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Phase sensitivity of a three-mode nonlinear interferometer
Optics Communications 452 (2019) 189–194
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Phase sensitivity of a three-mode nonlinear interferometer Chao-Ping Wei a ,∗, Zhao-Ming Wu a , Cheng-Zhi Deng a , Li-Yun Hu b ,∗ a
Jiangxi Province Key Laboratory of Water Information Cooperative Sensing and Intelligent Processing, Nanchang Institute of Technology, School of Information Engineering, Nanchang Institute of Technology, Nanchang 330022, China b Center for Quantum Science and Technology, Jiangxi Normal University, Nanchang 330022, China
ARTICLE
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Keywords: Nonlinear interferometry Phase sensitivity Photon losses
ABSTRACT In this paper, we construct the input–output relationships of a three-mode nonlinear beam splitter (TNBS) mathematically, and a set of special solutions for the input–output equations are given. By combining two TNBSs and a phase shifter, we propose a scheme for a three-mode nonlinear interferometer (TNI) and give the input–output expression of the whole TNI. Then, we study the phase sensitivity of the TNI by giving input states and detection scheme. In addition, we also calculate the phase sensitivity of a Mach–Zehnder interferometer (MZI), SU(1,1) interferometer (SU(1,1)I) and modified Mach–Zehnder interferometer (MMZI), and we discuss the relationships of phase sensitivity between the MZI, SU(1,1)I, MMZI and TNI. Finally, in the presence of photon losses, we discuss the phase sensitivity of the four different interferometers and make detailed comparisons. It is shown that the TNI has the best robustness against photon losses under certain conditions with the same parameters.
1. Introduction Phase estimation theory is a major branch in the field of quantum metrology [1], and optical interferometers have become the main tools of phase estimation [2]. How to improve the phase sensitivity of an interferometer is one of the main research focuses in phase estimation theory. At present, there are three main approaches to improving an interferometer’s phase sensitivity. The first one employs some special entanglement, squeezed and superposition [3–5], to increase directly the couple with phase information; the second involves using some ultra-sensitive detection schemes to achieve the maximum phase information [6]; the third straightly remoulds the structure of the interferometer, such as by adding some nonlinear effects including nonlinear parameter processes [7,8] or nonlinear phase shift operations [9– 11] to the original interferometer, thus enabling the realization of improvements to the intercoupling between the interferometer and phase information. At present, studies on the first and second methods are relatively mature, and physicists have completed numerous relevant research on improving the interferometer’s phase sensitivity by changing input states or detection schemes. For traditional two-mode interferometers, some investigations on improving the phase sensitivity have been done by the third method. However, there are relatively few studies for multi-mode interferometers up to now, especially for multi-mode nonlinear interferometers. The earliest theory model for multiple phase estimations can be found in Refs. [12,13]. Besides, it has been shown that, given a fixed total photon number, the precision of
estimating multiple phases outperforms individual quantum estimation schemes. However, it would be interesting to extend the traditional dual-mode nonlinear beam splitter to a multi-mode nonlinear beam splitter and to construct a model for the multi-mode nonlinear interferometer to study the influence of nonlinear operations on the interferometer’s phase sensitivity. These problems need to be further studied. In this paper, by extending a dual-mode nonlinear beam splitter, we construct a three-mode nonlinear beam splitter model, and we give its mathematical description, corresponding physical model and implementation scheme. By combining two three-mode nonlinear beam splitters (TNBSs) and a phase shifter (PS), a new frame diagram for a three-mode nonlinear interferometer (TNI) is proposed. In the interferometer, the effect of TNBS on improved phase estimation accuracy is studied. Subsequently, we discuss the relationship between the phase sensitivity of four interferometers, namely, a Mach–Zehnder interferometer (MZI), SU(1,1)I, modified MZI (MMZI), and TNI, under different parameters. Finally, we discuss the phase sensitivity of the four interferometers in the presence of photon losses. The organization of this paper is as follows. In Section 2, the architecture of the TNI is introduced in detail, and the corresponding input–output operator relationships are given. In Section 3, we study the phase sensitivity of the TNI and make detailed comparisons with three other familiar interferometers. In Section 4, we discuss the effect of photon losses on the phase sensitivity, and we present an analysis of
∗ Corresponding authors. E-mail addresses: [email protected] (C.-P. Wei), [email protected] (L.-Y. Hu).
https://doi.org/10.1016/j.optcom.2019.07.033 Received 28 April 2019; Received in revised form 17 June 2019; Accepted 16 July 2019 Available online 18 July 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.
C.-P. Wei, Z.-M. Wu, C.-Z. Deng et al.
Optics Communications 452 (2019) 189–194
𝑐1 = 𝑀31 𝑎†𝑖 + 𝑀32 𝑏𝑖 + 𝑀33 𝑐𝑖 .
(1)
These operators and their Hermitian conjugates satisfy the following boson commutation relations: [𝑥1 , 𝑦1 ] = [𝑥†1 , 𝑦†1 ] = 0, [𝑥1 , 𝑦†1 ] = 𝛿𝑖𝑗 , where 𝑥, 𝑦, 𝑖 and 𝑗 take on the values 𝑎, 𝑏 and 𝑐. When 𝑖 = 𝑗, 𝛿𝑖𝑖 = 1, otherwise 𝛿𝑖𝑗 = 0. Explicitly, by extending these commutation relationships, one can obtain the series of following expressions as |𝑀11 |2 − |𝑀12 |2 − |𝑀13 |2 = 1, | | | | | | |𝑀22 |2 − |𝑀21 |2 + |𝑀23 |2 = 1, | | | | | | |𝑀33 |2 − |𝑀31 |2 + |𝑀32 |2 = 1, | | | | | | 𝑀11 𝑀21 − 𝑀12 𝑀22 − 𝑀13 𝑀23 = 0, 𝑀11 𝑀31 − 𝑀12 𝑀32 − 𝑀13 𝑀33 = 0, Fig. 1. General sketch of a TNI with a series of input states |𝜓⟩𝑖 as inputs. The input states inject into the first TNBS, respectively. Then after applying a PS 𝑈 (𝜙) to 𝑎𝑖 mode of the path, these light are combined in the second TNBS. Finally, we can execute an adaptive detection in each output port.
∗ ∗ ∗ −𝑀21 𝑀31 + 𝑀22 𝑀32 + 𝑀23 𝑀33 = 0, ∗ ∗ ∗ −𝑀21 𝑀31 + 𝑀22 𝑀32 + 𝑀23 𝑀33 = 0.
(2)
A set of special solutions for the above equations are 𝑀11 = cosh2 𝑔, 𝑀12 = cosh 𝑔 sinh 𝑔, 𝑀13 = sinh 𝑔,
the robustness against photon losses for MZI, SU(1,1)I, MMZI and TNI. Section 5 presents the conclusions.
