PT -symmetric Feedback Induced Linewidth Narrowing

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Linewidth is one of the key factors that determine the performance of resonance systems, such as atoms, optical cavities and mechanical resonators.Especially, for precision measurement and sensing, we always strive for a narrow linewidth to achieve better measurement sensitivity.In various precision experiments such as atomic magnetometry [1,2], atomic gyroscopy [3], nuclear magnetic resonance spectroscopy [4], exploration of dark matter [5,6] and exotic forces [7], a very narrow linewidth enables one to detect extremely weak signals.On the other hand, narrow linewidth also represents long coherence time, which is beneficial for quantum storage and quantum information processing.In order to reduce the linewidth, various methods have been proposed, e.g., antirelaxation coating of vessel walls [8][9][10], the spinexchange-relaxation-free mechanism [11,12], the nonlinear magneto-optical rotation approach [13] and the coherent population trapping scheme [14] in atomic systems.However, these methods typically have specific and stringent requirements, e.g., highly demanding fabrication, strict magnetic shielding or specific energy levels.
We consider a generic dissipative resonance system, which can be described by the Hamiltonian H = ω 0 a † a and the corresponding quantum Langevin equation ȧ = (−iω 0 − κ)a − √ 2κa in , where a (a † ) is the annihilation (creation) operator of the resonance mode, ω 0 is the resonance frequency, κ is the amplitude dissipation rate with the associate noise operator being a in .The quadratures of the resonance mode are defined as X = (a + a † )/2 and Y = i(a † − a)/2, which correspond to the bases in the phase space.X in and Y in correspond to the input quadratures.The equations of motion for the quadratures are given by Ẋ Because of the dissipation, the evolution trajectory in the phase space is a spiral curve approaching the origin of coordinates, as shown in Fig. 1(a).To construct a PT -symmetric Hamiltonian, as sketched in Fig. 1(b), we use a feedback loop in which X component is measured with the outcome feedback to input X in component, i.e., X in → X in − ΓX/ √ 2κ, with Γ being the feedback parameter.Then the system dynamics is modified as In the bases of quadratures with the vector Ψ = (X Y ) T , the equations can be rewritten as Ψ = −iH eff Ψ with where I is the identity matrix.After dropping the identity matrix term that corresponds to a common gain or loss, the effective Hamiltonian is PT -symmetric.Here the parity operator P is the Pauli operator σ x representing the interchange between the two quadratures, and the time-reversal operator T denotes complex conjugation operation.Therefore, the feedback transfers the dissipative resonance system into a PT -symmetric system, with equal gain and loss in two quadratures of the same resonance mode.The key point is that the feedback breaks the symmetry between the quadratures, and thus the two quadratures of a single resonance mode behave like two different modes with gain and loss.As shown in Fig. 1(b), the effective coupling strength between the quadratures is equal to the resonance frequency ω 0 , as it is just the energy exchange frequency for different components within the system.For the Hermitian case without dissipation, the trajectory in the phase space is a closed circle.However, in the PT -symmetric case, the trajectory is squeezed into a oblique ellipse, as plotted in Fig. 1(c).In this case, the total effect of gain and loss for the ±π/4 quadratures X ± = (X ± Y )/ √ 2 are balanced, but the couplings between X + and X − are rescaled as a result of gain and loss, with Ẋ+ = − (ω 0 − Γ/2) X − and Ẋ− = (ω 0 + Γ/2) X + .Therefore, the trajectory along ±π/4 direction is squeezed (stretched) by a factor of 1 ± Γ/(2ω 0 ), resulting in an ellipse with an oblique angle of π/4.
The resultant PT -symmetric system possesses both PT -symmetric and PT -symmetry-broken phases, which can be tuned by the feedback parameter Γ.The eigenvalues of H eff are ω ± = i (Γ/2 − κ) ± ω 2 0 − Γ 2 /4, whose real (imaginary) parts represent the resonance frequency (dissipation rate) of the eigenmodes.In Fig. 1(d) and (e) we plot the real and imaginary parts of the eigenvalues as functions of feedback parameter Γ.When the feedback is weak (|Γ| < 2ω 0 ), the real parts of the eigenvalues are opposite to each other while the imaginary parts are equal,

