Abstract

Nonclassical beams in high order spatial modes have attracted much interest but they exhibit much less squeezing and entanglement than the fundamental spatial modes, limiting their applications. We experimentally demonstrate the relation between pump modes and entanglement of first-order Hermite Gauss modes (HG10 entangled states) in a type II OPO and show that the maximum entanglement of high order spatial modes can be obtained by optimizing the pump spatial mode. To our knowledge, this is the first time to report this. Utilizing the optimal pump mode, the HG10 mode threshold can be reached easily without HG00 oscillation and HG10 entanglement is enhanced by 53.5% over HG00 pumping. The technique is broadly applicable to entanglement generation in high order modes.

© 2017 Optical Society of America

1. Introduction

Continuous variable (CV) squeezed and entangled states are important in processes such as quantum computation, quantum communication and quantum metrology. Since the 1985 observation of CV squeezing by Slusher et al. [1], much research has followed on the generation and optimization of squeezing and entanglement in different systems. These include the optical parametric oscillator (OPO) [2, 3], four-wave mixing (FWM) [1], and the in-fiber optical Kerr effect [4, 5]. Among these tools, the OPO is the most widely used. In recent years, squeezing of up to 15 dB in type I OPOs [6] and entanglement of 8.4 dB in type II OPOs [7] were realized.

Traditionally most OPOs operate in the fundamental mode. However, higher order modes such as Hermite-Gauss (HG) and Laguerre-Gauss (LG) modes contain more spatial degrees of freedom and can give more information in applications than the fundamental mode. They can be used to enhance measurement precision of some physical quantities, such as lateral displacement [8] and transverse rotation angle of an optical beam [9]. They can also be applied in quantum imaging [10], quantum storage [11], quantum super-dense coding [12], and biological measurement [13]. In recent years, squeezing and entanglement have been expanded to higher order modes in OPOs. Lassen et al. generated quadrature squeezing of HG00, HG10 and HG20 modes separately with a type I OPO in 2006 [14, 15] and quadrature entanglement of first-order LG modes with a type I OPO in 2009 [16]. Multimode squeezing and entanglement can also be generated in a specially designed OPO [17–19]. Recently, a CV hyperentanglement state, wherein both spin and orbital angular momenta are entangled, was realized in a multimode type II OPO [20, 21].

To date the degree of squeezing and entanglement produced in higher order modes has been much lower than for the fundamental mode, which limits their applications. Almost all the above cited work adopted the fundamental mode as the pump for the higher order signal modes. This lead to low pump conversion efficiencies and crucially much higher oscillation thresholds than for the fundamental spatial mode, severely limiting the attainable squeezing and entanglement levels.

Lassen et al. presented the ideal pump for oscillation of the HG10 mode, a superposition of HG00 and HG20 modes, but synthesizing the multi-mode is experimentally very challenging [14–16]. In this Express paper, we experimentally demonstrate the relation between pump modes and entanglement of first-order HG modes (HG10 entangled states) in a type II OPO and show that the maximum entanglement of high order spatial modes can be obtained by optimizing the pump spatial mode. To our knowledge, this is the first time to report this. Using the optimal pump, the entanglement inseparability for HG10 mode is enhanced by 53.5% and the threshold is reduced by 66.7% relative to using HG00 in our result.

2. Theoretical model

For a type II OPO with an HG10 signal mode, we define vp(r) as the transverse distribution of the pump mode where r=(x,y) denotes the transverse coordinates. This can be expanded into a series of HG modes as

vp(r)=n=0cnvn0(r),
where vn0(r) denotes the transverse profile of the nth order HG mode and cn is its corresponding coefficient. The transverse profiles of the signal and idler modes can be described by us(r) and ui(r). The full Hamiltonian of the system can be written as
H^=iεp(a^pa^p)+iχΓ(a^pa^sa^ia^pa^sa^i),
where χ is the nonlinear coefficient of the crystal, âp, âs and âi are the annihilation operators of the pump, signal and idler fields, and εp is the pump parameter. Γ is the coupling coefficient of the three intracavity fields given by
Γ=+vp(r)us*(r)ui*(r)dr.

Additionally considering the quantum vacuum noise caused by the extra losses, the quantum Langevin equations of motion for the intracavity fields can be given by

τa^˙p(t)=γpa^p(t)χΓa^s(t)a^i(t)+εpeiθp+2μpb^inp(t)
τa^˙s(t)=γsa^s(t)+χΓa^p(t)a^i(t)+2γsa^ins(t)+2μsb^ins(t)
τa^˙i(t)=γia^i(t)+χΓa^p(t)a^s(t)+2γia^ini(t)+2μib^ini(t).

Here γk (k = p, s, i) are the transmission losses through the output coupler and μk are all other extra losses, γk=γk+μk(k=s,i) are the total losses. τ is the round-trip time of the three modes in the cavity, θp is the phase of the pump field, a^inl(t)(l=s,i) are the input signal and idler fields, and b^inm(t)(m=p,s,i) are the quantum vacuum noise of the three fields induced by the extra losses. Assuming the loss factors γp = 1, γs = γi = γ, μs = μi = μ and γs=γi=γ, then the oscillation threshold is obtained as

εpth=γ/(χΓ).

a^inl=αinleiθl(l=s,i), where θl are the phases of the input signal and idler fields. We introduce the amplitude quadrature X^=(a^+a^)/2 and phase quadrature Ŷ = −i (ââ)/2. When the relative phase between the pump and the seed φ = θp−(θs + θi) = 0, the system is in a parametric amplification state, and the correlation noise spectra can be given by

VX^sX^i=VY^s+Y^i=1ηesc4σ(1+σ)2+Ω2,
where ηesc = γ/γ′ is the escape efficiency, σ = εp/εpth is the normalized pump parameter, and Ω = ωτ/γ′ is the normalized analyzing frequency. When the relative phase between the pump and the seed φ = θp − (θs + θi) = π, the system is in a parametric deamplification state, and the correlation noise spectra can be given by
VX^s+X^i=VY^sY^i=1ηesc4σ(1+σ)2+Ω2,

Considering the total detection efficiency of the system, ηdet, Eq. (7) can be rewritten as

VX^s+X^i=VY^sY^i=1ηdetηesc4p/pth(1+p/pth)2+Ω2,
where ηdet = ηpropηhdηphot, ηprop is the propagation efficiency, ηhd is the homodyne detection efficiency and ηphot is the quantum efficiency of the photodiode. The normalized pump power is given by p/pth = σ2, where p is the actual pump power and pth = γ′2/(χ2Γ2) is the threshold pump power.

