Sub-Hertz resonance by weak measurement

Weak measurement (WM) with state pre- and post-selection can amplify otherwise undetectable small signals and thus has potential in precision measurement applications. Although frequency measurements offer the hitherto highest precision due to the stable narrow atomic transitions, it remains a long-standing interest to develop new schemes to further escalate their performance. Here, we demonstrate a WM-enhanced correlation spectroscopy technique capable of narrowing the resonance linewidth down to 0.1 Hz in a room-temperature atomic vapour cell. The potential of this technique for precision measurement is demonstrated through weak magnetic-field sensing. By judiciously pre- and post-selecting frequency-modulated input and output optical states in a nearly orthogonal manner, a sensitivity of 7 fT Hz−1/2 at a low frequency near DC is achieved using only one laser beam with 15 µW of power. Additionally, our results extend the WM framework to a non-Hermitian Hamiltonian and shed new light on metrology and bio-magnetic field sensing.


Supplementary Note 1. EIT with laser frequency modulation
We consider a generic three-level -type EIT configuration (Fig. 1), involving a continuous wave (cw)polarized laser with angular frequency 0 . Its right-circular-polarization ( ) component with Rabi frequency Ω r is near resonant with the atomic transition | ⟩ → | ⟩, while the left-circular-polarization ( ) component with Rabi frequency Ω l is near resonantly applied to | ⟩ → | ⟩. The one-photon detuning is . For Ω r = Ω l = Ω in , the atomic population is equally distributed in the two ground states | ⟩ and | ⟩. A magnetic field along the light propagation direction (B) is applied to introduce the twophoton detuning in the (2) (0) resonance measurement.
With frequency modulation (FM), the laser frequency can be written as where m , m and m m are respectively the modulation depth, modulation frequency, and modulation range. In the rotating frame, the atom-light interaction can be readily computed by applying the transformation matrix ) (2) to the normal Hamiltonian. This leads to the time-dependent one-photon detuning: ( ) = m m cos( m ).
After some algebra, the atom-light interaction Hamiltonian becomes . (4) Although the FM introduces complexity to the atom-light interaction, by expanding each atomic coherence into a Fourier series ( = ∑ ( ) − m ) and by matching the terms with the same order, the major master equations then take the following form: where is the Doppler broadened linewidth of the excited state | ⟩, and Ω in 2 2 = p0 is the optical pumping rate on one-photon resonance. Normally, the optical pumping rate p is dependent on one- . We note that when ≫ 2 p + 2 (where 2 is the groundstate decoherence rate), EIT is destroyed and bc approaches zero. In this regime, one can utilize Im[ r(l) ] to extract the light transmittance as − / , where where is the atomic density, is the dipole moment, is the input laser wavelength, and 0 is the spontaneous emission rate of the excited state. After passing through the atomic medium with length , the output fields take the following simple relation with respect to the inputs, in terms of the effective wave vectors r(l) , Ω out,r(l) = Ω in r(l) , with r(l) = Ω in r(l). (9)

Supplementary Note 2. Jones vector representation of light and pointer-system correlation
In the WM method, we choose the polarization and frequency degrees of freedom of light as the measuring device (pointer) and observable (system), respectively. The key to the problem is representing a polarized EM field in terms of the Jones vectors. For example, let (ϖ) represent a monochromatic, linearly polarized EM field with angular frequency ϖ, which can be described by the orthogonal right-circular and left-circular polarization components in a normalized vector form, (ϖ) = ( rr + ll ) √| l | 2 +| l | 2 − with r and l being the unit vectors for the right-circular-polarization and left-circular-polarization bases, respectively. In the notation of Jones vectors, (ϖ) can be simply cast as (ϖ) = 1 √| r | 2 +| l | 2 ( r l ) − ϖ . With this notation, identifying the polarization and the frequency degrees of freedom becomes straightforward. In the WM language, we can further retype this vector as the product of the polarization and frequency states, that is, Note that in the Jones vector form above, the polarization and frequency degrees of freedom are separable due to the absence of the physical coupling between them. In general, however, these two degrees of freedom can be coupled with one another. When this occurs, the factorization behaviour is no longer valid. For example, let ( ) represent a coherently superposed EM field by two frequencies at ϖ + 1 and ϖ + 2 , ( ) = 1 √| r | 2 +| l | 2 +| r ′ | 2 +| l ′ | 2 ( rr − (ϖ+ 1 ) + ll − (ϖ+ 1 ) + r ′̂r − (ϖ+ 2 ) + l ′̂l − (ϖ+ 2 ) ) (10) In the Jones representation, the field takes the form: Similarly, in the WM language, this EM field can be represented as With this form of the electric field, it is apparent that the polarization and frequency degrees of freedom now become non-separable, implying the formation of a classical analogue of an entangled state [1]. In our case, we have a polarization vector space and a frequency vector space, respectively, represented by the polarization Poincaré sphere and harmonic frequencies, and the classical analogue of entanglement is between these two degrees of freedom of a single system (i.e., intra-system correlation). The frequency states obey the following operations, ⟨ 1,2 | = (ϖ+ 1,2 ) , and | 2 ⟩⟨ 1 | = ( 1 − 2 ) .
Back to our experiment, the input -polarized cw laser field takes the full initial state as before the atom-light interaction. The application of FM creates a variety of new harmonic components ± m ( ∈ integer) away from 0 . In the Jones notation, the EM field now turns out to be ) and the leading AC components ± m (modelled by the projection operators ∑ |( ) through frequency-dependent intensity filtering. After carefully examining the weight of each frequency component in the converted AM, the contributions from these three projection operators, |0⟩⟨0|, | ± m ⟩⟨0|, and |0⟩⟨± m |, are proven to be dominant both theoretically and in the measured data. Throughout the work, therefore, the truncated frequency Hilbert space (spanned by |0⟩, | m ⟩ and | − m ⟩) will be of interest, while higher frequency components will not be taken into account. The post-selection on the frequency results in the post-selected system state as where the parameter close to unity ascribes the DC intensity filtering in the measurement. It can be proven that such a state selection on the electrical field of light is equivalent to keeping (1 − ) times the DC part plus all the AC part at m in the light intensity.

