Exceptional-point sensing with a quantum interferometer

Recently, multiple studies have suggested that exceptional points (EPs) in lossless nonlinear optical systems can minimize quantum noise arising from the material gain and loss in conventional non-Hermitian systems, offering the possibility of quantum EP sensing. Meanwhile, nonlinear SU(1,1) interferometers have been established as useful in sensing due to their reduced quantum noise. In this work, we demonstrate the existence of EPs in a dual-beam SU(1,1) interferometer with two nonlinear parametric amplifiers. Our analysis of the input-output matrix in terms of joint quadrature amplitudes shows that EPs can be linked to both high signal, through a zero matrix element, and low noise, through noise preservation, in sensing by selecting an appropriate operation gauge of the quadrature amplitudes. Additionally, for a multistage SU(1,1) interferometer, EPs of the overall input-output matrix form multiple bands of high signal-to-noise ratio (SNR) which further separate into two phases indicated by the EPs of the transfer matrix of a repeating unit. Our investigations demonstrate the significance of quantum EPs in quantum interferometer sensing and broaden the operating regimes from diabolical points in some of the conventional SU(1,1) interferometers to EPs while still maintaining a high SNR.


Introduction
Traditionally, lossless optical systems are considered Hermitian. With recent advancements incorporating material loss or gain into the system, these optical systems are now non-Hermitian with eigenvalues that are generally complex, and the eigenvectors of the Hamiltonian are no longer required to be orthogonal. The parity-time (PT) symmetric system is a significant type of non-Hermitian system in which the system remains unchanged under the combined operations of parity inversion (P operation, i.e. mirror image) and time reversal (T operation). Research has demonstrated that PT-symmetric systems can still exhibit real eigenvalues even when the system is non-Hermitian [1]. In an extreme case, these non-Hermitian eigenvalues and eigenvectors can coalesce, resulting in the Hamiltonian becoming defective at so-called exceptional points (EPs) with demonstrations in optics [2][3][4][5]. These EPs enable various intriguing phenomena, such as unidirectional zero reflection (UZR), which occurs when the scattering matrix is at an EP [6,7]. At this point, one of the off-diagonal elements of the scattering matrix becomes zero, indicating zero reflection from one side but not the other. EP can also be related to chiral mode conversion when the state circles an EP in parameter space [8][9][10]. Due to the special eigenvalue topology around the EP, the final state depends on the path orientation. This chiral behavior has been recently used to modify other optical phenomena such as chiral lasing [11,12] and its time-reversal counterpart, coherent perfect absorption [13][14][15]. Other notable applications include electromagnetically induced transparency [16], suppression of lasing with increasing gain [17], etc.
Recently, non-Hermitian systems and EPs have garnered attention for their implementation in the quantum realm. For instance, the quantum interference in a PT-symmetric waveguide has been shown to be different from that in Hermitian systems [18]. The use of UZR metasurface for photon state transformations [19,20] has also been explored. Chiral quantum state conversions have been experimentally observed in a cold atom system [21]. One of the most significant applications of EPs, both in classical and quantum regimes, is sensing due to their sensitivity to perturbations. At an EP, the eigenvalue splitting under small perturbationsâ 0 is of the order O(√δ) for a second-order EP, where two eigenvalues and eigenvectors coalesce. In contrast, at a diabolical point (DP), where only the eigenvalues degenerate and the eigenvectors remain distinct, the eigenvalue splitting is of the order O(δ) and is less sensitive to perturbations [22][23][24]. However, the utilization of EPs for sensing has been met with challenges, as the sensitivity of frequency splitting around EPs can be impaired by both classical and quantum noise [25][26][27]. The usefulness of EPs in sensing remains an open question [28][29][30][31], and may depend on whether the system is linear and also whether we should look for scattering anomalies near EP instead [22,24]. Towards the regime of quantum EP sensing, these developments make the ultimate application of EPs in quantum sensing elusive since quantum noise is hardly avoidable in the limited number of photons and also the way to set up a quantum EP is still not entirely clear. An alternative solution has been proposed, using systems with Hermitian second-quantized Hamiltonian and nonlinear interactions (e.g. a parametric amplifier (PA)) instead of conventional non-Hermitian systems driven by material gain and loss, to reduce quantum noise [32,33]. This has already been demonstrated in nonlinear optical materials and cold atom systems with four-wave mixing, showing the capability to capture EPs [34][35][36].
