and P. Di Trapani, “Detection of sub-shot-noise spatial correlation in high-gain parametric down conversion,” Phys. Rev. Lett. 93, 243601 (2004). [CrossRef]

8

8. J. Janousek, K. Wagner, J. F. Morizur, N. Treps, P. K. Lam, C. C. Harb, and H. A. Bachor, “Optical entanglement of co-propagating modes,” Nat. Photonics 3, 399–402 (2009). [CrossRef]

]. In particular, in the Continuous Variable regime, transport and processing of quantum information requires the use of quantum multimode quadrature-entangled light beams, which can be produced by the linear mixing of squeezed states in different optical modes [9

9. M. D. Reid, “Demonstration of the Einstein–Podolsky–Rosen paradox using nondegenerate parametric amplification,” Phys. Rev. A 40, 913–923 (1989). [CrossRef] [PubMed]

]. So far, the complexity of multimode squeezing and entanglement experiments has increased to up to four modes [10

10. M. Yukawa, R. Ukai, P. van Loock, and A. Furusawa, “Experimental generation of four-mode continuous-variable cluster states,” Phys. Rev. A 78, 012301 (2008). [CrossRef]

] with application to cluster-state based quantum computation. The quantum properties of a light beam consisting of several copropagating transverse modes of a monochromatic laser beam has recently raised a strong interest, both theoretically [2

2. L. Lopez, B. Chalopin, A. Rivière de la Souchère, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: squeezed vacuum and Einstein–Podolsky–Rosen beams generation,” Phys. Rev. A 80, 043816 (2009). [CrossRef]

, 11

11. M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 711539–1589 (1999). [CrossRef]

] and experimentally [8

8. J. Janousek, K. Wagner, J. F. Morizur, N. Treps, P. K. Lam, C. C. Harb, and H. A. Bachor, “Optical entanglement of co-propagating modes,” Nat. Photonics 3, 399–402 (2009). [CrossRef]

, 12

12. M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous variable entanglement and squeezing of orbital angular momentum states,” Phys. Rev. Lett. 102, 163602 (2009). [CrossRef] [PubMed]

, 13

13. B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Multimode nonclassical light generation through the optical-parametric-oscillator threshold,” Phys. Rev. A 81, 061804(R) (2010). [CrossRef]

], since transverse modes are easy to handle. We present in this paper a source for multimode squeezing of three transverse modes, which, according to the theoretical model developed in [2

2. L. Lopez, B. Chalopin, A. Rivière de la Souchère, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: squeezed vacuum and Einstein–Podolsky–Rosen beams generation,” Phys. Rev. A 80, 043816 (2009). [CrossRef]

], could be scalable to a much larger number of modes.

2. The self-imaging Optical Parametric Oscillator

Optical parametric amplifiers (OPA) and oscillators (OPO) are amongst the most reliable and efficient sources for generating non-classical beams [14

14. L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986). [CrossRef] [PubMed]

, 15

15. T. Eberle, S. Steinlechner, J. Bauchrowitz, V. Händchen, H. Vahlbruch, M. Mehmet, H. Müller-Ebhardt, and R. Schnabel, “Quantum enhancement of the zero-area Sagnac interferometer topology for gravitational wave detection,” Phys. Rev. Lett. 104, 251102 (2010). [CrossRef] [PubMed]

]. They use parametric down-conversion of an intense pump laser beam in a optical cavity. The photons created in pairs are strongly correlated, both temporally and spatially, but the single-mode cavity usually loses all the transverse correlations. The experiment we present here relies on the use of a multimode cavity, the self-imaging cavity. Introduced by Arnaud [16

16. J. A. Arnaud, “Degenerate optical cavities,” Appl. Opt. 8, 189–195 (1969). [CrossRef] [PubMed]

], this cavity allows the simultaneous resonance of several transverse modes of a monochromatic laser beam. This enables the cavity build-up of images instead of just Gaussian beams. It has been used for example to efficiently frequency double images of various shapes [17

17. B. Chalopin, A. Chiummo, C. Fabre, A. Maitre, and N. Treps, “Frequency doubling of low power images using a self-imaging cavity,” Opt. Express 188033–8042 (2010). [CrossRef] [PubMed]

]. The cavity, depicted in Fig. 1 consists of a plane mirror, a converging lens of focal length f and a spherical mirror of radius of curvature R. The cavity is self-imaging when the distances L1 and L2 between the optical elements verify the relations: L1 = L1,deg = f + f2/R and L2 = L2,deg = f + R, in which case the cavity is completely degenerate and allows the resonance of many transverse modes of a monochromatic laser beam, with a limit set by the transverse size and the aberration of the optical elements. If these conditions are not exactly fulfilled, one can still have the resonance of several transverse modes, as described in [17

