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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 8 — Apr. 16, 2007
  • pp: 5077–5082
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Generation of polarization squeezing with periodically poled KTP at 1064 nm

Mikael Lassen, Metin Sabuncu, Preben Buchhave, and Ulrik L. Andersen  »View Author Affiliations


Optics Express, Vol. 15, Issue 8, pp. 5077-5082 (2007)
http://dx.doi.org/10.1364/OE.15.005077


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Abstract

We report the experimental demonstration of directly produced polarization squeezing at 1064 nm from a type I optical parametric amplifier (OPA) based on a periodically poled KTP crystal (PPKTP). The orthogonal polarization modes of the polarization squeezed state are both defined by the OPA cavity mode, and the birefringence induced by the PPKTP crystal is compensated by a second, but inactive, PPKTP crystal. Stokes parameter squeezing of 3.6 dB and anti squeezing of 9.4 dB is observed.

© 2007 Optical Society of America

1. Introduction

The quantum properties of polarization states of light have recently received a great deal of attention. They are not only interesting from a fundamental point of view, but they also has some practical relevance since it facilitates the execution of various quantum information protocols: A future quantum information network will probably consist of nodes of atoms, where the quantum information is processed, linked by optical channels [1

1. J. I. Cirac and P. Zoller, “Quantum Computations with Cold Trapped Ions,” Phys. Rev. Lett. 74, 4091–4094 (1995). [CrossRef] [PubMed]

]. The transfer of quantum information from atoms to photons and photons to atoms is made possible using the quantum properties of the different polarization states [2

2. J. Hald, J. L. Sørensen, C. Schori, and E. S. Polzik, “Spin Squeezed Atoms: A Macroscopic Entangled Ensemble Created by Light,” Phys. Rev. Lett. 83, 1319–1322 (1999). [CrossRef]

].

Several methods for generating polarization squeezed states have been proposed and experimentally realized. For example, by making use of the nonlinearity provided by Kerr-like media, such as in nonlinear fibers and atomic media, or by combining a dim quadrature squeezed beam generated by an optical parametric amplifier (OPA) with a bright coherent beam on a polarizing beam splitter (PBS) [10

10. P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed light-enhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153–2156 (1987). [CrossRef] [PubMed]

, 2

2. J. Hald, J. L. Sørensen, C. Schori, and E. S. Polzik, “Spin Squeezed Atoms: A Macroscopic Entangled Ensemble Created by Light,” Phys. Rev. Lett. 83, 1319–1322 (1999). [CrossRef]

, 11

11. N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and Ch. Silberhorn, “Polarization squeezing and continuous-variable polarization entanglement,” Phys. Rev. A 65, 052306–052318 (2002). [CrossRef]

, 12

12. J. Heersink, T. Gaber, S. Lorenz, O. Glöckl, N. Korolkova, and G. Leuchs, “Polarization squeezing of intense pulses with a fiber-optic Sagnac interferometer,” Phys. Rev. A 68, 013815–013827 (2003). [CrossRef]

, 13

13. J. Heersink, V. Josse, G. Leuchs, and U. L. Andersen, “Efficient polarization squeezing in optical fibers,” Opt. Lett. 30, 1192–1194, (2005). [CrossRef] [PubMed]

, 14

14. W. P. Bowen, R. Schnabel, H.-A. Bachor, and P. K. Lam, “Polarization Squeezing of Continuous Variable Stokes Parameters,” Phys. Rev. Lett. 88, 093601–093605 (2002). [CrossRef] [PubMed]

, 15

15. V. Josse, A. Dantan, L. Vernac, A. Bramati, M. Pinard, and E. Giacobino, “Polarization squeezing with cold atoms,” Phys. Rev. Lett. 91, 103601–103604 (2003). [CrossRef] [PubMed]

, 16

16. J. F. Sherson and K. Mølmer, “Polarization squeezing by optical Faraday rotation”, Phys. Rev. Lett. 97, 143602–143606 (2006). [CrossRef] [PubMed]

, 17

17. U. L. Andersen and P. Buchhave, “Polarization squeezing and entanglement produced by a frequency doubler,” J. Opt. B. 5, 486–491 (2003). [CrossRef]

].

