Low-frequency vector electrometry with a Rydberg dipolar chain

Abstract

Low-frequency electric field sensors are essential for applications in geophysics, electrical engineering, aerospace, and medical technology. However, conventional technologies often suffer from intrinsic trade-offs among traceability, multidimensional vector detection, and miniaturization, which significantly hinder their scalability and deployment in compact platforms. To address these challenges, we propose a vector-resolved quasi-static electric field sensor based on a Rydberg dipolar chain, where the external field reorients the atomic quantization axis and thereby modulates the angle-dependent dipolar exchange interaction. Using a unified framework combining time-domain propagation, Ramsey-mode spectroscopy, and end-to-end Green’s-function analysis, we identify three complementary observables—arrival time, eigenmode frequency shifts, and transmission fringes—that encode both the amplitude and direction of the applied field. The approach operates at micrometer scales compatible with optical-tweezer arrays, offers tunable sensitivity near the magic angle, and provides multi-channel readout within a single platform. Our results establish a compact and experimentally feasible route toward high-resolution, vector-sensitive low-frequency electrometry with the potential for quantum-enhanced performance.

Graphical abstract

Keywords

Quantum optics / Rydberg atom array / Quantum sensing / Electric field sensor

1 Introduction

Low-frequency electric field measurement has widespread applications in geophysics, electrical engineering, aerospace, and medical technology [1,2]. Although existing traditional low-frequency electric field sensors—such as field mills, capacitive sensors, electro-optic crystals, and micro-electro-mechanical systems—can achieve sensitivity down to the mV/m level, they still face inherent performance constraints and trade-offs in terms of traceability, multidimensional vector detection, and miniaturization [1–5]. To address these limitations, quantum sensing approaches that exploit coherence, entanglement, and interference have emerged as an alternative route to improved performance [6]. Rydberg atoms possess very large electric dipole moments due to their highly excited valence electrons, and are therefore highly sensitive to weak electric fields. They also offer rich level structures, long excited-state lifetimes, tunable interactions, and good compatibility with optical platforms, which together make them a promising platform for electric field quantum sensing [7–9].

Currently, most Rydberg-atom-based electric field sensors rely on alkali-metal vapor cells, extracting external field information by probing electromagnetically induced transparency (EIT) signals [8–14]. By introducing signal-field-induced Stark shifts, ultra-low-frequency/DC electric fields can be detected [13–15]. However, in this vapor-cell-based sensing paradigm, macroscopic gas effects such as collisions and Doppler broadening reduce spectral resolution and thereby limit measurement accuracy. In addition, vapor-cell-based schemes typically rely on ensemble-averaged readout over a macroscopic sensing volume, which makes atom-scale spatial localization and site-resolved vector-field mapping challenging. With the advancement of experimental techniques, neutral atom arrays can now be assembled and precisely manipulated using optical tweezers [16–22]. High-fidelity preparation, control, and readout of atomic quantum states can be achieved via lasers. This makes it feasible to construct electric field sensors based on Rydberg atom arrays. In recent years, studies on exploiting many-body effects to surpass the standard quantum limit (SQL) have also indicated the unexplored potential of Rydberg atom arrays in quantum sensing [23–26].

Our work proposes and theoretically investigates a new paradigm for low-frequency vector electric field sensing using Rydberg dipole chains. We encode the signal electric field into the quantization-axis orientations of atoms in the Rydberg dipole chain. By leveraging the angular dependence of interatomic dipole-dipole interactions, the vector signal information can be extracted from the system's transport times, spectral structures, and frequency responses. Theoretical estimates suggest that this paradigm can produce significant responses to low-frequency vector electric fields on the order of tens of V/m, with spatial resolution reaching the micrometer scale at the atom chain level, and with potential scalability compatible with integrated photonic structures. Furthermore, the system’s response characteristics can be flexibly tuned by adjusting parameters such as the geometry of the atom chain, the choice of Rydberg states, and the orientation of bias fields. By incorporating nonclassical states or coupling with other sensing processes, the system exhibits strong scalability in both measurement precision and sensing frequency range. This work proposes a new approach to miniaturized, tunable, and vector-resolved low-frequency electric field quantum sensing.