The inseparability criterion can be expressed as [22]

V=VXs+Xi+VYsYi=2ηdetηesc8p/pth(1+p/pth)2+Ω2<2.

From Eqs. (3)(5), different pump modes correspond to different coupling coefficients and thus different nonlinear efficiencies, leading to different pump thresholds. The coupling coefficient for the HG00 signal mode u00(r) with HG00 pump mode is

Γ=+v00(r)[u00(r)]2dr=1,
so the oscillation threshold for the HG00 signal mode with HG00 pump is pth0000=γ2/(χ2Γ2)=γ2/χ2. For the HG10 signal mode u10(r) generation with all possible pump, we have the expression from Eq. (1)
Γ=n=0cn+vn0(r)[u10(r)]2dr=n=0cnΓn,
where Γn=+vn0(r)[u10(r)]2dr denotes the coupling coefficient of the nth order HG pump mode. These are
Γ0=+v00(r)[u10(r)]2dr=1/2,
Γ2=+v20(r)[u10(r)]2dr=1/2,
and Γn = 0 for all other n. The HG10 signal mode threshold with an HG00 pump mode (c0 = 1) is pth0010=γ2/(χ2Γ02)=4γ2/χ2, and with an HG20 pump mode (c2 = 1) it is pth2010=γ2/(χ2Γ22)=2γ2/χ2.

For the optimal pump mode, Γ=c0Γ0+c2Γ2=(1/2)c0+(1/2)c2, and c02+c22=1. The maximum value of Γ is 3/2, with c0=1/3 and c2=2/3, so the optimal pump mode is vp=1/3v00+2/3v20, a superposition of HG00 and HG20 modes. The HG10 signal mode threshold with the optimal pump mode is pthopt10=γ2/(χ2Γ2)=4γ2/3χ2.

Figure 1 gives the theoretical curves of the inseparabilities versus normalized pump power for the three different pump modes HG00, HG20, and the optimal superposition under ideal conditions. Under HG00 pumping, the HG00 signal mode threshold is pth0000, which is one-quarter that of the HG10 signal mode pth0010=4pth0000. When the pump power reaches the HG00 threshold pth0000, the system starts to oscillate in the HG00 mode, so the maximum HG10 entanglement cannot be obtained. However, with HG20 pumping, the HG00 signal mode will not be excited. The HG10 signal mode threshold pth2010=2pth0000 can be reached with enough pump power in theory, so the maximum HG10 entanglement can be obtained using an HG20 pump. With optimal superposition mode pumping, the HG00 pump mode comprises 1/3 the total pump power. The threshold for the HG10 signal mode is pthopt10=4pth0000/3. Hence the maximum power of the HG00 component of the pump is 4pth0000/9, which is much smaller than the HG00 signal mode threshold pth0000. The HG00 signal mode will therefore not oscillate in under optimal mode pumping. Moreover, since the HG10 signal mode threshold is much lower than for pure HG20 pumping, the maximum entanglement can be obtained at lower pump power.

 figure: Fig. 1

Fig. 1 Theoretical inseparabilities V against normalized pump power p/pth0000 for three pump modes, HG00 (blue solid line), HG20 (green dashed line) and the optimal pump mode HGopt (red dotted line) under ideal conditions. The parameters are ηdet = 1, ηesc = 1, Ω = 0.

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3. Experiment

The experimental setup is depicted in Fig. 2. A continuous wave all solid state laser source emits both infrared at 1080 nm and green light at 540 nm. The infrared beam passes through a mode converter (MC1), which converts the HG00 mode into the HG10 mode. A part of the HG10 mode is injected into a non-degenerate optical parametric amplifier (NOPA) as the seed beam, and the rest of it is used as the local oscillator for homodyne detection. The green beam is used as the pump beam. It is split into two, one beam pass through the mode converter MC2, which converts HG00 mode into HG20 mode, the other beam is still HG00 mode, then the two beams are combined by a beam splitter, generating the superposition pump mode. By this arrangement, we can choose to pass either the HG00, the HG20, or the superposition pump mode.

 figure: Fig. 2

Fig. 2 Schematic of the experimental setup. NOPA: non-degenerate optical parametric amplifier, KTP: type II KTP crystal, M1 and M2: cavity mirrors, BS: beam splitter, PBS: polarizing beam splitter, MCs: mode converters, SG: signal generator, Servo: servo amplifier circuit for feedback system, PZTs: piezoelectric transducers, DBS: dichroic beam splitter, Local: local oscillator, BHDs: balanced homodyne detectors, +/−: positive/negative power combiner, and SA: spectrum analyzer.

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To lock the relative phase between the HG00 and HG20 modes, we use an iris aperture to pass only the center of the beam profile to a photodiode. With a lock-in amplifier, the relative phase is locked to zero. The mode converters and the NOPA cavity are locked using the standard Pound-Drever-Hall (PDH) technique [23].

The NOPA cavity consists of two 30 mm radius of curvature plano-concave mirrors and a 3 × 3 × 10 mm3 type II KTP crystal in the center. The seed beam input mirror M1 is highly reflective (R>99.95%) at both 1080 nm and 540 nm. The transmittance T of the output coupler M2 is 6% at 1080 nm and T>95% at 540 nm. The cavity is nearly concentric with a length of 62.5 mm and has a waist of 41 μm in the infrared and 29 μm in the green. The NOPA has a finesse of 84 for the signal beam with a free spectral range of 2.4 GHz and a bandwidth of 28 MHz. We lock the relative phase between the seed and the pump beam in the parametric deamplification regime with PZT2.

The NOPA output beams and the green beam pass through a dichroic beam splitter (DBS), which reflects only the infrared beam to be measured. This is divided into two parts by a PBS. They are detected by two balanced homodyne detectors (BHDs). The photocurrents from the two BHDs feed a positive/negative combiner (+/−), and those outputs are recorded by a spectrum analyzer (SA). The correlation noise spectra of the amplitude sum and phase difference of the signal and idler beams are measured by scanning the phase of the local infrared beam using a mirror mounted on piezoelectric transducer PZT3.

4. Experimental results

The experimental parameters in our experiment are as follows. The analyzing frequency is 5 MHz, the resolution bandwidth (RBW) is 300 kHz, and the video bandwidth (VBW) is 1 kHz. The bandwidth of the NOPA is 28 MHz (from which Ω = 5 MHz/28 MHz = 0.18). The various efficiencies are ηprop = 0.89±0.02, ηphot = 0.90±0.01, ηhd = 0.81±0.02, and ηesc = 0.79±0.01, thus the total efficiency ηtotal = 0.51±0.04. The pump threshold for the HG00 signal mode with an HG00 pump is pth0000=510mW. From theoretical prediction, the oscillation threshold for the HG10 signal mode is pth0010=2.04W with HG00 pumping, it is pth2010=1.02W with HG20 pumping, and with the optimal superposition mode pumping pthopt10=680mW.

