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Abstract
Electromagnetically induced transparency in atomic systems involving Rydberg states is known to be a sensitive probe of incident microwave (MW) fields, in particular those resonant with Rydberg-to-Rydberg transitions. Here we propose an intelligible analytical model of a Rydberg atomic receiver’s response to amplitude- (AM) and frequency-modulated (FM) signals and compare it with experimental results, presenting a setup that allows sending signals with either AM or FM and evaluating their efficiency with demodulation. Additionally, the setup reveals a detection configuration using all circular polarizations for optical fields and allowing detection of a circularly polarized MW field, propagating colinearly with optical beams. In our measurements, we systematically show that several parameters exhibit local optimum characteristics and then estimate these optimal parameters and working ranges, addressing the need to devise a robust Rydberg MW sensor and its operational protocol.
Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
1. INTRODUCTION
In recent years Rydberg-atom based microwave (MW) sensors have attracted the attention of many researchers, owing to substantial sensitivity of Rydberg-to-Rydberg MW transitions and a straightforward measurement scheme. Atomic transitions have been proposed as easily reproducible electric field amplitude standards in the MW regime [1], and various realizations of MW modulation have led to enhanced detection sensitivity [2,3] (likewise for modulation of optical fields [4]), as well as to transmitting both analog [5–9] and digital [10–15] information via Rydberg atomic receivers. There were also successful attempts at further characterizing measured MW field properties, such as frequency [16], phase [16–19], polarization [20,21], and angle of arrival [22]. Additionally, it was shown that a wide, off-resonant spectrum can be covered with Rydberg-based detection [23,24]. Advances in the fabrication of vapor cells, crucial for operation of sensors, have led to miniaturization [25–27], and the potential for commercialization is considerable. This prospect is very promising, as Rydberg atoms have proved to be a medium suitable for various other applications, such as detection of an electric field (of lower frequency than MW) [28,29], where the quantum limit has been achieved [30], and even MW-to-optical conversion of electromagnetic fields [31–33].
The detection principle has been analyzed for the optimal choice of states employed [34,35]. However, as far as MW modulation is concerned, to our knowledge, little has been explored in terms of amplitude (AM) and frequency modulation (FM) transfer signal dependence on various parameters (barring the measurements of AM transfer bandwidth [6] and of optimal coupling field detuning [13] in specific scenarios), although the principle behind detecting both types of modulation has been discussed [9]. Here we concentrate on this topic, presenting a simple model of modulation transfer and then experimentally verifying parametric dependence of AM and FM transfer, which leads to a comparison between both types of modulation, as well as several estimates of optimal working ranges for MW modulation and probe field detuning. The results provided here may prove useful in designing a proper modulation-based Rydberg atomic receiver and its tuning protocol.
Furthermore, almost all of the previous experimental works employ $\pi$ transitions caused by a linearly polarized MW field propagating perpendicularly to laser beams as the work horse of a detection setup. This solution requires the MW field to be uniform to take full advantage of the detection sensitivity and is not suitable, e.g., for sub-wavelength imaging of material MW responses. As a proof-of-concept demonstration, we present an alternative design utilizing $\sigma$ transitions from a circularly polarized MW field propagating with laser beams along a common axis. This alternative configuration takes advantage of a larger dipole moment of MW transition ($2500{a_0}e$ compared to $1900{a_0}e$ of corresponding transitions in all-linear $\pi$ configuration). Consequently, we demonstrate detection of a MW field propagating through an extremely small surface (${\sim}4000\,\,{{\unicode{x00B5}{\rm m}}^2}$), confirming the prospect of non-disruptive MW field imaging.
Fig. 1. (a) Energy level scheme of $^{85}{\rm Rb}$, (b) transmission spectrum at ${{\rm D}_2}$ transition with EIT from ${5^2}{{\rm P}_{3/2}}({\rm F} = 4) \to {55^2}{{\rm D}_{5/2}}$ transition visible in the middle, (c) zoom on the EIT feature, (d) A-T splitting of EIT caused by MW field tuned to ${55^2}{{\rm D}_{5/2}} \to {54^2}{{\rm F}_{7/2}}$ transition, and (e) visible wideband FM transferred to probe transmission spectrum.
2. PRINCIPLE AND METHODS
A. Working Principle
Let us consider a four-level cascade energy level scheme of $^{85}{\rm Rb}$, pictured in Fig. 1(a). In the following setup, a field coupled to the ${{\rm D}_2}$ transition between ground state ${5^2}{{\rm S}_{1/2}}({\rm F} = 3)$ and excited state ${5^2}{{\rm P}_{3/2}}$ is employed as a probe $p$. The incident field is absorbed by warm atomic vapors, and the Doppler-broadened spectrum of the transition can be seen in a detuning-transmission plot in Fig. 1(b).