17. B. Chalopin, A. Chiummo, C. Fabre, A. Maitre, and N. Treps, “Frequency doubling of low power images using a self-imaging cavity,” Opt. Express 188033–8042 (2010). [CrossRef] [PubMed]

]. Placing a χ(2) nonlinear crystal inside this cavity, a self-imaging OPO is built, as proposed in [2

2. L. Lopez, B. Chalopin, A. Rivière de la Souchère, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: squeezed vacuum and Einstein–Podolsky–Rosen beams generation,” Phys. Rev. A 80, 043816 (2009). [CrossRef]

].

Fig. 1: Sketch of the self-imaging OPO. The cavity contains a plane mirror, a lens and a spherical mirror, and reaches complete transverse mode degeneracy for precise values of their relative distances. A non-linear crystal is placed inside the cavity. The parametric downconversion process of pump photons generate degenerate signal and idler photons in different spatial modes. The output of the OPO is a set of squeezed transverse modes.

Since the cavity is completely degenerate, one can choose any mode basis to describe the transverse modes resonating inside the cavity [17

17. B. Chalopin, A. Chiummo, C. Fabre, A. Maitre, and N. Treps, “Frequency doubling of low power images using a self-imaging cavity,” Opt. Express 188033–8042 (2010). [CrossRef] [PubMed]

]. In [2

2. L. Lopez, B. Chalopin, A. Rivière de la Souchère, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: squeezed vacuum and Einstein–Podolsky–Rosen beams generation,” Phys. Rev. A 80, 043816 (2009). [CrossRef]

], it was showed that the best basis to describe the non-classical states produced by a self-imaging OPO is the eigenbasis of the parametric down-conversion process of the corresponding pump profile, which can be found through the diagonalization of the coupling matrix. This matrix is given by the product between the phase-matching coefficient, the transverse profile of the pump and the longitudinal overlap between modes through diffraction. In many practical cases, this eigenbasis is very close to a set of Hermite-Gauss polynomials. Assuming equal loss rate γ for all modes, the evolution of the mode bosonic operator Ŝk associated to each eigenmode is given by:
dS^kdt=γS^k+ΛkS^k+2γS^k,in
(1)
where Λk is proportionnal to the eigenvalue associated to each eigenmode, and Ŝk,in the input operator for each mode. The output state of the self-imaging OPO is a then a set of independent squeezed modes [13

13. B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Multimode nonclassical light generation through the optical-parametric-oscillator threshold,” Phys. Rev. A 81, 061804(R) (2010). [CrossRef]

], each associated to a pump amplitude proportionnal to Λk. Hence, eigenvalues Λk determine the maximum amount of squeezing of each eigenmode k:
Vk,min=|Λ0||Λk||Λ0|+|Λk|
(2)
where Λ0 is the eigenvalue of largest modulus. The spectrum of the coupling matrix exhibits therefore the multimode non-classical capabilities of the self-imaging OPO: in particular, one sees that a highly multimode nonclassical beam is generated when many |Λk| are nonzero and close to |Λ0|.

The analysis of the coupling matrix in our present configuration shows that, to generate several non-classical modes, the waist of the pump beam of the OPO must be larger than the crystal coherence length lcoh=λlc/πn, as defined in [2

2. L. Lopez, B. Chalopin, A. Rivière de la Souchère, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: squeezed vacuum and Einstein–Podolsky–Rosen beams generation,” Phys. Rev. A 80, 043816 (2009). [CrossRef]

], where λ is the wavelength of the signal field, lc the crystal length, and ns its refractive index at λ. For a 10-mm long PPKTP crystal, we find lcoh ≈ 40 μm. With a pump waist larger than this value, one gets multimode squeezing at the output of the device. Moreover, if the pump profile is gaussian, the eigenmodes are very close to being a set of modes with orthogonal Hermite-Gauss profiles. The expected number of modes can be calculated using the cooperativity parameter [18

18. C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84, 5304–5307 (2000). [CrossRef] [PubMed]

], which depends quadratically on the waist of the pump. Note that the OPO threshold is also a quadratic function of the pump waist, so that the number of generated squeezed modes is only limited by the available pump power. To get more simultanuous squeezed modes, one would only need to engineer a coupling matrix with a flat spectrum. This can be done by modifying the crystal parameters, or the pump transverse profile. The squeezed modes are independent, and therefore increasing the number of modes will not affect the other modes.