In addition to the previously mentioned developments, nonlinear interactions in PAs have been found to be useful in constructing quantum interferometers, known as SU(1,1) interferometers [37,38]. In this case, the PAs mix two modes for interference, instead of using beam splitters in a traditional Mach-Zehnder interferometer. These SU(1,1) interferometers can be used for quantum sensing and have the potential to approach the fundamentally limited sensitivity imposed by quantum noise suppression [39]. They are potentially useful for detecting weak signals, such as gravitational waves [40]. Given their ability to suppress quantum noise, it raises the question of whether these lossless interferometers can also capture a quantum EP and whether quantum sensing around such a possible EP can benefit from both a large signal and small noise, which is not typically the case when quantum noise deteriorates the sensing capability at a quantum EP.
In this work, we focus on a dual-beam SU(1,1) interferometer as a platform to obtain a quantum EP. By operating the interferometer around a specific optical delay, the overall input-output relation matrix can become defective. We propose an EP sensing scheme that leverages the unique properties of an EP and assesses its sensitivity in terms of signal-to-noise ratio (SNR). Additionally, we examine a multi-stage variant of the SU(1,1) interferometer in which there are two types of EP: one for the overall input-output matrix and the other for the transfer matrix of a repeating unit. We explore their relationship and how they contribute to EP sensing. In some of the previous studies, the operation point of different SU(1,1) interferometer configurations, such as conventional, truncated, and dual-beam SU(1,1) interferometers, is often set at a DP of the input-ouput matrix [41][42][43]. These inspire our work to search and demonstrate the significance of quantum EPs in the context of quantum interferometer sensing. Our findings suggest that by designing SU(1,1) interferometers to operate around EPs, with high SNRs comparable to the optimal case can be achieved. This offers a new perspective on the design of SU(1,1) interferometers.

Dual-beam SU(1,1) interferometer
A SU(1,1) dual-beam interferometer is a type of quantum interferometer that uses quantum entanglement to measure small phase shifts [37]. A schematic setup of this interferometer is shown in figure 1. Two input beams are first mixed by a PA to replace the beam splitters in the classical Mach-Zehnder interferometer. Then two output beams from the PA are passed to a phase shift ϕ , with the goal of measuring this phase shift. Lastly, both arms are mixed by a PA again to perform interference.
The mixing effect of two PAs can be described by [A. Yariv, Quantum Electronics, 3rd ed (Wiley, New York, 1989).]: whereâ 0 ,b 0 , (â 1 ,b 1 ) are field operators of two input (output) arms of the PA, G = cosh β, g = sinh β are gain amplitudes and a positive β is a real parameter controlling the gain. We assume that two modes have the same frequency, ω, and that the pump has twice that frequency ω β = 2ω, for optimal phase matching. The phase of the gain, θ β , is related to the phase of the pump beam. Here we set both PAs having the same β and θ β = 0 for simplicity. We note that the matrix here is not unitary as a PA mixes creation and annihilation operators. From the canonical commutation relations = 0 for integer n = 0, 1, 2, this matrix belongs to a pseudo unitary group SU(1,1) [44] hence the name SU(1,1) Figure 1. Schematics of a dual-beam SU(1,1) interferometer. Two input beams with field operatorsâ0,b0 are first mixed by a PA into two output beams with field operatorsâ1,b1though equation (1). Then these two beams pass through a phase delay ϕ as the object. Lastly the beams again mixed in the second PA to perform the interference. Homodyne measurements are performed at each arm to retrieve quadrature amplitudesX a 2 andX b 2 of each arm. Lastly, the sensing observable joint quadrature amplitudeŝ X + 2 =X a 2 +X b 2 are obtained as the sum of individual quadrature amplitudes from each arm. We can also measure both amplitude and phase quadratures at output.