𝑀21 = sinh 𝑔, 𝑀22 = cosh 𝑔 sin 𝜃, 𝑀23 = cosh 𝑔 cos 𝜃, 𝑀31 = sinh 𝑔, 𝑀32 = cosh 𝑔 sin 𝜃, 𝑀33 = cosh 𝑔 cos 𝜃,
(3)
2. Three-mode nonlinear interferometer where, 𝑔 is regarded as the gain factor of the nonlinear process, and 𝜃, 𝑔 are constrained by 𝜃 = arcsin(cosh 𝑔 − 1). In addition, one can also calculate the total average photon number inside the interferometer, i.e., after the⟩TNBS, as the total average photon number 𝑁 = ⟨ 𝑎†1 𝑎1 + 𝑏†1 𝑏1 + 𝑐1† 𝑐1 = 𝛼 2 (cosh4 𝑔 + 2 sinh2 𝑔) + 4 sinh2 𝑔 + sinh4 𝑔, which
In some real scenarios, such as quantum imaging, magnetic imaging and gravitational field imaging [14,15], it is necessary to measure multiple parameters simultaneously. This makes multi-parameter estimation very attractive from a technological point of view. The physicists have conducted a great deal of theoretical research on multi-parameter estimation. For example, in Ref. [16], S. Adhikari et al. showed that this simple strategy can give asymptotically post-classical sensitivity for multi-parameter estimation even when the number of modes is large. Further, it has been shown that the quantum Cramér–Rao bound can be saturated for a multi-mode interferometer with a phase in each mode by using Gaussian inputs and passive elements [17]. In Ref. [18], the authors study a nonlinear interferometer consisting of two consecutive parametric amplifiers and achieve a Heisenberg scaling. Experimentally, multi-mode optical interferometers are the key devices for multiple phase estimations simultaneously, and these can be realized by cascading several multi-mode beam splitters (MBSs) and PSs [19]. Relevant research about multiple phase estimations and multi-mode interferometers can be seen in Refs. [20–22]. These results show that the simultaneous estimation of multiple phases in multiple interferometers can provide for better total precision than estimating them individually. Many studies, such as those described above, have shown that use of a multi-mode interferometer has enormous advantages and the potential for multiple phase estimations simultaneously. Such multi-mode nonlinear interferometers can be constituted by adding some nonlinear processes, including nonlinear phase shift operations or NMBS, to an TNI. Wei et al. proposed a multi-mode nonlinear interferometer and studied the phase sensitivity of three and four modes in the nonlinear interferometer with single photons as inputs [23]. In Ref. [24], the problem of two phase estimations simultaneously in a TNI was studied by the use of quantum Fisher information matrix and classical Fisher information matrix methods. In this section, we will construct a TNI by cascading two TNBSs and a PS, as shown in Fig. 1. It can also be seen as a modified MZI, in which the conventional linear beam splitters are replaced by TNBS. In the following, we mainly study the input–output transformation relationships, including TNBS and the whole TNI. According to the Lie group and SU(1,2) group theory method [25,26], we can define the relationships of the input and output modes of TNBS as follows:
is amplified by a cosh4 𝑔 scaling factor. After applying a conventional † PS 𝑈 (𝜙) = e𝑖𝜙𝑎1 𝑎1 to 𝑎1 modes, and using the basis of the transform relationship of 𝑎1 mode (𝑎1 → 𝑎1 𝑒𝑖𝜙 ) and the second TNBS, one can obtain the final input–output transform relationships of the whole interferometer as follows: 𝑎𝑜 = 𝑁11 𝑎𝑖 + 𝑁12 𝑏†𝑖 + 𝑁13 𝑐𝑖† , 𝑏𝑜 = 𝑁21 𝑏𝑖 + 𝑁22 𝑐𝑖 + 𝑁23 𝑎†𝑖 , 𝑐𝑜 = 𝑁31 𝑏𝑖 + 𝑁32 𝑐𝑖 + 𝑁33 𝑎†𝑖 ,
(4)
where, the corresponding coefficients are as follows: 𝑁11 = cosh4 𝑔𝑒𝑖𝜙 + cosh 𝑔 sinh2 𝑔 + sinh2 𝑔, 𝑁12 = cosh3 𝑔 sinh 𝑔𝑒𝑖𝜙 + cosh2 𝑔 sinh 𝑔 sin 𝜃 + cosh 𝑔 sinh 𝑔 sin 𝜃, 𝑁13 = cosh2 𝑔 sinh 𝑔𝑒𝑖𝜙 + cosh2 𝑔 sinh 𝑔 cos 𝜃 + sinh 𝑔 cosh 𝑔 cos 𝜃, 𝑁21 = cosh 𝑔 sinh2 𝑔𝑒−𝑖𝜙 + cosh2 𝑔 sin2 𝜃 + cosh2 𝑔 sinh 𝜃 cos 𝜃, 𝑁22 = sinh2 𝑔𝑒−𝑖𝜙 + cosh2 𝑔 sinh 𝜃 cos 𝜃 + cosh2 𝑔 cos2 𝜃, 𝑁23 = cosh2 𝑔 sinh 𝑔𝑒−𝑖𝜙 + cosh 𝑔 sinh 𝑔 sin 𝜃 + cosh 𝑔 sinh 𝑔 cos 𝜃, 𝑁31 = cosh 𝑔 sinh2 𝑔𝑒−𝑖𝜙 + cosh2 𝑔 sinh2 𝜃 + cosh2 𝑔 sinh 𝜃 cos 𝜃, 𝑁32 = sinh2 𝑔𝑒−𝑖𝜙 + cosh2 𝑔 sinh 𝜃 cos 𝜃 + cosh2 𝑔 cos2 𝜃, 𝑁33 = cosh2 𝑔 sinh 𝑔𝑒−𝑖𝜙 + cosh 𝑔 sinh 𝑔 sin 𝜃 + cosh 𝑔 sinh 𝑔 cos 𝜃.
(5)
So far, we have obtained the input–output relationships of the threemode nonlinear interferometer as shown in Eq. (4). Theoretically, the TNBS can be constructed by cascading four four-wave-mixers [26] and can be realized with current experimental technology [27]. As shown in Refs. [28,29], the three-mode linear beam splitter (tritter) can be realized by an ultrafast laser writing technique, and the twomode nonlinear beam splitter can be realized by an optical parametric amplification process [30] or spontaneous down-conversion [31]. Therefore, the TNBS may be integrated by combining the ultrafast laser writing technique with one of the optical parametric amplification processes and spontaneous down-conversion. In brief, we can obtain a three-mode nonlinear interferometer by cascading two TNBSs and a PS.
𝑎1 = 𝑀11 𝑎𝑖 + 𝑀12 𝑏†𝑖 + 𝑀13 𝑐𝑖† , 𝑏1 = 𝑀21 𝑎†𝑖 + 𝑀22 𝑏𝑖 + 𝑀23 𝑐𝑖 , 190
C.-P. Wei, Z.-M. Wu, C.-Z. Deng et al.
Optics Communications 452 (2019) 189–194
Fig. 2. The phase sensitivity of the three-mode nonlinear interferometer (TNI), Mach–Zehnder interferometer (MZI), SU(1,1)I, and modified MZI (MMZI) as a function of the average photon number in coherent states 𝑁𝛼 = |𝛼|2 . (a) g = 0; (b) g = 1.5. Black dot-dashed line: TNI; red dashed line: MZI; blue dotted: MMZI; green solid line: SU(1,1)I.
3. Phase sensitivity of the three-mode nonlinear interferometer with coherent state inputs
in Ref. [37], the relationship of output operator 𝑎𝑜 and input operators 𝑎𝑖 , 𝑏𝑖 is 𝑎𝑜 =
Recently, optical quantum metrology in linear multi-mode interferometers has been studied with single photon inputs, and it has been shown that the phase sensitivity can beat the shot-noise limit and approach to quantum Cramé r–Rao bound [32,33]. Experimentally, Szigeti et al. found that, pumped-up SU(1,1) interferometers with spinor Bose–Einstein condensates (BECs) (three-mode) or hybrid atom-lights (four-mode) are capable of surpassing the shot-noise limit with respect to the total number of input particles and are never worse than conventional SU(1,1) interferometry [34]. Su et al. observed the generalized Hong–Ou–Mandel effect with up to four photons in a quantum Fourier transform circuit and demonstrated optical phase supersensitivities deterministically [35]. In this section, we mainly study the phase sensitivity of the threemode nonlinear interferometer with coherent states inputs. Here, to obtain the phase information about 𝜙, a detection scheme should be executed in the output ports. There are several detection methods, for example, homodyne detection, parity detection, intensity detection, and so on. Here, we use the homodyne detection method at port 𝑎𝑜 . We define the quadrature operator 𝑋 as 𝑋=
𝑎𝑜 + 𝑎†𝑜 √ . 2
| 𝜕 ⟨𝑋⟩ | 𝛼 2 sin2 𝜙 | | , | 𝜕𝜙 | = 2 | | 1 (11) 𝛥2 𝑋 = (1 + cos 𝜙). 4 For a nonlinear SU(1,1)I, is composed of two optical parametric amplifiers and a phase shifter [38], the relationship of output operator 𝑎𝑜 and input operators 𝑎𝑖 , 𝑏𝑖 is 𝑎𝑜 = [𝑒𝑖𝜙 cosh 𝑔2 cosh 𝑔1 + 𝑒𝑖𝜃2 −𝜃1 sin 𝑔2 sinh 𝑔1 ]𝑎𝑖 − [𝑒𝑖𝜙 𝑒𝑖𝜃1 cosh 𝑔2 sin 𝑔1 + 𝑒𝑖𝜃2 sin 𝑔2 cosh 𝑔1 ]𝑏†𝑖 .
(12)
At equilibrium, i.e., 𝑔1 = 𝑔2 = 𝑔, 𝜃2 − 𝜃1 = 𝜋, 𝜃1 = 0, Eq. (12) reduces to 𝑎𝑜 = [𝑒𝑖𝜙 cosh2 𝑔 − sin2 𝑔]𝑎𝑖 − [𝑒𝑖𝜙 cosh 𝑔 sin 𝑔 − sin 𝑔 cosh 𝑔]𝑏†𝑖 .