FIG. 2. Experimental setup with a thermal atomic ensemble.
A 1 cm 3 cube glass cell is rich in 133 Cs, filled with 600 Torr 4 He and 20 Torr N2 and heated to 100 • C. The linearly polarized probe laser power is 25 µW and frequency is 40 GHz below the F = 4 to F ′ = 2 transition of the 133 Cs D2 line.The circularly polarized pump laser power is 1 mW and frequency is the F = 4 to F ′ = 3 transition of the 133 Cs D1 line.B0, z-axis static magnetic field (green arrow), λ/4, quarter wave plate; λ/2, half wavep late; WP, Wollaston prism; BPD, balanced photodetector.The output signal of the BPD is applied to a loop consisting of a feedback resistor and a y-axis feedback coil to achieve magnetic field feedback.The function generator is used to generate the weak driving magnetic field Bx.
and the system is in the PT -symmetric phase.In this case the time evolutions of the quadratures X and Y are still trigonometric functions, but the phase difference is no longer π/2 [Fig.1(f)], which is a result of the phase advance or lag induced by the effective gain or loss.If the feedback is strong enough (|Γ| > 2ω 0 ), the eigenvalues are purely imaginary, indicating that the eigenmodes are no longer harmonic modes.Then the amplitude of the quadratures increases exponentially with time, as shown in Fig. 1(g).
We demonstrate the PT -symmetric feedback mechanism in a thermal atomic ensemble, which is a typical example of magnetic resonance system.The experimental setup is sketched in Fig. 2. A ensemble of thermal cesium atoms is filled in a vapor cell, and the atoms can be described by a collective spin with spin polarization P = (P x , P y , P z ), where P µ=x,y,z is the spin polarization component along µ axis.A beam of circularly polarized laser propagating along the −z direction optically pumps the atomic ensemble to polarize the collective spin.A static magnetic field of B 0 = 2.2 µT is applied along z axis, then the collective spin undergoes a Larmor precession around z axis, with Larmor frequency being ω 0 = γB 0 , where γ is the gyromagnetic ratio of the atom.Thus, the transverse components P x,y oscillates in the xy plane, constituting a harmonic oscillator.We measure the spin polarization component P x using a probe laser via polarization homodyne detection, and the output signal is then fed into a loop that includes Amplitude (V) a feedback resistor and the y-axis feedback coil, which generates the feedback magnetic field B y = −Γ FB P x /γ with Γ FB being the feedback factor.No noise processing of the signal is necessary because the signal-to-noise ratio in our experiments is large enough.In this case the feedback magnetic field B y carries the information of the spin polarization component P x , which will result in PT -symmetric feedback.
When the feedback magnetic field B y is small compared with the static magnetic field B 0 , its effect on P z can be ignored, and P z remains equilibrium polarization P 0 .Starting from the Bloch equations, we obtain the simplified equations containing only two orthogonal components (P x , P y ) as where T 2 is the transverse relaxation time.The dynamics can be effectively described by which is equivalent to Eq. ( 2) with κ = 1/T 2 and Γ = Γ FB P 0 .Therefore, the collective spin oscillator constitutes a PT -symmetric system.Next we focus on the PT -symmetric phase (Γ FB P 0 < 2ω 0 ) and show the ability of linewidth narrowing.The imaginary part of the eigenvalues is As the feedback factor Γ FB increases, the system dissipation keeps reducing, and the resonance linewidth keeps narrowing.In our experiment, the feedback factor is inversely proportional to the feedback resistance (Γ FB ∝ 1/R).Therefore, our scheme enables flexible adjusting of the linewidth by changing the feedback factor Γ FB through the resistance R.
In order to measure the resonance linewidth, we apply a weak driving magnetic field B x = B 1 cos (ωt) along x axis, which corresponds to an additional term γB 1 cos (ωt) in the equation of P y in Eq. (3).Then the system undergoes forced oscillation, and we can use instantaneous drive with sudden tuning off to obtain the system evolution dynamics in the time domain, or use continuous drive with scanning frequency ω to obtain the system response in the frequency domain.In Fig. 3 shows clearly that the oscillation lasts longer, and the absorption linewidth becomes narrower, and the dispersion slope becomes sharper.The experimental results are in good agreement with the theoretical predictions, where the feedback delay has been taken into account (see Supplemental Material [102]).
The dependencies of linewidth and equivalent relaxation time on the feedback factor are plotted in Fig. 4. In the experiment, we have observed the reduction of the linewidth from 654 to 13.6 Hz, which is 48 times narrower.The equivalent relaxation time increases from 0.486 to 23.4 ms, which significantly extends the coherence time of the system.Further improvement is limited by the stability of the present experimental system, as it becomes more sensitive to the parameter variations when the linewidth is very narrow.
The PT -symmetric feedback induced linewidth narrowing method holds great potential for high-precision measurements.Our experimental system can be directly used to improve the measurement sensitivity of magnetic field, with the apparatus similar to the M x magnetometer [103,104].When the driving magnetic field is on resonance with the Larmor frequency ω 0 = γB 0 , the spin polarization P x reaches its maximum value.Thus the magnitude of the magnetic field B 0 can be obtained by scanning the frequency of the driving magnetic field.The measurement sensitivity of this M x magnetometer is [102][103][104] where To obtain the signal-to-noise ratio S/N , we measure the square root of the power spectral density (PSD 1/2 ) by feeding the output of time domain signals into the fast fourier transformation (FFT) spectrum analyzer (SR760).As compared in Fig. 5(a), the signal with feedback is significantly larger compared to that without feedback.As the background noise also increases, the signal-to-noise ratio stays almost unchanged.
The shift of resonance frequency originates from the relaxation and the feedback with delay (see Supplemental Material [102]), which does not affect the measurement sensitivity for small changes of magnetic field.According to Eq. ( 6), we can obtain the dependence of sensitivity on the feedback factor, as plotted in Fig. 5(b).The sensitivity of the M x magnetometer is enhanced up to 22 times, with the linewidth narrowing playing a significant role in this enhancement.Compared with the 48-fold reduction of the linewidth, this 22-times enhancement of the measurement sensitivity indicates some additional noise in the feedback process, which may be overcome by further stabilizing the feedback loop.
In summary, we propose a PT -symmetric feedback method in a general dissipative resonance system.By constructing a quadrature measurement-feedback loop in which one quadrature component is measured with feedback, a purely dissipative resonance system can be transformed into a PT -symmetric system, with tunable PTsymmetric phase and PT -symmetry-broken phase.Such a PT -symmetric system contains only a single resonance mode, without the requirement of two or more modes, as the feedback breaks the symmetry between the quadratures, and thus the two quadratures of a single resonance mode behave like two different modes.The method finds important applications in linewidth narrowing and enhancement of measurement sensitivity.We demonstrate the proposal in a thermal atomic ensemble and observe a 48-fold narrowing of the magnetic resonance linewidth.By applying the method in the M x magnetometer, we realize a 22-times enhancement of the magnetic field measurement sensitivity.It can also be directly applied to other precision measurement experiments limited by linewidth such as atomic gyroscopy.Our study provides a new perspective on using feedback to construct PTsymmetric systems, which form an excellent platform for studying non-Hermitian physics, with broad applications in high-precision measurement and sensing.In Eq.(S26), B 1 is small as it is a weak magnetic field applied to measure the system properties, so the transverse spin components produced by the driving field B 1 satisfy P x , P y ≪ 1.Thus, the driving term −P y ω 1 cos(ωt) can be neglected.When the feedback term Γ FB P x is much small than P z /T 1 and R op , the nonlinear term with noise −Γ FB P x P x − τ Ṗx + N can be ignored.Then P z reaches the steady state, corresponding to the equilibrium polarization P z = P 0 = R op T 1 , which keeps a constant in the experiment.
We deal with Eq.(S24) and Eq.(S25) in the same way as we do with the noiseless feedback equations Eq.(S5) and Eq.(S6),we can get where ) Due to the consideration of noise, Eq.(S27) has an additional noise term on the right side of the equation compared to Eq.(S13)It can be found that the noise term only corrects the driving term of the damped harmonic oscillator equation and does not affect the dissipation coefficient.Therefore, the noise term of the feedback does not affect the linewidth.
Next, we analyze the correction amplitude ε of the driver term generated by the noise term.Comparing the coefficients of the two terms, we get In our experiment, the minimum effective linewidth is Γ eff = 2π × 10 Hz, the Larmor frequency ω 0 = γB 0 = 2 π × 0.35 Hz/nT × 2200 nT = 2 π × 7.7 kHz, and the relaxation time T 2 = 0.486 ms, so And our signal-to-noise ratio (S/N ) is about 500.From Eq.(S19) we can get the magnitude of the signal in the near resonance condition ω = ω eff ≈ ω 0 and the delay τ is small, So the signal-to-noise ratio S/N is and we can obtain that Putting Eq.(S31) and (S34) to the expression of ε, Eq.(S30), It can be found that under our experimental conditions, the correction amplitude brought by noise is much smaller than that of the driving term.To be precise, because the signal-to-noise ratio of our experiment is large enough, we can safely ignore the effect of noise.