2 Theoretical framework

2.1 Sensing configuration

Figure 1a presents the sensing configuration employed in this work. We consider a one-dimensional chain of individually trapped neutral atoms with spacing a, each initially prepared in a Rydberg state |r_1⟩ from the ground state |g⟩ by preparation lasers ({\text{Upomega}}_{\text{prep}}) (Fig. 1b). A homogeneous electric field {\mathit{E}}_{\text{bias}} defines the initial quantization axis {\hat{\mathit{e}}}_{\mathrm{d}}\parallel {\mathit{E}}_{\text{tot}} for all atoms in the chain. A single Rydberg excitation |r_2⟩ is created at the left end of the chain by a local microwave (MW) pulse driving the |r_1⟩\leftrightarrow |r_2⟩ transition. The resonant dipole-dipole interaction between |r_1⟩ and |r_2⟩ is described by the exchange coefficient:

(1)J_n\left(\theta \right)={\displaystyle \frac{C_3}{(na{)}^3}}\left(1-3{\cos }^2\theta \right),

where n represents the number of atom spacing between interacting atoms, which is taken as 1 under the nearest-neighbor approximation. The C_3 coefficient characterizes the strength of resonant dipole-dipole interactions between Rydberg atoms. The angle \theta denotes the angle between the chain axis \hat{x} and the quantization axis {\hat{\mathit{e}}}_{\mathrm{d}}, which is parallel to the direction of the applied electric field.

To encode the external signal, a weak probing electric field {\mathit{E}}_{\text{sig}} is introduced. As illustrated in Fig. 2c, this vector signal electric field is combined with {\mathit{E}}_{\text{bias}}, resulting in a total electric field {\mathit{E}}_{\text{tot}}={\mathit{E}}_{\text{bias}}+{\mathit{E}}_{\text{sig}}, thereby rotating the dipole orientation from the initial working angle {\theta }_0 by an angle \delta \theta, which is incorporated into the exchange coefficient J(\theta ). For a transverse signal component relative to the bias field, the rotation satisfies \tan \left(\delta \theta \right)={\mathit{E}}_{\text{sig}}/{\mathit{E}}_{\text{bias}}. For instance, taking {\mathit{E}}_{\text{bias}}=10\text{V/cm} and {\mathit{E}}_{\text{sig}}=1\text{V/cm} gives \delta \theta \approx 5.7^{\circ }.

The rotated angle is thus encoded into the exchange coefficient J(\theta ), which can be further read out by different optical methods. As shown in Fig. 2d, by the laser and MW pulse, the chain system is prepared from the entire ground state |g,g,...,g⟩ to |r_2,r_1,...,r_1⟩. After the excitation is created, all driving fields are turned off and the system undergoes evolution T_evo, governed by the angle-related dipole-dipole interaction Hamiltonian H. At the end of the evolution, excitation can be detected at the rightmost site using a state-selective optical readout pulse [27].

2.2 Hamiltonian model

The dynamics of the chain is governed by the resonant dipole-dipole exchange interaction between Rydberg states \{|r_1⟩,|r_2⟩\}. For an atom pair separated by a distance R, the dipole-dipole interaction takes the form

(2)V_{dd}\left(\mathit{R}\right)={\displaystyle \frac{1}{4\text{π}{\epsilon }_0{R}^3}}\left[{\mathit{d}}_1\cdot {\mathit{d}}_2-3\left({\mathit{d}}_1\cdot \hat{\mathit{R}}\right)\left({\mathit{d}}_2\cdot \hat{\mathit{R}}\right)\right],

where {\mathit{d}}_{1,2} are the dipole operators. Because the pair states |r_2^{(i)}{r}_1^{(j)}⟩ and |r_1^{(i)}{r}_2^{(j)}⟩ are degenerate, this interaction resonantly couples them and produces coherent “flip-flop” exchange between sites i and j, resulting in the single Rydberg excitation |r_2⟩ propagating along the chain. Defining |j⟩ as the state with the single |r_2⟩ excitation localized on site j, the basis vectors of the system can be defined as |j⟩\equiv |r_{1,1},...,r_{2,j},...,r_{1,N}⟩. Under the restriction to the single-|r_2⟩-excitation manifold, the resonant dipole-dipole interaction between Rydberg atoms maps exactly onto an isotropic spin-1/2 XY (XX) model with long-range couplings. The effective Hamiltonian reads