The measured entanglement inseparabilities V are plotted against the normalized pump power p/pth0000 for the three different pump modes in Fig. 3. The corresponding theoretical curves in experimental conditions are also depicted.

 figure: Fig. 3

Fig. 3 The inseparabilities V versus normalized pump power p/pth0000, where pth0000=510mW. Data points from the experiment are blue squares for HG00 pumping, green circles for HG20 pumping and red triangles for the optimal pump mode HGopt. The solid curves are the theoretical values in experimental conditions for the three pump modes.

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From Fig. 3, the entanglement increases with the increasing pump power for the three pump modes and there is good agreement between theory and experiment. At a given pump power, the optimal pump mode HGopt outperforms the other two modes and the HG20 pump mode outperforms HG00. However, the minimum value of V is not close to zero as Fig. 1 due to the nonideal cavity and detection system. The maximum pump power for HG00 mode in our experiment is 500 mW, since the oscillating threshold of the HG00 signal mode is 510 mW, at higher power, the OPO will oscillate on the HG00 mode, so the maximum entanglement of HG10 mode can not be obtained with HG00 pump mode. However, with HG20 pump mode or the optimal pump mode HGopt, the maximum entanglement of HG10 mode can be obtained. Moreover, with the optimal pump mode HGopt, the maximum entanglement can be obtained at lower pump power compared with HG20 pumping.

Figure 4 gives the HG10 mode correlation noise spectra with the three different pump modes. For HG00 pumping at 500 mW, the amplitude sum squeezing was 2.36±0.07 dB and the phase difference squeezing was 2.56±0.06 dB. For HG20 pumping at 670 mW these were 2.92±0.08 dB and 2.76±0.10 dB. For HGopt pumping at 670 mW, the squeezings were 3.28±0.18 dB and 2.92±0.15 dB. Here in our experiment, the noise spectra were recorded in dB units, they can also be expressed in percentage form through the relation [3]

V(dB)=10log10V(per)
where V(dB) denotes the noise spectra in dB units, V(per) denotes the noise spectra in percentage form, then for HG00 pumping the amplitude sum squeezing in percentage form is VXs+Xi00=0.58±0.01 and the phase difference squeezing is VYsYi00=0.55±0.01. Similarly, for HG20 pumping at 670 mW, the amplitude sum and phase difference squeezings in percentage forms are 0.51 ± 0.01 and 0.53 ± 0.01. For HGopt pumping at 670 mW, the squeezings in percentage forms are 0.47 ± 0.02 and 0.51 ± 0.02. According to the inseperability criterion proposed by Duan et al [22] with the percentage forms of the noise spectra mentioned above, the entanglement inseparabilities for HG10 mode with the three pump modes are obtained as
VXs+Xi00+VYsYi00=1.13±0.02<2,
VXs+Xi20+VYsYi20=1.04±0.02<2,
VXs+Xiopt+VYsYiopt=0.98±0.04<2.
They are in well agreement with Fig. 3. Considering the total detection efficiency ηdet = ηpropηphotηhd = 0.65 ± 0.04, the inseparabilities of Eqs. (15)(17) become 0.66±0.03, 0.52±0.03 and 0.43±0.06. Compared with HG00 pumping, the inseperability is enhanced by η = 53.5% using the optimal pump mode.

 figure: Fig. 4

Fig. 4 The HG10 mode correlation noise powers for the amplitude sum VXs+Xi (a1–a3) and the phase difference VYsYi (b1–b3). The top row was taken using 500 mW of HG00 pumping, the middle row with 670 mW of HG20 pumping, and the bottom row with 670 mW of the superposition HGopt pumping.

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Summarizing the experimental results, we cannot obtain the maximum entanglement of the HG10 mode with HG00 pumping because of the low HG00 threshold. With HG20 pumping, this is not the case. Theoretically, the HG10 signal mode threshold can be reached and the maximum entanglement can be obtained, but in our experiment the laser-limited pump power is insufficient. With the optimal pump mode, the HG10 signal mode threshold is lower and the maximum entanglement can be obtained with lower pump power. Experimentally however, generating the optimal pump mode is relatively complicated and somewhat difficult. Using HG20 pumping is operationally much easier and with sufficient power we can obtain the same degree of entanglement as the optimal pump mode.

5. Conclusion

We experimentally studied HG10 mode entanglement in a type II OPO with three pump modes, HG00, HG20, and a superposition of the two modes. The superposition mode, a one-third HG00 and two-thirds HG20 combination, is theoretically optimal and experimentally shown to be able to obtain a higher entanglement at lower pump power. The experimental results match the theoretical prediction very well. The degree of entanglement is still relatively low resulting from extra losses and various inefficiencies in our experiment. The technique holds promise to obtain more than 10 dB squeezing for applications in quantum imaging [24, 25]. It is an efficient way to improve the squeezing of high-order spatial modes. Moreover, the method can be extended to high-dimension orbital angular momentum entanglement [26–28] to enhance the generation efficiency.

Funding

Ministry of Science and Technology of the People’s Republic of China (MOST) (2016YFA0301404); National Natural Science Foundation of China (NSFC) (91536222, 61405108,11674205); NSFC Project for Excellent Research Team (61121064); University Science and Technology Innovation Project in Shanxi Province (2015103).

References and links

1. R. Slusher, L. Hollberg, B. Yurke, J. Mertz, and J. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985). [CrossRef]   [PubMed]  

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

3. Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein-Podolsky-Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992). [CrossRef]   [PubMed]  

4. P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamamoto, “Quantum solitons in optical fibres,” Nature 365, 307–313 (1993). [CrossRef]  

5. S. Friberg, S. Machida, M. Werner, A. Levanon, and T. Mukai, “Observation of optical soliton photon-number squeezing,” Phys. Rev. Lett. 77, 3775–3778 (1996). [CrossRef]   [PubMed]  

6. H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 dB squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency,” Phys. Rev. Lett. 117, 110801 (2016). [CrossRef]  

7. Y. Zhou, X. Jia, F. Li, C. Xie, and K. Peng, “Experimental generation of 8.4 dB entangled state with an optical cavity involving a wedged type-II nonlinear crystal,” Opt. Express 23, 4952–4959 (2015). [CrossRef]   [PubMed]  