Supplementary Note 3. Stokes parameters and the "pointer" observable
The coherency matrix is a general description of light polarization. In our case, a 2 x 2 polarization coherency matrix ϱ can be attained that encodes all the available information about the light polarization, ( 1   1 ).

Supplementary Note 4. Physical meaning of 〈 〉 and 〈 〉
Since the measured (2) (0) linewidth highly depends on the expectation value 〈 〉, it would thus be interesting to look closely at this quantity with the notations defined above. We first start with the usual measurement without post-selection. For the electric field ( ), let us assume the weight of | 1 ⟩ is more prominent for the input -polarized electric field. We then have Assuming the post-selected frequency state is |Ψ sf ⟩ = 1 | 1 ⟩ + 2 | 2 ⟩ with the normalization | 1 | 2 + | 2 | 2 = 1, the Stokes parameter for ( ) now becomes thus revealing how the post-selection on the frequency alters the polarization. In contrast, 〈 〉 pf represents the expectation value after the post-selection in the frequency domain, where the subscript "p" means "pointer" and "f" means "final". Again, let's assume that the weight of | 1 ⟩ is more prominent for the input -polarized electric field. We then have The underlying physics here is that the post-selection is equivalent to Fourier decomposition on 〈 〉 and picking up the terms only associated with the frequency component m .

Supplementary Note 5. Characterization of
After interacting with the atoms, the polarization of the total output light without post-selection can be evaluated by .

(28)
In our experiment, the atoms are pumped into the dark state for small two-photon detuning, and consequently, ( Equation 28 can be simplified as ,out = 4 cos( m ). As seen, in our FM experiment the major contribution in ,out is from the first sidebands at frequencies ± m .

Supplementary Note 6. ( ) ( ) linewidth and derivation of the optical Hamiltonian
In quantum optics, the second-order temporal correlation function between two light fields defined by Glauber has the form with the intensities j ( ) = j (−) ( ) j (+) ( ) ( = 1,2) and being the time lag between the two photodetector triggers. In this correlation-resonance spectroscopy, we are interested in (2) ( ) at the zero-time lag ( = 0), (2) To use (2) (0) for spectroscopy and sensing, a modified expression is used, which is bound between −1 and 1: where r ( ) and l ( ) are respectively the recorded intensities for the right-circular polarization and left-circular polarization subject to the WM procedure. For simplicity, however, we will omit the subscript "WM" in all WM (2) (0) expressions hereafter in this paper.
According to previous results, when the two-photon detuning = 0, the light transmittance for leftcircular and right-circular polarizations is perfectly synchronized and yields the same values when varying the one-photon detuning . In this case, (2) (0) = 1. However, when ≠ 0, because of the breach of symmetry between left-and right-circular polarization, (2) (0) will decrease and become −1 for large . Since is introduced by an externally applied magnetic field B along the light propagation direction, the change of (2) (0) from 1 to −1 in turn reflects the change of B. The linewidth ℒ of (2) (0) therefore marks the ability of magnetic field sensing.
Since the (2)  is purely imaginary and non-Hermitian. The system operator (|− m ⟩⟨0| + | m ⟩⟨0|) accounts for the FM to AM conversion process that generates the first order sidebands frequency component in the rotating frame of the frequency-modulated laser. Let's recall that in the Jones vector representation, the full polarization information of light has to resort to both the polarization vector and the frequency vector in general, because they are correlated.