interferometer. This symmetry of the SU(1,1) matrix also implies the occurrence of EPs from the nonlinear system later. Together with the phase delay between the two PAs, the transfer matrix U can be written as: where: withâ 2 andb 2 are field operators at two output arms of the second PA and U is also a SU(1,1) matrix. Then the sensing process will be measuring observables at the output port to sense a small change in the phase ϕ . Here we choose the joint quadrature amplitudesX + n (θ) =X a n (θ) +X b n (θ) and Y + n (θ) =Ŷ a n (θ) +Ŷ b n (θ) as the measurement observables where the individual quadrature amplitudes arê for c = a, b, integer n = 0, 1, 2 and quadrature angle θ. Experimentally, those observables can be measured using Homodyne detection with a reference beam that has a phase θ relative to the detection path. This phase θ is usually locked relative to the input beam by using a piezoelectric stage to control the path length. An example of such a locking method can be found in [45]. We choose these observables since they are linear in the input-output relation which can be written as: where: The matrices for changing basis are D = 1 for m = in, out and θ in and θ out are the choices of quadrature angle of the input (n = 0) and output (n = 2) for analysis respectively. We note that the matrix S, a real matrix in relating input and output observables, is not SU(1,1) as the effect of the chosen quadrature angles in measurement is lumped into it while the physical system itself is SU(1,1). For a general SU(1,1) interferometer,X + 2 andŶ + 2 will 'cross' couple In this setup, they are decoupled due to the symmetry between the two arms therefore the dimension of the input-output relation matrix can be reduced to two for a simpler formulation.

Exceptional point
To establish an EP sensing scheme, we first write: due to the linearity of taking the average value ⟨·⟩. Here the quadrature angles are omitted for brevity. Equation (4) enables us to sense through one of the matrix elements by choosing the observable and input state. For example, if we want to sense through S 21 we can choose the observable to beŶ + 2 with coherent state input |αe iθ in , 0⟩. The first (second) slot of the state is the coherent amplitude in the arm a and b. The input coherent state in arm a is chosen as the same phase θ in . We have ⟨Ŷ + 2 ⟩ = √ 2S 21 α (for a positive α) while there is no input in arm b. We have used a convention on the coherent amplitude: ⟨X + 0 ⟩ = √ 2α to give later mathematical convenience. For sensing, it is also important to discuss the contribution of noise. In the current case, the noise is given by the quantum fluctuation given by ∆ 2X+ ( where the symmetrized covariance is defined by ∆(P,Q) = 1 2 ⟨PQ +QP⟩ − ⟨P⟩⟨Q⟩. Through such a form, we recognize that there is a 'gauge' degree of freedom to choose θ in so that SS T does not depend on θ in . Here, we choose a θ in so that S 11 = S 22 and then the EP discussion following is based on the S matrix in this gauge. For the current case of a symmetric SU(1,1) U matrix, θ in = π /2 − θ out is a convenient choice to have S 11 = S 22 . We should note that the inclusion of the pump phase, θ β , modifies the required phase relation to θ in + θ out − θ β = π /2. In this case, locking becomes more apparent, as after a small time δt the condition will change to (θ in − ω in δt) Here ω in represents the frequency of the input beam, ω out = ω in represents the frequency of the reference beam, which have the same frequency as the input beam, and ω β = 2ω in represents the frequency of the pump beam. The last term vanishes due to phase matching in time. Experimentally, the phases of two PAs are first locked to the same value, θ β , similar to the locking of homodyne phase θ out . This common phase can then be locked relative to θ in + θ out by controlling the common path length of pump beams for the two PAs using a piezoelectric stage. An example of locking the phase of the pump beam can be found in [43]. By using this phase relation together with det S = 1, we obtain the following formula for the relationship between noise and EP: where EP (S) = Tr(S) 2 − 4 det S = (λ 1 − λ 2 ) 2 = 4S 12 S 21 is the indicator of EP. Either S 12 = 0 or S 21 = 0 refer to a EP to have zero EP (S) (including DPs actually when both S 12 and S 21 are zero). Such a clear association of EP to S 12 and S 21 is possible due to the choice of gauge, in analogy to the PT symmetry consideration in classical systems to obtain UZR [7]. Equation (6) then implies either ∆ 2X+ 2 = 1 or ∆ 2Ŷ+ 2 = 1 at an EP. We refer to it as the noise preserving condition as the input noise remains constant and normalized at one for any coherent input channels.