(3)H={\displaystyle \frac{1}{2}}{\sum }_{i\ne j}{J}_{ij}\left(\theta \right)\left(b_i^{†}{b}_j+b_i{b}_j^{†}\right),

where b_j^{†}={|r_2⟩}_j{⟨r_1|}_j, b_j={\left(b_j^{†}\right)}^{†} act as spin-raising/lowering operators creating or annihilating the single |r_2⟩ excitation at site j. The lattice spacing a = 10 μm is chosen to set the overall interaction scale in a realistic experimental regime. Within this range, the dipole coupling is dominated by the nearest-neighbor term. We adopt the nearest-neighbor approximation and obtain the XY-type tight binding Hamiltonian

(4)H=J\left(\theta \right)\sum _{j=1}^{N-1}\left(b_j^{†}{b}_{j+1}+b_j{b}_{j+1}^{†}\right),J\left(\theta \right)={\displaystyle \frac{C_3}{a^3}}\left(1-3{\cos }^2\theta \right).

To analyze the system in momentum space, a Fourier transform to the real-space basis is applied to the infinite periodic chain, yielding the Bloch Hamiltonian H_k as a function of the quasi-momentum k:

(5)H_k={\sum }_{k}E(k)b_k^{†}{b}_k,E(k)=2J(\theta )\cos (ka).

2.3 Observable physical quantities and readout methods

In this system, the angular dependence of the dipole-dipole exchange interaction can be directly coded into several experimentally accessible observables. In this work, we employ three complementary readout methods, namely excitation arrival time, Ramsey spectroscopy, and end-to-end transmission response, each probing a different functional dependence of the dynamics on the exchange coupling. Together, these techniques constitute a comprehensive set of time-domain, energy-domain, and frequency-domain measurement tools for low-frequency vector electric field sensing based on the angle dependence of the dipole-exchange interaction. Additional computational details are provided in Online Resource 1.

2.3.1 Excitation arrival time t*

The angle-dependent exchange interaction can be reflected from the time it takes for the |r_2⟩ excitation to travel from one end to the other of the chain. Starting from the initial state |\psi (0)⟩=|1⟩ with the single |r_2⟩ excitation prepared from the leftmost site, the time-evolved state is |\psi (t)⟩={\mathrm{e}}^{-\mathrm{i}Ht/\hslash }|1⟩. The |r_2⟩ excitation probability at the chain endpoint is

(6)P_N\left(t\right)={\left|⟨N|\psi \left(t\right)⟩\right|}^2,

which is experimentally accessible through multiple repeated state-selective push-out measurements. Repeating this sequence for different evolution times gives access to the full time dynamics. The “arrival time” t* is defined as the location of the first dominant peak of P_N\left(t\right), under the periodic approximation

(7)t\ast \approx {\displaystyle \frac{\left(N-1\right)a}{v_g^{max}\left(\theta \right)}}={\displaystyle \frac{\left(N-1\right)\hslash }{2\left|J\left(\theta \right)\right|}},

where v_g={\displaystyle \frac{1}{\hslash }}{\displaystyle \frac{\mathrm{\partial }E\left(k\right)}{\mathrm{\partial }k}} is the group velocity, which provides a functional relationship between angle and time.

