We theoretically investigate an interferometer composed of four four-wave-mixers by Lie group method. Lie group characterizes the mode transformations of this kind of interferometer. With vacuum state inputs, the phase sensitivity of interferometer achieves the Heisenberg limit, and the absolute accuracy beats interferometer because of higher intensity of light inside the interferometer. For different input cases, the optimal combination of output photon number for detection to obtain the best phase sensitivity is calculated. Our research on interferometer sheds light on the performance of interferometer in quantum metrology.
pacs:42.50.St, 07.60.Ly, 02.20.Qs
Quantum metrology giovannetti2004quantum; giovannetti2011advances takes the advantages of quantum mechanics to realize the high resolution of parameter measurement in physical systems. The original motivation of studying quantum metrology comes from general relativity. General relativity predicts the existence of gravitational wave, which, however, is too weak to be experimentally detected. To enhance the sensitivity of interferometers, physicists feed high power laser into interferometers. But it seems that the sensitivity of this kind of laser interferometers is restricted by standard quantum limit (SQL) PhysRevD.23.1693. SQL indicates that the phase sensitivity is bounded by , where is the photon number. The best phase sensitivity which can be achieved with classical interferometer, like Mach-Zehnder interferometer (MZI) powered by a coherent state laser, cannot go beyond the SQL. But SQL is not the ultimate limit physicists can achieve, they develop new methods to beat the SQL and achieve the Heisenberg limit, which indicates that the phase sensitivity equals to , i.e., the reciprocal of the photon number. This kind of interferometers beating SQL are quantum interferometers.
With the development of modern mathematics, Lie group theory is widely used to describe symmetric structures in theoretical physics. In PhysRevA.33.4033, Lie group theory is used to describe the mode transformations in an interferometer. Specifically, the interferometers like MZI, consisting of beam splitters, can be characterized by the Lie group , while the interferometer using four wave mixers to replace beam splitters is characterized by the Lie group . This Lie group structure is determined by the Lie algebra formed by the Hamiltonian of the interferometer. With single Fock state input, the interferometer cannot go beyond the SQL. But fed with a special correlated Fock state , interferometer can attain the Heisenberg limit PhysRevA.33.4033. Besides, it is shown that if the second input of a MZI is fed with a squeezed vacuum state, the phase sensitivity can beat the SQL PhysRevD.23.1693. In contrast, interferometer can achieve the Heisenberg limit with vacuum state inputs. It will be much more easier to implement since it needs fewer optical devices experimentally. But since the photon number inside the interferometer with vacuum state inputs is relatively low, W. N. Plick et al. plick2010coherent suggest injecting coherent state light into interferometer to significantly enhance the phase sensitivity. Feeding strong coherent state laser makes the photon number in interferometer increased vastly and makes the phase sensitivity reaching far below the SQL. However, the phase sensitivity can only inversely scale with the square root of the intensity of the coherent state input light. D. Li et al. li2014phase proposes that injecting a coherent state and a squeezed vacuum state into interferometer with homodyne detection can decrease the noise and enhance the sensitivity. But this method cannot always attain the Heisenberg limit, either. In this paper, we present a new interferometer called SU(1,2) interferometer composed of four four-wave-mixers (FWMs) which is described by the Lie group . The absolute accuracy can not only beat the sensitivity of interferometer for the same input fields, but also make the phase sensitivity scale with the Heisenberg limit.