(13)
For the same input states |𝛼⟩𝑎𝑖 |0⟩𝑏𝑖 , we can obtain 𝜕 ⟨𝑋⟩ = 2𝛼 2 sin2 𝜙 cosh4 𝑔, 𝜕𝜙 1 (14) 𝛥2 𝑋 = (cosh4 𝑔 − 2 sinh2 𝑔 cosh2 𝑔 cos 𝜙 + sinh4 𝑔). 2 Similarly, the MMZI can be treated as a variation of the MZI or SU(1,1)I, and it is composed of a beam splitter, an OPA and a phase | 𝜕⟨𝑋⟩ |2 shifter [7]. We can calculate the value of | 𝜕𝜙 | and variance of ⟨𝑋⟩ | | for MMZI as follows: 𝜕 ⟨𝑋⟩ = 𝛼 2 sin2 𝜙 cosh2 𝑔, 𝜕𝜙 1 𝛥2 𝑋 = [cosh2 𝑔 + sinh2 𝑔 − 2 cosh 𝑔 sinh 𝑔 cos 𝜙]. (15) 4 Substituting Eqs. (9), (11), (14) and (15) into Eq. (7), we can obtain the phase sensitivity of TNI, MZI, SU(1,1)I and MMZI, respectively. We can also plot the phase sensitivity as a function of different parameters as shown in Figs. 2 and 3. We can see that the phase sensitivity of the TNI, SU(1,1) and MMZI is equal and better than that of the MZI in the case of gain factor 𝑔 = 0. For given 𝑔 = 1.5, the relationship of their phase sensitivity is: 𝛥𝜙𝑇 𝑁𝐼 < 𝛥𝜙𝑆𝑈 (1,1)𝐼 < 𝛥𝜙𝑀𝑀𝑍𝐼 < 𝛥𝜙𝑀𝑍𝐼 , which means the result of TNI is the best for the four interferometers. It is should be noted that the MZI, SU(1,1) and MMZI all have two input– output paths, but the TNI has three input–output paths. Therefore, in order to make a better comparison among the four interferometers, we also plot the phase sensitivity as a function of the total average photon number in the interferometer 𝑁, as shown in Fig. 3. It can be seen that, for a fixed total photon number and given gain factor 𝑔, the three nonlinear interferometers surpasses the MZI. This can be interpreted as the gain factor 𝑔 playing a crucial role in the enhancement of the
(6)
𝛥2 𝑋 , (7) |𝜕 ⟨𝑋⟩ ∕𝜕𝜙|2 ⟨ ⟩ where, 𝛥2 𝑋 = 𝑋 2 − ⟨𝑋⟩2 is the variance of 𝑋. We consider a coherent input state in mode 𝑎𝑖 and vacuum state in mode 𝑏𝑖 and 𝑐𝑖 (|𝛼⟩𝑎𝑖 |0⟩𝑏𝑖 |0⟩𝑐𝑖 ), and 𝛼 is real number; then, the average value of ⟨𝑋⟩ is (𝛥𝜙)2 =
(8)
| 𝜕⟨𝑋⟩ |2 Similarly, we can calculate the value of | 𝜕𝜙 | and variance of ⟨𝑋⟩ | | as follows: | 𝜕 ⟨𝑋⟩ |2 2 8 2 | | | 𝜕𝜙 | = 2𝛼 sin 𝜙 cosh 𝑔, | | 1 𝛥2 𝑋 = [cosh8 𝑔 + 2 cosh4 𝑔 sinh2 𝑔(cosh 𝑔 + 1) cos 𝜙 2 + sinh4 𝑔(cosh 𝑔 + 1)2 ].
(10)
| 𝜕⟨𝑋⟩ |2 Consider input states |𝛼⟩𝑎𝑖 |0⟩𝑏𝑖 , we can calculate the value of | 𝜕𝜙 | | | and variance of ⟨𝑋⟩ as
where, 𝑎𝑜 = 𝑁11 𝑎𝑖 +𝑁12 𝑏†𝑖 +𝑁13 𝑐𝑖† is the operator at the out-port 𝑎 mode. The phase sensitivity 𝛥𝜙 of an interferometer can be characterized by the error propagation formula [36]
𝛼 ⟨𝑋⟩ = √ [cosh4 𝑔(𝑒𝑖𝜙 + 𝑒−𝑖𝜙 ) + 2 cosh 𝑔 sinh2 𝑔 + 2 sinh2 𝑔]. 2
1 𝑖𝜙 [(𝑒 + 1)𝑎𝑖 + (𝑒𝑖𝜙 − 1)𝑏𝑖 ]. 2
(9)
The phase sensitivity of MZI, SU(1,1)I and MMZI can be calculated in the same manner. For a conventional MZI, which is composed with two beam splitter and a phase shifter, and it is introduced detailedly 191
C.-P. Wei, Z.-M. Wu, C.-Z. Deng et al.
Optics Communications 452 (2019) 189–194
Fig. 3. The phase sensitivity of the TNI, MZI, SU(1,1) and MMZI as a function of total average photon number 𝑁. Here, we choose 𝑁 values by giving a gain factor 𝑔 = 0.2. Black dot-dashed line: TNI; red dashed line: MZI; blue dotted: MMZI; green solid line: SU(1,1)I.
Fig. 5. Phase sensitivity 𝛥𝜙 as a function of the internal transmissivity rates T with 𝑔 = 1.5 and 𝑁𝛼 = 4. Black dot-dashed line: TNI; blue dotted: MMZI; green solid line: SU(1,1)I; red dashed line: MZI.
𝑐1 = 𝑀31 𝑎†𝑖 + 𝑀32 𝑏𝑖 + 𝑀33 𝑐𝑖 , √ √ 𝑎2 = 𝑇1 𝑎1 + 1 − 𝑇1 𝑣𝑎1 , √ √ 𝑏2 = 𝑇2 𝑏1 + 1 − 𝑇2 𝑣𝑏1 , √ √ 𝑐2 = 𝑇3 𝑐1 + 1 − 𝑇3 𝑣𝑐1 , †
†
𝑎𝑜 = 𝑀11 𝑎†2 + 𝑀12 𝑏2 + 𝑀13 𝑐2 , 𝑏𝑜 = 𝑀21 𝑎†2 + 𝑀22 𝑏2 + 𝑀23 𝑐2 , 𝑐𝑜 = 𝑀31 𝑎†2 + 𝑀32 𝑏2 + 𝑀33 𝑐2 .
(16)
Now, we refined the form of the input state as |𝛼00000⟩𝑎𝑖 𝑏𝑖 𝑐𝑖 𝑣𝑎1 𝑣𝑏1 𝑣𝑐1 . By using the same method in the previous section, we can obtain the | 𝜕⟨𝑋⟩ |2 value of | 𝜕𝜙 | and 𝛥2 ⟨𝑋⟩ as follows: | | | 𝜕 ⟨𝑋⟩ |2 8 2 2 | | | 𝜕𝜙 | = 2𝑇1 𝛼 cosh 𝑔 sin 𝜙, | | √ √ 1 𝛥2 𝑋 = [𝑇1 cosh8 𝑔 + 2 𝑇1 cosh4 𝑔( 𝑇2 cosh 𝑔 sinh2 𝑔 2 √ + 𝑇3 sinh2 𝑔) cos 𝜙 1 + 𝑇3 sinh4 𝑔] + [cosh4 𝑔(1 − 𝑇1 ) + 2 cosh2 𝑔 sinh2 𝑔(1 − 𝑇2 ) + sinh2 𝑔(1 − 𝑇3 )].
Fig. 4. Schematic diagram of the photon losses after the PS and before the second TNBS. The photon losses are modeled by three fictitious beam splitters (BSs).
phase sensitivity of the three nonlinear interferometers (TNI, SU(1,1)I and MMZI). These results of the SU(1,1)I and MMZI outperform the TNI exactly show that the magnification of energy works best in TNI.
(17)
In the case where the three modes have the same transmissivity, i.e.,𝑇1 = 𝑇2 = 𝑇3 = 𝑇 and at the optimal phase point 𝜙 = 𝜋2 , the phase sensitivity of a lossy TNI is
4. Effects of photon losses
(𝛥𝜙𝐿𝑜𝑠𝑠 )2 = (𝛥𝜙)2 + 𝛥𝜙𝐸𝑇 𝑁𝐼 ,
In real experimental environments, the photon losses and detection efficiency have great effects on the precision and phase sensitivity [39, 40]. In the presence of photon losses, especially with internal losses, the phase sensitivity decoheres very quickly because of the amplification of the vacuum noise [41]. Therefore, it is necessary to enhance the robustness of interferometers against photon losses. Theoretically, there exist intrinsic losses (material absorption, scattering and radiation losses) when the photons travel the TNBS. In this paper, we ignore the intrinsic losses and only consider the photon losses between the phase shifter and the second TNBS. Here, we use three fictitious beam splitters (BSs) to model photon losses in the three parallel paths (Fig. 4) [42]. For convenience, we consider the same internal transmissivity in every path. The three fictitious BSs are placed after the PS and before the second TNBS, and for the lossy TNI, the relationships among the operators in Fig. 4 are given by
(18)
where the first term represents the phase sensitivity when the photon losses are neglected, as discussed above. The second term 𝛥𝜙𝐸𝑇 𝑁𝐼 is the additional term induced by the photon losses. 𝛥𝜙𝐸𝑇 𝑁𝐼 =
(cosh4 𝑔 + sinh2 𝑔 cosh2 𝑔 + sinh2 𝑔) 4𝛼 2 𝑇 cosh8 𝑔
.