2. Polarization squeezing

The polarization state of light can be described by the four Stokes operators Ŝ 0, Ŝ 1, Ŝ 2 and Ŝ 3, where Ŝ 0 represents the beam intensity whereas Ŝ 1, Ŝ 2 and Ŝ 3 characterize its polarization and form a cartesian coordinate system. If the Stokes vector points in the direction of Ŝ 1, Ŝ 2 or Ŝ 3 the polarized part of the beam is horizontally, linearly at 45°, or right-circularly polarized, respectively [3

3. G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Camb. Phil. Soc. 9, 399Ű416 (1852).

, 4

4. A. E. Siegman, LASERS (University Science Books, 1986).

]. It is well known that the polarization of a light beam is a property which is limited by quantum noise and that it is possible to achieve polarization states below the standard quantum noise limit (QNL). This was first suggested in the work by Cirkin et al. in 1993 [5

5. A. S. Chirkin, A. A. Orlov, and D. Yu. Paraschuk, “Quantum theory of two-mode interactions in optically anisotropic media with cubic nonlinearities: generation of quadrature- and polarization-squeezed light,” Quantum Electron 23, 870–874 (1993). [CrossRef]

], where the Heisenberg inequalities for the Stokes operators were derived by making use of their cyclical commutation relations:

VŜiVŜjεijkŜk2,
(1)

where i,j,k= 1,2,3. This means that intrinsic quantum fluctuations of the different Stokes operators exist. The variances of the Stokes operators, Vi = 〈Ŝi 2〉 - 〈Ŝi 2〉, can be expressed in terms of the amplitude and phase quadrature operator variances, Vs,p ±, where ± refers to squeezing and anti-squeezing of the s- and p-polarized states, respectively. Assuming that the s- and p-polarized states are uncorrelated it can be shown that the variance of the different Stokes operators are given by [6

6. R. Schnabel, W. P. Bowen, N. Treps, T. C. Ralph, H.-A. Bachor, and P. K. Lam, “Stokes operator squeezed continuous variable polarization states,” Phys. Rev. A 67, 012316–012327 (2003). [CrossRef]

]:

V0=V1=αs2Vs++αp2Vp+
V2(θ)=cos2(θ)(αs2Vp++αp2Vs+)+sin2(θ)(αs2Vp+αp2Vs)
V3(θ)=V2(θπ2),
(2)

where θ is the phase between the s-polarized and p-polarized beams and αs, αp are the classical amplitudes. In our experiment we use a phase difference of θ= 0.

In the case of an s-polarized bright amplitude squeezed beam and vacuum in the orthogonal polarization mode we expect the variances to be: V 0 = V 1 < 1 and V 2 = V 3 = 1, where we have normalized the QNL and amplitudes. Note that this is not polarization squeezing, but simply amplitude squeezing. The generation of polarization squeezing requires the presence of vacuum squeezing along with a bright beam. For example, considering the case of a dim s-polarized squeezed beam and a strong p-polarized coherent beam the expected variances will be: V 0 = V 1 = 1, V 2 < 1 and V 3 > 1 (see Eq. 2), and thus the state is polarization squeezed.

The polarization states can also be visualized in a 3-dimensional diagram, the so-called Poincaré sphere. The mapping of the state onto the Poincaré sphere provides the full characterization. A recent example of such a mapping of the polarization state was demonstrated by Marquardt et al. [7

7. Ch. Marquardt, J. Heersink, R. Dong, M.V. Chekhova, A. B. Klimov, L. L. Sanchez-Soto, U. L. Andersen, and G. Leuchs, “Quantum reconstruction of an intense polarization squeezed optical state,” quant-ph/0701123, (2007).

