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Abstract
We theoretically study the angular rotation displacement based on the hybrid interferometers, which contain a beam splitter (BS) and an optical parameter amplication (OPA) for beam splitting and recombination. Two schemes with different orders of an OPA and a BS are discussed and both of them can realize the super resolving and sensitive angular rotation displacement. The sensitivity of angular rotation displacement can surpass the shot noise limit $\frac {1}{2l\sqrt {N}}$ with the orbital angular momentum input beams. The squeezing strength of an OPA and the reflectivity of the BS play a decisive role on the resolutions and sensitivities while the losses play a negative effect on the sensitivity.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
It is well known that the photons can carry not only spin angular momentum (SAM), but also orbital angular momentum (OAM). The SAM of $\hbar$ per photon depending on the handedness can form two-dimensional Hilbert space, where $\hbar$ is the reduced Plack constant. For OAM, described by an azimuthal phase dependence exp(i$l\theta$), every photon can carry $l\hbar$ OAM where $l$ is called topological charge and can take any integer value, so the OAM is defined in an unbounded infinite-dimensional space [1–5]. This feature makes OAM widely used in quantum information [6–8], quantum detection [9,10], and quantum communication [11–13].
In the precision measurements field, lots of researches have shown that OAM can be used to enhance the sensitivity of the angular rotation measurement. In 2011, Jha et.al employed a Mach-Zehnder interferometer to study the propagation of entangled OAM modes and showed that path-entangled state photons with nonzero OAM can increase the sensitivity of angular-rotation measurements, $\frac {1}{2\sqrt {N}l}$ for N-independent photons and $\frac {1}{2lN}$ for N-entangled photons [14]. In 2013, D’Ambrosio et.al used the photon orbital angular momentum as a ’photonics gear’ and amplified the mechanical rotation angle from $\theta$ to $l\theta$[15] while the resolution can be enhanced by $l$ times. Recently, Zhang $et$. $al$ proposed that the quantum detection can improve the resolution and sensitivity based on SU(2) and hybrid interferometers, and the sensitivity can approach the Heisenberg limit (HL) $\frac {1}{2lN}$[16,17]. Furthermore, in order to beat HL, Liu et.al proved that by using an SU(1,1) interferometer, the angular rotation sensitivity can be enhanced by a factor of 2, reaching $\frac {1}{4lN}$[18].
Here, the measurement of angular rotation displacement based on hybrid interferometers with OAM input beams is revisited and the angular rotation sensitivities can beat SNL. The hybrid interferometers are composed of a beam splitter (BS) and an optical parameter amplication (OPA). We investagate the visibility, resolution and sensitivity for two situations of hybrid interferometers that the BS and the OPA have reversed position. Both of them can realize the super resolving and sensitive angular rotation displacement. The sensitivity of angular rotation displacement can surpass the SNL $\frac {1}{2l\sqrt {N}}$ with OAM input beams. This paper is organized as follows. In the next part, the hybrid interferometers which can display the angular rotation measurement theoretically are explicitily introduced. In the third part, the visibility, resolution and sensitivity of these hybrid interferometers are discussed. The factors including the loss which can enhance the resolution and sensitivity are studied. Finally, we make a conclusion.
2. Model
Figure 1 illustrates the hybrid interferometers [19–21], which consist of a BS and an OPA [22–25] for beam splitting and recombination. The position of an OPA and a BS can be reversed and they are termed as Model A and Model B. A coherent beam with OAM $l\hbar$ per photon feeds into the interferometer from input $a_\textrm {in}$. Two Dove prisms are located in each arm to impose a phase shift to the injected beams. The phase shift for the injected beams is $2l\theta$ and $\theta$ is the angle of rotation of the prism.
Fig. 1. Sketch of the hybrid interferometers with input beam carrying orbital angular momentum. (a) an interferometer with an optical parametric amplifiers (OPA) as the beam splitting element and a beam splitter (BS) as the beam recombination element; (b) reversed order of OPA and BS1 from (a). M is mirror, B is beam block and DP is Dove-prism. The rotation angle of DP1 is 0 and the rotation angle of DP2 is $\theta$.
Note the differences of the two schemes. Model A: OPA+BS with the pump beam carrying OAM and the probe beam injected into port A does not carry OAM. According to the conservation of the OAM, the generated conjugate beam carries OAM $2l\hbar$ per photon. Model B: BS+OPA with the probe beam carrying OAM. In order to simplify the interferometer, the OPA process is assumed to be a degenerate process as [21].