II. PT -SYMMETRIC FEEDBACK IN THE OPTICAL CAVITY
Optical cavity is widely used in precision measurement experiments, and our PT -symmetric feedback scheme can also be applied to the system of optical cavity to realize the linewidth narrowing.As shown in the Fig. S1, the laser is incident into the optical cavity through the phase modulator.We detect the output laser by homodyne detection, and then the detection signal is fed back to the phase modulator to modulate the phase of the input optical field.a a † is the annihilation (creation) operator of the light field in the Fabry-Perot cavity, satisfying the Hamiltonian H = (ω c − ω L )a † a = −∆a † a, where ω c is the resonance frequency and ω L is the frequency of the light field, ∆ is the detuning.So the Langevin equation for the light field in the cavity is ȧ = (i∆ − κ) a − √ 2κ in a in , where κ in is the loss rate associated with the input coupling, κ is the total loss rate.Since the annihilation operator a can be written as two orthogonal components a = X + iY , the equation satisfied by the two orthogonal components X and Y is , which can be rewritten as Next, we introduce the measurement-feedback scheme.According to the input-output relation, the transmission light field satisfies a out = √ κ in a.We use strong local oscillating light a L and transmission light interference for homodyne detection, and the light intensity of the two channels after splitting through the beam splitter satisfies where α L = |α L | e iθ , θ is the phase of the local light.The intensity difference between the two channels is We can define so we obtain The signal from the photodetector is fed back to a phase modulator (e.g., electro-optic modulator, EOM) to add an additional phase φ to the input laser a in → a in e iφ , where φ ∝ I 1 − I 2 .Let η be the total conversion efficiency, including the photodetector and phase modulator conversion efficiency, we can obtain φ = ηQ θ .When φ is small, we use the Taylor expansion