These interferometers we mentioned are two input and two output interferometers, whereas three input and three output interferometer sanders1999vector; rowe1999representations; PhysRevLett.110.113603 and other multi-path interferometers PhysRevA.46.2840; PhysRevA.55.2267; PhysRevA.55.2564; BenAryeh20102863; spagnolo2012quantum are also investigated for different purposes. Multi-path interferometer generally owns more than two input and two output ports. Over two beams are mixed in a multi-path interferometer. B. C. Sanders et al. sanders1999vector investigates measuring vector phase with interferometer. interferometer, different from a two-path interferometer, can be used to estimate two phase shifts simultaneously. G. M. DAriano et al. PhysRevA.55.2267 propose a scheme of using multi-path interferometer to enhance phase sensitivity and asserts that the phase sensitivity can inversely scale with the number of beam paths in an interferometer. It seems that there are two advantages of multi-path interferometers in phase estimation. One is to estimate more phases simultaneously, the other is to enhance the phase sensitivity. These two advantages motivate us to investigate quantum metrology with multi-path interferometer.
As far as we know, most multi-path interferometers discussed previously uses linear optical devices like beam splitters, while multi-path nonlinear quantum interferometer utilizing nonlinear optical devices such as FWM has not been theoretically investigated. We want to use the Lie group theoretical method in PhysRevA.33.4033 to investigate a multi-path nonlinear quantum interferometer, denoted by interferometer. We’re interested in the sensitivity of interferometer in phase estimation and want to compare its phase sensitivity with interferometer PhysRevA.33.4033; plick2010coherent.
This paper mainly contains three parts. Sec. II introduces eight Hermitian operators spanning the Lie algebra . The mode transformations given by the conjugation operation form the group . Then we introduce the structure of interferometer. Sec. III shows that with vacuum state inputs, interferometer can attain the Heisenberg limit and beats the sensitivity of interferometer. The optimal combination of output photon numbers for detection is found to achieve the best sensitivity. Sec. IV analyzes the case that a coherent state beam is fed into interferometer. We give the optimal combination of output photon numbers for detection in different input cases. Furthermore, if the coherent state light is injected into the third input, the phase sensitivity can still approach the Heisenberg limit when the amplification gain of the FWMs is large. Sec. V gives the conclusion of this paper.
Ii Lie group and interferometer
In this section, based on the method of Lie group we introduce the interferometer possessing three inputs and three outputs. A general three input and three output interferometer is shown in Fig. 1. If we restrict the unitary operation to operation, then it is a interferometer. Then in Fig. 2 we use the nonlinear optical devices FWMs to form one type of interferometer. The connection between Lie group and the optical devices FWMs are presented. It can be constructed with current experimental technology PhysRevLett.113.023602.
A mode transformation is
where the creative and annihilation operators satisfy the boson commutation relations. The transformation matrix has the following relation:
All the matrices satisfying Eq. (2) form a matrix Lie group . To generate these mode transformations, we introduce eight Hermitian operators which are given by
Each operator () in Eqs. (3) is a Hamiltonian of some physical process generating mode transformations. and create or annihilate photons in pairs, which describe the process of generating photons in FWMs. and are the Hamiltonian of a FWM with beam and beam as inputs, and and are the Hamiltonian of a FWM with beam and beam as inputs. and annihilate a photon and create another photon simultaneously, maintaining the total photon number. It is a passive optical device, generally we call beam splitter. and are the Hamiltonian of a beam splitter with beam and beam as inputs. and are combinations of photon number operators and numbers. They can be the Hamiltonian of relative phase shifts.
Using Baker-Campbell-Hausdorff (BCH) formula, we can calculate the conjugation of and given by the exponential of the operators in Eqs. (3), which are given by
where . The transformations are in the form of Eq. (1), and satisfy Eq. (2), implying that the transformations are all matrices. From Eqs. (4), we can see that and amplify beam and beam and that and amplify beam and beam . They represent the operations of FWMs. and generate a combination of beam and beam , which is like a rotation in the two-dimensional space spanned by and . They represent the operations of beam splitters. Whereas and multiply and by a unit complex number. This process only changes the phase of each mode.
The eight operators in Eqs. (3) span the Lie algebra . From the Lie group theory cahn2006semi we know that the conjugation given by the exponential of a Lie algebra forms a Lie group. Specifically, in our case, ( is from to ) form the Lie algebra , the conjugation given by the exponential of in the vector space spanned by and form the Lie group .