In the scheme of Model A, the incoming coherent state is split by an OPA and the beam recombined on a BS1. The inupt-output relations of the OPA are
For BS1, according to [20], the relationships are
Here the balance homodyne detection (BHD) is employed for the measurement of angular rotation displacement [26]. And the quadrature amplitude operator $\hat {X}$ at output port $a_\textrm {out}$ is defined as below:
In order to get the quantum fluctuations of quadrature amplitude, first, the expected value $\langle {\hat {X}}\rangle$ is calculated which can be expressed as,Next, the visibility (V) which is employed as a criterion for the interferometer is calculated. The definition of $V$ is described by $V_\textrm {A}=\frac {\langle {\hat {X}_{a}}\rangle _\textrm {max}-\langle {\hat {X}_{a}}\rangle _\textrm {min}}{\langle {\hat {X}_{a}}\rangle _\textrm {max}+\langle {\hat {X}_{a}}\rangle _\textrm {min}}$. $\langle {\hat {X}_{a}}\rangle _\textrm {max}$ and $\langle {\hat {X}_{a}}\rangle _\textrm {min}$ mean the maximum and minimum values of $\langle {\hat {X}_{a}}\rangle$. So the V of scheme A is
For the scheme B where the beam is split by a BS1 and recombination on an OPA. Similar to the scheme A, the expected value, the corresponding visibility and the angular rotation displacement are shown as
3. Analysis and discussion
In this section, the visibility, resolutions and sensitivities of the two models are discussed. From Eqs. (5) and (8), the visibilities are the same for both schemes. Figure 2 shows the visibilities versus the squeezing parameter $r$ with $T=R=0.5$. The visibility is increasing with the increase of $r$ and it can be greater than 0.99 when $r$ reaches 3 or more.
Figure 3(a) shows the normalized beating signals obtained by the hybrid interferometers with the variety of rotation angle of the Dove prism. In order to have the comparison, the result of normalized beating signal $I=N_0\textrm {cos}^2(l\theta )$ is also showed which is defined as the SNL of resolution. Here, the better resolution means the narrower full widths at half maximum (FWHM). From Fig. 3(a), the resolution of model A is better and its FWHM is only half that of the SU(2) interferometer. In this case, compared with SU(2) and scheme B, the resolution of scheme A can be enhanced by a factor of 2.
Fig. 3. (a) Normalized beating signals versus the rotation angle. (b) FWHM versus the topological charge $l$. Other parameters are $R=T=0.5$, $l=1$, $\textrm {cosh}^2r=2$ and $N_0=100$.
Figure 3(b) represents the FWHM of the normalized beating signals versus the topological charge $l$ of the input beams using method of BHD according to Eqs. (4) and (7). The resolution is improved with the increase of topological charge $l$. Furthermore, while the topological charge $l$ is same, compared with the FWHM of the SU(2) interferometer, the FWHM of model B is the same and the model A always has the half FWHM. This character is coincident with Fig. 3(a).
In this section, the sensitivities of angular rotation measurement are discussed. The minimum value of sensitivity means the optimal sensitivity, which is $\frac {1}{2lN}$ for the SU(2) interferometer and $\frac {1}{4lN}$ for the SU(1,1) interferometer. From Figs. 4(a) and 4(b), the sensitivity of model A is better than SNL which means that with only a coherent beam and a vacuum beam, the sensitivity of the hybrid interferometer can beat SNL. For model B, it is even worse than SNL. And it can be explained by Eq. (9). When $2l\theta =\frac {\pi }{2}$, $\Delta \theta$ can be optimal. However, when consider $T=R=0.5$, it will be $\Delta \theta =\frac {\sqrt {2\textrm {cosh}^2r-1}}{2l\sqrt {N_0}\sqrt {2\textrm {cosh}^2r-2}} > \frac {1}{2l\sqrt {N_0}}$. So in this case, the sensitivity is worse than SNL. At the same time, for the models, the optimal angular rotation sensitivity occurs at different rotation angles. Furthermore, in model A, the $N$ in $\frac {1}{2lN}$ and $\frac {1}{4lN}$ is $N={\textrm {cosh}(2r)}N_0+2{\textrm {sinh}^2r}$. For model B, the $N$ in $\frac {1}{2lN}$ and $\frac {1}{4lN}$ is $N_0$. This is due to the definition of SNL and HL. The $N$ is the photon number inside the interferometer. For model A, the total photon number after OPA process is $N={\textrm {cosh}(2r)}N_0+2{\textrm {sinh}^2r}$. For model B, after BS1 process, the photon number is $N_0$.
Fig. 4. (a) Sensitivities versus the rotation angle for model A and (b) for model B. (c) Optimal sensitivity with the increase of topological charge. SNL is $\frac {1}{2l\sqrt {N}}$, $\frac {1}{2lN}$ is HL. $\frac {1}{4lN}$ is the best sensitivity so far. Other parameters are $T=R=0.5$, $l=1$, $\textrm {cosh}^2r=2$ and $N_0=100$. For model A, the $N$ in $\frac {1}{2lN}$ and $\frac {1}{4lN}$ is $N={\textrm {cosh}(2r)}N_0+2{\textrm {sinh}^2r}$. For model B, the $N$ in $\frac {1}{2lN}$ and $\frac {1}{4lN}$ is $N_0$.
