The phase sensitivity of an SU(1,1) interferometer with homodyne detection

We theoretically study the phase sensitivity of the SU(1,1) interferometer with a coherent light together with a squeezed vacuum input case using the method of homodyne. We find that the homodyne detection has better sensitivity than the intensity detection under this input case. For a certain intensity of coherent light (squeezed light) input, the relative phase sensitivity is not always better with increasing the squeezed strength (coherent light strength). The phase sensitivity can reach the Heisenberg limit only under a certain moderate parameter interval, which can be realized with current experiment ability.

We theoretically study the phase sensitivity of the SU(1,1) interferometer with a coherent state in one input port and a squeezed-vacuum state in the other input port using the method of homodyne. We find that the the relative phase sensitivity is not always better with increasing the input squeezed strength or increasing the input coherent light intensity. We give out the optimal condition to make the relative phase sensitivity to reach the Heisenberg limit, where the parameter requirements can be realized with current experiment technology. The interferometer is a fundamental physical apparatus that has been implemented using photons, neutrons [1], electrons [2], and atoms [3]. It has broad applications that a number of different interferometer configurations have been proposed to measure the orbital angular momentum of the photon [4], to study the berry phase [5], to encode qubits [6]. The optical interferometer is a very useful and flexible measuring tool for phase estimation [7][8][9][10][11][12][13]. The precision with which phase shifts can be determined in an interferometer is limited by shot noise or the standard quantum limit (SQL), △φ SQL ∼ 1/ √ N . Recent designed quantum procedures make it possible to reduce the classic noise to the level where the quantum noise becomes dominant that beat the SQL and reach the Heisenberg Limit (HL) △φ HL ∼ 1/N [7][8][9][10][11][12][13].
To beat the SQL, part of the research work based on the SU(2) type Mach-Zehnder interferometer (MZI) consider how to improve the input state. Different nonclassical input states have been suggested, such as, coherent ⊗ squeezed-vacuum light as input of a MZI suggested by Caves [7] and realized by Xiao et. al. [8], NOON states [10], etc. Another line of research follows the trail of improving measurement methods. Some measurement schemes have been proposed to improve the phase sensitivity, such as, parity measurements [14] which can be used to beat the HL [15], photon counts of two output ports [16], etc. Another part of research work pay attention to how to change the structure or hardware of the interferometer to enhance the phase sensitivity [12,13,[17][18][19]. For example, the nonlinear elements were introduced in the linear interferometers. In 1986, such a class of interferometers introduced by Yurke et. al. [19] is described by the group SU(1,1)-as opposed to SU (2). This class of interferometers may be realized by replacing the 50-50 beam splitters with four-wave mixers (FWMs) or parameter amplifiers in a traditional MZI [12,13,19,20]. Recently, our group firstly realized it in experiment with FWMs acting as beam splitters to split and recombine the incoming optical fields [21]. The phase sensitivity of the SU(1,1) interferometer with coherent state input has been studied using the intensity detection in detail [20,22]. The coherent light together with a squeezed vacuum injecting the SU(1,1) interferometer to break through the SQL has been proposed [12].
Here, we further study the phase sensitivity by using coherent ⊗ squeezed-vacuum light as input of a SU(1,1) interferometer in detail.
In this paper, we study the phase sensitivity of the SU(1,1) interferometer for a coherent light together with a squeezed vacuum light as input using the homodyne detection. For a certain strengths of squeezed and coherent lights, we describe the optimal parameter condition where the phase sensitivity can reach the Heisenberg limit. We also compare the homodyne detection with the intensity detection under this input case, and find that the phase sensitivity in the homodyne detection is better than that in the intensity detection. Here, the phase shift studied is not general phase but is sufficiently close to the optimal phase point [12,23].
An SU(1,1) interferometer proposed by Yurke et al. [19] is shown in Fig. 1. In our scheme, the detected variable is amplitude quadratureX other than the photon numberN which has been studied in Ref. [20,22]. Using the amplitude quadratureX, the phase sensitivity of the SU(1,1) interferometer is given by We firstly consider a lossless SU(1,1) interferometer with two input ports as shown in Fig. 1. After the first FWM, one output is retained as a reference, while the other experience a phase shift process. After the beam recombine in the second FWM with the reference light, the output fields are dependent on the phase difference φ between the two beams.â (â † ) andb (b † ) are the annihilation (creation) operators for the two mode, respectively. Then the output amplitude quadrature operator can be written asX Through the SU(1,1) transformation [20,22], we can obtainâ where U = cosh g 1 cosh g 2 + e −iφ e i(θ2−θ1) sinh g 1 sinh g 2 and V = e iθ1 sinh g 1 cosh g 2 + e −iφ e iθ2 cosh g 1 sinh g 2 , so |U| 2 − |V| 2 = 1. g 1 (g 2 ) and θ 1 (θ 2 ) describe the strength and phase shift in the process of FWM in the atomic cell 1 (2), respectively. The balanced situation is θ 2 − θ 1 = π and g = g 1 = g 2 that the second FWM will "undo" what the first did when the phase shift φ is 0. Simply for convenience, next we consider the balanced configuration (g 1 = g 2 = g and θ 2 − θ 1 = π). Here, we consider a coherent light |β together with a squeezed vacuum |0, ξ with ξ = re iη as input shown in Fig. 2, then the noise and phase sensitivity of the output amplitude quadratureX are given by where Φ = θ 2 − θ β − φ − π/2, Θ = η + 2θ U , θ U is given by U = |U| exp(iθ U ). When Θ = 0, the noise of Eq. (5) can be (△X) 2 = e −2r /2. Obviously e −2r < 1 when vacuum squeezed strength r > 0, therefore the noise can be reduced to below vacuum noise 1/2.