8. H. Sun, K. Liu, Z. Liu, P. Guo, J. Zhang, and J. Gao, “Small-displacement measurements using high-order Hermite-Gauss modes,” Appl. Phys. Lett. 104, 121908 (2014). [CrossRef]  

9. V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

10. G. Brida, M. Genovese, and I. R. Berchera, “Experimental realization of sub-shot-noise quantum imaging,” Nat. Photonics 4, 227–230 (2010). [CrossRef]  

11. A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8, 234–238 (2014). [CrossRef]  

12. J. T. Barreiro, T. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008). [CrossRef]  

13. M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H. A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013). [CrossRef]  

14. M. Lassen, V. Delaubert, C. 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]  

15. M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H. Bachor, P. Lam, N. Treps, P. Buchhave, C. Fabre, and C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98, 083602 (2007). [CrossRef]   [PubMed]  

16. 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]  

17. 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]  

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

19. B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Direct generation of a multi-transverse mode non-classical state of light,” Opt. Express 19, 4405–4410 (2011). [CrossRef]   [PubMed]  

20. B. dos Santos, K. Dechoum, and A. Khoury, “Continuous-variable hyperentanglement in a parametric oscillator with orbital angular momentum,” Phys. Rev. Lett. 103, 230503 (2009). [CrossRef]  

21. K. Liu, J. Guo, C. Cai, S. Guo, and J. Gao, “Experimental generation of continuous-variable hyperentanglement in an optical parametric oscillator,” Phys. Rev. Lett. 113, 170501 (2014). [CrossRef]   [PubMed]  

22. L. Duan, G. Giedke, J. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000). [CrossRef]   [PubMed]  

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

24. M. A. Taylor and W. P. Bowen, “Quantum metrology and its application in biology,” Phys. Rep. 615, 1–59 (2016). [CrossRef]  

25. M. Tsang, R. Nair, and X. M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” Phys. Rev. X 6, 031033 (2016).

26. K. Liu, J. Guo, C. Cai, J. Zhang, and J. Gao, “Direct generation of spatial quadripartite continuous variable entanglement in an optical parametric oscillator,” Opt. Lett. 41, 5178–5181 (2016). [CrossRef]   [PubMed]  

27. J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777–838 (2012). [CrossRef]  

28. Z. Y. Zhou, Y. Li, D. S. Ding, W. Zhang, S. Shi, B. S. Shi, and G. C. Guo, “Orbital angular momentum photonic quantum interface,” Light Sci. Appl. 5, e16019 (2016). [CrossRef]  

References

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  1. R. Slusher, L. Hollberg, B. Yurke, J. Mertz, and J. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
    [Crossref] [PubMed]
  2. L. A. Wu, H. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
    [Crossref] [PubMed]
  3. Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein-Podolsky-Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
    [Crossref] [PubMed]
  4. P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamamoto, “Quantum solitons in optical fibres,” Nature 365, 307–313 (1993).
    [Crossref]
  5. S. Friberg, S. Machida, M. Werner, A. Levanon, and T. Mukai, “Observation of optical soliton photon-number squeezing,” Phys. Rev. Lett. 77, 3775–3778 (1996).
    [Crossref] [PubMed]
  6. H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 dB squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency,” Phys. Rev. Lett. 117, 110801 (2016).
    [Crossref]
  7. Y. Zhou, X. Jia, F. Li, C. Xie, and K. Peng, “Experimental generation of 8.4 dB entangled state with an optical cavity involving a wedged type-II nonlinear crystal,” Opt. Express 23, 4952–4959 (2015).
    [Crossref] [PubMed]
  8. H. Sun, K. Liu, Z. Liu, P. Guo, J. Zhang, and J. Gao, “Small-displacement measurements using high-order Hermite-Gauss modes,” Appl. Phys. Lett. 104, 121908 (2014).
    [Crossref]
  9. V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).
  10. G. Brida, M. Genovese, and I. R. Berchera, “Experimental realization of sub-shot-noise quantum imaging,” Nat. Photonics 4, 227–230 (2010).
    [Crossref]
  11. A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8, 234–238 (2014).
    [Crossref]
  12. J. T. Barreiro, T. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).
    [Crossref]
  13. M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H. A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
    [Crossref]
  14. M. Lassen, V. Delaubert, C. 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]
  15. M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H. Bachor, P. Lam, N. Treps, P. Buchhave, C. Fabre, and C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98, 083602 (2007).
    [Crossref] [PubMed]
  16. 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]
  17. 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]
  18. B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Multimode nonclassical light generation through the optical-parametric-oscillator threshold,” Phys. Rev. A 81, 061804 (2010).
    [Crossref]
  19. B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Direct generation of a multi-transverse mode non-classical state of light,” Opt. Express 19, 4405–4410 (2011).
    [Crossref] [PubMed]
  20. B. dos Santos, K. Dechoum, and A. Khoury, “Continuous-variable hyperentanglement in a parametric oscillator with orbital angular momentum,” Phys. Rev. Lett. 103, 230503 (2009).
    [Crossref]
  21. K. Liu, J. Guo, C. Cai, S. Guo, and J. Gao, “Experimental generation of continuous-variable hyperentanglement in an optical parametric oscillator,” Phys. Rev. Lett. 113, 170501 (2014).
    [Crossref] [PubMed]
  22. L. Duan, G. Giedke, J. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
    [Crossref] [PubMed]
  23. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
    [Crossref]
  24. M. A. Taylor and W. P. Bowen, “Quantum metrology and its application in biology,” Phys. Rep. 615, 1–59 (2016).
    [Crossref]
  25. M. Tsang, R. Nair, and X. M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” Phys. Rev. X 6, 031033 (2016).
  26. K. Liu, J. Guo, C. Cai, J. Zhang, and J. Gao, “Direct generation of spatial quadripartite continuous variable entanglement in an optical parametric oscillator,” Opt. Lett. 41, 5178–5181 (2016).
    [Crossref] [PubMed]
  27. J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777–838 (2012).
    [Crossref]
  28. Z. Y. Zhou, Y. Li, D. S. Ding, W. Zhang, S. Shi, B. S. Shi, and G. C. Guo, “Orbital angular momentum photonic quantum interface,” Light Sci. Appl. 5, e16019 (2016).
    [Crossref]

2016 (5)

H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 dB squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency,” Phys. Rev. Lett. 117, 110801 (2016).
[Crossref]

M. A. Taylor and W. P. Bowen, “Quantum metrology and its application in biology,” Phys. Rep. 615, 1–59 (2016).
[Crossref]