]. The polarization plane on which the classical stokes parameter is perpendicular defines the polarisation squeezing of the state in case of large coherent excitation [13

13. J. Heersink, V. Josse, G. Leuchs, and U. L. Andersen, “Efficient polarization squeezing in optical fibers,” Opt. Lett. 30, 1192–1194, (2005). [CrossRef] [PubMed]

]. Therefore, if the light is classically S1 polarized, the “dark” polarization plane is spanned by the S2 and S3 parameters, and the squeezing and anti-squeezing of the quantum polarization is to be found in this plane. Using this simple observation, the analogy between polarization squeezing and quadrature squeezing is obvious [8

8. D. F. Walls and G. J. Milburn, Quantum Optics, 1st. edition, (Springer-Verlag, Berlin, 1994).

, 9

9. H-A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics, 2nd edition, (Wiley, 2004). [CrossRef]

].

3. Generation and detection of polarization squeezing

Our squeezing setup is depicted in Fig. 1 and consists of an empty cavity, an optical parametric amplifier and a detection scheme. The laser source, a cw solid-state monolithic YAG laser, Diabolo from Innolight, provides 800 mW at 532 nm and 450 mW power at 1064 nm. The 1064 nm beam from the laser is directed through an empty ring cavity, a so-called mode-cleaner (MC), which filters out the intensity and frequency noise of the laser above the bandwidth of the MC. In addition the MC defines a high quality spatial TEM00 mode. A bandwidth of 2.7 MHz is measured and a transmission greater than 90% is obtained for the TEM00 mode.

We used a bow-tie shaped cavity for the OPA in order to avoid possible destructive interference that may result from a double passed linear cavity. Furthermore the circulating beam encounters the passive loss in the crystal only once per round trip. The nonlinear crystal used is a 1×2×10 mm3 type I periodically poled KTP (PPKTP) crystal manufactured by Raicol Inc. Our bow-tie cavity consists of two curved mirrors of 25 mm radius of curvature and two plane mirrors. Three of the mirrors are highly reflective at 1064 nm, R > 99.9%, while the output coupler has a transmission of T = 15%. The transmittance of the mirrors at the pump wavelength, 532 nm, is more than 95%. The crystal is placed in the smallest beam waist, which is located between the two curved mirrors. In order to maximize the conversion efficiency, corresponding to an optimization of the Boyd-Kleinmann factor [25

25. G. D. Boyd and D. A. Kleinman, “Parametric interactions of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968). [CrossRef]

], a beam waist of 20 μm is chosen. This was enabled by using a cavity length of about 145 mm and setting the distance between the two curved mirrors to be 28 mm. The OPA has a measured finesse of approximately 20, a free spectral range (FSR) of 2 GHz and a cavity bandwidth of 100 MHz. From the finesse of the cavity we deduced an overall intra-cavity loss of L = 1.0±0.2%.

We use the half-wave plate in front of the OPA to inject a beam that has components of both s- and p-polarization, where the s-polarized beam is a seed for the squeezed beam, and the p-polarized beam is the coherent beam. Due to the birefringence of the nonlinear crystal the s- and p-polarization is not simultaneously resonant. A difference between the s- and p-polarization resonances are measured to 0.5 FSR at a crystal temperature of 32°C, which is the optimal phase-matching temperature. By changing the temperature of the crystal to 24°C a simultaneous resonance of the s- and p-polarization could be enabled. However, this temperature lies outside the phase-matching bandwidth of the nonlinear crystal, which was measured to be approximately 5°C. We therefore use a second, but identical, PPKTP crystal for compensating the birefringence. We tune the temperature of the second crystal so that the s- and p-polarization are simultaneously resonant. The cavity was locked to resonance using the Pound Drever Hall locking technique [26

26. R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, Laser phase and frequency stabilization using an optical resonator, Appl. Phys. B 31, 97–105 (1983). [CrossRef]

].

Depending on the relative phase between the pump and the seed, the seed is either amplified or de-amplified. We measure a maximum amplification of 14 and a de-amplification of 0.38. The relative phase is locked to de-amplification in order to generate an amplitude quadrature squeezed beam. This beam is then directed to the polarization measurement scheme [27

27. G. S. Agarwal and S. Chaturvedi, “Scheme to measure quantum Stokes parameters and their fluctuations and correlations,” J. Mod. Opt. 50, 711–716 (2003). [CrossRef]

].

