Measurement of Stokes-operator squeezing for continuous-variable orbital angular momentum

We demonstrate experimentally a measurement scheme for the Stokes operators for the continuous-variable squeezed states of orbital angular momentum (OAM). An OAM squeezed state is generated by coupling a dim Hermite-Gauss HG01-mode quadrature-squeezed light beam with a bright HG10-mode coherent light beam on a 98/2 beam splitter. Using an asymmetric Mach–Zehnder interferometer with an extra Dove prism in one arm, we measured the three orbital Stokes operators of the OAM squeezed states with a self-homodyne detection and finally characterized their positions and noise on the orbital Poincaré sphere.


Orbital angular momentum squeezed state
Similar to the polarization of light, the first-order spatial modes can also be characterized by the orbital Stokes operators and mapped onto the orbital Poincaré sphere 16,19 . Such a sphere is displayed in Fig. 1 for the first-order OAM modes. where Ô 0 , Ô 1 , Ô 2 and Ô 3 denote respectively operators corresponding to the total number of photons, the difference in photon number between modes HG 10 and HG 01 modes, and likewise between the pair of modes  HG 10 45 and  HG 10 135 , and modes + LG 0 1 and − LG 0 113, 15 . ˆ † a 10(01) and â 10(01) are the creation and annihilation operators for the HG 10 (HG 01 ) modes, ϕ is the phase difference between modes HG 10 and HG 01 . The definitions of Ô 2 and Ô 3 in Eq. (1) contain the product of annihilation operators relative to two different field modes. This in turn implies that to have an apparatus able to effectively realise this operator, spatial and temporal profiles of the two modes have to match optimally (perfectly in the ideal case) and a control on the relative phase between the two modes is needed.

Detection scheme
To measure the three orbital Stokes operators (see Fig. 3), we propose a scheme based on the asymmetric Mach-Zehnder interferometer with a Dove prism in one arm. The Dove prism is used to convert a HG 10 mode into a HG 01 mode or vice versa. In an asymmetric Mach-Zehnder interferometer, there are two mirrors M1 and M2 in arm a 16,20 that add an extra phase e iπ to the HG 10 mode but have no effect on the HG 01 mode. M1 and M2 are the same for the first three schemes (1), (2), and (3) used in the detection of ˆˆÔ O O , , 0 1 2 . In scheme (4) for Ô 3 , unlike the first three schemes, there is only a single mirror in a arm, and hence creates a symmetric Mach-Zehnder interferometer having a Dove prism in b arm.
Any first order spatial mode ψ can be expressed as i 01 01 10 10 where  u r ( ) 01 (10) are the normalized transverse beam amplitude functions for modes HG 01 (HG 10 ), and ϕ is the phase difference between modes HG 10 and HG 01 . An account of the quantum vacuum noise entering the setup through the unused port BS1 is introduced by defining the operator i 01 01 10 10 where ν a 01(10) are the quantum vacuum noise operators for HG 01 (HG 10 ) modes. With the presence of the two mirrors M1 and M2, in a arm for schemes (1) and (2), the HG 10 mode receives an extra phase e iπ , that is, u 10 → −u 10 , while HG 01 mode is not changed, u 01 → u 01 ; hence, a i i 01 01 10 10 01 01 10 10 whereas in b arm, we have b i i 01 01 10 10 01 01 10 10 The two output states of beam splitter BS2 are where θ is the relative phase between the two arms of the interferometer. When θ = 0, the sum of the two photocurrents is   (3), the equation is the same as Eq. (5), whereas in b arm, the Dove prism is used to rotate the mode by 90°, that is, u 01 → u 10 , u 10 → −u 01 , and therefore b i i 01 10 10 01 01 10 10 01 when θ = 0. The difference between the two photocurrents is  whereas in b arm, with the Dove prism inserted, the state is the same as Eq. (10).
i i 1 1 2 2 2 2 10 01 01 10 3 which is the fourth orbital Stokes operator [see Eq. (1)]. Therefore, we can use the detection scheme to measure the four orbital Stokes operators 16 , and the quantum vacuum noise has no effect on the results. Considering the imperfection of the setup and assuming the mode conversion efficiency η 1 of the two mirrors M1 and M2 and η 2 of the Dove prism, the modes through the conversions of mirrors M1 and M2 become  10 . Therefore, the imperfections will cause the coupling of different Stokes operators, degrading the detection accuracies for the three Stokes operators. On the other hand, considering the mode matching efficiency ξ 2 between the two arms caused by misalignment of Mach-Zehnder interferometer, the optical transmission efficiency η tr and the quantum efficiency of photodetectors η phot , the noise of Ô 1 becomes η η ηη det , so all these inefficiencies will introduce the vacuum noise and degrade the detection efficiencies for Ô 1 , Ô 2 and Ô 3 , further degrading the degree of the measured squeezing in experiment.