M. Tsang, R. Nair, and X. M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” Phys. Rev. X 6, 031033 (2016).

K. Liu, J. Guo, C. Cai, J. Zhang, and J. Gao, “Direct generation of spatial quadripartite continuous variable entanglement in an optical parametric oscillator,” Opt. Lett. 41, 5178–5181 (2016).
[Crossref] [PubMed]

Z. Y. Zhou, Y. Li, D. S. Ding, W. Zhang, S. Shi, B. S. Shi, and G. C. Guo, “Orbital angular momentum photonic quantum interface,” Light Sci. Appl. 5, e16019 (2016).
[Crossref]

2015 (1)

2014 (3)

H. Sun, K. Liu, Z. Liu, P. Guo, J. Zhang, and J. Gao, “Small-displacement measurements using high-order Hermite-Gauss modes,” Appl. Phys. Lett. 104, 121908 (2014).
[Crossref]

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8, 234–238 (2014).
[Crossref]

K. Liu, J. Guo, C. Cai, S. Guo, and J. Gao, “Experimental generation of continuous-variable hyperentanglement in an optical parametric oscillator,” Phys. Rev. Lett. 113, 170501 (2014).
[Crossref] [PubMed]

2013 (2)

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H. A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

2012 (1)

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777–838 (2012).
[Crossref]

2011 (1)

2010 (2)

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

G. Brida, M. Genovese, and I. R. Berchera, “Experimental realization of sub-shot-noise quantum imaging,” Nat. Photonics 4, 227–230 (2010).
[Crossref]

2009 (3)

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]

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]

B. dos Santos, K. Dechoum, and A. Khoury, “Continuous-variable hyperentanglement in a parametric oscillator with orbital angular momentum,” Phys. Rev. Lett. 103, 230503 (2009).
[Crossref]

2008 (1)

J. T. Barreiro, T. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).
[Crossref]

2007 (1)

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H. Bachor, P. Lam, N. Treps, P. Buchhave, C. Fabre, and C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98, 083602 (2007).
[Crossref] [PubMed]

2006 (1)

M. Lassen, V. Delaubert, C. 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]

2000 (1)

L. Duan, G. Giedke, J. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[Crossref] [PubMed]

1996 (1)

S. Friberg, S. Machida, M. Werner, A. Levanon, and T. Mukai, “Observation of optical soliton photon-number squeezing,” Phys. Rev. Lett. 77, 3775–3778 (1996).
[Crossref] [PubMed]

1993 (1)

P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamamoto, “Quantum solitons in optical fibres,” Nature 365, 307–313 (1993).
[Crossref]

1992 (1)

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein-Podolsky-Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[Crossref] [PubMed]

1986 (1)

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

1985 (1)

R. Slusher, L. Hollberg, B. Yurke, J. Mertz, and J. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[Crossref] [PubMed]

1983 (1)

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

Andersen, U. L.

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]

Aolita, L.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Bachor, H.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H. Bachor, P. Lam, N. Treps, P. Buchhave, C. Fabre, and C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98, 083602 (2007).
[Crossref] [PubMed]

Bachor, H. A.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H. A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

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]

M. Lassen, V. Delaubert, C. 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]

Barreiro, J. T.

J. T. Barreiro, T. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).
[Crossref]

Berchera, I. R.

G. Brida, M. Genovese, and I. R. Berchera, “Experimental realization of sub-shot-noise quantum imaging,” Nat. Photonics 4, 227–230 (2010).
[Crossref]

Bowen, W. P.

M. A. Taylor and W. P. Bowen, “Quantum metrology and its application in biology,” Phys. Rep. 615, 1–59 (2016).
[Crossref]

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H. A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

Brida, G.

G. Brida, M. Genovese, and I. R. Berchera, “Experimental realization of sub-shot-noise quantum imaging,” Nat. Photonics 4, 227–230 (2010).
[Crossref]

Buchhave, P.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H. Bachor, P. Lam, N. Treps, P. Buchhave, C. Fabre, and C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98, 083602 (2007).
[Crossref] [PubMed]

Cai, C.

K. Liu, J. Guo, C. Cai, J. Zhang, and J. Gao, “Direct generation of spatial quadripartite continuous variable entanglement in an optical parametric oscillator,” Opt. Lett. 41, 5178–5181 (2016).
[Crossref] [PubMed]

K. Liu, J. Guo, C. Cai, S. Guo, and J. Gao, “Experimental generation of continuous-variable hyperentanglement in an optical parametric oscillator,” Phys. Rev. Lett. 113, 170501 (2014).
[Crossref] [PubMed]

Chalopin, B.

B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Direct generation of a multi-transverse mode non-classical state of light,” Opt. Express 19, 4405–4410 (2011).
[Crossref] [PubMed]

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

Chen, Z. B.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777–838 (2012).
[Crossref]

Cirac, J.

L. Duan, G. Giedke, J. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[Crossref] [PubMed]

D’Ambrosio, V.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Danzmann, K.

H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 dB squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency,” Phys. Rev. Lett. 117, 110801 (2016).
[Crossref]

Daria, V.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H. A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

Dechoum, K.

B. dos Santos, K. Dechoum, and A. Khoury, “Continuous-variable hyperentanglement in a parametric oscillator with orbital angular momentum,” Phys. Rev. Lett. 103, 230503 (2009).
[Crossref]

Del Re, L.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Delaubert, V.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H. Bachor, P. Lam, N. Treps, P. Buchhave, C. Fabre, and C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98, 083602 (2007).
[Crossref] [PubMed]

M. Lassen, V. Delaubert, C. 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]

Ding, D. S.

Z. Y. Zhou, Y. Li, D. S. Ding, W. Zhang, S. Shi, B. S. Shi, and G. C. Guo, “Orbital angular momentum photonic quantum interface,” Light Sci. Appl. 5, e16019 (2016).
[Crossref]

dos Santos, B.

B. dos Santos, K. Dechoum, and A. Khoury, “Continuous-variable hyperentanglement in a parametric oscillator with orbital angular momentum,” Phys. Rev. Lett. 103, 230503 (2009).
[Crossref]

Drever, R. W. P.

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

Drummond, P. D.

P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamamoto, “Quantum solitons in optical fibres,” Nature 365, 307–313 (1993).
[Crossref]

Duan, L.

L. Duan, G. Giedke, J. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[Crossref] [PubMed]

Fabre, C.

B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Direct generation of a multi-transverse mode non-classical state of light,” Opt. Express 19, 4405–4410 (2011).
[Crossref] [PubMed]

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

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H. Bachor, P. Lam, N. Treps, P. Buchhave, C. Fabre, and C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98, 083602 (2007).
[Crossref] [PubMed]

Ford, G. M.

