Multi-parameter estimation with multi-mode Ramsey interferometry

Estimating multiple parameters simultaneously is of great importance to measurement science and application. For a single parameter, atomic Ramsey interferometry (or equivalently optical Mach–Zehnder interferometry) is capable of providing the precision at the standard quantum limit (SQL) using unentangled probe states as input. In such an interferometer, the first beam splitter represented by unitary transformation U generates a quantum phase sensing superposition state, while the second beam splitter U−1 recombines the phase encoded paths to realize interferometric sensing in terms of population measurements. We prove that such an interferometric scheme can be directly generalized to estimation of multiple parameters (associated with commuting generators) to the SQL precision using multi-mode unentangled states, if (but not iff) U is orthogonal, i.e. a unitary transformation with only real matrix elements. We show that such a U can always be constructed experimentally in a simple and scalable manner. The effects of particle number fluctuation and detection noise on such multi-mode interferometry are considered. Our findings offer a simple solution for estimating multiple parameters corresponding to mutually commuting generators.


Introduction
One of the central objectives of quantum metrology concerns improving measurement precision with finite sized ensembles [1][2][3][4]. Most previous investigations have focused on single parameter estimation, of which the standard quantum limit (SQL) or the classical limit, N 1 , represents the minimal phase uncertainty achievable in an interferometric measurement using an ensemble of N uncorrelated particles [5]. Recently, the problem of estimating multiple parameters has attracted much interests , where the focus shifts to finding efficient strategies for estimating parameters corresponding to multiple commuting or non-commuting generators as precisely as possible. Potential applications of such studies include quantum imaging [8,15,28], sensor networks [37,40], measurements of multidimensional fields [18], and joint measurements of multiple quadratures [10,[32][33][34], etc.
The main tasks in multi-parameter estimation are to generate an input quantum state capable of realizing the optimal precision limited by the laws of quantum mechanics, and to find a corresponding measurement scheme that achieves this precision. In the language of estimation theory, the former obtains a quantum state with the lowest quantum Cramér-Rao bound (QCRB) for a set of parameters to be estimated, while the latter provides measurement results of which the Cramér-Rao bound (CRB) equals the QCRB. For estimation of a single parameter, the latter can always be fulfilled using interferometry in which the second beam splitter acts as the inverse transformation (time reversed operation) to the first [42], such as in an atomic Ramsey interferometer and an optical Mach-Zehnder interferometer. However, the same does not apply in general for multi-parameter estimation.
This work considers the more specific case of multi-mode interferometry for estimating a set of parameters corresponding to mutually commuting generators using unentangled particles (as illustrated in figure 1). A probe state is generated by splitting a pure single-mode state ñ i | into multiple modes using a multi-mode beam splitter represented by a unitary transformation U 1 . The prepared (D+1)-mode probe state then undergoes phase accumulation, and is subsequently transformed by U 2 at the second beam splitter. The measured particle number distributions at the (D+1)-outputs are used to estimate the D parameters in the end. Unlike the case of single parameter estimation, setting = = -U U U 2 1 1 1 † does not guarantee CRB will be equal to the QCRB in general when D>1. Instead for a given U 1 , U 2 has to be optimized numerically to reach QCRB. This becomes a cumbersome and tedious job particularly when the number of parameters to be estimated is large.
As a main result to be reported in this paper, we prove that the Ramsey interferometric scheme can be straightforwardly generalized to estimation of multiple parameters (associated with commuting generators) using multi-mode pure states, if (but not iff) U is made orthogonal, i.e. when U is unitary and has only real matrix elements. We also illustrate how such orthogonal U can be constructed experimentally in a simple and scalable way. The influences of particle number fluctuation and detection noise will also be discussed.
So far, most measurement schemes which saturate the QCRB for multi-parameter estimation, if they exist, are found on a case by case basis. Important progresses have been made in this direction recently [8,27,32,36,39]. However many of the proposed measurement schemes are either not directly implementable or experimentally prohibitive, particularly when they involve measurements on entangled particles . Therefore, generalization of Ramsey interferometry to multi-parameter estimation represents an interesting and timely advance.
This article is organized as follows: section 2 defines the problem we consider and gives the QCRB of a multimode probe state. Section 3 proves that for an orthogonal U 1 , the CRB from setting = -U U 2 1 1 is equal to the QCRB. Section 4 illustrates how to determine the optimal probe state that gives the lowest QCRB. In section 5, we show how an orthogonal U can always be constructed experimentally in a simple and scalable manner in an optical or atomic system. Finally, we consider the influence of particle number fluctuation and detection noise on the multi-mode Ramsey interferometer in sections 6 and 7, respectively. The article ends with appendices A and B containing further calculation details.