Experimental set-up
Referring to the experimental set-up illustrated in Fig. 4, a two mode squeezed state of 1080 nm for HG 01 mode is generated from a NOPA 15,21 , then the bright mode which is an amplitude squeezed state is separated by the combination of a half wave plate and a polarizing beam splitter 22 . The HG 01 mode squeezed state is firstly detected by a balanced homodyne detection with the flip mirror F1 on, and the squeezing value is obtained. Then F1 is turned off, the HG 01 squeezed state with power of 30 μW is coupled with a bright coherent HG 10 mode of 1080 nm with power of 100 mW on a 98/2 beam splitter, ensuring the squeezing power of 98% is transmitted and the coherent power of 2% is reflected, generating the OAM squeezed state. As the definition in Eq. (1), the coupling of HG 01 mode and HG 10 mode requires mode matching and phase locking, since the HG 01 mode and the HG 10 mode are two orthogonal modes, we assess the mode matching by the interference between the HG 00 modes which are eigenmodes orthogonal with the HG 10 modes on the 98/2 beam splitter. In addition, we use an iris to acquire part of the interference of HG 01 and HG 10 modes to control the relative phase through a servo system and PZT1, generating different types of OAM squeezed states. We lock the relative phase ϕ to 0 or π 2 in the experiment. The OAM squeezed states are measured implementing the scheme discussed in the previous section (see Fig. 3). As shown in Fig. 4, we use a flip mirror (F2) to choose where the OAM squeezed state goes. If F2 is off/on, the state goes to the detection scheme for Ô 1 /Ô 2 and Ô 3 . The other two flip mirrors, F3 and F4, are used to determine whether Ô 2 or Ô 3 is detected; if F3 and F4 are both off/on, the asymmetric/symmetric Mach-Zehnder interferometer is active, and hence Ô 2 /Ô 3 is detected. PZT2 and PZT3 are used to lock the relative phases θ between the two arms of the Mach-Zehnder interferometers. When Ô 1 and Ô 2 are being measured, θ is locked to zero; when Ô 3 is being measured, θ is locked to π 2 . The two outputs of the interferometers enter two photodetectors, and the photocurrents feed a positive/negative combiner (+/−), and these outputs are recorded by a spectrum analyser (SA). A positive combiner (+) determines the SNL; a negative combiner (−) determines the noise of the orbital Stokes operators.
In our scheme, a local oscillator is unneeded, so it is more efficient in certain nonlocal quantum information protocols, such as free-space quantum state distribution 17,18 which has the potential to form a key component in future quantum networks. In the squeezing enhanced CV quantum key distribution (QKD) protocols, the decoherence as a result of phase relation variations and wave front distortions plays an important role in the degradation of the quantum states, thus standard homodyne measurements at the receiver are challenging 17 . Similar to polarization squeezed state, the OAM squeezed state based on our measurement scheme of Stokes operators is promising to supply a way to avoid the problem. Moreover, it can be expanded to high-dimensional CV QKD based on high-dimensional OAM. Figure 5 gives the squeezing curves for HG 01 mode, (a) is the squeezing and anti-squeezing values for HG 01 mode from 1 MHz to 30 MHz. The squeezing exists over a large frequency domain of 1-30 MHz; (b) is the noise power for HG 01 mode at 5 MHz. The squeezing value is −3.01 ± 0.03 dB at 5 MHz. Considering the overlap efficiency in balanced homodyne detection η hd = 0.93 ± 0.01, and the quantum efficiency of the photodiode η phot = 0.90 ± 0.02, the inferred squeezing is −3.95 ± 0.12 dB.