The field coupled to the second ${5^2}{{\rm P}_{3/2}}({\rm F} = 4) \to {55^2}{{\rm D}_{5/2}}$ transition (coupling field $c$) leads to the emergence of the widely studied electromagnetically induced transparency (EIT) effect [36–38], which can be seen in Fig. 1(b), as well as zoomed in Fig. 1(c). Strong coupling to the transitions results in a rise of dark states, which manifest as transparent resonances in the absorption spectrum. The use of counterpropagating probe and coupling fields has the advantage of partially canceling the Doppler effect in two-photon resonance—the example in Figs. 1(b) and 1(c) shows the resonance narrowed to ${\sim}5 \cdot 2\pi \,{\rm MHz}$ (compared to the full ${\sim}600 \cdot 2\pi\,{\rm MHz}$ absorption profile of the ${{\rm D}_2}$ line).
The third ${55^2}{{\rm D}_{5/2}} \to {54^2}{{\rm F}_{7/2}}$ transition is in the interesting MW regime (${f_M} = 13.9\;{\rm GHz} $). As a transition between two Rydberg states, it exhibits a very large dipole moment $|{d_M}| = 2500{a_0}e$ (for comparison, the dipole moment for probe field $|{d_p}| = 3{a_0}e$ and coupling field $|{d_c}| = 0.014{a_0}e$), resulting in great sensitivity. Namely, the MW field amplitude causes Autler–Townes (A-T) splitting of energy levels, which can be read directly as the splitting of the EIT feature, as seen in Fig. 1(d). Having taken the Doppler effect into consideration, the splitting in the scale of probe detuning ${\Delta _p}$ can be expressed as $\frac{{{\lambda _c}}}{{{\lambda _p}}}{\Omega _M}$, with the ${\Omega _M}$ Rabi frequency being linearly dependent on the MW electric field amplitude ${A_M}$:
When the MW field is amplitude (AM) or frequency modulated (FM), it naturally transfers its modulation to the transmission spectrum of the probe field. An exemplary wideband FM spectrum is presented in Fig. 1(e)—the modulation fringes can be seen due to aliasing effects. These modulations, in terms of their intensity dependence on various parameters, such as probe detuning, are our main focus in this research.
B. Susceptibility Model
Let us consider the semi-classical atom–light interaction described with the four-level cascade transition scheme in Fig. 1. Solving the GKSL equation (Lindbladian) in rotating wave approximation, in a steady state case with a weak probe field assumption, yields the following formula for the probe field’s electric susceptibility, in the form of a regular nested fraction [38]:
Fig. 2. Measured responses to AM (blue, upper row) and wideband (${h_{{\rm FM}}} = 10$) FM (red, lower row) as a function of probe detuning for various carrier intensities defined with Rabi frequency ${\Omega _M}$, and corresponding responses simulated in theoretical model (black).
Translation to a realistic model, with warm atoms affected by the Doppler effect and parameterized by the detuning of lasers, in a realization with a probe beam counterpropagating to other fields, requires the following substitutions:
Acquiring a full Doppler-broadened spectrum over all velocity classes is done by integration:
MW modulation transfer, denoted as $\eta$, for reasonably small modulation frequencies (in particular, smaller than ${\Omega _M}$) can be understood as a derivative of susceptibility over Rabi frequency ${\Omega _M}$ for AM, and derivative over detuning ${\Delta _M}$ for FM:
The imaginary parts of these modulation transfers (proportional to absorption spectra) as a function of probe detuning are compared to the experimental results in Fig. 2. These simulated modulation transfer spectra (black) show good agreement with the experimentally obtained (blue, red), in particular, concerning the optimal working points. A full model with arbitrarily large or fast modulation calls for a much more complex treatment, for example, using the Floquet expansion in the MW frequency, or simulating the full time evolution of the atomic density matrix. This highlights the need for the simpler approach of still considerable predictive power we employ here.
As the parameters introduced are general Rabi frequencies and detunings, the model is applicable not only to the described $^{85}{\rm Rb}$ transitions, but also more generally to other configurations involving modulated field detection in a three-step ladder system, as long as the weak probe and counterpropagation conditions are met.