(19)
Similarly, for MZI, SU(1,1)I and MMZI, we can calculate their additional terms in the presence of photon losses, and the results are as follows: 1−𝑇 , 𝛥𝜙𝐸𝑀𝑍𝐼 = 𝛼2 𝑇 2 (cosh 𝑔 + sinh2 𝑔)(1 − 𝑇 ) 𝛥𝜙𝐸𝑆𝑈 (1,1)𝐼 = , 𝛼 2 𝑇 cosh4 𝑔 (1 − 𝑇 ) 𝛥𝜙𝐸𝑀𝑀𝑍𝐼 = . (20) 2𝛼 2 𝑇 cosh2 𝑔
𝑎1 = 𝑒𝑖𝜙 (𝑀11 𝑎𝑖 + 𝑀12 𝑏†𝑖 + 𝑀13 𝑐𝑖† ),
According to Eqs. (18), (19) and (20), we plot the phase sensitivity as a function of the internal transmissivity rates 𝑇 under photon losses
𝑏1 = 𝑀21 𝑎†𝑖 + 𝑀22 𝑏𝑖 + 𝑀23 𝑐𝑖 , 192
C.-P. Wei, Z.-M. Wu, C.-Z. Deng et al.
Optics Communications 452 (2019) 189–194
for four different interferometers, as shown in Fig. 5. The relationship of phase sensitivity among four interferometers is 𝛥𝜙𝑇 𝑁𝐼 < 𝛥𝜙𝑆𝑈 (1,1)𝐼 < 𝛥𝜙𝑀𝑀𝑍𝐼 < 𝛥𝜙𝑀𝑍𝐼 in a large transmission area. Furthermore, we can see that the TNI has very good robustness against photon losses when the transmissivity 𝑇 > 0.1. When comparing the mathematical expression of additional term (19) of TNI and other three interferometers (20), we can see that the gain g play an decisive role on addition term, as a result, the addition term of TNI has the smallest value among all addition term. In addition, the internal transmissivity rate T has a little effect on addition term of TNI, which lead to the phase sensitivity looks constant from nearly T = 0.05 to 1. These results indicate that the TNI has the best robustness against photon losses among the four interferometers. Therefore, the TNI might be a better choice when estimating a single phase in the presence of photon losses.
𝑀22 𝑀32 − 𝑀23 𝑀33 = 0, ∗ 𝑀13 𝑀23 = 0, ∗ 𝑀12 𝑀32 = 0, ∗ 𝑀21 𝑀31 = 0.
(23)
We can find out that a set of special solutions for the above equations are 𝑀11 = cosh2 𝑔, 𝑀12 = cosh 𝑔 sinh 𝑔, 𝑀13 = sinh 𝑔, 𝑀21 = sinh 𝑔, 𝑀22 = cosh 𝑔 sin 𝜃, 𝑀23 = cosh 𝑔 cos 𝜃, 𝑀31 = sinh 𝑔, 𝑀32 = cosh 𝑔 sin 𝜃, 𝑀33 = cosh 𝑔 cos 𝜃,
(24)
where, 𝑔 is regarded as the gain factor of the nonlinear process, and 𝜃, 𝑔 are constrained by 𝜃 = arcsin(cosh 𝑔 − 1). By repeated application of Eq. (21) and the transform relationship of the phase shifter in 𝑎1 mode (𝑎1 → 𝑎1 𝑒𝑖𝜙 ), we can obtain the transform relation between the operators of the output modes (𝑎𝑜 , 𝑏𝑜 ) and the input modes (𝑎1 , 𝑏1 ) as
5. Conclusions In summary, by extending a traditional two-mode beam splitter, we constructed the input–output relationships of a three-mode nonlinear beam splitter (TNBS) mathematically and gave the special solutions for the input–output equations. Then, we proposed architecture for a three-mode nonlinear interferometer (TNI) by combining two TNBSs and a (PS). Thereafter, we discussed the phase sensitivity of the TNI by inputting coherent states and homodyne detection method, and detailed comparisons were made among the MZI, SU(1,1)I, MMZI and TNI by fixing the gain factor g. These results showed that the TNI had the highest phase sensitivity. Finally, we discussed the phase sensitivity and make detailed comparisons for the four different interferometers in the presence of photon losses. It was shown that the TNI had the best robustness against photon losses under certain conditions with the same parameters. Our research results may be valuable for future theoretical and experimental research in the field of multiple phase estimation.
𝑎𝑜 = 𝑀11 𝑎1 𝑒𝑖𝜙 + 𝑀12 𝑏†1 + 𝑀13 𝑐1† , 𝑏𝑜 = 𝑀21 𝑎†1 𝑒−𝑖𝜙 + 𝑀22 𝑏1 + 𝑀23 𝑐1 , 𝑐𝑜 = 𝑀31 𝑎†1 𝑒−𝑖𝜙 + 𝑀32 𝑏1 + 𝑀33 𝑐1 .
(25)
Substituting Eqs. (21) and (24) into Eq. (25) and simplifying the results, we can obtain 𝑎𝑜 = (cosh4 𝑔𝑒𝑖𝜙 + cosh 𝑔 sinh2 𝑔 + sinh2 𝑔)𝑎𝑖 + (cosh3 𝑔 sinh 𝑔𝑒𝑖𝜙 + cosh2 𝑔 sinh 𝑔 sin 𝜃 + cosh 𝑔 sinh 𝑔 sin 𝜃)𝑏†𝑖 + (cosh2 𝑔 sinh 𝑔𝑒𝑖𝜙 + cosh2 𝑔 sinh 𝑔 cos 𝜃 + sinh 𝑔 cosh 𝑔 cos 𝜃)𝑐𝑖† , 𝑏𝑜 = (cosh 𝑔 sinh2 𝑔𝑒−𝑖𝜙 + cosh2 𝑔 sin2 𝜃 + cosh2 𝑔 sinh 𝜃 cos 𝜃)𝑏𝑖 + (sinh2 𝑔𝑒−𝑖𝜙 + cosh2 𝑔 sinh 𝜃 cos 𝜃 + cosh2 𝑔 cos2 𝜃)𝑐𝑖 + (cosh2 𝑔 sinh 𝑔𝑒−𝑖𝜙 + cosh 𝑔 sinh 𝑔 sin 𝜃 + cosh 𝑔 sinh 𝑔 cos 𝜃)𝑎†𝑖 , 𝑐𝑜 = (cosh 𝑔 sinh2 𝑔𝑒−𝑖𝜙 + cosh2 𝑔 sinh2 𝜃 + cosh2 𝑔 sinh 𝜃 cos 𝜃)𝑏𝑖
Acknowledgments
+ (sinh2 𝑔𝑒−𝑖𝜙 + cosh2 𝑔 sinh 𝜃 cos 𝜃 + cosh2 𝑔 cos2 𝜃)𝑐𝑖 This work was supported by the Youth Science Fund of the Jiangxi Province Education Department, China (Grant No. GJJ171012), the National Natural Science Foundation of China (Grant No. 11664017), and the Outstanding Young Talent Program of Jiangxi Province, China (Grant No. 20171BCB23034).
+ (cosh2 𝑔 sinh 𝑔𝑒−𝑖𝜙 + cosh 𝑔 sinh 𝑔 sin 𝜃 + cosh 𝑔 sinh 𝑔 cos 𝜃)𝑎†𝑖 . (26) So far, we obtain the Eq. (4), and these corresponding coefficients are given in Eq. (5) in Section 2. References
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As shown in Section 2, the relationships of the input and output modes of the first TNBS as follows: 𝑎1 = 𝑀11 𝑎𝑖 + 𝑀12 𝑏†𝑖 + 𝑀13 𝑐𝑖† , 𝑏1 = 𝑀21 𝑎†𝑖 + 𝑀22 𝑏𝑖 + 𝑀23 𝑐𝑖 , 𝑐1 = 𝑀31 𝑎†𝑖 + 𝑀32 𝑏𝑖 + 𝑀33 𝑐𝑖 .