We show two eigenvalues λ 1 , λ 2 of the S against ϕ in figure 2(a) and the noises ∆ 2X+ 2 and ∆ 2Ŷ+ 2 from equation (5) in figure 2(b) for the case of β = 1/2, homodyne phase θ out = 0.3 π and input phase θ in = π /2 − 0.3π = 0.2π . The matrix elements of S against ϕ are also shown in figure 2(c). We found that in this setup there are four EPs and we denote them as ϕ EP,i for i = 1, 2, 3, 4. We confirm that two of the EPs correspond to ∆ 2X+ 2 = 1 and the other two correspond to ∆ 2Ŷ+ 2 = 1. In figure 2(b), the EP preserves the noise of only one observable because the condition of EP (S) = 0 requires either S 12 = 0 or S 21 = 0. Figure 2(c) shows that when S 12 = 0, the noise ofX + observable is preserved, and when S 21 = 0 the noise of Y + observable is preserved. When the noise of one observable is preserved, the other observable always exhibit the same or higher noise, as the uncertainty principle (∆X + ∆Ŷ + ⩾ 1 in our notation) prevents a simultaneous reductions of noise in both observables. We note that preservation of noise at the EP does not guarantee it is minimized as noise can be lower elsewhere. In the above, we first fix θ in , and look for θ out to give an EP (defective S), meaning zero S 12 S 21 and equivalently noise preservation either onX + orŶ + . Alternatively, we can start by fixing θ out to preserve noise onX + orŶ + . Then, S 12 S 21 = 0 equivalently means an EP (defective S) with the gauge θ in = ±π /2 − θ out . We can also start by having S 12 S 21 = 0. Then, noise preservation equivalently means an EP (defective S) with the same gauge. In the next section, we will see that, S 21 = 0 is crucial in setting up an EP sensing scheme.

Revealing EPs of SU(1,1) interferometer
For a classical EP sensor relying on frequency splitting, one has to extract the frequency splitting or directly the sensing parameter from the scattering parameters and hope to obtain a large signal around EP [22][23][24][25][26]. The quantum fluctuation or noise of measuringX + 2 orŶ + 2 minus 1, at each EP the noise of either ∆ 2X+ 2 or ∆ 2Ŷ+ 2 is equal to one as predicted by equation (6). (c) The matrix elements of S in equation (3). At each EP, either one of the off-diagonal elements S12 or S21 is zero.
Here, by working at the above EP, we directly set up a sensing scheme so that either observable ⟨X + 2 ⟩ or ⟨Ŷ + 2 ⟩ (with a similar role to the scattering parameters in classical EP sensors) is expected to be amplified but the noise is not. To sense a small change in ϕ , the signal is defined as the squared derivative of the mean measurement outcome ( d dϕ ⟨X + 2 ⟩) 2 or ( d dϕ ⟨Ŷ + 2 ⟩) 2 depending on which observable we choose to measure. Without losing generality, we choose to observeŶ + 2 . Figure 2(c) show that S 21 is actually a sinusoidal function fluctuating around zero in varying ϕ and hence the signal will be approximately largest if we choose an operational point ϕ at zero S 21 , i.e. at an EP. As a side note, the eigenvalues have similar features to those of a PT-symmetric scattering matrix of a classical system. In the region of ϕ EP,1 < ϕ < ϕ EP,2 and ϕ EP,3 < ϕ < ϕ EP,4 , both eigenvalues form a conjugate pair and become unimodular, i.e. λ 1 = λ * 2 and |λ 1 | = |λ 2 | which is in analogy to the PT-symmetric phase of a scattering matrix. While in other regions, both eigenvalues become real, i.e. Im (λ 1 ) = Im (λ 2 ) = 0 which is in analogy to the PT-broken phase.