3. EXPERIMENTAL DETAILS
The scheme of the experimental setup used in this research is pictured in Fig. 3. The probe (coupling) laser beam of power ${P_p} = 210\,\,{\rm nW}$ (${P_c} = 37\;{\rm mW} $) is focused inside the rubidium vapor cell with waist ${w_p} = 35\,\,{\unicode{x00B5}{\rm m}}$ (${w_c} = 40\,\,{\unicode{x00B5}{\rm m}}$), resulting in Rabi frequency ${\Omega _p} = 1.7 \cdot 2\pi\,{\rm MHz}$ (${\Omega _c} = 2.9 \cdot 2\pi\,{\rm MHz}$). Both lasers exhibit short-time spectral stability ${\lt}100 \cdot 2\pi \,{\rm kHz}$. The counterpropagation of the laser beams results in partial cancellation of the Doppler effect and makes it easier to separate the optical fields with dichroic mirrors. The length of the cell is 50 mm; however, the atom–light interaction length can be considered shorter, as Rayleigh lengths for the laser beams are ${z_{R,p}} = 4.9\;{\rm mm} $ and ${z_{R,c}} = 10.5\;{\rm mm} $.
The MW helical antenna generates a right-hand circularly polarized (RHCP) MW field at the $13.9\;{\rm GHz} $ ($2.15\;{\rm cm} $) transition. The antenna is fabricated with a hole in its backplate, so that the beams propagate through it, and the whole setup includes field propagation along only one axis. For polarization consideration, the coupling beam is made RHCP with a quarter-wave plate and the probe beam (counterpropagating to the other fields) is made left-hand circularly polarized (LHCP). This setup, concerning the energy level structure [Fig. 1(a)], assures sign-matched $\sigma$ transitions, which were observed to result in the most prominent EIT and A-T splitting effects in this configuration due to the largest transition dipole moments.
Fig. 3. Experimental setup for detecting modulated MW field. QWP, quarter-wave plate; DM, dichroic mirror; APD, avalanche photodiode; VCO, voltage controlled oscillator; LO, local oscillator.
To isolate the setup from spurious MW fields, the vapor cell and the antenna are placed inside a shield made from MW-absorbing foam. The apertures where optical beams pass through are considered sub-wavelength to the MW field. To avoid interference, all elements inside the shield, including the cell and its holder, are non-metallic. As the cell is not thermally controlled, the shield also provides thermal insulation assuring relatively constant ${22.5^\circ}{\rm C}$ temperature, which results in constant atomic number density.
The MW electrical signal is generated via the LMX2820 phase-locked loop (PLL) frequency synthesizer and can be regulated in terms of amplitude and frequency. The modulation of the MW field is introduced on the electrical level, FM is realized with $\Delta f = 1\;{\rm MHz} $ injected into PLL feedback of the synthesizer, and AM is realized with an external mixer (Fig. 3), having the same $1\;{\rm MHz} $ signal in one of the inputs. The strength of modulation (modulation index) is regulated with the amplitude of modulation signals, produced via the Red Pitaya STEMlab 125-14 multifunction measurement tool.
Control over probe detuning is performed with laser driver scanning of laser current and piezo. After passing through the vapor cell, the probe beam is focused on an avalanche photo-diode (APD, Thorlabs APD120A) aperture where transmission is measured. The APD signal is then transferred to the STEMlab 125-14, which acts both as analog-to-digital converter and demodulator. The signal is high-pass filtered and quadrature-demodulated, with both operations implemented in the FPGA programmable logic. The data containing raw APD signal, as well as both quadratures of the demodulated signal, are then transferred to a PC for further processing.
Data after demodulation are averaged ($n = 10$) for clarity. Demodulation phase is chosen with respect to minimizing one of the quadratures—the other one is then presented as the proper result of modulation transfer parameterized with probe detuning. Exemplary measurements with both quadratures are shown in Fig. 4.
Fig. 4. Exemplary measured scan trace with superimposed (a) full amplitude (${h_{{\rm AM}}} = 1$, ${\Omega _M} = 7.3 \cdot 2\pi\,{\rm MHz}$), (b) comparable frequency (${h_{{\rm FM}}} = 1$, ${\Omega _M} = 8.8 \cdot 2\pi \,{\rm MHz}$), and (c) wideband frequency modulation (${h_{{\rm FM}}} = 10$, ${\Omega _M} = 8.8 \cdot 2\pi {\rm MHz}$). Black: probe transmission spectrum; red: demodulated signal’s proper quadrature; blue: minimized residual quadrature.