At the optimal point φ = 0, the best phase sensitivity of our scheme is where Φ = 0 is used. From Eq. (7), we find that the sensitivity △φ ′ can be improved by the factor e −r from the squeezed light and by the factor sinh(2g) from the FWM process. Now, we compare the phase sensitivity △φ ′ with the HL of △φ HL = 1/N Tot where N Tot (≡ â † 1â 1 +b † 1b 1 ) is the total photon number inside the SU(1,1) interferometer, then △φ HL = 1 cosh(2g)(|β| 2 + sinh 2 r) + 2 sinh 2 g .
When g = 0, the above result is △φ HL = 1/(|β| 2 +sinh 2 r) of the traditional MZI [7]. The performances of phase  Fig. 3(a) and Fig. 3(b), respectively. From Fig. 3(a), we obtain that for a certain intensity of coherent light input, the phase sensitivity is not always better with increasing squeezed strength r. In a similar way, for a certain intensity of squeezed light input, the phase sensitivity is not always better with increasing input coherent light strength, which is shown in Fig. 3(b). With Eqs. (7) and (8), the optimal condition to reach HL is |β| ≃ e r 2 tanh(2g).
Therefore, when the condition |β| ≃ e r tanh(2g)/2 is met between the input coherent state, the input squeezed vacuum state and the FWM process, the phase sensitivity can reach HL. As has been previously pointed out that the loss is the limiting factor in precision measurement [12,24,25]. Next, we investigate the effects of photon losses on the phase sensitivity of this coherent ⊗ squeezed-vacuum light as input case. Firstly we consider that both arms of the interferometer have the same internal losses L 1 and outside loss L 2 . After taking account into these losses we introduce that X L , compared with Eq. (2), can be written as where the subindex L means that the losses are considered. As shown in Fig. 2, in the balanced case, consid- ering the losses the sensitivity is given by where △φ is from Eq. (6), and the second term on the right-hand side is the extra noise term from loss. The inside losses make more impact on the sensitivity of phase measurement than the outside losses, which can be seen from the term cosh(2g)L 1 (1 − L 2 ) + L 2 ]/(1 − L 1 )(1 − L 2 ) due to cosh(2g). When L 1 = 0 and L 2 = 0, the effect is slight larger than that of the case L 1 = 0 and L 2 = 0 due to cosh(2g) > 1. We calculate the internal loss by making L 2 = 0 in Eq. (11), then it can be written as Similarly, when only consider the outside loss that letting L 1 = 0, such that, The comparison of Eq. (12) and Eq. (13) is shown in Fig.  4. Obviously, one can see that the effect of the internal loss is slightly greater than the outside loss. Finally, we will give a brief comparison between amplitude quadrature detectionX and output light intensity detectionN (≡n a2 +n b2 ) which has been discussed in Ref. [20,22]. Using the light intensity detection, the sensitivity of SU(1,1) interferometer is [20,22] where the superscript N means light intensity detection. We consider the case that a coherent light |β together with a squeezed vacuum |0, ξ input, without losses, the phase sensitivity of intensity detection is given by A = Λ 16(|β| 2 + 1 + sinh 2 r) 2 sin 2 (φ) sinh 2 g cosh 2 g , where ∆φ N s and △φ N c are the phase sensitivities of coherent ⊗ vacuum light as input and coherent ⊗ squeezedvacuum light as input, respectively. Under the same condition, the phase sensitivities by the homodyne detection and intensity detection are shown in Fig. 5. Obviously, one can obtain that the performance of the homodyne detection is slightly better than that of the intensity detection.
In summary, we investigated the phase sensitivity of the SU(1,1) interferometer for a coherent state and a squeezed vacuum state as input. We obtain that the the relative phase sensitivity is not always better with increasing the input squeezed strength or increasing the input coherent light intensity, and give out the optimal condition to reach the HL. For the loss effect, the inside losses make more impact on the sensitivity of phase measurement than the outside losses. Compared to the intensity detection, our scheme shows a slightly better phase sensitivity in precision phase measurement involving squeezing state input case.