(21)
These operators and their Hermitian conjugates satisfy the following boson commutation relations: [𝑥1 , 𝑦1 ] = [𝑥†1 , 𝑦†1 ] = 0, [𝑥1 , 𝑦†1 ] = 𝛿𝑖𝑗 , where 𝑥, 𝑦, 𝑖 and 𝑗 take on the values 𝑎, 𝑏 and 𝑐. When 𝑖 = 𝑗, 𝛿𝑖𝑖 = 1, otherwise 𝛿𝑖𝑗 = 0. According to the commutation relations of [𝑎1 , 𝑎†1 ] = 1, [𝑏1 , 𝑏†1 ] = 1, [𝑐1 , 𝑐1† ] = 1, we can get |𝑀11 |2 − |𝑀12 |2 − |𝑀13 |2 = 1, | | | | | | |𝑀22 |2 − |𝑀21 |2 − |𝑀23 |2 = 1, | | | | | |
|𝑀33 |2 − |𝑀31 |2 − |𝑀32 |2 = 1. | | | | | |
(22)
Similarly, According to the commutation relations [𝑎1 , 𝑏1 ] = 0, [𝑎1 , 𝑐1 ] = 0, [𝑏1 , 𝑐1 ] = 0 and [𝑎𝑜 , 𝑏†𝑜 ] = 0, [𝑎𝑜 , 𝑐𝑜† ] = 0, [𝑏𝑜 , 𝑐𝑜† ] = 0, we have 𝑀11 𝑀21 − 𝑀12 𝑀22 = 0, 𝑀11 𝑀31 − 𝑀13 𝑀33 = 0, 193
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Phase sensitivity of a three-mode nonlinear interferometer Chao-Ping Wei a ,∗, Zhao-Ming Wu a , Cheng-Zhi Deng a , Li-Yun Hu b ,∗ a
Jiangxi Province Key Laboratory of Water Information Cooperative Sensing and Intelligent Processing, Nanchang Institute of Technology, School of Information Engineering, Nanchang Institute of Technology, Nanchang 330022, China b Center for Quantum Science and Technology, Jiangxi Normal University, Nanchang 330022, China
ARTICLE
INFO
Keywords: Nonlinear interferometry Phase sensitivity Photon losses
ABSTRACT In this paper, we construct the input–output relationships of a three-mode nonlinear beam splitter (TNBS) mathematically, and a set of special solutions for the input–output equations are given. By combining two TNBSs and a phase shifter, we propose a scheme for a three-mode nonlinear interferometer (TNI) and give the input–output expression of the whole TNI. Then, we study the phase sensitivity of the TNI by giving input states and detection scheme. In addition, we also calculate the phase sensitivity of a Mach–Zehnder interferometer (MZI), SU(1,1) interferometer (SU(1,1)I) and modified Mach–Zehnder interferometer (MMZI), and we discuss the relationships of phase sensitivity between the MZI, SU(1,1)I, MMZI and TNI. Finally, in the presence of photon losses, we discuss the phase sensitivity of the four different interferometers and make detailed comparisons. It is shown that the TNI has the best robustness against photon losses under certain conditions with the same parameters.
1. Introduction Phase estimation theory is a major branch in the field of quantum metrology [1], and optical interferometers have become the main tools of phase estimation [2]. How to improve the phase sensitivity of an interferometer is one of the main research focuses in phase estimation theory. At present, there are three main approaches to improving an interferometer’s phase sensitivity. The first one employs some special entanglement, squeezed and superposition [3–5], to increase directly the couple with phase information; the second involves using some ultra-sensitive detection schemes to achieve the maximum phase information [6]; the third straightly remoulds the structure of the interferometer, such as by adding some nonlinear effects including nonlinear parameter processes [7,8] or nonlinear phase shift operations [9– 11] to the original interferometer, thus enabling the realization of improvements to the intercoupling between the interferometer and phase information. At present, studies on the first and second methods are relatively mature, and physicists have completed numerous relevant research on improving the interferometer’s phase sensitivity by changing input states or detection schemes. For traditional two-mode interferometers, some investigations on improving the phase sensitivity have been done by the third method. However, there are relatively few studies for multi-mode interferometers up to now, especially for multi-mode nonlinear interferometers. The earliest theory model for multiple phase estimations can be found in Refs. [12,13]. Besides, it has been shown that, given a fixed total photon number, the precision of
estimating multiple phases outperforms individual quantum estimation schemes. However, it would be interesting to extend the traditional dual-mode nonlinear beam splitter to a multi-mode nonlinear beam splitter and to construct a model for the multi-mode nonlinear interferometer to study the influence of nonlinear operations on the interferometer’s phase sensitivity. These problems need to be further studied. In this paper, by extending a dual-mode nonlinear beam splitter, we construct a three-mode nonlinear beam splitter model, and we give its mathematical description, corresponding physical model and implementation scheme. By combining two three-mode nonlinear beam splitters (TNBSs) and a phase shifter (PS), a new frame diagram for a three-mode nonlinear interferometer (TNI) is proposed. In the interferometer, the effect of TNBS on improved phase estimation accuracy is studied. Subsequently, we discuss the relationship between the phase sensitivity of four interferometers, namely, a Mach–Zehnder interferometer (MZI), SU(1,1)I, modified MZI (MMZI), and TNI, under different parameters. Finally, we discuss the phase sensitivity of the four interferometers in the presence of photon losses. The organization of this paper is as follows. In Section 2, the architecture of the TNI is introduced in detail, and the corresponding input–output operator relationships are given. In Section 3, we study the phase sensitivity of the TNI and make detailed comparisons with three other familiar interferometers. In Section 4, we discuss the effect of photon losses on the phase sensitivity, and we present an analysis of
∗ Corresponding authors. E-mail addresses: [email protected] (C.-P. Wei), [email protected] (L.-Y. Hu).
https://doi.org/10.1016/j.optcom.2019.07.033 Received 28 April 2019; Received in revised form 17 June 2019; Accepted 16 July 2019 Available online 18 July 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.
C.-P. Wei, Z.-M. Wu, C.-Z. Deng et al.
Optics Communications 452 (2019) 189–194
𝑐1 = 𝑀31 𝑎†𝑖 + 𝑀32 𝑏𝑖 + 𝑀33 𝑐𝑖 .
(1)
These operators and their Hermitian conjugates satisfy the following boson commutation relations: [𝑥1 , 𝑦1 ] = [𝑥†1 , 𝑦†1 ] = 0, [𝑥1 , 𝑦†1 ] = 𝛿𝑖𝑗 , where 𝑥, 𝑦, 𝑖 and 𝑗 take on the values 𝑎, 𝑏 and 𝑐. When 𝑖 = 𝑗, 𝛿𝑖𝑖 = 1, otherwise 𝛿𝑖𝑗 = 0. Explicitly, by extending these commutation relationships, one can obtain the series of following expressions as |𝑀11 |2 − |𝑀12 |2 − |𝑀13 |2 = 1, | | | | | | |𝑀22 |2 − |𝑀21 |2 + |𝑀23 |2 = 1, | | | | | | |𝑀33 |2 − |𝑀31 |2 + |𝑀32 |2 = 1, | | | | | | 𝑀11 𝑀21 − 𝑀12 𝑀22 − 𝑀13 𝑀23 = 0, 𝑀11 𝑀31 − 𝑀12 𝑀32 − 𝑀13 𝑀33 = 0, Fig. 1. General sketch of a TNI with a series of input states |𝜓⟩𝑖 as inputs. The input states inject into the first TNBS, respectively. Then after applying a PS 𝑈 (𝜙) to 𝑎𝑖 mode of the path, these light are combined in the second TNBS. Finally, we can execute an adaptive detection in each output port.
∗ ∗ ∗ −𝑀21 𝑀31 + 𝑀22 𝑀32 + 𝑀23 𝑀33 = 0, ∗ ∗ ∗ −𝑀21 𝑀31 + 𝑀22 𝑀32 + 𝑀23 𝑀33 = 0.
(2)
A set of special solutions for the above equations are 𝑀11 = cosh2 𝑔, 𝑀12 = cosh 𝑔 sinh 𝑔, 𝑀13 = sinh 𝑔,
the robustness against photon losses for MZI, SU(1,1)I, MMZI and TNI. Section 5 presents the conclusions.
