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Enhanced Sensing Mechanism Based on Shifting an Exceptional Point
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Xuan Mao1, Guo-Qing Qin2, Hao Zhang3, Bo-Yang Wang1, Dan Long1, Gui-Qin Li1, 4, Gui-Lu Long1, 4, 5, *
Research. Vol 6 Article ID 0260
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Research. Vol 6 Article ID 0260
• Research Article •
Enhanced Sensing Mechanism Based on Shifting an Exceptional Point
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Xuan Mao1, Guo-Qing Qin2, Hao Zhang3, Bo-Yang Wang1, Dan Long1, Gui-Qin Li1, 4, Gui-Lu Long1, 4, 5, *
Affiliations
  • 1Department of Physics, State Key Laboratory of Low-Dimensional Quantum Physics, Tsinghua University, Beijing 100084, China.
  • 2Beijing Institute of Radio Measurement, The Second Academy of China Aerospace Science and Industry Corporation (CASIC), Beijing 100854, China.
  • 3Purple Mountain Laboratories, Nanjing 211111, China.
  • 4Frontier Science Center for Quantum Information, Beijing 100084, China.
  • 5Beijing Academy of Quantum Information Sciences, Beijing 100193, China.
About Author:

† These authors contributed equally to this work.

Published: 2023-10-20 doi: 10.34133/research.0260
Outline
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Non-Hermitian systems associated with exceptional points (EPs) are expected to demonstrate a giant response enhancement for various sensors. The widely investigated enhancement mechanism based on diverging from an EP should destroy the EP and further limits its applications for multiple sensing scenarios in a time sequence. To break the above limit, here, we proposed a new enhanced sensing mechanism based on shifting an EP. Different from the mechanism of diverging from an EP, our scheme is an EP nondemolition and the giant enhancement of response is acquired by a slight shift of the EP along the parameter axis induced by perturbation. The new sensing mechanism can promise the most effective response enhancement for all sensors in the case of multiple sensing in a time sequence. To verify our sensing mechanism, we construct a mass sensor and a gyroscope with concrete physical implementations. Our work will deepen the understanding of EP-based sensing and inspire designing various high-sensitivity sensors in different physical systems.

Xuan Mao, Guo-Qing Qin, Hao Zhang, Bo-Yang Wang, Dan Long, Gui-Qin Li, Gui-Lu Long. Enhanced Sensing Mechanism Based on Shifting an Exceptional Point[J]. Research, 2023 , 6 (10) : 0260 - . DOI: 10.34133/research.0260
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Introduction
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Non-Hermitian quantum systems present attractive properties such as exceptional points (EPs) [1–6] at which 2 or more eigenvalues and the corresponding eigenstates coalesce simultaneously [7]. The existence of EPs has been verified in various physical systems such as optomechanical systems [8–10], coupled atom–cavity systems [11], and plasmonic systems [12–14]. The exotic properties of non-Hermitian systems [15–18] open potential possibilities for diverse advanced applications including phonon lasers [9,19–21], chiral mode conversion [8,22–26], and enhanced sensing [3,4,27–32]. The response enhancement at EPs has been widely investigated in microcavities [3,4,29,33], optomechanical systems [31,34,35], and circuits [28,36–38].
Owing to the complex response near an nth-order EP, any small perturbation destroys the degeneracy that results in an nth-root response for the frequency splitting [4]. For sufficiently small perturbation, the splitting is much larger than observed in conventional schemes where the response is proportional to the perturbation strength. The giant enhancement of EPs has been verified both theoretically and experimentally in various detection schemes [3,4,32,39–41] ranging from mass sensor [31] to nanoparticle detector [4,29,32], from magnetometer [34,39] to gyroscope [3,33,35,42–44]. Not only the presence of perturbation will cause the frequency splitting, but also the degeneracy of the linewidths disappears after introducing perturbation. The EP condition is totally destroyed by the perturbation, and the EP no longer exists which will limit the enhancement for multiple sensing schemes in a time sequence. In this case, it is necessary to develop an EP condition nondemolition enhanced sensing mechanism to break the limit.