General framework and the QCRB of a given probe state
In this section,we define the problem we consider and give the QCRB for a given probe state. As shown in figure 1,the parameters we consider are encoded into quantum states with D+1 modes,which can be implemented with photons split into multiple paths,or atoms with large spins. For unentangled particles,the interferometry can be discussed in terms of self-interference of individual particles [43]. Therefore,we consider an arbitrary single particle initial state ñ i | ,and a probe state y a ñ = ñ = å ñ = U i k k D k p 1 0 | | | after transformation U 1 ,with a k being the probability amplitude in mode k. We assume that the probe state is pure for the time being and the interferometry is noiseless. The phase accumulation evolves the probe state into y a ñ = å ñ Interference from the first order coherence allows D (out of the D+1) phases to be measured in the absence of an external reference. This is often carried out by choosing an arbitrary mode,say ñ 0 | ,as the reference,and measuring the relative phase shifts q f f º - . According to multi-parameter Figure 1. A standard (D+1)-mode interferometer for unentangled particles. The interferometer starts with a pure single mode state ñ i | followed by a unitary transformation U 1 (linear beam splitter), phase accumulation, and a second unitary transformation U 2 (combining), and ends with particle number detection in every mode. quantum estimation theory [44,45], the lower bound of Q D 2 ( ) with an unbiased estimator is determined by the trace of the inverse of quantum Fisher information matrix (QFIM)  Q : where N M is the number of experiments repeated (set to 1 hereafter for simplicity). Note that, the choice of a figure of merit for precision as the trace of the inverse of  Q in equation (1) is fully general, since the weight of each parameter f k can be adjusted by changing the coefficients of the linear combinations in f For a pure state y ñ f | , the matrix elements of  Q are explicitly given by [44,45] y y y y y y = 1 † followed by particle number detection afterwards gives the best precision allowed by the QCRB when Q~0.
For Ramsey interferometry, the state after the full interferometric protocol (before particle number detection) is represented by y ñ =  ñ . The CRB, which sets the minimal Q D 2 ( ) given a measurement scheme, can be calculated for any U using the classical Fisher information matrix (CFIM) [46] å Q Q denotes the probability of finding a particle in mode ñ m | for a given Q.
To show that a Ramsey interferometric scheme can be used to estimate multiple parameters to the SQL precision, we need to prove the CFIM given by equation (4) equals to QFIM given by equation (1) (since they correspond to the CRB and the QCRB, respectively). For small Q, omitting the third order corrections, a Taylor series expansion around Q~0 gives In [27], Pezzè et al found the necessary and sufficient conditions (iff) for projective measurements which saturate the QCRB of a probe state. In their language, our measurement can be described by a set of projectors ¡ ñá¡ | . In the limit Q~0 and given that all elements of U are real, the projectors ¡ ñá¡ k k {| |}indeed satisfy the required condition given by their equation (7) in [27].

Determining the optimal probe state
In this section, we demonstrate how to determine the optimal probe state. As an illustration, we consider the most common choice of In this case, the generator of parameter θ k is proportional to ñá k k | |. Computing equation (3) and taking the trace of gives the optimal probe state described by and the QCRB of This precision can be reached in the asymptotic regime of large N M . For comparison, we consider an individual estimation scheme which divides the N particles into D equal partitions, and uses each partition for measuring one q k through two-mode interferometry between ñ 0 | and ñ k | . Since the SQL of each θ k in this case is N D 1 , the lowest bound for the phase variance becomes For D=1 as in single parameter estimation, both equations (10) and (11) reduce to N 1 as expected (i.e. the SQL). For larger D, the simultaneous estimation scheme (equation (10)) always outperforms the individual estimation scheme (equation (11)).
We note that the results of equations (9) and (10) resemble an earlier study [8], where Humphreys et al considered a multi-mode entangled NOON state y a a a ñ = ¼ ñ + They found an optimal probe defined also by equation (9) and a QCRB N times smaller than equation (10), in agreement with the typical ratio between the SQL and the Heisenberg limit (HL). Their results reduce to ours when a multimode NOON state for N=1 is considered.
The value of the reference mode a 0 2 | | in equation (9) is D times larger than the other modes. Because f 0 is referenced to by all q f f º k k 0 , the measurement variance of f 0 therefore contributes D times more to q D 2 ( ) than any other uncorrelated phases {f k }. Such a bias results from the choice of parameters. Consequently, equations (9) and (10) cannot be always optimal if the parameters of interest are different. For instance, should we consider a different set of parameters of interest, say, j

Experimental realization of U
We now illustrate how a multi-mode Ramsey interferometer can be realized experimentally in a simple and scalable way. Here, the task reduces to designing an orthogonal U that generates the optimal probe state. We would illustrate our scheme first for an optical interferometer and then an atomic interferometer.