Figure 4(c) displays the optimal sensitivity of angular rotation measurement with the increase of topological charge. For both model A and model B, when topological charge $l$ is increasing, the optimal sensitivities are better. In addition, compared with model B, the optimal sensitivity of model A is always better. So for the sensitivity, model A is a more optimal scheme.
Next, the optimal sensitivities with the variation of BS reflectivity are investigated. For the models, as shown in Figs. 5(a) and 5(b), with the increase of reflectivity, the optimal sensitivities are better. For model B, the value of optimal sensitivities goes down dramatically as the increase of reflectivity of BS. The sensitivity is worse than SNL when the reflectivity is less or equal to 0.5 while it can beat SNL with reflectivity more than 0.8. Even reflectivity reaches 0.9, the optimal sensitivity is still worse than $\frac {1}{2lN}$. For model A, as the increase of reflectivity, from 0.1 to 0.5, the optimal sensitivity goes downly rapidly. However, with the sustained increase of $R$, the sensitivity can be enhanced smoothly and almost the same. Same to model B, with the increase of reflectivity, the optimal sensitivity can beat $\frac {1}{2l\sqrt {N}}$ and approach $\frac {1}{2lN}$.
Fig. 5. (a) Optimal sensitivity with the reflectivity for model A and (b) for model B. (c) Optimal sensitivity with gain G for model A and (d) for model B. Other parameters are same to Fig.4. Here, $G=\textrm {cosh}^2r$. For better comparision, $\textrm {log}_{10}\Delta \theta$ are used in this figure.
From Figs. 5(c) and 5(d), as predicted, even the gain is much bigger when $R$ is 0.5, the optimal sensitivity of model B never can reach SNL. For model A, the sensitivity can beat $\frac {1}{2l\sqrt {N}}$. And optimal sensitivities are better with the increase of gain $G$ while they all change smoothly. However, only when $G=2$ or $G=3$, the optimal sensitivity of model A can beat SNL.
4. Effect of loss on the sensitivity
Here, the effect of loss on the sensitivity of angular rotation measurement are studied. By assuming that there are two fictitious beam splitters [27,28] in the interferometers and the transmissivities are $T_1$ and $T_2$ as shown in Fig. 6. For the fictitious beam splitters, the inupt-output relations $\hat {a}_{2}=\sqrt {T_2}\hat {a}_{1}+\sqrt {1-T_2}\hat {v}_{0}$ and $\hat {b}_{2}=\sqrt {T_1}\hat {b}_{1}+\sqrt {1-T_1}\hat {v}_{0}$. $\hat {v}_{0}$ is the annihilation operator from the vacuum. Then the sensitivities of angular rotation measurement including the loss can be written as
Fig. 6. Sketch of the hybrid interferometers with input beam carrying OAM with transmissivity $T_1$ and $T_2$. FBS are fictitious beam splitters.
Figures 7(a) and 7(b) show the optimal sensitivities which vary with different transmissivities for model A and model B. In the experiments, the transmissivity usually is much bigger than 0.5. So the condition that both $T_1$ and $T_2$ are from 0.5 to 1 is considered. For model A and model B, when $T_1=1$ and $T_2=1$, the sensitivities are optimal. When $T_1=0.5$ and $T_2=0.5$, the sensitivities are the worst. Notice that for model A, when $T_1=T_2=0.5$, the sensitivity is still better than that of model B with $T_1=T_2=1$. At the same time, $T_1$ and $T_2$ do not play the same role on the sensitivity. For model A, when $T_1=0.85$ and $T_2=1$, the sensitivity still can be optimal. And the sensitivity is not optimal when $T_1=1$ and $T_2=0.85$. Meanwhile, for model B, when $T_2 \geq 0.95$ and $T_1=1$, the sensitivity can be optimal. For model A, it can be optimal only with $T_2 \geq 0.9$ and $T_1=1$. In addition, the sensitivity ranges of model A are narrower which means that compared with model B, model A is a more robust scheme.
Fig. 7. Optimal sensitivities with the variance of transmissivities $T_1$ and $T_2$. (a) is model A and (b) is model B. Other parameters are $T=R=0.5$, $l=1$, $\textrm {cosh}^2r=2$ and $N_0=100$.
5. Conclusion
In summary, this paper proposes two hybrid interferometers which can realize the super-resolving and ultrasensitive angular rotation displacement. When the input beams are a coherent beam and a vacuum beam, the visibilities of both hybrid interferometers can be close to 1. The hybrid interferometer, whose order is an OPA and a BS with the pump beam carrying OAM, can enhance the resolution by a factor of 2. It is also possible to realize ultra-sensitive angular rotation measurement by altering the reflectivity and squeezing strength $r$. Different from the previous schemes, these hybrid interferometers use a degenerate OPA and do not need a frequency converter. The schemes extend the application of optical interferometers and have potential applications in precision measurements.
Funding
National Natural Science Foundation of China (11534008, 11574247, 11605126, 11774286); Natural Science Foundation of Shaanxi Province (2019JM-279).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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