In this paper, we propose a novel EP-based enhanced sensing mechanism that relies on the shift of an EP in contrast to the conventional method of diverging from an EP. For the enhanced mechanism based on diverging from an EP, the perturbation breaks the EP condition, and it further deteriorates the EP enhance performance for implementing multiple sensing in a time sequence. Compared with diverging from an EP mechanism, the enhanced sensing mechanism based on shifting an EP demonstrates a slight shift along the parameter axis induced by the perturbation resulting in a remarkable enhancement for the frequency splitting. For the shift of an EP, there exists an EP in parameter space after perturbation, and one can prepare the system back to an EP state that promises the most effective enhancement for every sensing. Furthermore, we propose a mass sensor and a gyroscope scheme based on shifting an EP to verify the enhanced sensing mechanism. This work opens up paths to design ultrasensitive sensors and verifies potential applications in various fields including precision measurement and quantum metrology.
Results
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Two enhanced sensing mechanisms based on an EP
Considering a 2-level coupled system composed of 2 modes represented by a1 and a2 as illustrated by the inset of Fig. 1A, the dynamical equations of the general coupled system can be written as
ddta1a2=−iω1−iγ1JJω2−iγ2a1a2,
where t can be the evolution time and the propagation distance. The resonance frequency and the decay of a1 (a2) are represented by ω1 (ω2) and γ1 (γ2), respectively. J denotes the coupling coefficient, and it can be real or complex. For the purpose of conciseness, we only consider the reciprocal coupling case and the similar conclusion can be obtained for the nonreciprocal coupling. Assuming that the solutions have the harmonic form a1, 2 = a1, 2e−iλt, the eigenvalues of the coupled system can be expressed as
λ±=ωmean−iγmean±ωdiff+iγdiff2+J2.
ωmean = (ω1 + ω2)/2 and γmean = (γ1 + γ2)/2 represent the mean values of the resonance frequency and the loss. ωdiff = (ω1 − ω2)/2 and γdiff = (γ1 − γ2)/2 are the differences of the frequency and the decay. It can be indicated that when the parameter values meet the requirement ±iJ = ωdiff + iγdiff, the system is prepared on an EP state.
The perturbation introduced to the sensor systems can induce frequency shift [3,31], linewidth broadening [45,46], or both. For the sake of simplicity, we consider the case that the perturbation only induces frequency shift to illustrate the difference between the 2 enhanced sensing mechanisms based on an EP.
If the coupling coefficient J is a real number and the enhanced sensing mechanism is well known as the eigenvalues diverging from an EP, the enhanced sensing mechanism based on the eigenvalues diverging from an EP after the perturbation ε is illustrated by Fig. 1B and C. The solid lines and the dashed lines represent the evolution of the 2 eigenvalues along with the system parameter J before and after introducing the perturbation. Usually, before introducing perturbation, the sensing system is prepared near an EP and one can monitor the frequency shift Re[Δλ] and the linewidth change Im[Δλ] after the perturbation. According to the frequency shift, the strength of the perturbation ε can be indicated. Due to the singularity of EPs, the frequency shifts of the eigenvalues obtain ε1/2 (the red dashed line in Fig. 1D) enhancement comparing to the diabolic point (DP) (the blue solid line in Fig. 1D) in the condition of ε ≪ 1. In contrast to EP, some microcavities support modes also with degenerate eigenvalues, but the associated eigenvectors can always be chosen to be orthogonal to each other, also known as DP. For the traditional cavity-based sensing schemes, the DP is utilized and the perturbation-induced frequency shift response is proportional to the perturbation strength, which is shown by the blue solid line in Fig. 1D. The inset of Fig. 1D indicates the frequency shift value is proportional to ε1/2 as the slope is 1/2 in a logarithmic scale. To demonstrate the enhancement of EPs, the enhancement factor can be defined as
η=∣ReΔλε∣.
Figure 1E illustrates the enhancement factor of an EP (the red dashed line) and a DP (the blue solid line) versus the perturbation strength. It can be indicated that an EP enhance performance is much better in the weak perturbation regime, which is consistent with the analysis above. It can be inferred that the EP condition is destroyed by the perturbation and the EP no longer exists in the parameter space for the enhanced sensing mechanism based on diverging from an EP. In the case of implementing multiple sensing in a time sequence, the enhancing performance deteriorates as the number of sensingincreases as every next detection is conducted on the basis of being destroyed by the previous detection.