𝑀21 = sinh 𝑔, 𝑀22 = cosh 𝑔 sin 𝜃, 𝑀23 = cosh 𝑔 cos 𝜃, 𝑀31 = sinh 𝑔, 𝑀32 = cosh 𝑔 sin 𝜃, 𝑀33 = cosh 𝑔 cos 𝜃,
(3)
2. Three-mode nonlinear interferometer where, 𝑔 is regarded as the gain factor of the nonlinear process, and 𝜃, 𝑔 are constrained by 𝜃 = arcsin(cosh 𝑔 − 1). In addition, one can also calculate the total average photon number inside the interferometer, i.e., after the⟩TNBS, as the total average photon number 𝑁 = ⟨ 𝑎†1 𝑎1 + 𝑏†1 𝑏1 + 𝑐1† 𝑐1 = 𝛼 2 (cosh4 𝑔 + 2 sinh2 𝑔) + 4 sinh2 𝑔 + sinh4 𝑔, which
In some real scenarios, such as quantum imaging, magnetic imaging and gravitational field imaging [14,15], it is necessary to measure multiple parameters simultaneously. This makes multi-parameter estimation very attractive from a technological point of view. The physicists have conducted a great deal of theoretical research on multi-parameter estimation. For example, in Ref. [16], S. Adhikari et al. showed that this simple strategy can give asymptotically post-classical sensitivity for multi-parameter estimation even when the number of modes is large. Further, it has been shown that the quantum Cramér–Rao bound can be saturated for a multi-mode interferometer with a phase in each mode by using Gaussian inputs and passive elements [17]. In Ref. [18], the authors study a nonlinear interferometer consisting of two consecutive parametric amplifiers and achieve a Heisenberg scaling. Experimentally, multi-mode optical interferometers are the key devices for multiple phase estimations simultaneously, and these can be realized by cascading several multi-mode beam splitters (MBSs) and PSs [19]. Relevant research about multiple phase estimations and multi-mode interferometers can be seen in Refs. [20–22]. These results show that the simultaneous estimation of multiple phases in multiple interferometers can provide for better total precision than estimating them individually. Many studies, such as those described above, have shown that use of a multi-mode interferometer has enormous advantages and the potential for multiple phase estimations simultaneously. Such multi-mode nonlinear interferometers can be constituted by adding some nonlinear processes, including nonlinear phase shift operations or NMBS, to an TNI. Wei et al. proposed a multi-mode nonlinear interferometer and studied the phase sensitivity of three and four modes in the nonlinear interferometer with single photons as inputs [23]. In Ref. [24], the problem of two phase estimations simultaneously in a TNI was studied by the use of quantum Fisher information matrix and classical Fisher information matrix methods. In this section, we will construct a TNI by cascading two TNBSs and a PS, as shown in Fig. 1. It can also be seen as a modified MZI, in which the conventional linear beam splitters are replaced by TNBS. In the following, we mainly study the input–output transformation relationships, including TNBS and the whole TNI. According to the Lie group and SU(1,2) group theory method [25,26], we can define the relationships of the input and output modes of TNBS as follows:
is amplified by a cosh4 𝑔 scaling factor. After applying a conventional † PS 𝑈 (𝜙) = e𝑖𝜙𝑎1 𝑎1 to 𝑎1 modes, and using the basis of the transform relationship of 𝑎1 mode (𝑎1 → 𝑎1 𝑒𝑖𝜙 ) and the second TNBS, one can obtain the final input–output transform relationships of the whole interferometer as follows: 𝑎𝑜 = 𝑁11 𝑎𝑖 + 𝑁12 𝑏†𝑖 + 𝑁13 𝑐𝑖† , 𝑏𝑜 = 𝑁21 𝑏𝑖 + 𝑁22 𝑐𝑖 + 𝑁23 𝑎†𝑖 , 𝑐𝑜 = 𝑁31 𝑏𝑖 + 𝑁32 𝑐𝑖 + 𝑁33 𝑎†𝑖 ,
(4)
where, the corresponding coefficients are as follows: 𝑁11 = cosh4 𝑔𝑒𝑖𝜙 + cosh 𝑔 sinh2 𝑔 + sinh2 𝑔, 𝑁12 = cosh3 𝑔 sinh 𝑔𝑒𝑖𝜙 + cosh2 𝑔 sinh 𝑔 sin 𝜃 + cosh 𝑔 sinh 𝑔 sin 𝜃, 𝑁13 = cosh2 𝑔 sinh 𝑔𝑒𝑖𝜙 + cosh2 𝑔 sinh 𝑔 cos 𝜃 + sinh 𝑔 cosh 𝑔 cos 𝜃, 𝑁21 = cosh 𝑔 sinh2 𝑔𝑒−𝑖𝜙 + cosh2 𝑔 sin2 𝜃 + cosh2 𝑔 sinh 𝜃 cos 𝜃, 𝑁22 = sinh2 𝑔𝑒−𝑖𝜙 + cosh2 𝑔 sinh 𝜃 cos 𝜃 + cosh2 𝑔 cos2 𝜃, 𝑁23 = cosh2 𝑔 sinh 𝑔𝑒−𝑖𝜙 + cosh 𝑔 sinh 𝑔 sin 𝜃 + cosh 𝑔 sinh 𝑔 cos 𝜃, 𝑁31 = cosh 𝑔 sinh2 𝑔𝑒−𝑖𝜙 + cosh2 𝑔 sinh2 𝜃 + cosh2 𝑔 sinh 𝜃 cos 𝜃, 𝑁32 = sinh2 𝑔𝑒−𝑖𝜙 + cosh2 𝑔 sinh 𝜃 cos 𝜃 + cosh2 𝑔 cos2 𝜃, 𝑁33 = cosh2 𝑔 sinh 𝑔𝑒−𝑖𝜙 + cosh 𝑔 sinh 𝑔 sin 𝜃 + cosh 𝑔 sinh 𝑔 cos 𝜃.
(5)
So far, we have obtained the input–output relationships of the threemode nonlinear interferometer as shown in Eq. (4). Theoretically, the TNBS can be constructed by cascading four four-wave-mixers [26] and can be realized with current experimental technology [27]. As shown in Refs. [28,29], the three-mode linear beam splitter (tritter) can be realized by an ultrafast laser writing technique, and the twomode nonlinear beam splitter can be realized by an optical parametric amplification process [30] or spontaneous down-conversion [31]. Therefore, the TNBS may be integrated by combining the ultrafast laser writing technique with one of the optical parametric amplification processes and spontaneous down-conversion. In brief, we can obtain a three-mode nonlinear interferometer by cascading two TNBSs and a PS.
𝑎1 = 𝑀11 𝑎𝑖 + 𝑀12 𝑏†𝑖 + 𝑀13 𝑐𝑖† , 𝑏1 = 𝑀21 𝑎†𝑖 + 𝑀22 𝑏𝑖 + 𝑀23 𝑐𝑖 , 190
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Fig. 2. The phase sensitivity of the three-mode nonlinear interferometer (TNI), Mach–Zehnder interferometer (MZI), SU(1,1)I, and modified MZI (MMZI) as a function of the average photon number in coherent states 𝑁𝛼 = |𝛼|2 . (a) g = 0; (b) g = 1.5. Black dot-dashed line: TNI; red dashed line: MZI; blue dotted: MMZI; green solid line: SU(1,1)I.
3. Phase sensitivity of the three-mode nonlinear interferometer with coherent state inputs
in Ref. [37], the relationship of output operator 𝑎𝑜 and input operators 𝑎𝑖 , 𝑏𝑖 is 𝑎𝑜 =
Recently, optical quantum metrology in linear multi-mode interferometers has been studied with single photon inputs, and it has been shown that the phase sensitivity can beat the shot-noise limit and approach to quantum Cramé r–Rao bound [32,33]. Experimentally, Szigeti et al. found that, pumped-up SU(1,1) interferometers with spinor Bose–Einstein condensates (BECs) (three-mode) or hybrid atom-lights (four-mode) are capable of surpassing the shot-noise limit with respect to the total number of input particles and are never worse than conventional SU(1,1) interferometry [34]. Su et al. observed the generalized Hong–Ou–Mandel effect with up to four photons in a quantum Fourier transform circuit and demonstrated optical phase supersensitivities deterministically [35]. In this section, we mainly study the phase sensitivity of the threemode nonlinear interferometer with coherent states inputs. Here, to obtain the phase information about 𝜙, a detection scheme should be executed in the output ports. There are several detection methods, for example, homodyne detection, parity detection, intensity detection, and so on. Here, we use the homodyne detection method at port 𝑎𝑜 . We define the quadrature operator 𝑋 as 𝑋=
𝑎𝑜 + 𝑎†𝑜 √ . 2
| 𝜕 ⟨𝑋⟩ | 𝛼 2 sin2 𝜙 | | , | 𝜕𝜙 | = 2 | | 1 (11) 𝛥2 𝑋 = (1 + cos 𝜙). 4 For a nonlinear SU(1,1)I, is composed of two optical parametric amplifiers and a phase shifter [38], the relationship of output operator 𝑎𝑜 and input operators 𝑎𝑖 , 𝑏𝑖 is 𝑎𝑜 = [𝑒𝑖𝜙 cosh 𝑔2 cosh 𝑔1 + 𝑒𝑖𝜃2 −𝜃1 sin 𝑔2 sinh 𝑔1 ]𝑎𝑖 − [𝑒𝑖𝜙 𝑒𝑖𝜃1 cosh 𝑔2 sin 𝑔1 + 𝑒𝑖𝜃2 sin 𝑔2 cosh 𝑔1 ]𝑏†𝑖 .
(12)
At equilibrium, i.e., 𝑔1 = 𝑔2 = 𝑔, 𝜃2 − 𝜃1 = 𝜋, 𝜃1 = 0, Eq. (12) reduces to 𝑎𝑜 = [𝑒𝑖𝜙 cosh2 𝑔 − sin2 𝑔]𝑎𝑖 − [𝑒𝑖𝜙 cosh 𝑔 sin 𝑔 − sin 𝑔 cosh 𝑔]𝑏†𝑖 .