C. Energy level diagram and laser frequencies
The energy level diagrams and laser frequencies in the experiment are shown in Fig. S2.The linearly polarized probe laser power is 25 µW and its frequency is 40 GHz below the F = 4 to F ′ = 2 transition of the 133 Cs D2 line (wavelength 852 nm).The circularly polarized pump laser power is 1 mW and its frequency is on resonance with the F = 4 to F ′ = 3 transition of the 133 Cs D1 line (wavelength 894 nm).

D. Principle of measuring the spin polarizability Px
The far detuning linearly polarized probe light incident on the atomic vapor cell along the x direction.The right and left circularly polarized components of linearly polarized light have different refractive indices due to the spin polarization P of the atoms in the x direction.After passing through the same distance, the phase difference between left circularly polarized light and right circularly polarized light will change, thus rotating the polarization plane of probe light.This is the Faraday rotation effect.For example, we consider the D1 transition of the Cs atom in the Fig. S3.The ground state of cesium has an angular momentum of +1/2 or −1/2.The transition rule determines that electrons with different orientations of spin angular momentum projection will absorb photons with different polarization states.The atom with an angular momentum of +1/2 only absorbs σ − circularly polarized light, while the atom with an angular momentum of −1/2 only absorb σ + circularly polarized light.The atomic polarization (the difference in population of the ground states) makes the atomic vapor absorb light at different rates for the two different circularly polarized states, which gives rise to circular dichroism.We know that different absorbance means different refractive indices for the two circularly polarized light.This results in a phase difference between the two types of circularly polarized light emitted, which leads to a rotation of the polarization plane of the final synthesized linearly polarized light.We use a half waveplate and a polarization beam splitter to split the probe light.The intensity of the two component light fields (I 1 , I 2 ) is detected with a balanced photodetector, and then the angle of rotation of the polarization plane of the probe light is detected by subtraction (I 1 − I 2 ).Because the rotation angle of the polarization plane of the probe light is proportional to the spin polarization in the x-direction, we can obtain the information of the spin polarization P x .Figure S4 shows the sequence diagram of the drives for the two linewidth measurement schemes.The main difference is x-axis magnetic field B x , which has different forms and different roles in the two measurement schemes.In the frequency domain measurement, the driving magnetic field B x is a continuous RF magnetic field, as shown in Fig. S4  (a).We constantly change the frequency of B x and scan through the resonance frequency of spin precession.And then we use the lock-in amplifier to demodulate the resonance lineshape and obtain the resonance linewidth.In the time domain measurement, we use a pulsed magnetic field B x to rotate the atomic spins to the x-direction, as shown in Fig. S4 (b).Then the spins undergo Larmor precession and gradual relaxation under the action of the z-direction bias magnetic field, and the linewidth can be obtained by measuring the relaxation time.

F. Discussion of measuring magnetic field and sensitivity
The discussion of measuring magnetic field and sensitivity is divided into three parts (1) Method of measuring the magnetic field in the z-axis.(2) Definition of sensitivity and derivation of sensitivity formula (3) Experimental measurement procedures for signal-to-noise ratio S/N and sensitivity.And the reason why the feedback method improves the sensitivity is analyzed.