The shift of an EP mechanism requires the coupling coefficient is an imaginary number and can also exhibit ε1/2 enhancement. The imaginary coupling between the 2 modes have been experimentally realized in dissipative coupled systems including atomic systems [47,48], optical systems [3,49,50], mechanical systems [9], circuits [51], and thermal materials [52]. Under some specific parameter values, the existence of the EP can be maintained along with a shift in the parameter space after the perturbation which is denoted by Fig. 1F and G. Due to the topological structure near an EP, the shift of an EP will lead to similar enhancement results to the sensing performance based on the eigenvalues diverge. The response of eigenvalues to the strength of perturbation and the enhancement factor of shifting an EP mechanism are demonstrated in Fig. 1H and I, respectively. On the other hand, the linewidth changes of the 2 eigenstates demonstrate different behaviors. In the case of an EP shift sensing, the 2 linewidths of the eigenstates stay the same as before the perturbation, which may further improve the enhancement performance in the perspective of experiments. The unbroadened linewidths after the perturbation may improve the precision of dispersion measurement. Another interesting point is there also exists an EP in the parameter space after the perturbation for the sensors based on shifting an EP which is an EP condition nondemolition enhanced sensing mechanism. One can prepare the system back to an EP state by tuning corresponding parameters after introducing the perturbation. In the case of the influence of the perturbation to the sensor that is hard to eliminate such as mass sensor and nanoparticle sensing, it is a challenging task to separate the deposited mass or nanoparticle and the sensing system, and the sensors based on shifting an EP will always obtain the most efficient enhancement as the system can be prepared back to an EP before the next sensing. Thus, for the task of implementing multiple sensing in a time sequence, sensors based on shifting an EP will always demonstrate the most effective EP enhancement for every detection.
As a conclusion, we demonstrate that the 2 enhanced sensing mechanisms based on an EP for the perturbation only induces frequency shift, and the case that the linewidth broadening induced by the perturbation is similar to the case we consider here. The conclusions for the 2 different systems can be found in Table.
Mass sensor based on shifting an EP
Optomechanics [53] has been a promising platform to investigate fundamental phenomena [54–58] and applied science [59–64]. Benefiting from abundant manipulation methods, realization of ultrasensitive sensors operating at EPs in optomechanical systems has gained wide attention over the last few decades. To illustrate the enhanced mechanism based on shifting an EP, here, we propose an optomechanical mass sensor. The mass sensor system as illustrated by Fig. 2A consists of 2 mechanical vibrators b1 and b2 coupling to the same optical field a with frequency ωa and loss κa. The Hamiltonian of the system is (ℏ = 1)
Hm=Σj=1,2‍[ωmjbj†bj+gja†abj+bj†]+ωaa†a+iκexϵa†e−iωdt−aeiωdt
The mechanical resonator bj is characterized by the frequency ωmj and the damping rate γmj (j = 1, 2). gj is the single-photon optomechanical coupling strength between the mechanical mode bj and the optical mode a. κex represents the coupling loss. Figure 2B represents the driving scheme of the mass sensor in frequency domain. The pump laser owns the driving frequency ωd and the amplitude ϵ. Following the standard optomechanical linearization procedure and adiabatically eliminate the optical field, one can obtain the effective Hamiltonian of the system in the matrix form
Hmeff=Ω0−iγm12+2G12κa−i2G1G2κa−i2G1G2κa−Ω0−iγm22+2G22κa,
where Ω0 = (ωm1 − ωm2)/2. Gi=gin is the effective optomechanical coupling strength with n denoting the intracavity photon number. κa = κex + κ0 represents the total loss rate of the optical mode. κ0 indicates the intrinsic loss of the cavity mode. For convenience, we can set G1 = G1 = G and γm1 = γm2 = γm. The eigenvalues of Hmeff are given by
λm±=−iγm/2+Γ±Ω02−Γ2,
where Γ = 2G2/κa. Note that the Hamiltonian form in Eq. 5 satisfies anti-parity time (PT) symmetry with an EP at Γ =  ∣ Ω0∣. The evolution of the eigenvalues under different optomechanical coupling strengths are illustrated by the solid lines in Fig. 2C to E. Note that the locations of EPs are marked by the green dots in Fig. 2C.
For a traditional mass sensor, the relationship between the deposited mass δm and the caused frequency shift δω can be given by [65]
δω=ωm22mδm=Rδm.
R = ωm2/2m represents the mass responsivity. ω2 and m are the resonant frequency and the effective mass of the mechanical resonator supporting the deposition.