In order to perform the measurement of the Stokes operators the state propagates through a sequence of wave-plates and is subsequently projected on a PBS. The intensities of the two outputs are then measured and by subtracting the resulting photocurrents any Stokes parameter can be accessed. If the waveplates are aligned so that the polarization state is unchanged Ŝ 1 is accessed. To measure Ŝ 2 the polarization of the beam was rotated by 45° with a half-wave plate, and, if in addition to the half wave plate rotation, a quarter wave plate introduces a π/2 phase shift between s- and p-polarization, Ŝ 3 is measured. The spectral densities of the resulting difference currents are measured with an electronic spectrum analyser (ESA).

Fig. 1. Schematics of the experimental setup to generate amplitude and polarization squeezing. PBS: polarizing beam splitter. DM: dichroic mirror. λ/2: half-wave plate. λ/4: quarter-wave plate. Pol. SQL: polarization squeezed light. θ: phase between pump and seed. The PPKTP1 crystal is the squeezing crystal while PPKTP2 is the birefringence compensating crystal.

The measurements were performed at a detection frequency of 14 MHz. The detectors used are designed to be resonant at 14 MHz having a 1 MHz bandwidth. The variances have all been corrected for dark-noise of the detectors, which is more than 12 ± 0.2 dB below the QNL, thus had a negligible effect on the noise spectrum. Each trace depicted in Fig. 2 is normalised to the QNL and measured with a resolution bandwidth (RBW) of the ESA set to 300 kHz and with a video bandwidth (VBW) of 300 Hz. The calibration of the QNL is done by operating the OPA without the pump. We then adjusted the seed power to be equal to that of the squeezed beam when the pump is turned on.

The total detection efficiency of our experiment is given by: ηtotal = ηcavηpropηdetηOL, where ηprop = 0.97 ± 0.02 is the propagation efficiency from the cavity to the detectors, ηdet = 0.95 ± 0.02 is the quantum efficiency of the photo-detectors, ηcav = T/(T+L) = 0.94 is the quantum escape efficiency and ηOL is the overlap efficiency. Since both the seed and coherent beam are injected into the cavity the efficiency is ηOL ≈ 1. The total detection efficiency of our system is therefore ηtotal = 0.87 ± 0.02.

We first measured the polarization state of a single bright amplitude squeezed beam without a coherent beam in the orthogonal polarization mode. This is achieved by injecting all the OPA seed light into the squeezed p-polarization mode and subsequently measuring the amplitude of output with a single detector. The variance of the measurement outcomes is displayed in Fig. 2(a), and we see that amplitude squeezing of -3.8±0.2 dB relative to the quantum noise limit is observed. Taking into account the detection efficiency we infer an amplitude squeezing of -5.0±0.4 dB.

Fig. 2. a) Bright amplitude squeezed. b) Ŝ 0 and Ŝ 1. c) Ŝ 2. d) Ŝ 3. The measurements are taken at 14 MHz in zero span mode with RBW=300 kHz and VBW=300 Hz. CB: coherent beam. SQL: amplitude squeezed light.

4. Summary

We have demonstrated the generation of polarization squeezed light using PPKTP. We generated -3.6±0.2 dB squeezing in Ŝ 2 and +9.4±0.2 dB anti-squeezing in Ŝ 3 using a dim amplitude squeezed beam combined with a coherent beam. To ensure an efficient spatial mode overlap between the coherent state and the squeezed state, both modes were defined in the same cavity in orthogonal polarization modes. We also produced bright amplitude squeezing and measured -3.8±0.2 dB squeezing.

We thank Søren Hjort for technical support. This work was supported by the Danish Technical Research Council (STVF Project No. 26-03-0304) and the EU project COVAQIAL (Project No. FP6-511004).

References and links

1.

J. I. Cirac and P. Zoller, “Quantum Computations with Cold Trapped Ions,” Phys. Rev. Lett. 74, 4091–4094 (1995). [CrossRef] [PubMed]

2.

