Performance of antenna-based and Rydberg quantum RF sensors in the electrically small regime

The radio frequency (RF) spectrum spans more than nine orders of magnitude in frequency, from near-DC to $1012$ Hertz. Due to an increasingly crowded spectrum, modern technologies demand more agility and sensitivity over this large, but finite, usable phase space. Recently, RF sensors based on highly excited atomic Rydberg states have achieved sensitivity across the RF spectrum, demonstrating their broad tunability.^1–5 Rydberg quantum sensors operate based on the quantum physics of atom–photon interactions and are limited by quantum projection noise instead of internal thermal noise.^6–8 Therefore, they do not obey the same limits as traditional antenna-based sensors. In this work, we derive and compare idealized and practical sensitivity limits of Rydberg sensors and antenna-based RF sensors with classical electronic readout.

Here, we focus on signal detection in the electrically small regime, where the system size is much smaller than the carrier wavelength $λ$⁠. In this limit, matched antenna-based sensors that efficiently absorb the field energy are often impractical or even impossible to build. This is because a lossless and matched antenna must be resonant with a large quality factor Q resulting in low bandwidth $ωBW∼1/Q$⁠.^9,10 Significant research has been conducted on alternative passive and active antenna-based systems to enable more efficient sensing in the electrically small regime. Examples include active non-Foster circuits,^11,12 optimal matching schemes,^13,14 and active receiver systems.¹⁵ The relationship between Rydberg sensors and antenna-based RF sensors, including active circuits, is an area of ongoing study.^2,16,17

In this work, we first derive the RF detection sensitivities of electrically small antennas coupled to optimal active and passive electronic receiver backends. Next, we calculate the detection sensitivity of an ideal Rydberg sensor operating at the standard quantum limit (acronymized RSQL) at frequencies below 30 MHz. We also derive a prediction for the sensitivity achieved by warm Rydberg vapors and electromagnetically induced transparency (REIT sensor) at the same frequencies. We show that unmatched, active antenna-based circuits outperform REIT sensors, and that both can approach atmospheric noise levels. RSQL sensors offer significant opportunity for improvement beyond state-of-the-art sensitivity, which could enable applications such as precision RF metrology in shielded environments. We present equivalent circuit models that facilitate comparisons between quantum and antenna-based sensors in the electrically small regime, and we conclude by plotting an example comparison for 1 cm-sized sensors in the frequency range of $3×104$ to $3×107$ Hz. We observe a large performance gap between the sensitivity of current warm-atom-based Rydberg sensors and the fundamental sensitivity predicted by the standard quantum limit, motivating further research in Rydberg sensors.

We begin by calculating the minimum detectable incoming RF field for each sensor type. We consider an incoming plane wave RF field as well as external noise characterized by the amplitude of their respective Poynting vectors S and $S̃$⁠, as depicted in Fig. 1. We use $∼$ over symbols to represent noise processes, which are written in spectral density units. For example, $S̃$ has units of W/(m² Hz).

We treat the scenario depicted in Fig. 1. Because a sensor does not differentiate between incoming noise and signal, the only way to improve $SNR̃ex$ is to increase the antenna (or Rydberg sensor) directivity. Electrically small antennas with high directivity are theoretically possible¹⁸ although typically impractical. The antenna gain profile of a Rydberg sensor and a dipole antenna is comparable since we consider atomic transitions where the atom-field interaction is dipolar. Therefore, all sensors considered in this work—including the Rydberg sensor—are assumed to have a directivity equal to that of a dipole, namely, 3/2. Thus, all sensors we consider are equivalent with respect to $SNR̃ex$⁠.

Given these assumptions, we narrow focus to the internal signal-to-noise ratio, $SNR̃in$⁠, that quantifies our ability to precisely measure $Vin$⁠. We calculate the input-referred noise voltage $Ṽin$⁠. $Ṽin$ is also referred to as noise equivalent voltage (NEV) and is related to the noise equivalent input field (NEF) $Ẽmin$ or $B̃min$ (for a loop) by applying the appropriate field conversion factor.