After depositing mass δm into the mechanical resonator b2, the effective Hamiltonian of the system is
Hmeff′=Ω0′−iγm12+2G12κa−i2G1G2κa−i2G1G2κa−Ω0′−iγm22+2G22κa,
where Ω0′=ωm1−ωm2−δω/2 . After introducing the mass, the EP will move from the former point in the parameter space as Fig. 2C to E shown. The dashed lines demonstrate the evolution of the eigenvalues after the perturbation. In Fig. 2D and E, the dotted lines show the difference between the eigenvalues before and after, i.e., Re[λm]after −  Re [λm]before and Im[λm]after −  Im [λm]before. It can be noticed that the deposited mass causes the shift of the EP and further lead to the largest frequency shift near the EP. Meanwhile, the introduction of the deposited mass will not change the bandwidth of the eigenstate, which theoretically boosts the sensitivity enhancement for EP-based sensors.
One can implement multiple mass sensing in a time sequence utilizing the mass sensor based on shifting an EP as it can be tuned back to an EP after each sensing period. Figure 2F demonstrates the differences of the real parts and the imaginary parts. There are 2 periods, sensing period and tuning period, during the time sequence mass sensing. During the sensing periods, the frequency shifts obtain the most effective enhancement and present mass-dependent response. The corresponding frequency shifts due to the deposited mass are δω = 0.0001ω1, 0.0002ω1, and 0.0005ω1, respectively. During the tuning periods, the system can be prepared back to an EP by tuning the optomechanical coupling strength. Note that no matter the system experiences sensing periods or tuning periods, the linewidths of the sensor are always the same as the unperturbed system as shown in Fig. 2F.
Gyroscope based on shifting an EP
The gyroscope scheme based on shifting an EP is illustrated by Fig. 3A. Two whispering-gallery-mode (WGM) cavities are coupled to 2 drop-filter waveguides with center-to-center distance L. On the basis of the 2 resonator modes Ψ = (a1,  a2)⊤, the effective Hamiltonian in matrix form can be expressed as [66]
Hgeff=ωa1−iγeff12−ieiφκ1κ2−ieiφκ1κ2ωa2−iγeff22,
where ωa1 and ωa2 are the resonance frequencies of the 2 WGM resonators. γeffj = Γj + 2κj denotes the effective loss and Γj represents the intrinsic loss rate in the jth resonator. κj is the coupling rate between the jth resonator and the drop-filter waveguide. The phase delay factor is φ = 2πnwL/λ, with nw being the refractive index of waveguides and λ representing the wavelength of the light.
Benefiting from the tunability of the phase delay, the indirectly coupling system can exhibit anti-PT symmetry [9,47,52,67]. Without losing of generality, we can assume the 2 resonators have the equal losses Γ1 = Γ2 and the waveguide-resonator coupling rates are equal κ1 = κ2 = κ. Thereby, the effective Hamiltonian in anti-PT symmetry form is
Hgeff′=Ω−−iΓ1+Γ24+κiκiκ−Ω−−iΓ1+Γ24+κ.
Here, Ω− = (ωa1 − ωa2)/2. When introducing rotation frequency into the first WGM resonator, the resonance frequency experiences a Sagnac frequency shift associated with the rotation direction
ΔSag=nRΩωa1c1−1n2−λndndλ,
where n and R are the refractive index and the radius of the resonator, respectively. c denotes the speed of the light, and the dispersion term dn/dλ is relatively small in typical materials (∼1%) [68]. For convenience, we define the sign of the rotation frequency depending on the rotation direction. As shown in Fig. 3A and B, in the case of counterclockwise (CCW) rotation the sign of the rotation frequency is defined as positive associating with a positive Sagnac frequency shift and vice versa. For the case gyroscope experience clockwise (CW) rotation, the resonant frequency of a1 has a red shift as indicated in Fig. 3B. Due to the definition of the sign of the rotation frequency, one can simply add the perturbation matrix into Eq. 10 after introducing rotation
Hgeff′′=Ω_+ΔSag−iΓ1+Γ24+κiκiκ-Ω_−iΓ1+Γ24+κ
And the eigenvalues of the effective Hamiltonian are
λg±=ΔSag2−iΓ1+Γ24+κ±ΔSag+2Ω−22−κ2.