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

Friberg, S.

S. Friberg, S. Machida, M. Werner, A. Levanon, and T. Mukai, “Observation of optical soliton photon-number squeezing,” Phys. Rev. Lett. 77, 3775–3778 (1996).
[Crossref] [PubMed]

Friberg, S. R.

P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamamoto, “Quantum solitons in optical fibres,” Nature 365, 307–313 (1993).
[Crossref]

Gao, J.

K. Liu, J. Guo, C. Cai, J. Zhang, and J. Gao, “Direct generation of spatial quadripartite continuous variable entanglement in an optical parametric oscillator,” Opt. Lett. 41, 5178–5181 (2016).
[Crossref] [PubMed]

K. Liu, J. Guo, C. Cai, S. Guo, and J. Gao, “Experimental generation of continuous-variable hyperentanglement in an optical parametric oscillator,” Phys. Rev. Lett. 113, 170501 (2014).
[Crossref] [PubMed]

H. Sun, K. Liu, Z. Liu, P. Guo, J. Zhang, and J. Gao, “Small-displacement measurements using high-order Hermite-Gauss modes,” Appl. Phys. Lett. 104, 121908 (2014).
[Crossref]

Genovese, M.

G. Brida, M. Genovese, and I. R. Berchera, “Experimental realization of sub-shot-noise quantum imaging,” Nat. Photonics 4, 227–230 (2010).
[Crossref]

Giacobino, E.

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8, 234–238 (2014).
[Crossref]

Giedke, G.

L. Duan, G. Giedke, J. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[Crossref] [PubMed]

Giner, L.

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8, 234–238 (2014).
[Crossref]

Guo, G. C.

Z. Y. Zhou, Y. Li, D. S. Ding, W. Zhang, S. Shi, B. S. Shi, and G. C. Guo, “Orbital angular momentum photonic quantum interface,” Light Sci. Appl. 5, e16019 (2016).
[Crossref]

Guo, J.

K. Liu, J. Guo, C. Cai, J. Zhang, and J. Gao, “Direct generation of spatial quadripartite continuous variable entanglement in an optical parametric oscillator,” Opt. Lett. 41, 5178–5181 (2016).
[Crossref] [PubMed]

K. Liu, J. Guo, C. Cai, S. Guo, and J. Gao, “Experimental generation of continuous-variable hyperentanglement in an optical parametric oscillator,” Phys. Rev. Lett. 113, 170501 (2014).
[Crossref] [PubMed]

Guo, P.

H. Sun, K. Liu, Z. Liu, P. Guo, J. Zhang, and J. Gao, “Small-displacement measurements using high-order Hermite-Gauss modes,” Appl. Phys. Lett. 104, 121908 (2014).
[Crossref]

Guo, S.

K. Liu, J. Guo, C. Cai, S. Guo, and J. Gao, “Experimental generation of continuous-variable hyperentanglement in an optical parametric oscillator,” Phys. Rev. Lett. 113, 170501 (2014).
[Crossref] [PubMed]

Hage, B.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H. A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

Hall, J.

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

Hall, J. L.

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

Harb, C.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H. Bachor, P. Lam, N. Treps, P. Buchhave, C. Fabre, and C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98, 083602 (2007).
[Crossref] [PubMed]

Harb, C. C.

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]

M. Lassen, V. Delaubert, C. 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]

Hollberg, L.

R. Slusher, L. Hollberg, B. Yurke, J. Mertz, and J. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[Crossref] [PubMed]

Hough, J.

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

Janousek, J.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H. A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

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]

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H. Bachor, P. Lam, N. Treps, P. Buchhave, C. Fabre, and C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98, 083602 (2007).
[Crossref] [PubMed]

Jia, X.

Khoury, A.

B. dos Santos, K. Dechoum, and A. Khoury, “Continuous-variable hyperentanglement in a parametric oscillator with orbital angular momentum,” Phys. Rev. Lett. 103, 230503 (2009).
[Crossref]

Kimble, H.

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

Kimble, H. J.

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein-Podolsky-Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[Crossref] [PubMed]

Knittel, J.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H. A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

Kowalski, F. V.

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

Kwek, L. C.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Kwiat, P. G.

J. T. Barreiro, T. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).
[Crossref]

Lam, P.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H. Bachor, P. Lam, N. Treps, P. Buchhave, C. Fabre, and C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98, 083602 (2007).
[Crossref] [PubMed]

Lam, P. K.

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]

M. Lassen, V. Delaubert, C. 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]

Lassen, M.

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]

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H. Bachor, P. Lam, N. Treps, P. Buchhave, C. Fabre, and C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98, 083602 (2007).
[Crossref] [PubMed]

M. Lassen, V. Delaubert, C. 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]

Laurat, J.

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8, 234–238 (2014).
[Crossref]

Leuchs, G.

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]

Levanon, A.

S. Friberg, S. Machida, M. Werner, A. Levanon, and T. Mukai, “Observation of optical soliton photon-number squeezing,” Phys. Rev. Lett. 77, 3775–3778 (1996).
[Crossref] [PubMed]

Li, F.

Li, Y.

Z. Y. Zhou, Y. Li, D. S. Ding, W. Zhang, S. Shi, B. S. Shi, and G. C. Guo, “Orbital angular momentum photonic quantum interface,” Light Sci. Appl. 5, e16019 (2016).
[Crossref]

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Liu, K.

K. Liu, J. Guo, C. Cai, J. Zhang, and J. Gao, “Direct generation of spatial quadripartite continuous variable entanglement in an optical parametric oscillator,” Opt. Lett. 41, 5178–5181 (2016).
[Crossref] [PubMed]

K. Liu, J. Guo, C. Cai, S. Guo, and J. Gao, “Experimental generation of continuous-variable hyperentanglement in an optical parametric oscillator,” Phys. Rev. Lett. 113, 170501 (2014).
[Crossref] [PubMed]

H. Sun, K. Liu, Z. Liu, P. Guo, J. Zhang, and J. Gao, “Small-displacement measurements using high-order Hermite-Gauss modes,” Appl. Phys. Lett. 104, 121908 (2014).
[Crossref]

Liu, Z.

H. Sun, K. Liu, Z. Liu, P. Guo, J. Zhang, and J. Gao, “Small-displacement measurements using high-order Hermite-Gauss modes,” Appl. Phys. Lett. 104, 121908 (2014).
[Crossref]

Lu, C. Y.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777–838 (2012).
[Crossref]

Lu, X. M.

M. Tsang, R. Nair, and X. M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” Phys. Rev. X 6, 031033 (2016).

Machida, S.