Next, we tackle the NEV/NEF of a Rydberg sensor. We first consider the case of an ideal quantum sensor operating at the standard quantum limit (RSQL).^6,20 We consider a single Rydberg transition with frequency $ω0$ and resonant dipole moment d. We take the incoming field with frequency $ω$ to be resonant, $ω=ω0$⁠. We display the Rydberg sensing circuit model in Fig. 2(c). The results of the circuit model yield the same result, by construction, that would be achieved by considering canonical models for quantum sensors.^2,20

We take the treatments described in preceding paragraphs and compare the predicted noise equivalent RF fields in Fig. 3. Each of the sensors considered assume a sensing region of approximately 1 cm. Furthermore, to calculate concrete numbers in Fig. 3, several additional assumptions are made including antenna wire dimensions, noise characteristics, and atom number limitations. The supplementary material is included with figure-generating Python source code to allow these choices to be modified for broad exploration.

In order to construct the results for antenna-based sensors, we assume that the antenna resistance arises solely from Ohmic losses. The conductor that makes up the loop and dipole antenna conductor is chosen to be copper with the diameter $a=1$ mm. The loop is given a diameter $d=$ 1 cm, and the dipole is similarly assigned full length $l=$ 1 cm. The copper conductor resistivity is $Rs=1.8×10−8$  $Ω$m, and the magnetic permeability of the loop core is permeability of free space, $μ$⁠. For the active circuit, we assume the parameters of a low-noise pre-amp with high input impedance, $Ṽamp=1 nV/Hz$ and $Ramp=100 MΩ$⁠. These realistic values are commonly achieved with available amplifiers.

We use Eq. (5) to add the results for matched/passive antennas to Fig. 3 (orange dash, red dot). Because the single-turn loop that we consider only detects the magnetic field derivative, given by Faraday's law, the loop is significantly less sensitive to the incoming Poynting vector at the frequencies considered here. However, loop antennas (often with multiple turns and ferrite cores) offer several practical advantages including easily tunable resonance and relatively easy matching. Conversely, the NEF for the matched dipole starts out frequency independent but rises slightly with frequency due to wire skin depth decreasing with frequency. Although the matched dipole is the most sensitive of the antenna-based sensors, this device suffers significant difficulty in the practicality of lossless matching and the fundamental limitation in bandwidth that results due to the Chu-Harrington limit.⁹ 

Next, we plot the NEFs for unmatched dipoles with passive [blue dash, Eq. (6)] and active (green dot-dash) receiver backends. The unmatched, passive sensor, similar to what was considered in a previous comparison work,² is the least sensitive of those considered. The active receiver significantly improves performance by reducing the voltage division effect caused by impedance mismatch due to the amplifier's high input impedance. For the values chosen, this active system becomes flat in sensitivity above 100 kHz and is limited solely by the amplifier's input noise. In the near term, this approach is practical and can surpass the sensitivity of current state-of-the-art Rydberg sensors.

To add RSQL sensors to our comparison, we must apply Eqs. (10) and (8), which amounts to choosing an appropriate dipole moment, decoherence rate, and atom number. In choosing these values, we assume tightly packed, cold atoms. The result is plotted as purple dots in Fig. 3. We apply the following process. First, we map out the possible Rydberg transitions in rubidium resonant in the VLF-HF bands, restricting ourselves to quantum numbers $l≤15$ and $n<100$⁠, above which operation is non-optimal. Using the Atomic Rydberg Calculator (ARC),²⁴ we calculate each transition's dipole moment and decoherence rate $Γ$⁠, including black body induced decay at room temperature. The decoherence rate $Γ$ is of order $2π×1$ kHz over the range of the plot, and exact values may be found (and modified) in the supplementary material.

Last, we define the atom number by confining ourselves to a 1 cm³ volume and restricting the density. The density is chosen to be that which limits van der Waals interactions between atoms to less than a 1 MHz frequency shift. This shift is calculated using C₆ coefficients from ARC. The atom number is optimized at each frequency, but is of order $108$ over the range of the plot. This gives us an idealized view of the capability of Rydberg sensors in this frequency regime, yielding a sensitivity $∼10−11$ V/(m $Hz)$ shown as a purple dotted line in Fig. 3. Numerous effects, including resonant dipole–dipole interactions, plasma formation,²⁵ or other collective effects²⁶ may further limit the ideal sensor. Much like the line for an ideal matched dipole, the RSQL sensor is likely to be physically unrealizable but shows a bound of what is possible. Atomic entanglement may lead to additional improvements.²⁰ 