J. Hald, J. L. Sørensen, C. Schori, and E. S. Polzik, “Spin Squeezed Atoms: A Macroscopic Entangled Ensemble Created by Light,” Phys. Rev. Lett. 83, 1319–1322 (1999). [CrossRef]

3.

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Camb. Phil. Soc. 9, 399Ű416 (1852).

4.

A. E. Siegman, LASERS (University Science Books, 1986).

5.

A. S. Chirkin, A. A. Orlov, and D. Yu. Paraschuk, “Quantum theory of two-mode interactions in optically anisotropic media with cubic nonlinearities: generation of quadrature- and polarization-squeezed light,” Quantum Electron 23, 870–874 (1993). [CrossRef]

6.

R. Schnabel, W. P. Bowen, N. Treps, T. C. Ralph, H.-A. Bachor, and P. K. Lam, “Stokes operator squeezed continuous variable polarization states,” Phys. Rev. A 67, 012316–012327 (2003). [CrossRef]

7.

Ch. Marquardt, J. Heersink, R. Dong, M.V. Chekhova, A. B. Klimov, L. L. Sanchez-Soto, U. L. Andersen, and G. Leuchs, “Quantum reconstruction of an intense polarization squeezed optical state,” quant-ph/0701123, (2007).

8.

D. F. Walls and G. J. Milburn, Quantum Optics, 1st. edition, (Springer-Verlag, Berlin, 1994).

9.

H-A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics, 2nd edition, (Wiley, 2004). [CrossRef]

10.

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed light-enhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153–2156 (1987). [CrossRef] [PubMed]

11.

N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and Ch. Silberhorn, “Polarization squeezing and continuous-variable polarization entanglement,” Phys. Rev. A 65, 052306–052318 (2002). [CrossRef]

12.

J. Heersink, T. Gaber, S. Lorenz, O. Glöckl, N. Korolkova, and G. Leuchs, “Polarization squeezing of intense pulses with a fiber-optic Sagnac interferometer,” Phys. Rev. A 68, 013815–013827 (2003). [CrossRef]

13.

J. Heersink, V. Josse, G. Leuchs, and U. L. Andersen, “Efficient polarization squeezing in optical fibers,” Opt. Lett. 30, 1192–1194, (2005). [CrossRef] [PubMed]

14.

W. P. Bowen, R. Schnabel, H.-A. Bachor, and P. K. Lam, “Polarization Squeezing of Continuous Variable Stokes Parameters,” Phys. Rev. Lett. 88, 093601–093605 (2002). [CrossRef] [PubMed]

15.

V. Josse, A. Dantan, L. Vernac, A. Bramati, M. Pinard, and E. Giacobino, “Polarization squeezing with cold atoms,” Phys. Rev. Lett. 91, 103601–103604 (2003). [CrossRef] [PubMed]

16.

J. F. Sherson and K. Mølmer, “Polarization squeezing by optical Faraday rotation”, Phys. Rev. Lett. 97, 143602–143606 (2006). [CrossRef] [PubMed]

17.

U. L. Andersen and P. Buchhave, “Polarization squeezing and entanglement produced by a frequency doubler,” J. Opt. B. 5, 486–491 (2003). [CrossRef]

18.

U. L. Andersen and P. Buchhave, “Green bright squeezed light from a cw periodically poled KTP second harmonic generator,” Opt. Express 10, 887–896 (2002). [PubMed]

19.

T. Tanimura, D. Akamatsu, Y. Yokoi, A. Furusawa, and M. Kozuma “Generation of a squeezed vacuum resonant on Rubidium D1 line with periodically-poled KTiOPO4,” Opt. Lett. 31, 2344–2346 (2006). [CrossRef] [PubMed]

20.

G. Hétet, O. Glöckl, K. A. Pilypas, C. C. Harb, B. C. Buchler, H-A. Bachor, and P. K. Lam, “Squeezed light for bandwidth-limited atom optics experiments at the rubidium D1 line,” J. Phys. B: At. Mol. Opt. Phys. 40, 221–226 (2007). [CrossRef]

21.