S. Friberg, S. Machida, M. Werner, A. Levanon, and T. Mukai, “Observation of optical soliton photon-number squeezing,” Phys. Rev. Lett. 77, 3775–3778 (1996).
[Crossref] [PubMed]

Marrucci, L.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Maxein, D.

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8, 234–238 (2014).
[Crossref]

Mehmet, M.

H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 dB squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency,” Phys. Rev. Lett. 117, 110801 (2016).
[Crossref]

Mertz, J.

R. Slusher, L. Hollberg, B. Yurke, J. Mertz, and J. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[Crossref] [PubMed]

Morizur, J. F.

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]

Mukai, T.

S. Friberg, S. Machida, M. Werner, A. Levanon, and T. Mukai, “Observation of optical soliton photon-number squeezing,” Phys. Rev. Lett. 77, 3775–3778 (1996).
[Crossref] [PubMed]

Munley, A. J.

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

Nair, R.

M. Tsang, R. Nair, and X. M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” Phys. Rev. X 6, 031033 (2016).

Nicolas, A.

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8, 234–238 (2014).
[Crossref]

Ou, Z. Y.

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein-Podolsky-Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[Crossref] [PubMed]

Pan, J. W.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777–838 (2012).
[Crossref]

Peng, K.

Peng, K. C.

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein-Podolsky-Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[Crossref] [PubMed]

Pereira, S. F.

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein-Podolsky-Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[Crossref] [PubMed]

Scazza, F.

B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Direct generation of a multi-transverse mode non-classical state of light,” Opt. Express 19, 4405–4410 (2011).
[Crossref] [PubMed]

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

Schnabel, R.

H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 dB squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency,” Phys. Rev. Lett. 117, 110801 (2016).
[Crossref]

Sciarrino, F.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Shelby, R. M.

P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamamoto, “Quantum solitons in optical fibres,” Nature 365, 307–313 (1993).
[Crossref]

Shi, B. S.

Z. Y. Zhou, Y. Li, D. S. Ding, W. Zhang, S. Shi, B. S. Shi, and G. C. Guo, “Orbital angular momentum photonic quantum interface,” Light Sci. Appl. 5, e16019 (2016).
[Crossref]

Shi, S.

Z. Y. Zhou, Y. Li, D. S. Ding, W. Zhang, S. Shi, B. S. Shi, and G. C. Guo, “Orbital angular momentum photonic quantum interface,” Light Sci. Appl. 5, e16019 (2016).
[Crossref]

Slusher, R.

R. Slusher, L. Hollberg, B. Yurke, J. Mertz, and J. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[Crossref] [PubMed]

Slussarenko, S.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Spagnolo, N.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Sun, H.

H. Sun, K. Liu, Z. Liu, P. Guo, J. Zhang, and J. Gao, “Small-displacement measurements using high-order Hermite-Gauss modes,” Appl. Phys. Lett. 104, 121908 (2014).
[Crossref]

Taylor, M. A.

M. A. Taylor and W. P. Bowen, “Quantum metrology and its application in biology,” Phys. Rep. 615, 1–59 (2016).
[Crossref]

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H. A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

Treps, N.

B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Direct generation of a multi-transverse mode non-classical state of light,” Opt. Express 19, 4405–4410 (2011).
[Crossref] [PubMed]

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

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]

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H. Bachor, P. Lam, N. Treps, P. Buchhave, C. Fabre, and C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98, 083602 (2007).
[Crossref] [PubMed]

M. Lassen, V. Delaubert, C. 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]

Tsang, M.

M. Tsang, R. Nair, and X. M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” Phys. Rev. X 6, 031033 (2016).

Vahlbruch, H.

H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 dB squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency,” Phys. Rev. Lett. 117, 110801 (2016).
[Crossref]

Valley, J.

R. Slusher, L. Hollberg, B. Yurke, J. Mertz, and J. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[Crossref] [PubMed]

Veissier, L.

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8, 234–238 (2014).
[Crossref]

Wagner, K.

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]

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H. Bachor, P. Lam, N. Treps, P. Buchhave, C. Fabre, and C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98, 083602 (2007).
[Crossref] [PubMed]

Walborn, S. P.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Ward, H.

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

Wei, T.

J. T. Barreiro, T. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).
[Crossref]

Weinfurter, H.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777–838 (2012).
[Crossref]

Werner, M.

S. Friberg, S. Machida, M. Werner, A. Levanon, and T. Mukai, “Observation of optical soliton photon-number squeezing,” Phys. Rev. Lett. 77, 3775–3778 (1996).
[Crossref] [PubMed]

Wu, H.

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

Wu, L. A.

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

Xie, C.

Yamamoto, Y.

P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamamoto, “Quantum solitons in optical fibres,” Nature 365, 307–313 (1993).
[Crossref]

Yurke, B.

R. Slusher, L. Hollberg, B. Yurke, J. Mertz, and J. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[Crossref] [PubMed]

Zeilinger, A.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777–838 (2012).
[Crossref]

Zhang, J.

K. Liu, J. Guo, C. Cai, J. Zhang, and J. Gao, “Direct generation of spatial quadripartite continuous variable entanglement in an optical parametric oscillator,” Opt. Lett. 41, 5178–5181 (2016).
[Crossref] [PubMed]

H. Sun, K. Liu, Z. Liu, P. Guo, J. Zhang, and J. Gao, “Small-displacement measurements using high-order Hermite-Gauss modes,” Appl. Phys. Lett. 104, 121908 (2014).
[Crossref]

Zhang, W.

Z. Y. Zhou, Y. Li, D. S. Ding, W. Zhang, S. Shi, B. S. Shi, and G. C. Guo, “Orbital angular momentum photonic quantum interface,” Light Sci. Appl. 5, e16019 (2016).
[Crossref]

Zhou, Y.

Zhou, Z. Y.

Z. Y. Zhou, Y. Li, D. S. Ding, W. Zhang, S. Shi, B. S. Shi, and G. C. Guo, “Orbital angular momentum photonic quantum interface,” Light Sci. Appl. 5, e16019 (2016).
[Crossref]

Zoller, P.