Finally, we briefly consider REIT sensors and state-of-the-art experimental demonstrations based on warm vapors. Previous work has calculated the optimal NEF that may be achieved with such sensors,²⁷ and the results are consistent with experimental results,³ resulting in a bounding sensitivity for warm-atom EIT schemes of approximately $10−6 V/(mHz)$⁠. Recent experiments have further shown that the optimum resonant sensitivity, achieved at microwave frequencies, may be scaled to lower frequencies using high angular momentum states.^28–31 

The estimate of optimal NEF for REIT-based sensors is shown as a pink line in Fig. 3. To calculate this line, we linearly scale the optimum REIT sensitivity of approximately $10−6 V/(mHz)$^3,27 by the dipole transition strengths of available resonant transitions in rubidium 85. The maximum principal quantum number n is capped at 100, and the angular momentum is capped at quantum number $l=15$⁠. Optimizing within this range results in the jagged nature of the line. Nonetheless, the prediction remains of order $10−6$ V/(m $Hz$⁠) across the plot range. We consider this prediction to be optimistic and bounding for REIT-based readout, and practical limitations such as cell shielding³² may further limit sensitivity. We also include two data points corresponding to published experimental results within the plotted range.^22,23

It is valuable to view Fig. 3 as being made up of two regions. The lower two dotted lines in Fig. 3, corresponding to the ideal matched dipole and the RSQL sensor, are idealizations, and both fall beneath typical atmospheric noise levels. Although these lines will not likely be realized by a physical platform, they, nonetheless, set an important fundamental bound for the technology. In contrast, the lines for the REIT sensor and amplified dipole antenna can likely be achieved in practice and meet the 10% exceedance level for atmospheric noise (blue band) that nominally sets a limit for outdoor communications. Our analysis makes a strong case that unmatched dipole antennas with amplifier readout yield superior performance to what can currently be achieved with REIT sensors. Nonetheless, the large amount of available sensitivity between the two regions of the plot is promising.

While sensors can approach atmospheric noise levels, sufficient for many applications, Rydberg RF sensors may become useful for reasons apart from their sensitivity. Recent demonstrations including simultaneous phase and amplitude detection,³³ multi-band reception,³⁴ large dynamic range,³⁵ accurate calibration,³⁶ Tera-Hertz imaging,³⁷ and sub-thermal detection at microwave frequencies³⁸ serve as justification and motivation for further research. New sensing schemes that use a larger effective atom number and reach the standard quantum limit (or beyond, by using entanglement) are an important area for future study. The fact that current Rydberg atom systems are far from the idealized interaction-limited sensitivity is a significant motivation for future research.

See the supplementary material for a zipped folder containing a Jupyter notebook and Python source code to re-create Fig. 3 and modify the parameters to create custom comparisons.

The authors acknowledge useful discussions with Fredrik Fatemi.

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The views, opinions, and/or findings expressed are those of the authors and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government.

The authors from The MITRE Corporation also acknowledge support from the MITRE Independent Research and Development Program.

The authors have no conflicts to disclose.

K. M. Backes: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Writing – original draft (lead); Writing – review & editing (lead). P. K. Elgee: Conceptualization (supporting); Formal analysis (equal); Writing – original draft (supporting); Writing – review & editing (supporting). K.-J. LeBlanc: Conceptualization (supporting); Formal analysis (supporting). C. T. Fancher: Conceptualization (supporting); Supervision (supporting); Writing – review & editing (supporting). D. H. Meyer: Conceptualization (equal); Supervision (supporting); Writing – review & editing (supporting). P. D. Kunz: Conceptualization (supporting); Project administration (supporting); Supervision (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). N. Malvania: Conceptualization (supporting); Investigation (supporting). K. L. Nicolich: Conceptualization (supporting); Writing – review & editing (supporting). J. C. Hill: Conceptualization (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). B. L. Schmittberger Marlow: Conceptualization (supporting); Supervision (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). K. C. Cox: Conceptualization (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Project administration (lead); Supervision (lead); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are available within the supplementary material.