Performance of EP sensing
As discussed before, sensing throughŶ + 2 with the input state |αe iθ in , 0⟩ at the EP will have a large potential for sensing due to the preservation of noise and the large signal (slope of S 21 ) at the EP. We will refer to this sensing scheme as the EP sensing scheme later in this paper. To quantify the performance of sensing, we calculate the SNR [46,47] of this scheme using: where the signal d dϕ ⟨Ŷ + 2 ⟩ = √ 2α dS21 dϕ can be obtained from differentiating equation (4) and the noise ∆ 2Ŷ+ 2 is given by equation (5). The SNR is normalized to (△ϕ) 2 here. The noise and the SNR are shown in Here the SNR ratio is plotted as the one from equation (7) normalized by the highest possible SNR that can be theoretically obtained [48]. The phase of the input beam is set to θ in = 0.075π and homodyne phase is set to θ out = 0.425π to satisfy the gauge condition. The white dashed line indicates the EP condition with S 21 = 0. The solid black line indicates the noise preserving condition, i.e. ∆ 2Ŷ+ 2 = 1, which separates the noise amplifying regime and noise suppressing regime. In fact, one set of the black lines (the curved ones) completely overlaps with the white dashed line as a result of equation (6). We note that at ϕ = ±0.5π , the noise is also equal to unity because the S matrix becomes a rotation matrix with SS T = I. When β becomes large, the EP line converges to ϕ = ±0.5π and hence the noise suppressing regime is shrinking. The noise around ϕ = −0.5π in high gain regime is shown in figure 3(b). Here the noise is suppressed by EP similar to the low gain regime, however, small perturbations to the ϕ now result in high noise due to high gain of the system. For SNR in figure 3(c), we observe the SNR is generally higher in the low noise region, i.e. the noise suppressing regime located between the two black lines shown in figure 3(a). Moreover, the SNR peaks around the EP line, due to the EP line having a high signal (highest for the same β) and low noise. It is noteworthy that higher SNR with lower noise can be achieved by sacrificing some signal instead of sensing exactly at the EP. However, sensing at the EP may be the preferred operating point when external noise sources are present, as it can offset the advantage of lower noise levels. Nevertheless, operating at the EP is very close to the highest SNR point. Compared to the original sensing scheme for the interferometer [48], our EP scheme has an SNR ratio of approximately 90% of the highest theoretically possible value, indicating that sensing at the EP is a viable alternative to the original sensing scheme and expands the possible operating points of SU(1,1) interferometers. The SNR ratio approaches 1 only at small β and starts to decrease as the gain β increases. Figure 3(d) shows the SNR around ϕ = −0.5π in the high gain regime, which demonstrates that the SNR remains high for large β. For β within figure 3, the SNR ratio stays above 90% of the highest theoretically possible value.  1) interferometer. Two input beams are passing through PAs and phase delays ϕ as the sensing object alternativity. The same joint homodyne measurements are performed at each arm to retrieve joint quadrature amplitudesX + n =X a n +X b n . (b) and (c) Noise and sensitivity for the multi-stage dual beam interferometer in (a) for four stages, i.e. both arm passes through four PAs. The EP sensing scheme is measuringŶ + n with an input coherent state |αe iθin , 0⟩. Here θout = 0.425π , θ in = π /2 − θout and all PA have the same gain β. Black curves denote the noise is being preserved in the interferometer, i.e. ∆ 2Ŷ+ n = ∆ 2Ŷ+ 0 = 1. White dashed curve represents the EP we are utilizing i.e. (Sn) 12 = 0. (d) and (e) Noise and sensitivity for the multi-stage dual beam interferometer in (a) for six stages with the same sensing scheme in (b) and (c). The color bar for noise is saturated at 10 as we are not interested in very high noise. In (c), (e) SNR are renormalized by |α| 2 n 2 e 4β for n stages interferometer and the color is also saturated to provide a rough indicator for the place of SNR peaks.

Multistage SU(1,1) interferometer
Here we discuss a variant of the SU(1,1) interferometer introduced in figure 1 which also relates to EP. The variant is called a multistage SU(1,1) interferometer described in figure 4(a) in which both beams pass through repeating units of the phase object and PA multiple times while we still want to sense a small change in ϕ , which is common to the whole system at every stage. Now, we denoteâ n andb n for the field operator of the arm a and b immediately after n-th PA, then the overall transfer matrix U n for n stages interferometer is expressed as: where the transfer matrix of a single repeating unit consisting of one phase object and one PA is: ) .