Multi-mode optical interferometer
We consider a design that employs a series of 2×2 non-polarizing beam splitters (BS k , for k=1, 2, 3, L) for splitting particles into the optimal distributions a k 2 | | as shown in figure 2. In this case, each U ( k) which represents the transformation due to beam splitter BS k acts only on two of the adjacent modes, leaving other modes untouched. The overall transformation =  must be unitary since each lossless physical splitter U ( k) is unitary.
To ensure that the resulting U and U † constructed from these beam splitters are real (orthogonal), the most straightforward way is to make sure that each of the beam splitters behaves as a real 2×2 transformation. This criterion, which requires zero (or multiple of 2π) phase shifts for both the transmitted and reflected beams with respect to both input beams, is not automatically satisfied for any beam splitters. Fortunately, it is always possible to fulfil this criterion by adding respective phase compensating waveplate to each port of a beam splitter. After compensation, the matrix elements of a real U ( k) become h represents the reflectance (transmittance) of BS k . Given an optimal distribution a k 2 | | , the reflectance of BS k should be chosen as h a = a cos , 12 In addition, extra phase compensators are needed in every arm of the interferometer to null out the difference in optical path lengths and to tune every phase shift f k to the region where Q can be measured most sensitively. If there is no detection noise, this region is Q close to zero, otherwise, it is shifted away from Q~0 (see section 7). We emphasize that compensating for f k to give the optimal sensitivity does not represent a flaw, in fact, as such tuning is needed in practically all real interferometric measurements near the SQL precision.

Multi-mode atomic interferometer
In interferometry of atoms with hyperfine spin = F D F , 2 different parameters can be estimated. Analogous to the optical scheme, an arbitrary spin distribution can be constructed using a sequence of Rabi rotations between two adjacent Zeeman sublevels. Such rotations can be realized, for instance, using a two-photon Raman transition through an intermediate state as illustrated in figure 3. As long as the intermediate hyperfine levels have a different Landé g-factor from those involved in interferometry, one could perform Rabi rotations between any two adjacent sublevels by selectively detuned to a suitable intermediate states. To make sure that the individual transformation is orthogonal, every rotation should be performed along the s = - , within the two-level subspace. However, as each of the Zeeman sublevel exhibits different shift inside a magnetic field and thus different phase accumulation rate, one would need to keep track of the phases of every levels and to account for them when performing individual Rabi rotations. While this is possible with current technologies in cold atom experiments, the process is perhaps too cumbersome to be practical, especially when atomic spin is large. For the aforementioned reasons, we restrict the transformation in the following to a single-pulse multimode Rabi rotation over an angle χ along the F y direction (since the corresponding matrix c = -U F exp i y ( ) is always orthogonal for any atomic spin F), and study the performance of the Ramsey interferometric protocol for measuring q. Experimentally, such a F y rotation can be realized using a radio-frequency resonant with adjacent Zeeman sublevels, when the quadratic Zeeman shift is negligible. It transforms the initial state is chosen as the reference mode. Figures 4 (a)-(c) present the values of equation (13) for F=1, 3, and 5, respectively (for m 0 =0). This one-step-rotation scheme (OSRS), which employs the limited family of a single SU(2) transformation, is found to always outperform the individual measurement scheme (equation (11), grey dashed horizontal line) using a suitable initial state ñ F m , i | and a rotation angle χ, at least up to F=5 ( figure 4(d)). The same conclusion is reached for parameters {j k }. Figure 3. Preparation of the optimal probe state with a sequence of two-photon Raman pulses between two adjacent states. This example starts from the state -ñ F F , | , although more generally the state preparation can start from any Zeeman sublevels to reach the same final probability amplitude distribution of the optimal state. Such multi-parameter estimation scheme can be useful when atoms are subjected to different sources of phase shifts simultaneously, as for example, with spin-1 87 Rb atoms dressed by near-resonant microwaves while under a static magnetic field [47,48], or spin-9/2 87 Sr atoms placed in an optical lattice with polarization dependent light shifts, and collisions with background or non-condensed atoms.

