Weak electric-field detection with sub-1 Hz resolution at radio frequencies using a Rydberg atom-based mixer

Rydberg atoms¹ have been demonstrated as quantum sensors for electric(E)-field metrology over the radio frequency (RF) range of approximately 500 MHz-1 THz, and have properties not found in classical E-field sensors, such as sub RF-wavelength size,^2–5 self calibration,^6,7 and system international (SI) traceability to Plank’s constant.⁸ Electromagneticlly induced transparency^9–11(EIT), and Autler-Townes (AT) splitting¹² used to realize the Rydberg atom E-field sensor, reduce an RF E-field measurement to an optical frequency measurement. Progress has been made using Rydberg atoms to characterize classical properties of RF E-fields including magnitude,^6,7,13,14 polarization,¹⁵ phase¹⁶ and, power.¹⁷ More recently the concept of the Rydberg E-field sensor has been expanded in the form of the “Rydberg Atom Receiver” and “Rydberg Atom Radio”^5,18–22 which have been used to detect time varying fields of common modulation schemes such as QPSK, AM, and FM.

The detection of weak RF fields (i.e. below 1 mV/m) is important for practical applications if the Rydberg atom RF field sensor is to compete with traditional circuit based sensors. Several techniques have been proposed to improve signal to noise levels for weak RF E-field measurements such as, using optical cavities²³ to narrow the EIT line width and improve AT splitting resolution, homodyne detection with a Mach Zehnder interferometer²⁴ and frequency modulated spectroscopy.²⁵ Previously, we reported on the Rydberg atom mixer¹⁶ for determining the phase of an RF field. Here, in a continuation of exploring the Rydberg atom mixer, we show how this mixer effect can also be applied for the detection of weak RF fields that are well below AT splitting with the added benefit of isolation of signals at adjacent frequencies, and frequency selectivity of ∼ 10⁸ better than that provided by the Rydberg transition alone. Using the Rydberg atom mixer we demonstrate a weakest detectable field of ≈ 46 μV/m ± 2 μV/m with a sensitivity of 79 μVm⁻¹Hz^−1/2 without the need for cavities or inteferometers with better than ∼ 1 Hz resolution.

The setup for this work is shown in Fig. 1. Rydberg atoms are produced using a 75 mm×25 mm (Length×Diameter) cylindrical glass atomic vapor cell filled with cesium (¹³³Cs) atoms. A probe laser tuned to the the D2 transition wavelength of λ_p=852 nm excites the ¹³³Cs from the ground state to the first excited state (6S_1/2 → 6P_3/2). A counter propagating coupling laser is tuned to λ_c=511.148 nm, and further excites the ¹³³Cs atoms to the Rydberg state 34D_5/2 thus producing a transparency region in the probe laser spectrum. The probe laser beam has a full-width half-maximum (FWHM) of 425 μm and a power of 49 μW, the coupling laser has a FWHM of 620 μm and a power of 60.6 mW. Under these conditions an incident RF field operating near the frequency of 19.626 GHz drives the 34D_5/2 → 35P_3/2 transition. With the probe laser frequency fixed on resonance with the D2 transition, the transmission through the vapor cell is in general reduced when in the presence of the applied RF field. For appreciable field strengths the atoms are driven to the Autler-Towns regime¹² which splits the observed EIT peak in the probe laser transmission spectrum. The frequency separation Δf_AT of the two AT peaks is given^6,14 by,

Where ℘_RF is the dipole matrix element of the RF Rydberg transition and ℏ is Plank’s constant. The dipole moment for the resonant RF transition is ℘ = 723.3739ea₀ (which includes a radial part of 1476.6048ea₀ and an angular part of 0.48989, which correspond to co-linear polarized optical and RF fields, where e is the elementary charge; a₀ = 0.529177 × 10⁻¹⁰ m and is the Bohr radius). AT splitting as a method for E-field sensing becomes less effective for E-fields too weak to cause resolvable AT peak separation. The work described below overcomes this weak E-field limitation through the Rydberg atom mixer effect with the added benefit of narrow band frequency selection and tuning. Here, we define the minimum detectable RF field capable of being detected with AT splitting as that which causes an AT peak separation equivalent to the EIT line width Γ_EIT. From (1) this is,

As determined from the EIT spectrum shown in Fig. 2, Γ_EIT ≈ 4 MHz and E_AT=0.72 V/m for the above mentioned Rydberg states.