(13)
For the same input states |𝛼⟩𝑎𝑖 |0⟩𝑏𝑖 , we can obtain 𝜕 ⟨𝑋⟩ = 2𝛼 2 sin2 𝜙 cosh4 𝑔, 𝜕𝜙 1 (14) 𝛥2 𝑋 = (cosh4 𝑔 − 2 sinh2 𝑔 cosh2 𝑔 cos 𝜙 + sinh4 𝑔). 2 Similarly, the MMZI can be treated as a variation of the MZI or SU(1,1)I, and it is composed of a beam splitter, an OPA and a phase | 𝜕⟨𝑋⟩ |2 shifter [7]. We can calculate the value of | 𝜕𝜙 | and variance of ⟨𝑋⟩ | | for MMZI as follows: 𝜕 ⟨𝑋⟩ = 𝛼 2 sin2 𝜙 cosh2 𝑔, 𝜕𝜙 1 𝛥2 𝑋 = [cosh2 𝑔 + sinh2 𝑔 − 2 cosh 𝑔 sinh 𝑔 cos 𝜙]. (15) 4 Substituting Eqs. (9), (11), (14) and (15) into Eq. (7), we can obtain the phase sensitivity of TNI, MZI, SU(1,1)I and MMZI, respectively. We can also plot the phase sensitivity as a function of different parameters as shown in Figs. 2 and 3. We can see that the phase sensitivity of the TNI, SU(1,1) and MMZI is equal and better than that of the MZI in the case of gain factor 𝑔 = 0. For given 𝑔 = 1.5, the relationship of their phase sensitivity is: 𝛥𝜙𝑇 𝑁𝐼 < 𝛥𝜙𝑆𝑈 (1,1)𝐼 < 𝛥𝜙𝑀𝑀𝑍𝐼 < 𝛥𝜙𝑀𝑍𝐼 , which means the result of TNI is the best for the four interferometers. It is should be noted that the MZI, SU(1,1) and MMZI all have two input– output paths, but the TNI has three input–output paths. Therefore, in order to make a better comparison among the four interferometers, we also plot the phase sensitivity as a function of the total average photon number in the interferometer 𝑁, as shown in Fig. 3. It can be seen that, for a fixed total photon number and given gain factor 𝑔, the three nonlinear interferometers surpasses the MZI. This can be interpreted as the gain factor 𝑔 playing a crucial role in the enhancement of the
(6)
𝛥2 𝑋 , (7) |𝜕 ⟨𝑋⟩ ∕𝜕𝜙|2 ⟨ ⟩ where, 𝛥2 𝑋 = 𝑋 2 − ⟨𝑋⟩2 is the variance of 𝑋. We consider a coherent input state in mode 𝑎𝑖 and vacuum state in mode 𝑏𝑖 and 𝑐𝑖 (|𝛼⟩𝑎𝑖 |0⟩𝑏𝑖 |0⟩𝑐𝑖 ), and 𝛼 is real number; then, the average value of ⟨𝑋⟩ is (𝛥𝜙)2 =
(8)
| 𝜕⟨𝑋⟩ |2 Similarly, we can calculate the value of | 𝜕𝜙 | and variance of ⟨𝑋⟩ | | as follows: | 𝜕 ⟨𝑋⟩ |2 2 8 2 | | | 𝜕𝜙 | = 2𝛼 sin 𝜙 cosh 𝑔, | | 1 𝛥2 𝑋 = [cosh8 𝑔 + 2 cosh4 𝑔 sinh2 𝑔(cosh 𝑔 + 1) cos 𝜙 2 + sinh4 𝑔(cosh 𝑔 + 1)2 ].
(10)
| 𝜕⟨𝑋⟩ |2 Consider input states |𝛼⟩𝑎𝑖 |0⟩𝑏𝑖 , we can calculate the value of | 𝜕𝜙 | | | and variance of ⟨𝑋⟩ as
where, 𝑎𝑜 = 𝑁11 𝑎𝑖 +𝑁12 𝑏†𝑖 +𝑁13 𝑐𝑖† is the operator at the out-port 𝑎 mode. The phase sensitivity 𝛥𝜙 of an interferometer can be characterized by the error propagation formula [36]
𝛼 ⟨𝑋⟩ = √ [cosh4 𝑔(𝑒𝑖𝜙 + 𝑒−𝑖𝜙 ) + 2 cosh 𝑔 sinh2 𝑔 + 2 sinh2 𝑔]. 2
1 𝑖𝜙 [(𝑒 + 1)𝑎𝑖 + (𝑒𝑖𝜙 − 1)𝑏𝑖 ]. 2
(9)
The phase sensitivity of MZI, SU(1,1)I and MMZI can be calculated in the same manner. For a conventional MZI, which is composed with two beam splitter and a phase shifter, and it is introduced detailedly 191
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Fig. 3. The phase sensitivity of the TNI, MZI, SU(1,1) and MMZI as a function of total average photon number 𝑁. Here, we choose 𝑁 values by giving a gain factor 𝑔 = 0.2. Black dot-dashed line: TNI; red dashed line: MZI; blue dotted: MMZI; green solid line: SU(1,1)I.
Fig. 5. Phase sensitivity 𝛥𝜙 as a function of the internal transmissivity rates T with 𝑔 = 1.5 and 𝑁𝛼 = 4. Black dot-dashed line: TNI; blue dotted: MMZI; green solid line: SU(1,1)I; red dashed line: MZI.
𝑐1 = 𝑀31 𝑎†𝑖 + 𝑀32 𝑏𝑖 + 𝑀33 𝑐𝑖 , √ √ 𝑎2 = 𝑇1 𝑎1 + 1 − 𝑇1 𝑣𝑎1 , √ √ 𝑏2 = 𝑇2 𝑏1 + 1 − 𝑇2 𝑣𝑏1 , √ √ 𝑐2 = 𝑇3 𝑐1 + 1 − 𝑇3 𝑣𝑐1 , †
†
𝑎𝑜 = 𝑀11 𝑎†2 + 𝑀12 𝑏2 + 𝑀13 𝑐2 , 𝑏𝑜 = 𝑀21 𝑎†2 + 𝑀22 𝑏2 + 𝑀23 𝑐2 , 𝑐𝑜 = 𝑀31 𝑎†2 + 𝑀32 𝑏2 + 𝑀33 𝑐2 .
(16)
Now, we refined the form of the input state as |𝛼00000⟩𝑎𝑖 𝑏𝑖 𝑐𝑖 𝑣𝑎1 𝑣𝑏1 𝑣𝑐1 . By using the same method in the previous section, we can obtain the | 𝜕⟨𝑋⟩ |2 value of | 𝜕𝜙 | and 𝛥2 ⟨𝑋⟩ as follows: | | | 𝜕 ⟨𝑋⟩ |2 8 2 2 | | | 𝜕𝜙 | = 2𝑇1 𝛼 cosh 𝑔 sin 𝜙, | | √ √ 1 𝛥2 𝑋 = [𝑇1 cosh8 𝑔 + 2 𝑇1 cosh4 𝑔( 𝑇2 cosh 𝑔 sinh2 𝑔 2 √ + 𝑇3 sinh2 𝑔) cos 𝜙 1 + 𝑇3 sinh4 𝑔] + [cosh4 𝑔(1 − 𝑇1 ) + 2 cosh2 𝑔 sinh2 𝑔(1 − 𝑇2 ) + sinh2 𝑔(1 − 𝑇3 )].
Fig. 4. Schematic diagram of the photon losses after the PS and before the second TNBS. The photon losses are modeled by three fictitious beam splitters (BSs).
phase sensitivity of the three nonlinear interferometers (TNI, SU(1,1)I and MMZI). These results of the SU(1,1)I and MMZI outperform the TNI exactly show that the magnification of energy works best in TNI.
(17)
In the case where the three modes have the same transmissivity, i.e.,𝑇1 = 𝑇2 = 𝑇3 = 𝑇 and at the optimal phase point 𝜙 = 𝜋2 , the phase sensitivity of a lossy TNI is
4. Effects of photon losses
(𝛥𝜙𝐿𝑜𝑠𝑠 )2 = (𝛥𝜙)2 + 𝛥𝜙𝐸𝑇 𝑁𝐼 ,
In real experimental environments, the photon losses and detection efficiency have great effects on the precision and phase sensitivity [39, 40]. In the presence of photon losses, especially with internal losses, the phase sensitivity decoheres very quickly because of the amplification of the vacuum noise [41]. Therefore, it is necessary to enhance the robustness of interferometers against photon losses. Theoretically, there exist intrinsic losses (material absorption, scattering and radiation losses) when the photons travel the TNBS. In this paper, we ignore the intrinsic losses and only consider the photon losses between the phase shifter and the second TNBS. Here, we use three fictitious beam splitters (BSs) to model photon losses in the three parallel paths (Fig. 4) [42]. For convenience, we consider the same internal transmissivity in every path. The three fictitious BSs are placed after the PS and before the second TNBS, and for the lossy TNI, the relationships among the operators in Fig. 4 are given by
(18)
where the first term represents the phase sensitivity when the photon losses are neglected, as discussed above. The second term 𝛥𝜙𝐸𝑇 𝑁𝐼 is the additional term induced by the photon losses. 𝛥𝜙𝐸𝑇 𝑁𝐼 =
(cosh4 𝑔 + sinh2 𝑔 cosh2 𝑔 + sinh2 𝑔) 4𝛼 2 𝑇 cosh8 𝑔
.