It’s interesting that the combination of photon number operators commute with all the . It means that throughout any mode transformation, the photon number difference remains the same. It makes sense because the FWMs generate photon pairs in beam and beam , or in beam and beam , while beam splitter remain the total photon number in beam and beam invariant. Going through any combination of these devices, the same amount of photons will be added into beam and the combination of beam and beam .
A interferometer consisting of four FWMs is shown in Fig. 2. The two inputs of a FWM are different: one is called probe beam, the other one is idler beam. We use beam and beam to denote the probe beam and the idler beam of the first FWM respectively. The two output beams are denoted by beam and beam :
where and are the amplification gain and the phase parameter of the first FWM. Later we use and ( is from to ) to denote the phase parameters and the amplification parameters of each FWM from left to right. Then beam is injected into the second FWM as the probe beam, and beam injected as the idler beam. The two output beams are denoted by beam and beam :
After that, beam , and experience three independent phase shifts, , and , respectively. We denote the three beams after phase shifting as beam to beam . Then they pass through the third and the fourth FWMs. The transformations of the annihilation operators in each beam can be written as
In the end, one can detect the sum of the output photon numbers in beam and and the photon number in beam .
In the following, we are going to investigate the phase sensitivity of interferometer. The phase sensitivity is defined by
where can be or , are real numbers, and , similar for and , denoting the photon numbers in beam . Phase sensitivity is the standard deviation of the estimation result of the phase shift. Choose the output photon number as the estimator. The phase sensitivity equals to the standard deviation of the estimator divided by the derivative of the mean value of the estimator with respect to the corresponding phase shift. Various estimators result in different phase sensitivities for the same estimation target.
Without loss of generality, we set the phase parameters of the FWMs and . The change of the phase parameters makes the third and fourth FWMs perform the inverse operations of the first and second FWMs, respectively.
Iii Phase sensitivity with vacuum state inputs
This section investigates the phase sensitivity of interferometer when all the three inputs are vacuum states.
The detailed calculation of the phase sensitivity is omitted. Here we only focus on analysis of the results. We find that the optimal sensitivity is achieved when the sum of the photon numbers in beam and beam is detected. The comparison of the phase sensitivity of interferometer with interferometer shows its advantage of enhancing phase sensitivity.
In interferometer, each phase sensitivity depends on the amount of the phase shift in each beam. Each phase sensitivity achieves its minimum when all the three phase shifts vanish, when the output state approaches the input state. Fig. 3 shows how the phase sensitivity varies with respect to the other two phase shifts. The impact of shifting on the sensitivity is much greater than shifting . Because
For large , is nearly equal to . Thus shifting with is nearly equivalent to shifting with , which significantly affect .
Since the phase shift can be controlled by feedback loops yonezawa2012quantum to keep the phase sensitivity optimal, we assume during each single phase estimation, the other two phase shifts vanish. We will focus on the phase sensitivity or . As for , when , the interferometer is similar to the interferometer, but the light intensities insider the interferometers are different.
At the output side, detecting different combinations of photon numbers in three outputs may lead to different phase sensitivities, among which, one combination is optimal. Fig. 4 shows the phase sensitivity as a function of the ratios and . Any combination of the output photon numbers can be written as a combination of the linearly independent operators and . The first two operators can be obtained from and at the output side, and the last one is the invariant operator equaling to . For vacuum state input, . So we only need to find the optimal ratio between and . It’s found that the best combination is the sum of the output photon numbers in beam and beam , i.e., , which is achieved when and in Fig. 4.
By taking the limitations of the expression of phase sensitivity at zero phase shifts, we obtain the optimal phase sensitivity which is given by
and are not exactly the same, but when , they are nearly equal to each other. The phase sensitivity of interferometer is , where is the amplification gain of the FWMs in interferometer. Because of a second optical parametric amplification, the phase sensitivity and are improved compared to the interferometer. With vacuum states input, the absolute accuracy of interferometer beats that of interferometer.