4. RESULTS AND DISCUSSION
To estimate the MW Rabi frequency, we performed standard measurement of A-T splitting using a non-modulated MW field. Then we translated electrical signal amplitude generated by the LMX (checked independently on an electrical spectrum analyzer) to Rabi frequencies. Transit-time broadening caused by atoms flying through the interaction volume was measured by fitting the model to a three-level EIT shape and yielded ${\gamma _{{\rm t} - {\rm t}}} = 3.9 \cdot 2\pi \,{\rm MHz}$. This seems to be in good agreement with a crude yet simple estimation, where $2\pi \frac{{{v_{{\rm avg}}}}}{{2{w_p}}} \approx 3.9 \cdot 2\pi\, {\rm MHz}$ for Maxwell’s distribution average velocity ${v_{{\rm avg}}}$ in temperature ${22.5^\circ}{\rm C}$. As the Rydberg state natural linewidths are in kHz range, we assumed ${\gamma _2} = {\gamma _3} = {\gamma _{{\rm t} - {\rm t}}}$ in Eq. (2) governing our model. As for the decay rate of the first excited level, we assumed ${\gamma _1} = \frac{\Gamma}{2} + {\gamma _{{\rm t} - {\rm t}}}$, where $\Gamma = 6.06 \cdot 2\pi\,{\rm MHz}$ is the natural linewidth of the rubidium ${D_2}$ transition.
If not noted otherwise, all the following results were obtained in the case of resonant coupling and MW fields, and the modulation frequency was $\Delta f = 1\;{\rm MHz} $. The modulation frequency $\Delta f$ had to be smaller than MW Rabi frequency ${\Omega _M}$ for the experiment to operate in the regime where our model applies. On the other hand, $\Delta f$ has to be much larger than laser scanning frequency to avoid undesirable aliasing effects. We decided on $1\;{\rm MHz} $, as it satisfied both conditions and was convenient to work with our electronic setup. However, a brief analysis of other frequencies has shown little to no dependence of modulation transfer on modulation frequency in the range of $100\;{\rm kHz}{-}10\;{\rm MHz}$.
The results measured in this setup serve as a proof-of-concept demonstration of a new $\sigma$ transition-based setup, suitable for detection of circularly polarized MW fields. The weakest measured AM field (Fig. 2, ${\Omega _M} = 1.6 \cdot 2\pi\,{\rm MHz}$) has amplitude ${A_M} = 490\frac{{{\unicode{x00B5} \rm V}}}{{{\rm cm}}}$, which does not come close to ${\lt}10\frac{{{\unicode{x00B5} \rm V}}}{{{\rm cm}}}$ achieved by other groups [1,14,16,29,39]; however, as the area of detection is extremely small, ${\sim}4000\,\,{{\unicode{x00B5}{\rm m}}^2}$ (effective area of probe Gaussian beam cross section $\pi w_p^2$), in terms of MW photons interacting with atoms, we achieve ${\sim}{10^9}$ MW photons per second, meaning that very few MW photons need to interact with the receiver to result in a measurable signal.
A. Comparison between AM and FM
A quick comparison between AM and FM measurements is shown in Fig. 4. The AM signal is more prominent than FM [Figs. 4(a) and 4(b)] for a similar modulation index $h$, which can be understood as a measure of how much energy is transferred to sideband frequencies in the spectral image of a modulated signal. In the case of AM, modulation index ${h_{{\rm AM}}} = 1$ can be considered full modulation, as the electric field decreases to zero. However, for the FM index, ${h_{{\rm FM}}} \gt 1$ can be achieved, and then the demodulated signal becomes the most prominent for modulation index ${h_{{\rm FM}}} = 10$ [Fig. 4(c)]. Additionally, the shape of the probe transmission spectrum (Fig. 4, black) is dependent on modulation type and modulation index.
The interesting aspect of modulated signal detection is optimal probe detuning. It can be seen in Fig. 4 that for AM, the optimal detuning is ${\Delta _p} = 0$, but for wideband FM, we measured ${\Delta _p} = \pm 2.4 \cdot 2\pi\,{\rm MHz}$. However, this value is dependent on ${\Omega _M}$ and becomes larger particularly for stronger MW fields, as can be seen in Fig. 2 (lower row, red). The same figure shows that for stronger (${\Omega _M} \gt 8 \cdot 2\pi\,{\rm MHz}$) MW fields, the AM response changes as well (upper row, blue), with detuning ${\Delta _p} = 0$ no longer being optimal and the overall response being worse than for FM—contrary to what was found for smaller MW field intensity. The FM response does not change its shape except for broadening, which may be more useful in potential demodulation tuning protocols. The strongest AM response was estimated for ${\Omega _M} = 7 \cdot 2\pi\,{\rm MHz}$ and the strongest FM response for ${\Omega _M} = 14 \cdot 2\pi\,{\rm MHz}$.