Supplementary Note 7. Weak value and its induced linewidth reduction
Given the effective Hamiltonian (Supplementary Equation 41), we are now ready to look at the weak value. With the pre-and post-selected frequency states shown in Supplementary Equation 13 and Supplementary Equation 14, respectively, the final polarization of the output fields evolves to The formal normalization of Supplementary Equation 42 gives where the weak value W (originally defined by Aharonov, Albert, and Vaidman) is real and takes the value of With |Φ pf ⟩ given in Supplementary Equation 43, the output polarization can be computed as signalling the anomalous amplification of 〈 〉 outside of the conventional inequality |〈 〉| ≤ 1. With the use of Supplementary Equations 6, 7, 35, 37, and 45, the linewidth ℒ now assumes the expression clearly indicating that the anomalous WV amplification results in the narrowing of the correlationresonance linewidth.

Supplementary Note 8. Magnetometer Sensitivity
In our scheme, near (2) (0) = 0 of a Lorentzian resonance profile, the magnetic field sensitivity can be simply calculated by where SNR is the signal-to-noise ratio of the (2) (0) correlation resonance, with the signal amplitude equal to 2 and the noise amplitude equal to (2) (0). For experiments without any technical noises, we can obtain the photon-shot-noise limited SNR: where ph is the photon-number rate. Conversely, SNR degrades by the same factor W by which the linewidth is narrowed, rendering a sensitivity independent of WV. However, as shown in the following section, in the presence of technical noise, the sensitivity can be optimized by choosing the proper postselection parameter in the WM process.
From Supplementary Equation 52, one can see that, for a certain amount of RAM, the practical sensitivity can be optimized by choosing a suitable D value, as illustrated in Supplementary Figure 1. The practical sensitivity improves for smaller D values and then stays nearly constant below a certain D value. This predicted trend agrees qualitatively with the experiment, as shown in Fig. 4 in the main text.
In practice, however, the sensitivity becomes slightly worse when D is smaller because the linewidth in this region is beyond the small two-photon detuning approximation (i.e., assuming that Im[ cb ] is linear with B) in the simplified model. Additionally, a similar amount of degradation in sensitivity requires less RAM than that calculated in the theory, which is likely due to the absence of other noise sources in the model, including the magnetic field noise.

Supplementary Figure 1 | Practical sensitivity calculated in the presence of laser intensity noise.
For different amounts of residual amplitude modulation (RAM), the theory shows that the optimal sensitivity is achievable by choosing a proper D value. The parameters involved in the numerical simulations are the same as those used in Fig. 4 in the main text.

Supplementary Note 10. Bandwidth of the magnetometer
The bandwidth of the magnetometer is associated with the ground state dynamics of the atoms and is mainly determined by the optical pumping process establishing the dark state for EIT. In Fig. 3b in the main text, the sensitivity spectrum is obtained from the (2) (0) noise spectrum and the resonance profile's slope at different magnetic field frequencies. To compute the slope, i.e., the response curve, we set the two-photon detuning (via the applied magnetic field) to be = 0 + 1 sin ( B ), where 0 is the HWHM of the (2) (0) resonance, and 1 is set to be small to remain in the linear region of the slope. Then, we numerically solve the master equation to obtain the transmission signal, from which the oscillation amplitude ( ) of (2) (0) and the slope 1 ⁄ can be sequentially obtained. As shown in Supplementary Figure 2, the calculated response can be well fitted by the function BW/√BW 2 + 2 , with a BW value of 9.2 Hz, giving an HWHM of approximately 16 Hz, close to the power broadened EIT linewidth in the theoretical model. In the calculation of this response curve, no noise has been included. In fact, the presence of noise can affect the linewidth of this curve. For instance, if the noise level is higher (hence a broader (2) (0) resonance and a smaller slope) at a lower frequency than at higher frequency, such as that from the 1/ noise, the measured response will be slightly broadened, resulting in a broader response curve than what was calculated, as shown in Fig. 3b in the main text. In this work, we found our magnetometer still offers an appreciable sensitivity, even in the frequency range beyond 60 Hz. As an example, Supplementary Figure 3 reports the typical experimental results of the magnetometer sensitivity for the range from near DC up to 200 Hz. As one can see, in the range of 2 -100 Hz the measured sensitivity can remain below 20 fT Hz -1/2 . Although, along with the growth in the frequency, the sensitivity becomes worse. However, even near 200 Hz, the sensitivity only drops to approximately 40 fT Hz -1/2 .