(19)
Similarly, for MZI, SU(1,1)I and MMZI, we can calculate their additional terms in the presence of photon losses, and the results are as follows: 1−𝑇 , 𝛥𝜙𝐸𝑀𝑍𝐼 = 𝛼2 𝑇 2 (cosh 𝑔 + sinh2 𝑔)(1 − 𝑇 ) 𝛥𝜙𝐸𝑆𝑈 (1,1)𝐼 = , 𝛼 2 𝑇 cosh4 𝑔 (1 − 𝑇 ) 𝛥𝜙𝐸𝑀𝑀𝑍𝐼 = . (20) 2𝛼 2 𝑇 cosh2 𝑔
𝑎1 = 𝑒𝑖𝜙 (𝑀11 𝑎𝑖 + 𝑀12 𝑏†𝑖 + 𝑀13 𝑐𝑖† ),
According to Eqs. (18), (19) and (20), we plot the phase sensitivity as a function of the internal transmissivity rates 𝑇 under photon losses
𝑏1 = 𝑀21 𝑎†𝑖 + 𝑀22 𝑏𝑖 + 𝑀23 𝑐𝑖 , 192
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for four different interferometers, as shown in Fig. 5. The relationship of phase sensitivity among four interferometers is 𝛥𝜙𝑇 𝑁𝐼 < 𝛥𝜙𝑆𝑈 (1,1)𝐼 < 𝛥𝜙𝑀𝑀𝑍𝐼 < 𝛥𝜙𝑀𝑍𝐼 in a large transmission area. Furthermore, we can see that the TNI has very good robustness against photon losses when the transmissivity 𝑇 > 0.1. When comparing the mathematical expression of additional term (19) of TNI and other three interferometers (20), we can see that the gain g play an decisive role on addition term, as a result, the addition term of TNI has the smallest value among all addition term. In addition, the internal transmissivity rate T has a little effect on addition term of TNI, which lead to the phase sensitivity looks constant from nearly T = 0.05 to 1. These results indicate that the TNI has the best robustness against photon losses among the four interferometers. Therefore, the TNI might be a better choice when estimating a single phase in the presence of photon losses.
𝑀22 𝑀32 − 𝑀23 𝑀33 = 0, ∗ 𝑀13 𝑀23 = 0, ∗ 𝑀12 𝑀32 = 0, ∗ 𝑀21 𝑀31 = 0.
(23)
We can find out that a set of special solutions for the above equations are 𝑀11 = cosh2 𝑔, 𝑀12 = cosh 𝑔 sinh 𝑔, 𝑀13 = sinh 𝑔, 𝑀21 = sinh 𝑔, 𝑀22 = cosh 𝑔 sin 𝜃, 𝑀23 = cosh 𝑔 cos 𝜃, 𝑀31 = sinh 𝑔, 𝑀32 = cosh 𝑔 sin 𝜃, 𝑀33 = cosh 𝑔 cos 𝜃,
(24)
where, 𝑔 is regarded as the gain factor of the nonlinear process, and 𝜃, 𝑔 are constrained by 𝜃 = arcsin(cosh 𝑔 − 1). By repeated application of Eq. (21) and the transform relationship of the phase shifter in 𝑎1 mode (𝑎1 → 𝑎1 𝑒𝑖𝜙 ), we can obtain the transform relation between the operators of the output modes (𝑎𝑜 , 𝑏𝑜 ) and the input modes (𝑎1 , 𝑏1 ) as
5. Conclusions In summary, by extending a traditional two-mode beam splitter, we constructed the input–output relationships of a three-mode nonlinear beam splitter (TNBS) mathematically and gave the special solutions for the input–output equations. Then, we proposed architecture for a three-mode nonlinear interferometer (TNI) by combining two TNBSs and a (PS). Thereafter, we discussed the phase sensitivity of the TNI by inputting coherent states and homodyne detection method, and detailed comparisons were made among the MZI, SU(1,1)I, MMZI and TNI by fixing the gain factor g. These results showed that the TNI had the highest phase sensitivity. Finally, we discussed the phase sensitivity and make detailed comparisons for the four different interferometers in the presence of photon losses. It was shown that the TNI had the best robustness against photon losses under certain conditions with the same parameters. Our research results may be valuable for future theoretical and experimental research in the field of multiple phase estimation.
𝑎𝑜 = 𝑀11 𝑎1 𝑒𝑖𝜙 + 𝑀12 𝑏†1 + 𝑀13 𝑐1† , 𝑏𝑜 = 𝑀21 𝑎†1 𝑒−𝑖𝜙 + 𝑀22 𝑏1 + 𝑀23 𝑐1 , 𝑐𝑜 = 𝑀31 𝑎†1 𝑒−𝑖𝜙 + 𝑀32 𝑏1 + 𝑀33 𝑐1 .
(25)
Substituting Eqs. (21) and (24) into Eq. (25) and simplifying the results, we can obtain 𝑎𝑜 = (cosh4 𝑔𝑒𝑖𝜙 + cosh 𝑔 sinh2 𝑔 + sinh2 𝑔)𝑎𝑖 + (cosh3 𝑔 sinh 𝑔𝑒𝑖𝜙 + cosh2 𝑔 sinh 𝑔 sin 𝜃 + cosh 𝑔 sinh 𝑔 sin 𝜃)𝑏†𝑖 + (cosh2 𝑔 sinh 𝑔𝑒𝑖𝜙 + cosh2 𝑔 sinh 𝑔 cos 𝜃 + sinh 𝑔 cosh 𝑔 cos 𝜃)𝑐𝑖† , 𝑏𝑜 = (cosh 𝑔 sinh2 𝑔𝑒−𝑖𝜙 + cosh2 𝑔 sin2 𝜃 + cosh2 𝑔 sinh 𝜃 cos 𝜃)𝑏𝑖 + (sinh2 𝑔𝑒−𝑖𝜙 + cosh2 𝑔 sinh 𝜃 cos 𝜃 + cosh2 𝑔 cos2 𝜃)𝑐𝑖 + (cosh2 𝑔 sinh 𝑔𝑒−𝑖𝜙 + cosh 𝑔 sinh 𝑔 sin 𝜃 + cosh 𝑔 sinh 𝑔 cos 𝜃)𝑎†𝑖 , 𝑐𝑜 = (cosh 𝑔 sinh2 𝑔𝑒−𝑖𝜙 + cosh2 𝑔 sinh2 𝜃 + cosh2 𝑔 sinh 𝜃 cos 𝜃)𝑏𝑖
Acknowledgments
+ (sinh2 𝑔𝑒−𝑖𝜙 + cosh2 𝑔 sinh 𝜃 cos 𝜃 + cosh2 𝑔 cos2 𝜃)𝑐𝑖 This work was supported by the Youth Science Fund of the Jiangxi Province Education Department, China (Grant No. GJJ171012), the National Natural Science Foundation of China (Grant No. 11664017), and the Outstanding Young Talent Program of Jiangxi Province, China (Grant No. 20171BCB23034).
+ (cosh2 𝑔 sinh 𝑔𝑒−𝑖𝜙 + cosh 𝑔 sinh 𝑔 sin 𝜃 + cosh 𝑔 sinh 𝑔 cos 𝜃)𝑎†𝑖 . (26) So far, we obtain the Eq. (4), and these corresponding coefficients are given in Eq. (5) in Section 2. References
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As shown in Section 2, the relationships of the input and output modes of the first TNBS as follows: 𝑎1 = 𝑀11 𝑎𝑖 + 𝑀12 𝑏†𝑖 + 𝑀13 𝑐𝑖† , 𝑏1 = 𝑀21 𝑎†𝑖 + 𝑀22 𝑏𝑖 + 𝑀23 𝑐𝑖 , 𝑐1 = 𝑀31 𝑎†𝑖 + 𝑀32 𝑏𝑖 + 𝑀33 𝑐𝑖 .
(21)
These operators and their Hermitian conjugates satisfy the following boson commutation relations: [𝑥1 , 𝑦1 ] = [𝑥†1 , 𝑦†1 ] = 0, [𝑥1 , 𝑦†1 ] = 𝛿𝑖𝑗 , where 𝑥, 𝑦, 𝑖 and 𝑗 take on the values 𝑎, 𝑏 and 𝑐. When 𝑖 = 𝑗, 𝛿𝑖𝑖 = 1, otherwise 𝛿𝑖𝑗 = 0. According to the commutation relations of [𝑎1 , 𝑎†1 ] = 1, [𝑏1 , 𝑏†1 ] = 1, [𝑐1 , 𝑐1† ] = 1, we can get |𝑀11 |2 − |𝑀12 |2 − |𝑀13 |2 = 1, | | | | | | |𝑀22 |2 − |𝑀21 |2 − |𝑀23 |2 = 1, | | | | | |
|𝑀33 |2 − |𝑀31 |2 − |𝑀32 |2 = 1. | | | | | |
(22)
Similarly, According to the commutation relations [𝑎1 , 𝑏1 ] = 0, [𝑎1 , 𝑐1 ] = 0, [𝑏1 , 𝑐1 ] = 0 and [𝑎𝑜 , 𝑏†𝑜 ] = 0, [𝑎𝑜 , 𝑐𝑜† ] = 0, [𝑏𝑜 , 𝑐𝑜† ] = 0, we have 𝑀11 𝑀21 − 𝑀12 𝑀22 = 0, 𝑀11 𝑀31 − 𝑀13 𝑀33 = 0, 193
C.-P. Wei, Z.-M. Wu, C.-Z. Deng et al.
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