Experimental Results
The measured noise powers for the OAM Stokes operators are depicted in Fig. 6. The quantum noise for the first Stokes operator Ô 1 is almost shot noise limited over the frequency domain 1-30 MHz for both ϕ = 0 and  ϕ = π/2. At 5 MHz, the quantum noise for Ô 1 is −0.07 ± 0.25 dB for ϕ = 0 and 0.12 ± 0.22 dB for ϕ = π/2. When ϕ = 0, the quantum noise for the second Stokes operator Ô 2 is squeezed. The squeezing exists over a large frequency domain of 1-30 MHz; the squeezing of −1.70 ± 0.15 dB at 5 MHz is obtained. The third Stokes operator Ô 3 is anti-squeezed, the anti-squeezing noise is 5.06 ± 0.06 dB at 5 MHz. When ϕ = π/2, the quantum noise for the third Stokes operator Ô 3 is squeezed,and the squeezing also exists over a large frequency domain of 1-30 MHz, and the squeezing of-1.96 ± 0.16 dB at 5 MHz is obtained. The second Stokes operator Ô 2 is anti-squeezed, the anti-squeezing noise is 5.06 ± 0.03 dB at 5 MHz. The peak at 18 MHz is a modulation signal for phase locking.
In experiment, the maximum coupling efficiency on the 98/2 beam splitter for HG 01 and HG 10 modes is η coup = 0.93 ± 0.02, which is the mode matching efficiency of the HG 00 modes, and considering the loss of 2% of the squeezed state, the total efficiency is η tot = 0.91 ± 0.02. After the 98/2 beam splitter, the squeezing of the Stokes operators Ô 2 and Ô 3 should be −3.41 ± 0.2 dB at 5 MHz, which is inferred from the squeezing of −3.95 ± 0.12 dB for HG 01 mode.
Here, we use the HG 00 mode interference to estimate the mode matching efficiency of the 98/2 beam splitter, the practical coupling efficiency for HG 01 and HG 10 modes on the 98/2 beam splitter should be lower than the estimate value, so the measured squeezing for Ô 2 and Ô 3 in experiment are lower than the inferred squeezing values. But for balanced homodyne detection 13,15 , the measurement results of squeezing are independent of the mode matching efficiency between HG 01 and HG 10 modes on the 98/2 beam splitter, and it can't infer accuracy results for Stokes operators.
The OAM squeezed states at 5 MHz were mapped onto the orbital Poincaré sphere (Fig. 7); (a) is the orbital Poincaré sphere for ϕ = 0, (b) is the orbital Poincaré sphere for ϕ = π/2, (a 1 ) and (b 1 ) show positions and forms of the OAM squeezed states for ϕ = 0 and ϕ = π/2. As the HG 10 mode is a bright coherent state, the OAM states are therefore positioned on the positive part of the Ô 1 axis. (a 2 ) shows the sphere for the quantum noise of the OAM squeezed state with ϕ = 0. Here ∆Ô 1 is SNL, ∆Ô 2 is squeezed, and ∆Ô 3 is anti-squeezed; hence it is pancake shaped. Similarly (b 2 ) shows the sphere for the state with ϕ = π/2. Note ∆Ô 1 is still SNL, with ∆Ô 2 anti-squeezed and ∆Ô 3 squeezed, and therefore also pancake shaped. The experimental results agree with Eq. (3) well.

Conclusion
The CV OAM squeezed states have great potential in high-dimensional quantum information processing, super-resolution quantum imaging, quantum precise measurement, and quantum storage. We demonstrated experimentally a new measurement scheme for the Stokes operators of the first-order OAM squeezed state. The OAM squeezed states are generated by coupling a HG 01 squeezed state with a bright coherent HG 10 mode on a 98/2 beam splitter. With the scheme, we measured the squeezing of the Stokes operators. The experiment demonstrates that the scheme is effective and efficient. The CV OAM states with the detection scheme is promising for applications in nonlocal quantum information, and the scheme may be extended to high-order CV OAM states for high-dimensional quantum information processing.