A schematic of the Rydberg atom mixer¹⁶ is shown in Fig. 3. A heterodyne detection scenario is used where two different RF fields are incident on the vapor cell, $E1=ELO⁡cosωLOt+ϕLO$⁠, and $E2=ESig⁡cosωSigt+ϕSig$⁠. One is tuned to f_LO=ω_LO/2π=19.626000 GHz such that it is on resonance with 34D_5/2 → 35P_3/2 Rydberg transition. This field acts as a local oscillator (LO). The second field E₂ is the signal field (Sig) that is to be sensed and is tuned to f_Sig=ω_Sig/2π=19.626090 GHz such that it is detuned by +90 kHz from the LO field. Here, we explore the case when both E₁ and E₂ are co-polarized and considered weak where E₁ ≈ E_AT and E₂ ≤ E_AT.

The interference occurring from the superposition of these fields results in a high frequency component E_res and low frequency component E_mod. With $ω¯=(ωLO+ωSig)/2$⁠, Δω = ω_LO − ω_Sig, and Δϕ = ϕ_LO − ϕ_Sig, for small relative detuning where $Δω/ω¯≪1$ the total field at the atoms E_atoms can be shown to be,

Where E_res oscillates at ω_LO and E_mod oscillates at Δω. The magnitude of the total field is given by,

For weak fields where E_Sig ≪ E_LO, (6) becomes,

The Rydberg atoms have a naturally different response to E_res and E_mod. Since E_res oscillates at ω_LO it is resonant with the Rydberg transition, where as E_mod oscillates at a frequency that is well below the Rydberg transition frequency and results in a modulation of the EIT spectrum and thus the probe laser intensity on the photodiode (see Fig 1). The effect being the down conversion of the incident field E₂ from the base band RF frequency of ω_Sig to an intermediate frequency (IF) of f_IF=Δω/(2π) (see Fig. 3),

In this case the probe laser intensity on the photodiode varies at f_IF=90 kHz. A detectable IF signal is produced even for E_sig well below E_AT. Fig. 4 shows time domain plots of the IF signal out of the photodiode for various E_sig levels. The 90 kHz modulation is easily seen as is the changing modulation amplitudes following the behavior of (8). For the final stage of detection the output of the photodiode is passed to a lock-in amplifier with a reference set equal to the IF frequency, f_REF=f_IF. The lock-in output voltage (V_LI) is thus proportional to weak field, V_LI ∝ E_Sig.

Two identical source antennas (Narda 638 horns were used, however mentioning this product does not imply an endorsement by NIST, but only serves to clarify the equipment used) were used to produce E_LO and E_Sig fields. The antennas were placed 385 mm from the ¹³³Cs vapor cell such that they were beyond the 2a²/λ_RF = 305 mm far field distance.²⁶ Where a = 48.28 mm is taken as the diagonal length of the antenna aperture and λ_RF = 15.286 mm. Two separate RF signal generators synced via a 10 MHz reference were used to feed the two antennas at frequencies of f_LO=19.62600 GHz, and f_Sig=19.626090 GHz. A calibrated power meter and vector network analyzer were used to account for cable loss from the RF signal generator and horn reflection coefficient and to determine the RF power at the horn antennas P_RF. For powers down to -70 dBm the power meter was used. For weak field generation P_RF was <-70 dBm and thus well below the dynamic range of an RF power meter. To overcome this, the signal generator was operated within the range of the power meter from +10 to -60 dBm and additional calibrated attenators were added providing up to − 111 dB of additional loss. With this configuration accurate control of power levels could be achieved down to ≈ -180 dBm.

To accurately determine the E-field within the vapor cell for low RF powers into the horns, AT splitting was used to calibrate and correct errors imparted on the E-field due to the presence of the vapor cell. As has been shown in Refs. 2, 27, and 28 for an RF field incident on a vapor cell, scattering off of the glass walls can cause internal resonances and alter the E-field amplitude inside the vapor cell from that which would exist given the vapor cell were not there. The E-field at the horn-to-laser beam distance R =385 mm was calculated using^26,29 the far-field formula $EFF=59.9585PRFG/R$ where the antenna gain is G =15.55 dB ± 0.4 dB. For a given distance R and RF frequency there is a fixed ratio of the E-field inside the vapor cell E_cell to the E-field in the absence of vapor cell E_FF. This is given by the cell factor C_f=E_cell/E_FF. Calibration data for E_cell was determined from the conventional AT splitting technique (1) for a range of P_RF strong enough to cause AT splitting. Cell factor calibration data comparing E_cell and E_FF is shown in Fig. 5. Given the uncertainty in G, power meter, and operating within the linear response³⁰ of the AT regime (1), weak E-fields detected by the Rydberg mixer could be known for a given P_RF to within an estimated uncertainty of ± % 5. For the configuration used here C_f=0.90 and thus for a given P_RF,