3. Experimental setup and results

Fig. 2: Sketch of the experimental setup.

The quantum fluctuations of the different modes contained in the output state of the OPO are measured with a homodyne detection whose local oscillator can be switched from one mode to another through a cavity mode-converter [19

19. M. Lassen, V. Delaubert, C. Harb, P. K. Lam, N. Treps, and H. A. Bachor, “Generation of squeezing in higher order Hermite–Gaussian modes with an optical parametric amplifier,” J. Eur. Opt. Soc. Rapid Publ. 1, 06003 (2006). [CrossRef]

]. Unlike other experiments [19

19. M. Lassen, V. Delaubert, C. Harb, P. K. Lam, N. Treps, and H. A. Bachor, “Generation of squeezing in higher order Hermite–Gaussian modes with an optical parametric amplifier,” J. Eur. Opt. Soc. Rapid Publ. 1, 06003 (2006). [CrossRef]

], the OPO cavity remains locked on the same resonance while the mode converter is switched from one mode to another. The alignment procedure is the following: we first mode-match the local oscillator to the transmitted seed, when there is no pump beam in the cavity. The local oscillator is filtered by a ring-cavity mode cleaner, as shown on Fig.2. Changing the resonance of this cavity from one transverse mode to another enables us to change the transverse mode of the local oscillator, thus changing the output mode analyzed by the homodyne detection, but without changing the alignment. This feature ensures that the different modes we analyzed with the homodyne detection are orthogonal.

The results are presented on Fig.3. We observe that at least three copropagating transverse modes have fluctuations below shot noise, with 1.2 dB of squeezing for the TEM00, 0.6 dB for the TEM10 and 0.3 dB for the TEM01. These fluctuations were measured at a frequency of 3 MHz. The higher order modes were analyzed, but the amount of squeezing was too low to clearly distinguish it from the shot noise.

Fig. 3: a) Output of the self-imaging OPO, three transverse modes of the same wavelength are squeezed. b) Fluctuations of three orthogonal output modes of the OPO measured with a homodyne detection. The phase of the local oscillator is swept, so that the fluctuations are measured on all different quadratures. The blue line indicates the shot noise level. The TEM00 (red curve) shows 1.2 dB of squeezing below shot noise, the TEM10 (cyan curve) 0.6 dB and the TEM01 (green curve) 0.3 dB.

The amount of squeezing we measure is in agreement with theoretical expectations taking into account all different parameters of our experiment, which cannot all be optimized for maximum squeezing. For example the lens inside the cavity added significant losses, which decreased greatly the escape efficiency of the system and therefore the maximum amount of squeezing that can be reached in our set-up. We compared our multimode OPO with an equivalent single-mode OPO by simply slightly decreasing the cavity length. We found then that the pure TEM00, single mode output of the OPO had the same squeezing as the most squeezed mode of the multimode OPO, which is in agreement with the theoretical prediction [2

2. L. Lopez, B. Chalopin, A. Rivière de la Souchère, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: squeezed vacuum and Einstein–Podolsky–Rosen beams generation,” Phys. Rev. A 80, 043816 (2009). [CrossRef]

].

The squeezing of the two other modes can also be compared to the theory. If we set the losses of the system at a value where the maximum squeezing is 1.2 dB, given the size of the pump beam and the length of the crystal, the squeezing on both these modes should be 0.9 dB. The difference with our measurement can be explained by the fact that the cavity is not completely degenerate, and that a slight cavity detuning is present on both of these modes. Finally, the difference between the TEM10 and the TEM01 can be explained by a non-homogenous transverse phase shift in the non-linear crystal due to the periodic poling itself, which is done by applying a large voltage between two facets of the crystal. This obviously introduces a limitation to the multimode performances of our system. However, several technical aspects can be improved. For example, with higher available pump power, one can decrease the cavity finesse and therefore increasing the escape efficiency and the cavity degeneracy, without increasing the thermal lens effects, because the pump beam is taken with a larger waist. One may also use another type of crystal, non-periodically poled, which would enable more degeneracy between the two transverse axes.