Then the corresponding input-output relation matrix S n (n stages) relating the joint quadrature amplitude after nth PA is:  figure 3, i.e. measuringŶ + n with an input coherent state |αe iθin , 0⟩ and θout = 0.425π , θ in = π /2 − θout. For non-integer n we replace S21 → Re (S21) in evaluating signal and S T → S † in evaluating noise as a method of analytical continuation due to the matrix S may not be real anymore. The white dashed curve represents the EP we are utilizing i.e. S12 = 0. Black dashed curve represents the EP of U unit . In (b) SNR are renormalized by |α| 2 n 2 e 4β . where: Again we have assumed all PAs have the same gain β and θ β = 0. In figures 4(b)-(e), we evaluate the noise and SNR for our EP sensing scheme on four and six stage interferometers using equations (5) and (7) with S being replaced by S 4 and S 6 in equation (10). Again, the white dashed lines indicate (S n ) 21 = 0 at an EP, which overlaps one set of the black lines, indicating the noise preserving condition (∆ 2Ŷ+ n = 1). We should note that the presence of an EP still implies the noise preserving condition as U n is still a symmetric SU(1,1) matrix and follows the same analysis as in the two stage SU(1,1) interferometer. However, we find that there are more additional EPs of S n and high-SNR bands for more stages with a pattern that there are 2 (n − 1) bands for a n-stage SU(1,1) interferometer. In each band the noise is more susceptible to small perturbation when compared to a two stage interferometer in figure 3 as the overall gain is larger for the same β. Similar to the twostage SU(1,1) interferometer for every band of high SNR, there is an accompanying EP because the measurement outcome ⟨Ŷ + n ⟩ = √ 2(S n ) 21 |α| is fluctuating around zero at EP and thus having a significant signal there (however may not be exactly maximum for 3 or more stages). As a result, sensing at EP leads to a high SNR.
One interesting feature is that, as the number of stages increase, the number of low noise and high SNR bands increase. However, the placement of these bands is restricted. For example, the additional low noise band observed in figure 4(d) does not expand to ϕ = 0 in high gain regime. To have a deeper understanding, we show the SNR in figure 5(b) for different n with the white dashed lines indicate the EP of S n and the black dashed line indicates the EP of U unit . At a large number of stages, the EPs in the current work for S n for a finite system will form two phases separated by the EPs of U unit . The EPs are shown in figure 5(a), matching well with the phase boundaries in figure 5(b). One phase has highly oscillating behavior in SNR while another has no such oscillations. This oscillating behavior is indeed inherent from the eigenvalue behavior of U unit . A similar situation occurs for four-wave mixing in a cold-atom system [35]. Therefore, the role of an EP of the transfer matrix of the repeating unit U 1 is for separating the oscillatory and non-oscillatory phases for the signal/SNR. The role of an EP for the overall input-output relation matrix S n is for revealing a high-SNR band. We also emphasize that such EP reveals a high SNR band locally by varying the phase-space parameter (e.g. ϕ in our case) but some of the EPs can have higher SNR than others while the SNR at EPs only guarantees local enhancement rather than guaranteeing the highest SNR globally in the whole phase space.

Discussion and conclusion
For multistage SU(1,1) interferometers, one of its applications is mode engineering. This is achieved by using a frequency-dispersive material as the phase object and obtaining a frequency-dependent phase between two PAs. It results in a shaped spectral wave function of a two-photon state to achieve high purity without using a frequency filter [49]. Meanwhile, EPs have applications in state engineering as well. For example, an EP-induced UZR can provide novel two-photon interference to control two-photon states [19,20]. EP also enables optimal ways to engineer quantum states in a two level systems [50]. We believe that transferring the state engineering capabilities of EPs in these systems to SU(1,1) interferometers will be a promising area of future work. In conclusion, we introduced a dual-beam SU(1,1) interferometer that comprises two PAs and an optical delay, as a platform for obtaining a quantum EP of the overall input-output relation matrix. We found that at this EP, the signal is amplified while the quantum fluctuation noise is preserved. To take advantage of this noise preserving property, we developed an EP sensing scheme. The results showed that the sensitivity in terms of SNR will reach its maximum around the quantum EP of the overall input-output relation matrix. Additionally, we studied a multistage variant of the SU(1,1) interferometer and found that the EP sensing scheme extends to this setting since the preserving property is universal to the number of stages. Finally, we found that the EP of the overall input-output relation matrix for an increasing number of stages will separate into two phases, indicated by the EPs of the transfer matrix of a single stage.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.