2.3.2 Ramsey-spectrum {\omega }_{\alpha }

The angle-dependent energy eigenvalues of the open finite Rydberg chain,

(8)E_{\alpha }\left(\theta \right)=2J\left(\theta \right)\cos \left({\displaystyle \frac{\text{π}\alpha }{N+1}}\right),\alpha =1,...,N,

can be obtained from an endpoint Ramsey spectrum {\omega }_{\alpha }=E_{\alpha }/\hslash. The Ramsey scheme naturally maps the coherent propagation of the single |r_2⟩ excitation to an interference signal whose Fourier components correspond to the discrete eigenenergies of the finite dipole chain. Experimentally, at the end of the dark free evolution, a single-site Ramsey sequence is applied to the last atom, which prepares a superposition by a π/2 pulse, performs free evolution under H, and finally applies a second π/2 pulse to readout the excitation probability [17,28,29]. Evolution time scanning and Fourier transformation yields the discrete Ramsey spectrum {\omega }_{\alpha }.

2.3.3 End-to-end transmission spectrum S\left(\omega \right)

The third observable is the end-to-end transmission amplitude S\left(\omega \right), defined as the linear response of the last site to a weak, monochromatic drive applied at the first site. This transmission response probes the frequency-resolved Green’s function of the chain [30,31]:

(9)G\left(\omega \right)={\left(\omega -H/\hslash +\mathrm{i}\eta \right)}^{-1}.

Here, we include a broadening \mathrm{i}\eta with \eta =\hslash \gamma /2, where \gamma is related to the Rydberg-state lifetime.

Experimentally, after preparing all the atoms on the chain to |r_1⟩, we apply a weak, continuous-wave microwave drive on the first atom, and scan the frequency \omega across the |r_1⟩\leftrightarrow |r_2⟩ transition. This drive creates a small, steady-state excitation amplitude at site 1, which subsequently propagates along the chain under the XY Hamiltonian. The response of the atom on the site N—either its excitation probability or its induced fluorescence—can be measured as

(10)S\left(\omega \right)\propto {\left|G_{N1}\left(\omega \right)\right|}^2.

For the nearest-neighbor open chain, the endpoint Green’s function is known analytically [31,32]:

(11)G_{N1}\left(\omega \right)={\displaystyle \frac{{\left(-1\right)}^{N-1}}{\hslash J\left(\theta \right)}}{\displaystyle \frac{1}{U_N\left({\displaystyle \frac{\omega +\mathrm{i}\eta }{2J\left(\theta \right)}}\right)}},

where U_N is the second kind Chebyshev polynomial. This type of model that utilizes the Green’s function to solve the transmission response has been extensively studied in systems such as magnetic spin chains.

3 Results and discussion

Based on the theoretical model developed above, we numerically analyze the response of a Rydberg chain to a low-frequency vector electric field in terms of its eigenstructure and three different observables. Except in special circumstances, the following parameters are adopted throughout this section: the chain length is N=100, the lattice spacing is a=10\text{μm}, the dipole-dipole interaction coefficient C_3=5.0\times 10^{-42}\text{J}\cdot {\mathrm{m}}^3, and the amplitude of the bias electric field is \left|E_{\text{bias}}\right|=10\text{V/cm}. In the end-to-end transmission model, we introduce an effective linewidth \eta =1.0\times 10^{-30}\text{J}. The signal field {\mathit{E}}_{\text{sig}} is treated as a classical vector characterized by its magnitude \left|E_{\text{sig}}\right| and the angle {\varphi }_{\text{sig}} relative to the chain axis \hat{x}.