In Fig. 2, we also compare results obtained experimentally (blue, red) to a simulated theoretical model (black). The model exhibits good prediction of optimal probe detunings and even optimal Rabi frequencies, but slight discrepancies arise when comparing signal transfer amplitudes for different Rabi frequencies. These may be attributed to the modulation frequency no longer being small enough (compared to ${\Omega _M}$) for smaller Rabi frequencies, and as for larger Rabi frequencies, additional decoherence from strong A-T splitting may be unaccounted for.
B. FM Bandwidth Consideration
Having analyzed modulation responses in terms of probe detuning and carrier MW field intensity, let us focus on the bandwidth of FM that Rydberg atoms acting as a receiver can efficiently respond to. The bandwidth of modulation can be understood as twice modulation deviation $2{f_\Delta}$, where ${f_\Delta} = {h_{{\rm FM}}}\Delta f$. We measured FM demodulated signal amplitude for a wide range of bandwidths, and the results are shown in Fig. 5. We estimated the optimal bandwidth to be $2{f_\Delta} = 20\;{\rm MHz} $, but it can be seen that the modulation signal is received well in a range of bandwidths, having ${\gt}0.5$ relative efficiency in range $2{f_\Delta} = 5{-}100\;{\rm MHz}$. This determines what bandwidth/modulation index should be chosen to maximize the signal transmission when designing a Rydberg receiver and transmission protocol.
Fig. 5. Measured response to FM as a function of modulation bandwidth. The amplitude is normalized to the highest value. Carrier Rabi frequency ${\Omega _M} = 8.8 \cdot 2\pi\,{\rm MHz}$.
C. Wideband FM Detuning
Now let us consider FM with a set bandwidth but carrier frequency detuned from the exact atomic resonance, which addresses the problem of employing various detuned communication channels near the resonant frequency. The demodulated signal amplitudes in the function of both probe detuning and MW field detuning are presented as a colormap in Fig. 6 (left), compared with a colormap obtained from computer simulation (right). The colormaps show two resonances: two-photon resonance and three-photon resonance, consolidating in the MW zero-detuning case. It is shown that demodulated signal amplitude weakens for large MW detunings. However, this does not strictly correspond to FM bandwidth, as in this case, ${\gt}0.5$ efficiency is estimated in ${\Delta _M} = \pm 17\;{\rm MHz} $ range, and our other measurements did not show straightforward dependence on modulation bandwidth.
Fig. 6. Measured (left) and simulated (right) responses to FM as a function of probe detuning and MW field detuning. The amplitude is normalized to the highest value, with respect to zero to show phase transitions. Modulation bandwidth $2{f_\Delta} = 20\;{\rm MHz} $ (${h_{{\rm FM}}} = 10$) and carrier Rabi frequency ${\Omega _M} = 8.8 \cdot 2\pi\,{\rm MHz}$.
5. SUMMARY AND PERSPECTIVES
The simple model presented in Eq. (5) has proved to be reliable in terms of predictions of optimal working point dependence on MW field amplitude and detuning, which is particularly important for designing a receiver working in both weak and standard fields.
The estimated optimal working parameters and ranges apply directly only to the employed detection setup. However, as most of them rely directly on the theoretical model, an adequate change of parameter values in Eq. (2) can straightforwardly lead to arriving at optimal working points in other configurations, such as different Rydberg transitions, utilizing different alkali elements, and using the well-developed linear polarization of the MW field as a modulation medium.
We have demonstrated a setup sensitive to MW fields driving $\sigma$ transitions and shown detection of MW propagating through a micrometric surface. Nevertheless, our realization remains relatively simple and can be further improved in demanding applications. For example, the highest sensitivity to AM or FM is achieved with an additional probe-reference field [12,15,29], or if atoms are used as a mixer, with an external local oscillator (LO) [14,39,40], which increases the complexity of the entire setup. Additionally, it was proposed to combine a Rydberg sensor with a conventional antenna [41]. Alternative approaches to measurement of MW fields with Rydberg atoms are also possible, e.g., utilizing collective Rabi splitting in a setup with a cavity [42]. The measurement scheme can be repeated in cold atoms, trapped in MOT, as MW field amplitude measurements have already been performed [43,44].
Funding
Narodowe Centrum Nauki (2021/43/D/ST2/03114); Fundacja na rzecz Nauki Polskiej (MAB/2018/4 “Quantum Optical Technologies”); European Regional Development Fund; Office of Naval Research Global (N62909-19-1-2127).
Acknowledgment
The “Quantum Optical Technologies” project is carried out within the International Research Agendas programme of the Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund. MM was also supported by the Foundation for Polish Science via the START scholarship. We thank K. Banaszek and W. Wasilewski for support and discussions. This research was funded in whole or in part by National Science Centre, Poland 2021/43/D/ST2/03114.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available in Ref. [45].
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