Y. Takeno, M. Yukawa, H. Yonezawa, and A. Furusawa, “Observation of -9 dB quadrature squeezing with improvement of phase stability in homodyne measurement,” quant-ph/0702139, (2007).

22.

T. Aoki, G. Takahashi, and A. Furusawa “Squeezing at 946nm with periodically-poled KTiOPO4,” Opt. Express 14, 6930–6935 (2006). [CrossRef] [PubMed]

23.

K. Goda, E. E. Mikhailov, O. Miyakawa, S. Saraf, S. Vass, A. Weinstein, and N. Mavalvala, “Generation of a stable low-frequency squeezed vacuum field with periodically-poled KTiOPO4 at 1064 nm,” quant-ph/0703001, (2007).

24.

S. Zhang, Y. Li, J. Liu, and K. Zhang, “Investigation of fundamental and second harmonic squeezed lights from a singly resonant PPKTP frequency doubler,” J. Phys. B: At. Mol. Opt. Phys. 39, 4163–4168 (2006) [CrossRef]

25.

G. D. Boyd and D. A. Kleinman, “Parametric interactions of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968). [CrossRef]

26.

R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, Laser phase and frequency stabilization using an optical resonator, Appl. Phys. B 31, 97–105 (1983). [CrossRef]

27.

G. S. Agarwal and S. Chaturvedi, “Scheme to measure quantum Stokes parameters and their fluctuations and correlations,” J. Mod. Opt. 50, 711–716 (2003). [CrossRef]

OCIS Codes
(270.0270) Quantum optics : Quantum optics
(270.6570) Quantum optics : Squeezed states

ToC Category:
Quantum Optics

History
Original Manuscript: March 5, 2007
Revised Manuscript: April 4, 2007
Manuscript Accepted: April 7, 2007
Published: April 11, 2007

Citation
Mikael Lassen, Metin Sabuncu, Preben Buchhave, and Ulrik L. Andersen, "Generation of polarization squeezing with periodically poled KTP at 1064 nm," Opt. Express 15, 5077-5082 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-5077