Weak E-field data (blue squares) are plotted in Fig. 6 for lock-in amplifier output voltage-vs-$PRF$ along with the corresponding E-field strength. For these data a 3 s time constant (bandwidth of f_c=0.33 Hz) and 24 dB/octave low pass filter slope was used. As P_RF approaches powers < -100 dBm the lock-in signal approaches the noise floor which shows up by the flattening out of the data curve. To estimate the repeatably of the measurement, 3 sets of data were acquired for each E-field level. This showed a 5 % variation. Also shown in Fig. 6 are the higher E-fields that were used for cell factor calibration and acquired from AT splitting. These data (red circles) follow the linear behavior predicted by equations (1) and (9). The weak E-field data remains linear up until E_AT is reached. The cross over between the weak field regime and AT regime shows up as a roll off of the weak field data near E_AT. This roll off is due to the EIT peak center frequency shifting away from the probe laser frequency as AT splitting begins to take place. The weakest detectable E-field is taken as the value at where the lock-in voltage curves to the noise floor. This corresponds to ≈ 46 μV/m and a sensitivity of ≈ 79 μVm⁻¹Hz^−1/2.

Another aspect of the Rydberg mixer is its ability to isolate and discriminate between signals of differing RF frequencies with a frequency resolution orders of magnitude finer than the response bandwidth of the Rydberg transition. As was shown in Ref. 31, through the generalized Rabi frequency, RF E-fields that are off-resonance with the Rydberg transition will still affect the EIT spectrum over a large continuum of frequencies of hundreds of MHz. For an RF frequency detuning of δ_RF, and on-resonance Rabi frequency of Ω_o, the generalized Rabi frequency becomes, $Ω′=Ωo2+δRF2$⁠. For example in the AT regime, splitting will still occur for off-resonance E-fields for a large range of δ_RF, where now the splitting Δf_AT → Ω′/(2π). As such, discriminating between E-fields of different RF frequencies through purely observing the EIT spectrum becomes difficult and ambiguous. The Rydberg atom mixer provides a means to overcome this so that E-fields differing in frequency by as little as 1 Hz can be discriminated. For this, the lock-in amplifier is tuned to the desired IF frequency corresponding to the desired down converted RF frequency. Simply tuning f_REF allows for signals at different RF frequencies to be discriminated and isolated.

We demonstrate this and examine the leakage in the lock-in signal for E-fields at neighboring frequencies and various strengths relative to the “in-tune” E-field. First, an in-tune IF signal was produced where the RF signal generator power was set to roughly middle of range at P_RF=-40 dBm and f_IF=90 kHz. This signal we denote as E_o=181 μV/m. The lock-in reference was also tuned to f_REF=90 kHz, and a time constant of 3 s, giving a cut off frequency of f_c=0.33 Hz. Three other signals denoted as E_Δf that were out of tune by Δf=0.1 Hz, Δf=1 Hz, Δf=10 Hz were also produced. For these three signals P_RF was then varied such that E_Δf/E_o ranged from 0 dB to greater than 60 dB. Fig. 7 shows a plot of the lock-in output for the three detuned signals normalized to the level produced by E_o. The lock-in noise floor is depicted as well. As can be seen there is a range of relative strengths for each detuned signal where the lock-in signal is at the noise floor and then rises up to equal the level of E_o. All three detunings show maximum isolation when equal to E_Δf/E_o = 0 dB. Where even for sub-Hz detuning of Δf=0.1 Hz, E_Δf does not rise above the noise floor. The isolation threshold in dB for each detuning is taken for the value of E_Δf/E_o that crosses -3 dB level of the lock-in signal. Isolation degrades more quickly for smaller detunings for E_Δf/E_o > 1. For a detuning of Δf=1 Hz the -3 dB crossing happens for E_Δf/E_o ≈ 60 dB.

This work shows E-field strengths -84 dB below the AT limit E_AT can be detected using the Rydberg atom mixer.¹⁶ Furthermore, the Rydberg atom mixer allows specific RF frequencies to be selected, isolated and rejected with resolution better than 1 Hz. This is a ∼ 10⁸ improvement in RF frequency resolution over that provided by the frequency bandwidth³¹ of the Rydberg transition alone. These attributes along with the ability to measure phase,¹⁶ and polarization¹⁵ allow for the development of a quantum-based sensor to fully characterize the RF E-field in one compact vapor cell.