3.1 Exchange coefficient and dispersion

We first examine the eigenstructure of the dipolar chain as a function of the angle between the total electric field and the chain axis. To quantify the angular sensitivity of the coupling, we introduce a normalized variation rate \mathrm{\Lambda }\left(\theta \right)={\displaystyle \frac{1}{J\left(\theta \right)}}{\displaystyle \frac{\mathrm{\partial }J\left(\theta \right)}{\mathrm{\partial }\theta }}, which characterizes the relative change of J(\theta ) introduced by a small change in \theta. Numerical results (Fig. 2a) show that J(\theta ) increases monotonically as \theta is rotated from 0° to 90°, crossing zero at {\theta }_{\mathrm{m}}=\arccos \sqrt{{\displaystyle \frac{1}{3}}}\approx {54.7}^{\circ }, the so-called “magic angle”. For , the chain effectively realizes an “antiferromagnetic-type” coupling with J < 0, whereas for \theta >{\theta }_{\mathrm{m}}, it corresponds to a “ferromagnetic-type” coupling. \left|\mathrm{\Lambda }\right| exhibits a sharp maximum in the vicinity of {\theta }_{\mathrm{m}}, indicating that small angular perturbations can lead to significant variations in the magnitude of J(\theta ). From the perspective of sensing, this provides pronounced angular sensitivity, but also implies that noise and drifts are strongly amplified near {\theta }_{\mathrm{m}}. Therefore, in practice the working point should be chosen close to but not exactly at the magic angle.

The calculated E\left(k,\theta \right) dispersion map (Fig. 2b) shows that, at \theta ={\theta }_{\mathrm{m}}, the entire band collapses to zero energy for all k, which means that the group velocity vanishes identically. This results in a fully localized |r_2⟩ excitation with formally infinite arrival time in an ideal chain. As \theta is detuned from the magic angle, the dispersion gradually recovers the familiar cosine-shaped tight binding band. The amplitude, proportional to J(\theta ), increases monotonically with \left|\theta -{\theta }_{\mathrm{m}}\right|, implying that the overall transport speed can be continuously controlled via the angle. In addition, the dispersions on the two sides of the magic angle are mirror images in k-space: for J>0 the band minimum is at k=0, while for it is at k=\text{π}/a. In summary, the analysis of the eigenstructure provides a unified physical picture: the low-frequency vector field modifies the direction \theta of the total field, thereby tuning the angle-dependent exchange coupling J\left(\theta \right) and dispersion E\left(k,\theta \right), which are imprinted in various time- and frequency-domain observables.

3.2 Readout 1: Excitation arrival time t*

We now focus on the time required for an excitation |r_2⟩ to propagate from one end of the chain to the other. The initial state is prepared with a single |r_2⟩ excitation at the first site, while all other atoms are in the |r_1⟩ state. Under the given total Hamiltonian in Eq. (4), the system evolves freely. We monitor the time-dependent excitation probability P_N\left(t\right) at the last site to extract the arrival time t*. Numerically, we define the arrival time as t^{\ast }=min\{t|P_N\left(t\right)>P_{\text{th}}\}, where the threshold is chosen as P_{\text{th}}=0.1. This quantity directly reflects the excitation transport velocity.

In the absence of the signal field, we first scan the bias angle {\theta }_0 and compute P_N\left(t\right) for each angle (Fig. 3a). The resulting traces show a sequence of nearly equally spaced peaks in time for a given {\theta }_0, corresponding to multiple “arrival events” caused by reflections at the boundaries of the finite chain. As {\theta }_0 approaches the magic angle, the peaks shift to later times and the spacing between them increases, indicating slower propagation. The first major peak defines t*, which is found to closely follow the trend of 1/\left|J\left({\theta }_0\right)\right|. This behavior is consistent with the Eq. (7). Balancing angular sensitivity and practical measurability, we choose {\theta }_0=45{}^{\circ } as a representative working point in the following, where the response remains strong but the arrival time is still within a reasonable observation window.

We then fix the bias electric field and investigate the two-dimensional response t^{\ast }\left(\left|E_{\text{sig}}\right|,{\varphi }_{\text{sig}}\right) (Fig. 3b). The numerical results show a pronounced anisotropy in the signal amplitude-angle plane. The sensitivity to the signal amplitude is maximal when the signal field is approximately orthogonal to the bias field, because in this configuration, the vector electric field sum is most efficiently pulled toward the magic angle, so that \theta has the largest derivative with respect to \left|E_{\text{sig}}\right|, and consequently both J(\theta ) and t* have the steepest slope as functions of the signal amplitude. This indicates that Readout 1 can simultaneously provide information on both the amplitude and the direction of the signal field, with particularly high sensitivity along certain optimal directions.