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References

  1. J. I. Cirac and P. Zoller, "Quantum Computations with Cold Trapped Ions," Phys. Rev. Lett. 74, 4091-4094 (1995). [CrossRef] [PubMed]
  2. J. Hald, J. L. Sørensen, C. Schori and E. S. Polzik, "Spin Squeezed Atoms: A Macroscopic Entangled Ensemble Created by Light," Phys. Rev. Lett. 83, 1319-1322 (1999). [CrossRef]
  3. G. G. Stokes, "On the composition and resolution of streams of polarized light from different sources," Trans. Camb. Phil. Soc. 9, 399˝U416 (1852).
  4. A. E. Siegman, LASERS (University Science Books, 1986).
  5. A. S. Chirkin, A. A. Orlov and D. Yu. Paraschuk, "Quantum theory of two-mode interactions in optically anisotropic media with cubic nonlinearities: generation of quadrature- and polarizationsqueezed light," Quantum Electron 23, 870-874 (1993). [CrossRef]
  6. R. Schnabel, W. P. Bowen, N. Treps, T. C. Ralph, H.-A. Bachor and P. K. Lam, "Stokes operator squeezed continuous variable polarization states," Phys. Rev. A 67, 012316-012327 (2003). [CrossRef]
  7. Ch. Marquardt, J. Heersink, R. Dong, M.V. Chekhova, A. B. Klimov, L. L. Sanchez-Soto, U. L. Andersen, G. Leuchs, "Quantum reconstruction of an intense polarization squeezed optical state," quant-ph/0701123, (2007).
  8. D. F. Walls and G. J. Milburn, Quantum Optics, 1st. edition, (Springer-Verlag, Berlin, 1994).
  9. H-A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics, 2nd edition, (Wiley, 2004). [CrossRef]
  10. P. Grangier, R. E. Slusher, B. Yurke and A. LaPorta, "Squeezed light-enhanced polarization interferometer," Phys. Rev. Lett. 59, 2153-2156 (1987). [CrossRef] [PubMed]
  11. N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph and Ch. Silberhorn, "Polarization squeezing and continuous-variable polarization entanglement," Phys. Rev. A 65, 052306-052318 (2002). [CrossRef]
  12. J. Heersink, T. Gaber, S. Lorenz, O.Gl¨ockl, N. Korolkova and G. Leuchs, "Polarization squeezing of intense pulses with a fiber-optic Sagnac interferometer,"Phys. Rev. A 68, 013815-013827 (2003). [CrossRef]
  13. J. Heersink, V. Josse, G. Leuchs, and U. L. Andersen, "Efficient polarization squeezing in optical fibers," Opt. Lett. 30, 1192-1194, (2005). [CrossRef] [PubMed]
  14. W. P. Bowen, R. Schnabel, H.-A. Bachor and P. K. Lam, "Polarization Squeezing of Continuous Variable Stokes Parameters," Phys. Rev. Lett. 88, 093601-093605 (2002). [CrossRef] [PubMed]
  15. V. Josse, A. Dantan, L. Vernac, A. Bramati, M. Pinard and E. Giacobino, "Polarization squeezing with cold atoms," Phys. Rev. Lett. 91, 103601-103604 (2003). [CrossRef] [PubMed]
  16. J. F. Sherson and K. Mølmer, "Polarization squeezing by optical Faraday rotation", Phys. Rev. Lett. 97, 143602-143606 (2006). [CrossRef] [PubMed]
  17. U. L. Andersen and P. Buchhave, "Polarization squeezing and entanglement produced by a frequency doubler," J. Opt. B. 5, 486-491 (2003). [CrossRef]
  18. U. L. Andersen and P. Buchhave, "Green bright squeezed light from a cw periodically poled KTP second harmonic generator," Opt. Express 10, 887-896 (2002). [PubMed]
  19. T. Tanimura, D. Akamatsu, Y. Yokoi, A. Furusawa, and M. Kozuma "Generation of a squeezed vacuum resonant on Rubidium D1 line with periodically-poled KTiOPO4," Opt. Lett. 31, 2344- 2346 (2006). [CrossRef] [PubMed]
  20. G. H’etet, O. Gl¨ockl, K. A. Pilypas, C. C. Harb, B. C. Buchler, H-A. Bachor and P. K. Lam, "Squeezed light for bandwidth-limited atom optics experiments at the rubidium D1 line," J. Phys. B: At. Mol. Opt. Phys. 40, 221-226 (2007). [CrossRef]
  21. Y. Takeno, M. Yukawa, H. Yonezawa and A. Furusawa, "Observation of -9 dB quadrature squeezing with improvement of phase stability in homodyne measurement," quant-ph/0702139, (2007).
  22. T. Aoki, G. Takahashi and A. Furusawa "Squeezing at 946nm with periodically-poled KTiOPO4," Opt. Express 14, 6930-6935 (2006). [CrossRef] [PubMed]
  23. K. Goda, E. E. Mikhailov, O. Miyakawa, S. Saraf, S. Vass, A. Weinstein and N. Mavalvala, "Generation of a stable low-frequency squeezed vacuum field with periodically-poled KTiOPO4 at 1064 nm," quant-ph/0703001, (2007).
  24. S. Zhang, Y. Li, J. Liu and K. Zhang, "Investigation of fundamental and second harmonic squeezed lights from a singly resonant PPKTP frequency doubler," J. Phys. B: At. Mol. Opt. Phys. 39, 4163-4168 (2006) [CrossRef]
  25. G. D. Boyd, D. A. Kleinman, "Parametric interactions of focused Gaussian light beams," J. Appl. Phys. 39, 3597-3639 (1968). [CrossRef]
  26. R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley and H. Ward, Laser phase and frequency stabilization using an optical resonator, Appl. Phys. B 31, 97-105 (1983). [CrossRef]
  27. G. S. Agarwal and S. Chaturvedi, "Scheme to measure quantum Stokes parameters and their fluctuations and correlations," J. Mod. Opt. 50, 711-716 (2003). [CrossRef]

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Figures

Fig. 1. Fig. 2.
 

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