4. Conclusion

To conclude, we have presented an experiment that showed for the first time the direct generation of a three-mode squeezed state, on three well-defined orthogonal transverse modes. The results can also be interpreted as a verification of the theory developed in [2

2. L. Lopez, B. Chalopin, A. Rivière de la Souchère, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: squeezed vacuum and Einstein–Podolsky–Rosen beams generation,” Phys. Rev. A 80, 043816 (2009). [CrossRef]

], which stated that the self-imaging OPO can produce a set of squeezed copropagating transverse modes. Moreover, these modes are Hermite-Gauss modes, and their number and size can be engineered through the crystal and pump parameters [2

2. L. Lopez, B. Chalopin, A. Rivière de la Souchère, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: squeezed vacuum and Einstein–Podolsky–Rosen beams generation,” Phys. Rev. A 80, 043816 (2009). [CrossRef]

]. Increasing the number of squeezed modes will naturally increase the necessary pump power, but will not cause the squeezing to be lower, as the squeezing of a mode only depends on its distance to maximum eigenvalue. This makes this self-imaging OPO a potentially scalable source for multimode squeezing and entanglement.

Acknowledgments

We acknowledge the financial support of the Future and Emerging Technologies (FET) programme within the seventh Framework Programme for Research of the European Commission, under the FET-Open grant agreement HIDEAS FP7-ICT-221906.

References and links

1.

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005.) [CrossRef]

2.

L. Lopez, B. Chalopin, A. Rivière de la Souchère, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: squeezed vacuum and Einstein–Podolsky–Rosen beams generation,” Phys. Rev. A 80, 043816 (2009). [CrossRef]

3.

A. S. Coelho, F. A. S. Barbosa, K. N. Cassemiro, A. S. Villar, M. Martinelli, and P. Nussenzveig, “Three-color entanglement,” Science 326, 823–826 (2009). [CrossRef] [PubMed]

4.

N. C. Menicucci, S. T. Flammia, and O. Pfister, “One-way quantum computing in the optical frequency comb,” Phys. Rev. Lett. 101, 130501 (2008). [CrossRef] [PubMed]

5.

A. Eckstein and C. Silberhorn, “Broadband frequency mode entanglement in waveguided parametric downconversion,” Opt. Lett. 33, 1825–1827 (2008). [CrossRef] [PubMed]

6.

O. Jedrkiewicz, Y-K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. A. Lugiato, and P. Di Trapani, “Detection of sub-shot-noise spatial correlation in high-gain parametric down conversion,” Phys. Rev. Lett. 93, 243601 (2004). [CrossRef]

7.

J-L. Blanchet, F. Devaux, L. Furfaro, and E. Lantz, “Measurement of sub-shot-noise correlations of spatial fluctuations in the photon-counting regime,” Phys. Rev. Lett. 101, 233604 (2008). [CrossRef] [PubMed]

8.

J. Janousek, K. Wagner, J. F. Morizur, N. Treps, P. K. Lam, C. C. Harb, and H. A. Bachor, “Optical entanglement of co-propagating modes,” Nat. Photonics 3, 399–402 (2009). [CrossRef]

9.

M. D. Reid, “Demonstration of the Einstein–Podolsky–Rosen paradox using nondegenerate parametric amplification,” Phys. Rev. A 40, 913–923 (1989). [CrossRef] [PubMed]

10.

M. Yukawa, R. Ukai, P. van Loock, and A. Furusawa, “Experimental generation of four-mode continuous-variable cluster states,” Phys. Rev. A 78, 012301 (2008). [CrossRef]

11.

M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 711539–1589 (1999). [CrossRef]

12.

M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous variable entanglement and squeezing of orbital angular momentum states,” Phys. Rev. Lett. 102, 163602 (2009). [CrossRef] [PubMed]

13.

B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Multimode nonclassical light generation through the optical-parametric-oscillator threshold,” Phys. Rev. A 81, 061804(R) (2010). [CrossRef]

14.

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986). [CrossRef] [PubMed]

15.

T. Eberle, S. Steinlechner, J. Bauchrowitz, V. Händchen, H. Vahlbruch, M. Mehmet, H. Müller-Ebhardt, and R. Schnabel, “Quantum enhancement of the zero-area Sagnac interferometer topology for gravitational wave detection,” Phys. Rev. Lett. 104, 251102 (2010). [CrossRef] [PubMed]

16.

J. A. Arnaud, “Degenerate optical cavities,” Appl. Opt. 8, 189–195 (1969). [CrossRef] [PubMed]

17.

B. Chalopin, A. Chiummo, C. Fabre, A. Maitre, and N. Treps, “Frequency doubling of low power images using a self-imaging cavity,” Opt. Express 188033–8042 (2010). [CrossRef] [PubMed]

18.

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84, 5304–5307 (2000). [CrossRef] [PubMed]

19.