To better understand the behavior near the magic angle, we fix the signal direction at the most sensitive configuration {\varphi }_{\text{sig}}={\theta }_0+90{}^{\circ }, and scan \left|E_{\text{sig}}\right| for several bias angles {\theta }_0 (Fig. 3c). We plot both the arrival times obtained from full time evolution and the curves predicted by the approximate relation t^{\ast }\propto 1/\left|J\left({\theta }_{\text{tot}}\right)\right|. When , increasing \left|E_{\text{sig}}\right| pushes \theta toward the magic angle, and t* grows monotonically before abruptly diverging as the total angle crosses {\theta }_{\mathrm{m}}. This divergence corresponds to a regime where the propagation speed is strongly suppressed and the first arrival falls outside the accessible time window. In contrast, when {\theta }_0>{\theta }_{\mathrm{m}}, the signal field drives \theta away from the magic angle, and t* decreases with increasing \left|E_{\text{sig}}\right|. In all cases, the slope \mathrm{d}{t}^{\ast }/\mathrm{d}\left|E_{\text{sig}}\right| is larger when {\theta }_0 is closer to {\theta }_{\mathrm{m}}, reflecting enhanced angular sensitivity. Thus, t* exhibits a switch-like behavior in the vicinity of the magic angle: once the signal field pushes \theta across {\theta }_{\mathrm{m}}, the arrival time diverges beyond the accessible temporal window, suggesting a threshold-type sensing functionality.

We further fix {\theta }_0=45{}^{\circ } and scan the signal angle {\varphi }_{\text{sig}} for several values of \left|E_{\text{sig}}\right|. Resulting curves display a clear peak-valley pattern: t* is maximal when {\varphi }_{\text{sig}}={\theta }_0+90{}^{\circ }, where \theta is closest to the magic angle, and minimal when {\varphi }_{\text{sig}}={\theta }_0-90{}^{\circ }, where \theta is furthest away. As the signal amplitude increases, the peak-valley contrast grows rapidly and can reach several microseconds for \left|E_{\text{sig}}\right|\sim 0.3-1.0\text{V/cm}. In contrast, in the weak-signal regime , the total variation of t* over all angles is only on the order of 0.1\text{μs}, which may be challenging to resolve in realistic experiments given finite timing resolution and technical noise. Overall, Readout 1 is particularly well-suited for detecting moderate and strong signals with unknown direction, and for exploiting the switch-like response near the magic angle, whereas high-precision estimation of very weak signals is better addressed by other readouts.

3.3 Readout 2: Ramsey-spectrum {\omega }_{\alpha }

The second readout is based on Ramsey interferometry at the chain boundary. In the present work, we focus on the lowest mode \alpha =1. The calculated {\omega }_1\left(\theta \right) (Fig. 4a) shows that, as the electric field angle is rotated from 0° to 90°, {\omega }_1 increases monotonically from negative to positive values and crosses zero near the magic angle. At {\theta }_0={\theta }_{\mathrm{m}}, {\omega }_1=0, reflecting the flat-band limit discussed in Section 3.1. The slope \mathrm{d}{\omega }_1/\mathrm{d}{\theta }_0 reaches its maximum near {\theta }_0=45{}^{\circ }, indicating that this angle also maximizes the sensitivity of the Ramsey frequency to angular perturbations.

Fixing {\theta }_0=45{}^{\circ } as the working point, we compute the two-dimensional response {\omega }_1\left(\left|E_{\text{sig}}\right|,{\varphi }_{\text{sig}}\right) (Fig. 4b). The resulting pattern closely resembles that of t*: the sensitivity to \left|E{}_{\mathrm{s}\mathrm{i}\mathrm{g}}\right| is maximal when {\varphi }_{\text{sig}}={\theta }_0+90{}^{\circ } and minimal when {\varphi }_{\text{sig}}={\theta }_0-90{}^{\circ }, and the overall frequency shift increases with increasing amplitude. In the small-signal regime, {\omega }_1 depends approximately linearly on \left|E_{\text{sig}}\right|, allowing straightforward extraction of the field amplitude from a linear fit. Compared with the observable t*, the Ramsey frequency is not limited by a finite observation window and can benefit from high-resolution Fourier analysis and phase-sensitive detection, making it a natural choice for high-precision metrology.