In order to know whether the phase sensitivity can achieve the Heisenberg limit, we need to find how the phase sensitivities scale with the photon number. The total photon number in the interferometry is the sum of the photon numbers in beam , beam and beam as shown in Fig. 2, which is given by
The phase sensitivities (solid red line) and (dot-dashed green curve) as functions of the total photon number are shown in Fig. 5. The dashed blue line represents the Heisenberg limit. It can be seen that both and nearly equal to the Heisenberg limit.
Iv Phase sensitivity with coherent state input
This section still investigates the interferometer in Fig. 2. Whereas the inputs become one coherent state and two vacuum states. There are three different cases since a coherent state can be injected into any one of the three input ports. We will find that with one coherent state input, the phase sensitivity of interferometer can be further enhanced and it’s still possible for the phase sensitivity to achieve the Heisenberg limit.
When a coherent state beam is injected into input , to achieve the optimal phase sensitivity at zero phase shifts, one cannot choose the photon number as the estimator to estimate any phase shift, otherwise the phase sensitivity will diverge. So in Eq. 6 and we can only detect the combination of photon numbers in beam and beam . Which linear combination of and leads to the optimal phase sensitivity depends on the amplification gain of the FWMs and the intensity of the coherent state light input. Fig. 6 shows the optimal value of to obtain the best phase sensitivity when detecting the combination . When the intensity of the coherent state input is low or the amplification gain of the FWMs is high, the best combination is . Otherwise, the optimal will be larger than . If the coherent state light is injected into the input , then one need to detect the combination of photon numbers in beam and beam . It’s similar for the case that the coherent state is in input . Fig. 7 shows the optimal ratio when a coherent state light is fed into input .
Lastly, let’s see whether the phase sensitivity of interferometer with one coherent state input can achieve the Heisenberg limit. In the case of one coherent state light input, the total photon number will be different. When the coherent state light is fed into input or input , the phase sensitivity and are respectively shown in Fig. 8 as a function of photon number . It’s shown that if the coherent state light is injected into the first input, the phase sensitivity cannot achieve the Heisenberg limit, while if the coherent state light is fed into the third input, it can attain the Heisenberg limit when the amplification gains are large. Comparing the values of the phase sensitivities in these two cases, we find injecting coherent state light into the third input leads to better phase sensitivity.
V Discussion and Conclusion
This paper discusses the ideal model of interferometer, since photon number loss error has not been considered PhysRevA.86.023844. But the phenomenon of photon loss widely exists in practical FWMs. interferometers utilizes two more FWMs than interferometers, which necessarily introduces higher probability of photon loss. In practical experiments, whether the enhancement of phase sensitivity of interferometer is robust to the photon loss error has not been studied now.
In this paper, we show that interferometer can enhance the phase sensitivity compared with interferometer. This work sheds light on the performance of interferometer. We believe by adding more FWMs, interferometer can further enhance the phase sensitivity. What’s more, in this work, we use the Lie group theoretical method to investigate quantum interferometers. The way of our investigation in this paper can be generalized into any interferometer.
In conclusion, this paper uses Lie group and Lie algebra theory to analyze one multi-path nonlinear optical quantum interferometer, denoted by interferometer. Eight Hermitian operators spanning Lie algebra are introduced to describe the Hamiltonian of interferometer. We calculate and analyze the phase sensitivities in interferometer. With all vacuum state inputs, the optimal phase sensitivity is achieved when the sum of the photon numbers in beam and beam is detected. This phase sensitivity of interferometer can achieve the Heisenberg limit and beats the sensitivity of interferometer, because interferometer amplifies the intensity of the input beams twice. As for the case of one coherent state light input, we analyze the optimal detection combination when coherent state light is fed into different input ports. It is shown that the Heisenberg limit can be approached only when the coherent state light is injected into the third input port.