L. Duan, G. Giedke, J. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[Crossref] [PubMed]

Zukowski, M.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777–838 (2012).
[Crossref]

Appl. Phys. B (1)

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

Appl. Phys. Lett. (1)

H. Sun, K. Liu, Z. Liu, P. Guo, J. Zhang, and J. Gao, “Small-displacement measurements using high-order Hermite-Gauss modes,” Appl. Phys. Lett. 104, 121908 (2014).
[Crossref]

J. Eur. Opt. Soc. Rapid Publ. (1)

M. Lassen, V. Delaubert, C. 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]

Light Sci. Appl. (1)

Z. Y. Zhou, Y. Li, D. S. Ding, W. Zhang, S. Shi, B. S. Shi, and G. C. Guo, “Orbital angular momentum photonic quantum interface,” Light Sci. Appl. 5, e16019 (2016).
[Crossref]

Nat. Commun. (1)

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Nat. Photonics (4)

G. Brida, M. Genovese, and I. R. Berchera, “Experimental realization of sub-shot-noise quantum imaging,” Nat. Photonics 4, 227–230 (2010).
[Crossref]

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8, 234–238 (2014).
[Crossref]

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H. A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

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]

Nat. Phys. (1)

J. T. Barreiro, T. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).
[Crossref]

Nature (1)

P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamamoto, “Quantum solitons in optical fibres,” Nature 365, 307–313 (1993).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rep. (1)

M. A. Taylor and W. P. Bowen, “Quantum metrology and its application in biology,” Phys. Rep. 615, 1–59 (2016).
[Crossref]

Phys. Rev. A (1)

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

Phys. Rev. Lett. (10)

B. dos Santos, K. Dechoum, and A. Khoury, “Continuous-variable hyperentanglement in a parametric oscillator with orbital angular momentum,” Phys. Rev. Lett. 103, 230503 (2009).
[Crossref]

K. Liu, J. Guo, C. Cai, S. Guo, and J. Gao, “Experimental generation of continuous-variable hyperentanglement in an optical parametric oscillator,” Phys. Rev. Lett. 113, 170501 (2014).
[Crossref] [PubMed]

L. Duan, G. Giedke, J. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[Crossref] [PubMed]

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H. Bachor, P. Lam, N. Treps, P. Buchhave, C. Fabre, and C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98, 083602 (2007).
[Crossref] [PubMed]

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]

S. Friberg, S. Machida, M. Werner, A. Levanon, and T. Mukai, “Observation of optical soliton photon-number squeezing,” Phys. Rev. Lett. 77, 3775–3778 (1996).
[Crossref] [PubMed]

H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 dB squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency,” Phys. Rev. Lett. 117, 110801 (2016).
[Crossref]

R. Slusher, L. Hollberg, B. Yurke, J. Mertz, and J. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[Crossref] [PubMed]

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

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein-Podolsky-Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[Crossref] [PubMed]

Phys. Rev. X (1)

M. Tsang, R. Nair, and X. M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” Phys. Rev. X 6, 031033 (2016).

Rev. Mod. Phys. (1)

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777–838 (2012).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1 Theoretical inseparabilities V against normalized pump power p / p t h 00 00 for three pump modes, HG00 (blue solid line), HG20 (green dashed line) and the optimal pump mode HGopt (red dotted line) under ideal conditions. The parameters are ηdet = 1, ηesc = 1, Ω = 0.
Fig. 2
Fig. 2 Schematic of the experimental setup. NOPA: non-degenerate optical parametric amplifier, KTP: type II KTP crystal, M1 and M2: cavity mirrors, BS: beam splitter, PBS: polarizing beam splitter, MCs: mode converters, SG: signal generator, Servo: servo amplifier circuit for feedback system, PZTs: piezoelectric transducers, DBS: dichroic beam splitter, Local: local oscillator, BHDs: balanced homodyne detectors, +/−: positive/negative power combiner, and SA: spectrum analyzer.
Fig. 3
Fig. 3 The inseparabilities V versus normalized pump power p / p t h 00 00, where p t h 00 00 = 510 mW. Data points from the experiment are blue squares for HG00 pumping, green circles for HG20 pumping and red triangles for the optimal pump mode HGopt. The solid curves are the theoretical values in experimental conditions for the three pump modes.
Fig. 4
Fig. 4 The HG10 mode correlation noise powers for the amplitude sum V X s + X i (a1–a3) and the phase difference V Y s Y i (b1–b3). The top row was taken using 500 mW of HG00 pumping, the middle row with 670 mW of HG20 pumping, and the bottom row with 670 mW of the superposition HGopt pumping.

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

v p ( r ) = n = 0 c n v n 0 ( r ) ,
H ^ = i ε p ( a ^ p a ^ p ) + i χ Γ ( a ^ p a ^ s a ^ i a ^ p a ^ s a ^ i ) ,
Γ = + v p ( r ) u s * ( r ) u i * ( r ) d r .
τ a ^ ˙ p ( t ) = γ p a ^ p ( t ) χ Γ a ^ s ( t ) a ^ i ( t ) + ε p e i θ p + 2 μ p b ^ i n p ( t )
τ a ^ ˙ s ( t ) = γ s a ^ s ( t ) + χ Γ a ^ p ( t ) a ^ i ( t ) + 2 γ s a ^ i n s ( t ) + 2 μ s b ^ i n s ( t )
τ a ^ ˙ i ( t ) = γ i a ^ i ( t ) + χ Γ a ^ p ( t ) a ^ s ( t ) + 2 γ i a ^ i n i ( t ) + 2 μ i b ^ i n i ( t ) .
ε p t h = γ / ( χ Γ ) .
V X ^ s X ^ i = V Y ^ s + Y ^ i = 1 η e s c 4 σ ( 1 + σ ) 2 + Ω 2 ,
V X ^ s + X ^ i = V Y ^ s Y ^ i = 1 η e s c 4 σ ( 1 + σ ) 2 + Ω 2 ,
V X ^ s + X ^ i = V Y ^ s Y ^ i = 1 η det η e s c 4 p / p t h ( 1 + p / p t h ) 2 + Ω 2 ,
V = V X s + X i + V Y s Y i = 2 η det η e s c 8 p / p t h ( 1 + p / p t h ) 2 + Ω 2 < 2 .
Γ = + v 00 ( r ) [ u 00 ( r ) ] 2 d r = 1 ,
Γ = n = 0 c n + v n 0 ( r ) [ u 10 ( r ) ] 2 d r = n = 0 c n Γ n ,
Γ 0 = + v 00 ( r ) [ u 10 ( r ) ] 2 d r = 1 / 2 ,
Γ 2 = + v 20 ( r ) [ u 10 ( r ) ] 2 d r = 1 / 2 ,
V ( d B ) = 10 log 10 V ( p e r )
V X s + X i 00 + V Y s Y i 00 = 1.13 ± 0.02 < 2 ,
V X s + X i 20 + V Y s Y i 20 = 1.04 ± 0.02 < 2 ,
V X s + X i o p t + V Y s Y i o p t = 0.98 ± 0.04 < 2 .

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