Method of measuring the magnetic field in the z-axis
There is a magnetic field B 0 to be measured in the z-direction.We apply an x-axis driving field B x = B 1 cos ωt and scan the frequency of the driving field ω.Then, we measure the signal of the photodetector and use the power spectral density (PSD, S PSD (ω)) to represent the magnitude of the signal.According to the expression of spin polarization P x (Eq.S19), the expression of signal power spectral density S PSD (ω) can be obtained that where ω 1 = γ |B x |, P 0 ≈ 1 and ∆ω FWHM is the linewidth.The delay τ is small, so ω eff ≈ ω 0 , 1 + Γ FB P 0 τ ≈ 1.As shown in Fig. S5, the blue curve shows the variation of the signal power spectral density S PSD (ω) with the frequency ω of the driving magnetic field.When the signal reaches its maximum, it means that the driving magnetic field frequency ω and the resonance center frequency ω 0 are equal (ω = ω 0 = γB 0 ).According to the resonance center frequency, the magnetic field to be measured can be calculated is B 0 = ω0 γ .And the max amplitude is Suppose there is a weak change in the magnetic field to be measured B new = B 0 + δB, then the new resonance frequency is ω new = ω 0 + δω, and the corresponding resonance curve will be shifted, as shown in the red curve in Fig. S5.Because the frequency of the driving magnetic field remains constant (ω = ω 0 ) and is not equal to the new resonance center frequency ω new , the signal becomes smaller, as shown by point B on the red curve.As the change δω of the magnetic field causes the signal amplitude to change from point A to point B, the change value of the signal is δS PSD .When change value of the signal δS PSD and noise N PSD satisfy δS PSD ≥ N PSD , we can distinguish that the signal has changed and know that the magnetic field to be measured has changed.When δS PSD = N PSD , we can just distinguish that the signal has changed, and the corresponding weak change of frequency δω (magnetic field δB) is our sensitivity.The value of the change in the signal can be written as When weak frequency is much small than linewidth (δω ≪ ∆ω FWHM ), we can obtain from Eq.(S47) and Eq.(S48) that FIG. 1.(a) Typical trajectory of a dissipative resonance system in the phase space.(b) Schematic diagram of the PT -symmetric feedback system.(c) Phase-space trajectory of the PT -symmetric feedback system (red solid curve) and Hermitian resonance system (black dashed curve).(d),(e) Real and imaginary parts of the eigenvalues for Γ/κ varied from -10 to 10 and ω0/κ = 2.The purple dots represent the case of Γ = 2κ, and the orange squares represent the case of Γ = 6κ, which are the eigenvalues of Figs.1(f) and (g), respectively.(f),(g) Typical time evolution of quadratures X (blue solid curve) and Y (orange dashed curve) in PT -symmetric phase (|Γ| < 2ω0) and symmetry-broken phase (|Γ| > 2ω0).

FIG. 3 .
FIG. 3. From (a) to (e) are the output signals of the BPD after turning off the driving magnetic field Bx.The inset in the (d) is a zoomed-in view of the orange area.From (e) to (h) and (i) to (l) are the out of phase and in phase output signals of the lock-in amplifier, respectively.The inset in the (e) shows a larger frequency range.The scatters are experimental results and the gray solid curves are theoretical results.From top to bottom, the feedback resistance decreases with the values labeled in the figure.

FIG. 4 .
FIG.4.Dependence of linewidth and equivalent relaxation time on the feedback factor.The scatters are experimental results and the red and blue solid line is the theoretical result.

FIG. 5 .
FIG. 5. (a) Square root of power spectral density (PSD 1/2 ) for feedback resistance R = 409 Ω (red curves) and without feedback (black curves).The solid (dashed) curves are the results in the presence (absence) of driving magnetic field Bx.(b) Sensitivity enhancement factor of the Mx magnetometer as a function of the feedback factor.
This work is supported by the Key-Area Research and Development Program of Guangdong Province (Grants No. 2019B030330001), the National Natural Science Foundation of China (NSFC) (Grants No. 12275145, No. 92050110, No. 91736106, No. 11674390, and No. 91836302), and the National Key R&D Program of China (Grants No. 2018YFA0306504).
FIG. S1.Schematic diagram of an optical cavity using PT feedback laser and pump laser used in the experiment.BS, beam splitter; PD, photodetector.
FIG. S2.Energy level diagrams of 133Cs atom and the frequencies of probe laser and pump laser used in the experiment.

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FIG. S4. (a), (b) Linewidth measurement in the frequency and time domain.
FIG. S5.Schematic diagram of the power spectral density