An important advantage of the Ramsey readout is that it is directly connected to standard quantum metrology theory for phase estimation, enabling a clear discussion of the SQL and the potential benefits of nonclassical many-body states. For an ensemble of N uncorrelated atoms prepared in a coherent spin state (CSS), the uncertainty of a frequency estimate after a total interrogation time T obeys the SQL \text{Updelta}{\omega }_{\text{SQL}}\propto {\displaystyle \frac{1}{\sqrt{NT}}} [23–26], which translates into a signal electric field amplitude uncertainty \text{Updelta}{E}_{\text{SQL}}\propto {\displaystyle \frac{{\displaystyle \frac{1}{\sqrt{NT}}}}{\left|{\displaystyle \frac{\mathrm{\partial }{\omega }_1}{\mathrm{\partial }{E}_{\text{sig}}}}\right|}}. Using the numerically obtained slope {\displaystyle \frac{\mathrm{\partial }{\omega }_1}{\mathrm{\partial }{E}_{\text{sig}}}}, we evaluate this scaling and compare it with two idealized classes of nonclassical many-body states: spin-squeezed states with squeezing parameter , for which \text{Updelta}{E}_{\text{sq}}=\xi \text{Updelta}{E}_{\text{SQL}}, and maximally entangled GHZ states, for which the Heisenberg-limited scaling \text{Updelta}{E}_{\text{GHZ}}\propto {\displaystyle \frac{1}{\left(NT\right)}} can in principle be achieved (Fig. 4c). The numerical results show that, as the chain length N increases, the gain of the squeezed state relative to the SQL remains approximately constant, whereas the gain of the GHZ state grows linearly with N. Recent experiments in optical-tweezer Rydberg systems have demonstrated Rydberg-mediated spin squeezing and multipartite GHZ-type entanglement [25,33], indicating that nonclassical many-body resources are becoming experimentally accessible in neutral-atom arrays. Although these results are obtained under idealized conditions neglecting decoherence and technical noise, they demonstrate that, among the three readout schemes, the Ramsey-frequency readout is the most natural candidate for implementing quantum-enhanced low-frequency vector electrometry on the same Rydberg chain platform.

3.4 Readout 3: End-to-end transmission spectrum S\left(\omega \right)

The third readout probes the end-to-end transmission of the chain in the frequency domain. Unlike the previous readouts, which focus on a single time scale or a single eigenfrequency, S\left(\omega \right) reflects the collective effect of the entire band structure and the interference of all modes contributing to the end-to-end transport. As shown in Fig. 5a, in the absence of the signal field, we compute S\left(\omega \right) as a function of the bias angle {\theta }_0. The transmission peaks form a series of well-defined stripes, resulting in a characteristic “butterfly-like” pattern: near the magic angle, all stripes collapse into a bright, narrow region around \omega =0, with minimal spacing and maximal intensity; as {\theta }_0 moves away from the magic angle, the stripes fan out toward higher frequencies, their spacing increases, and their contrast gradually decreases. Dispersion analysis in Section 3.1 indicates that these stripes are the discrete eigenmodes of the tight binding band projected onto the end-to-end Green’s function: at the magic angle, the bandwidth vanishes and all modes pile up at zero frequency; away from the magic angle, the modes separate, forming nearly equally spaced peaks. To illustrate the behavior around the working point, Fig. 5b shows cuts of S\left(\omega \right) for three bias angles {\theta }_0 as 43°, 45°, and 47°. The spectrum exhibits a nearly symmetric set of transmission peaks, while detuning the angle to either side leads to visible shifts and changes in the relative peak heights. These one-dimensional spectra make explicit how small angular variations around the working point distort the transmission fringes. Results indicate that the stripe spacing and the stripe intensity naturally provide two complementary metrics for sensing the angle and amplitude of the external field.