The effects of particle number fluctuation
In this section, we discuss the influence of particle number fluctuation of the probe state. Since quantum states with a definite large particle number are often difficult to prepare, we consider the situation when the particle number of the probe state fluctuates. Due to the superselection rule, such input state represents nothing but an incoherent superposition of different Fock state r r = Å = +¥ Q N N N in 0 ( ) in the absence of number coherences in the probe state and/or in the measurement strategy [49][50][51], where r N ( ) is the density matrix of the N-particle state and Q N the probability of having N particles. For coherent light of photons or an atomic Bose-Einstein condensate, the particle number obeys Poisson distribution with the probability , where N denotes the mean particle number. Since QFIM is additive under a direct sum of density matrix r N ( ) in orthogonal subspaces [2], , with  Q single being the QFIM of the single particle probe state r single . One has therefore Similarly, it can be readily shown that ]¯for input state r in [52] if particle numbers in all output ports are measured without detection noise. Since the CFIM of a single particle probe,  (10)) and q D ind 2 ( ) (equation (11)), respectively. The legends show the corresponding initial state ñ F m , i | before applying U. Irrespective of m i , the phase shifts q are always defined with respect to the reference mode ñ F, 0 | . (d) Comparison between the optimal q D 2 ( ) from OSRS to q D ind 2 ( ) and q D opt 2 ( ) for = F 1, 2, 3, 4, 5 { } . The OSRS is found to be on par with the optimal simultaneous scheme only for F=1, but it always performs better than the individual measurement scheme. N=1 for all figures.
scheme,  C also equals approximately to  Q for probe state with fluctuating particle number and all our conclusions outlined above remain intact.

The influence of detection noise
The conclusions in section 3 and appendix A are reached assuming noiseless particle number detections. When detection noise is taken into consideration, the optimal sensitivity typically shifts away from Q~0. For example, for single parameter estimation using Ramsey interferometry in an atomic clock, the measurement is usually performed near θ∼π/2, a region least sensitive to detection noise.
Here, we study numerically the effects of detection noise to the multi-parameter Ramsey interferometry using the example of two parameter estimation. We consider estimation of θ 1 and θ 2 using the optimal probe state given by equation (9). Starting from the initial state ñ = 0 0, 1, 0 | ( ) † , we choose an orthogonal U given by the SU (2) Here, χ is set to 0.2774π to give the optimal probe state. The simulated procedure consists of applying U, phase accumulation q q ñá + -ñáexp i 1 1 1 1 , and U † , followed by population detection with or without including noise.
When there exists no detection noise, the CFIM (equation (4)) of the aforementioned protocol is directly computed and the trace of its inverse is used to obtain the CRB of q D 2 ( ) . Figure 5(a) compares the value of the corresponding result to the QCRB of the individual measurement scheme (equation (11)) for {θ 1 , θ 2 }ä(0, π), illustrated by the parameter q q z = -D D 10 log 10 The region surrounded by the white dashed curve represents the {θ 1 , θ 2 }-space where the proposed scheme outperforms the individual estimation scheme. It shows that the proposed scheme works well even for q far away from zero. We emphasize that the probe state defined by equation (9) gives always the best QCRB for any q. However, application of the reversed transformation U † followed by a population measurement is not necessary the optimal measurement scheme when q is away from zero, which explains the deficiency of the scheme over some parameter space.
When detection noise is present, we numerically simulate the estimation process of the two parameters q q , 1 2 { } using 10 4 three-mode (spin-1) atoms with a detection resolution (noise) of 14 atoms (typical numbers achievable in cold-atom experiments [47,48]). For each pair of {θ 1 , θ 2 }, we first compute the probability of detecting an atom in the output mode q q q q = á ñá + -ñá-ñ m p m m U U , , e x p i 1 1 1 1 0 For each run, we perform Monte-Carlo simulation on the outcome for each of the 10 4 atoms according to the distribution of q q p m , 1 2 ( | )and obtain ¢ N m (the total number of particles in mode m without detection noise). We then add to ¢ N m a random detection noise featuring a normal distribution with an average of zero and a standard deviation of 14 to obtain N m . The maximal likelihood method (which can saturate the CRB [46] in the asymptotic limit and is unbiased) is then used to estimate {θ 1 , θ 2 }.  the individual measurement scheme using the same number of particles, is shown in figure 5(b). Although detection noise degrades the sensitivity of the proposed scheme, the discussed scheme is seen to maintain its advantage over the individual measurement scheme over a large parameter space. Similar to typical single parameter estimation scenario, the position of the minimum q D 2 ( ) is seen to shift away from zero. The star in figure 5(b) (near q q p = = 0.3 1 2 ) denotes the position of the maximum precision for the scenario we consider, where it is z 0.6 dB  more sensitive than the individual measurement scheme, but is 0.77dB less than the optimal z 1.37 dB  for noiseless detection. In short, the influence of detection noise to a multi-mode Ramsey interferometer is similar to that to a single-mode Ramsey interferometer.

Summary
In summary, we show that the Ramsey interferometric scheme can be extended to estimation of multiple parameters (associated with commuting generators) using multi-mode pure states, if (but not iff) the multimode beam splitter U is orthogonal, i.e. all matrix elements of U are real and = UU 1 † . We then discuss how to obtain the optimal probe state, and how to construct U experimentally in a simple and scalable manner. We find that the proposed scheme remains intact even under particle number fluctuation and detection noise. The results of this study can be useful to applications in multi-mode optical sensing and quantum phase imaging.