M. Lassen, V. Delaubert, C. Harb, P. K. Lam, N. Treps, and H. A. Bachor, “Generation of squeezing in higher order Hermite–Gaussian modes with an optical parametric amplifier,” J. Eur. Opt. Soc. Rapid Publ. 1, 06003 (2006). [CrossRef]

OCIS Codes
(270.6570) Quantum optics : Squeezed states
(270.5585) Quantum optics : Quantum information and processing

ToC Category:
Quantum Optics

History
Original Manuscript: January 19, 2011
Revised Manuscript: February 14, 2011
Manuscript Accepted: February 14, 2011
Published: February 22, 2011

Virtual Issues
April 14, 2011 Spotlight on Optics

Citation
Benoît Chalopin, Francesco Scazza, Claude Fabre, and Nicolas Treps, "Direct generation of a multi-transverse mode non-classical state of light," Opt. Express 19, 4405-4410 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-5-4405


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References

  1. S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005.) [CrossRef]
  2. L. Lopez, B. Chalopin, A. Rivière de la Souchère, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: squeezed vacuum and Einstein–Podolsky–Rosen beams generation,” Phys. Rev. A 80, 043816 (2009). [CrossRef]
  3. A. S. Coelho, F. A. S. Barbosa, K. N. Cassemiro, A. S. Villar, M. Martinelli, and P. Nussenzveig, “Three-color entanglement,” Science 326, 823–826 (2009). [CrossRef] [PubMed]
  4. N. C. Menicucci, S. T. Flammia, and O. Pfister, “One-way quantum computing in the optical frequency comb,” Phys. Rev. Lett. 101, 130501 (2008). [CrossRef] [PubMed]
  5. A. Eckstein, and C. Silberhorn, “Broadband frequency mode entanglement in waveguided parametric downconversion,” Opt. Lett. 33, 1825–1827 (2008). [CrossRef] [PubMed]
  6. O. Jedrkiewicz, Y.-K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. A. Lugiato, and P. Di Trapani, “Detection of sub-shot-noise spatial correlation in high-gain parametric down conversion,” Phys. Rev. Lett. 93, 243601 (2004). [CrossRef]
  7. J.-L. Blanchet, F. Devaux, L. Furfaro, and E. Lantz, “Measurement of sub-shot-noise correlations of spatial fluctuations in the photon-counting regime,” Phys. Rev. Lett. 101, 233604 (2008). [CrossRef] [PubMed]
  8. J. Janousek, K. Wagner, J. F. Morizur, N. Treps, P. K. Lam, C. C. Harb, and H. A. Bachor, “Optical entanglement of co-propagating modes,” Nat. Photonics 3, 399–402 (2009). [CrossRef]
  9. M. D. Reid, “Demonstration of the Einstein–Podolsky–Rosen paradox using nondegenerate parametric amplification,” Phys. Rev. A 40, 913–923 (1989). [CrossRef] [PubMed]
  10. M. Yukawa, R. Ukai, P. van Loock, and A. Furusawa, “Experimental generation of four-mode continuous-variable cluster states,” Phys. Rev. A 78, 012301 (2008). [CrossRef]
  11. M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1539–1589 (1999). [CrossRef]
  12. M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous variable entanglement and squeezing of orbital angular momentum states,” Phys. Rev. Lett. 102, 163602 (2009). [CrossRef] [PubMed]
  13. B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Multimode nonclassical light generation through the optical-parametric-oscillator threshold,” Phys. Rev. A 81, 061804(R) (2010). [CrossRef]
  14. L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986). [CrossRef] [PubMed]
  15. T. Eberle, S. Steinlechner, J. Bauchrowitz, V. Händchen, H. Vahlbruch, M. Mehmet, H. Müller-Ebhardt, and R. Schnabel, “Quantum enhancement of the zero-area Sagnac interferometer topology for gravitational wave detection,” Phys. Rev. Lett. 104, 251102 (2010). [CrossRef] [PubMed]
  16. J. A. Arnaud, “Degenerate optical cavities,” Appl. Opt. 8, 189–195 (1969). [CrossRef] [PubMed]
  17. B. Chalopin, A. Chiummo, C. Fabre, A. Maitre, and N. Treps, “Frequency doubling of low power images using a self-imaging cavity,” Opt. Express 18, 8033–8042 (2010). [CrossRef] [PubMed]
  18. C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84, 5304–5307 (2000). [CrossRef] [PubMed]
  19. M. Lassen, V. Delaubert, C. Harb, P. K. Lam, N. Treps, and H. A. Bachor, “Generation of squeezing in higher order Hermite–Gaussian modes with an optical parametric amplifier,” J. Eur. Opt. Soc. Rapid Publ. 1, 06003 (2006). [CrossRef]

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