It is clear that no matter the system experiences rotation or not, there is always a threshold coupling strength at (ΔSag + 2Ω−)/2. The real parts and the imaginary parts of λg± are illustrated by Fig. 3C and D. In the absence of the rotation, there are 3 regimes for different values of κ and the threshold is κEP = Ω− = γg. In the domain of κ < γg, the eigenstates share the same linewidth while possessing different resonance frequency and the system is in the anti-PT broken phase. In the case of κ > γg, the 2 eigenvalues have equal real part and different imaginary parts, which indicates that the system is in the anti-PT symmetry phase. The EP is the phase transition point of anti-PT symmetry phase and anti-PT broken phase. It is indicated that the EP is shifting along the parameter axis κ/γg after introducing a rotation. With the presence of the rotation, one can tune the system back to an EP state by adjusting the distance between the waveguides and the WGM cavities. Figure 3E shows the differences of the real parts and the imaginary parts of the eigenvalues before and after rotation, i.e., Re[λg]after −  Re [λg]before and Im[λg]after −  Im [λg]before. The solid lines and the dashed lines denote the real parts and the imaginary parts, respectively. One can see near an EP that the system demonstrates a great frequency splitting and meanwhile maintains the linewidth.
After introducing rotation into the system, the reactions of the frequency splitting Re[Δλ] =  ∣  Re[λg+] −  Re[λg−]∣ in 3 regimes are not the same. Figure 3F illustrates the logarithmic behavior of the frequency splitting under different rotations in the case of anti-PT symmetry phase (κ > γg), anti-PT broken phase (κ < γg), and at the EP (κEP). In the case of the anti-PT broken phase, the frequency splitting is largest while the slope is smallest for the same perturbation strengths. However, for a small rotation frequency, the slope of the response at an EP is 1/2, which is predicted by the perturbation theory [4]. For a relatively large rotation frequency, the perturbation theory no longer works, and therefore, the slope of the response is larger than 1/2.
Conclusion
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We propose a novel enhanced sensing mechanism based on perturbation-induced shift of an EP in contrast to the widely investigated method of diverging from an EP. Diverging from an EP mechanism destroys the EP condition in the parameter space and obtains ε1/2 enhancement in the weak perturbation regime. In this case, the EP condition is totally destroyed by the perturbation and the EP no longer exists, which will limit the enhancement for multiple sensing schemes in a time sequence. To overcome the challenge, we propose the EP condition nondemolition enhanced sensing mechanism based on shifting an EP, which demonstrates a slight shift along the parameter axis induced by the perturbation and also exhibits remarkable enhancement. The linewidths will maintain after the perturbation for the EP shift mechanism, which may improve the precision of dispersion measurement. To implement the EP shift sensing mechanism, we construct a mass sensor and a gyroscope sensing scheme for the mechanical modes and the optical mode pair, respectively. Combining other enhancing methods such as exceptional surfaces [30,69,70] and nonlinearity [38,71,72], the performance of sensors will be further improved. The enhanced sensing mechanism proposed in this paper provides a comprehensive understanding of EPs enhancing and may inspire technological development in various schemes.
Methods
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In our numerical calculation, the frequencies and the linewidths of the modes are corresponding to the real parts and the imaginary parts of the eigenvalues of the system. These are crucial to the demonstration of EP and the influence of perturbation. For different types of perturbation, one can calculate the frequency and the linewidth of the system before and after introducing perturbation. The perturbation strength can be inferred by monitoring the corresponding changes.
Funding
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  • National Natural Science Foundation of China(6213100)
  • The key research and development program of Guangdong provience(2018B030325002)
  • Beijing Innovation Center for Future Chip
  • Tsinghua Initiative Scientific Research Program
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Article Info
doi: 10.34133/research.0260
  • Receive Date：2023-08-10
  • Online Date：2026-10-28
  • Published：2023-10-20
Article Data
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History
  • Received：2023-08-10
  • Accepted：2023-10-10
Funding
National Natural Science Foundation of China(6213100)
The key research and development program of Guangdong provience(2018B030325002)
Beijing Innovation Center for Future Chip
Tsinghua Initiative Scientific Research Program
Affiliations
    1Department of Physics, State Key Laboratory of Low-Dimensional Quantum Physics, Tsinghua University, Beijing 100084, China.
    2Beijing Institute of Radio Measurement, The Second Academy of China Aerospace Science and Industry Corporation (CASIC), Beijing 100854, China.
    3Purple Mountain Laboratories, Nanjing 211111, China.
    4Frontier Science Center for Quantum Information, Beijing 100084, China.
    5Beijing Academy of Quantum Information Sciences, Beijing 100193, China.

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表12种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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