Taking into account both stripe visibility and sensitivity, we choose {\theta }_0=45{}^{\circ } as a representative working point for Readout 3. At this angle, the stripes are already well separated and easy to resolve, while the signal remains reasonably strong. Around this working point, we select a pair of symmetric peaks near \omega =0 (approximately corresponding to the 50th and 51st eigenmodes in a chain with N = 100) and define their frequency spacing as D. Denoting by D₀ the spacing in the absence of the signal field, we introduce \text{Updelta}D=D-D_0 as a geometric measure of the distortion of the stripe pattern induced by the low-frequency vector field. Figure 5c shows \text{Updelta}D\left(\left|E_{\text{sig}}\right|,{\varphi }_{\text{sig}}\right) at {\theta }_0=45{}^{\circ }. The spacing shift grows monotonically with the signal amplitude and reaches values on the order of 0.3 MHz for \left|E_{\text{sig}}\right|\sim 1.0\text{V/cm}. We also analyze how the peak amplitudes in the transmission spectrum respond to the signal field (Fig. 5d). For the same pair of peaks, we define the amplitude change as {\log }_{10}\left({\displaystyle \frac{S}{S_0}}\right), where S and S₀ are the peak amplitudes with and without signal field. For both of the sensing quantities, the angular dependence exhibits a peak-valley structure with extrema near {\varphi }_{\text{sig}}={\theta }_0\pm 90{}^{\circ }, and the corresponding response amplitude at different angles will increase as the amplitude of the signal field increases, fully consistent with Readout 1 and Readout 2.

Overall, Readout 3 provides a spectroscopic route to low-frequency vector field sensing. The stripe spacing encodes the geometry of the band structure, while the amplitude change captures how the participation of different modes in the end-to-end Green’s function is modified. Like t* and the Ramsey-mode frequency, the transmission spectrum probes the same angle-dependent dipolar dynamics, but it emphasizes how the chain responds in the frequency domain.

4 Conclusion and outlook

By exploiting the angle-dependent dipolar exchange interactions in a Rydberg atom chain, we have developed a sensing framework that encodes the external field information into the many-body dynamics of the chain. We theoretically analyzed three experimentally accessible observables: the first-arrival time, the Ramsey spectrum, and the end-to-end transmission spectrum, and showed that each provides complementary access to the characteristics of the applied low-frequency vector electric field. Our analysis is based on the nearest-neighbor approximation. Beyond-nearest-neighbor dipolar couplings introduce quantitative corrections to the spectrum but do not modify the fundamental angle-sensitive exchange mechanism; a qualitative discussion is provided in Online Resource 2.

In realistic implementations, residual atomic motion, finite Rydberg-state lifetime, laser/microwave phase noise, and bias-field fluctuations introduce additional dephasing and spectral broadening. These effects can be captured phenomenologically by an effective dephasing rate. Experiments in optical-tweezer Rydberg platforms have demonstrated coherence times on the order of tens of microseconds [34,35]. Looking ahead, advances in optical-tweezer array engineering [19,22], dynamical decoupling [28,29,36], and in situ field compensation techniques [37,38] provide a realistic route for overcoming these limitations.

Compared with vapor-cell-based Rydberg electrometers, the present approach encodes vector electric-field information into site-resolved many-body dynamics, enabling micrometer-scale spatial localization, intrinsic vector sensitivity, and multi-observable readout. Moreover, optical-tweezer Rydberg arrays operate at tens-of-kilohertz linewidths, whereas vapor-cell Rydberg-EIT resonances are typically tens of megahertz wide, enabling orders-of-magnitude stronger spectral discrimination. Combined with the flexible tunability of array geometry, interatomic spacing, and driving fields, this establishes a complementary sensing paradigm and a scalable route toward localized, vector-resolved quantum electrometry of low-frequency electric fields.

5 Data availability

The data that support the results of this work are available from the corresponding author, upon reasonable request.

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