Interfacing Rydberg atoms with a chip-based superconducting microwave resonator using an ac Stark shifted single-photon transition

TL;DR

This paper demonstrates coupling between helium Rydberg atoms and a superconducting microwave resonator via ac Stark tuning, advancing toward strong single-photon interactions for quantum information applications.

Contribution

It introduces a method to resonantly couple Rydberg atoms to a superconducting resonator using ac Stark shifts, with experimental and theoretical analysis of the coupling mechanism.

Findings

Achieved atom-resonator coupling with a Rabi frequency of ~2π×100 Hz.

Successfully tuned the atomic transition into resonance using ac Stark shifts.

Provided insights into the effects of residual electric fields near the superconducting chip.

Abstract

Helium atoms in the 1s50s $^3$S$_1$ Rydberg level have been resonantly coupled to the $2\pi\times11.721$ GHz second harmonic mode of a chip-based superconducting coplanar waveguide microwave resonator. To achieve this, the single-photon electric-dipole-allowed 1s50s $^3$S$_1\rightarrow$ 1s50p $^3$P$_J$ transition was tuned into resonance with the resonator mode through the ac Stark shift induced by a second strong $2\pi\times3.350$ GHz microwave dressing field. The effects of this dressing field, and residual uncanceled dc electric fields at the location of the atoms close to the superconducting chip surface were interpreted with support from Floquet calculations of the energy level structure of the Rydberg states. To observe appreciable population transfer in the $1~\mu$s atom-resonator interaction time using this transition, which had an electric dipole moment of $1500 e a_0$, pulsed microwave fields were injected into the resonator. From the photon occupation number in the resonator mode under these conditions, the single-photon Rabi frequency associated with the coupling of the atoms to the resonator was estimated to be $\sim2\pi\times100$ Hz. These results represent an important step toward operation of this Rydberg-atom-superconducting-circuit interface in the single-photon strong coupling regime.

Equations

$$H(t)=H_0+e{F}_{\mathrm{dc}}z+e{F}_{\mathrm{dress}}\cos ({\omega }_{\mathrm{dress}}t)z,$$

$$⟨n^{\prime }{\ell }^{\prime }{m}_{\ell }^{\prime }{q}^{\prime }|H_F|n\ell m_{\ell }q⟩$$

$$P_{\mathrm{circ}}=P_{\mathrm{inc}}Q\frac{10^{-\frac{L_{\mathrm{ins}}}{20}}}{m\pi },$$

Full text

††preprint: APS/123-QED

I Introduction

The strong single-photon electric-dipole transitions that occur at microwave frequencies between highly-excited Rydberg states in atoms, are well suited to interfacing with solid-state qubits in superconducting circuits. At angular frequencies close to $2\pi\times 10$ GHz, the electric dipole transition moments between these Rydberg states, and between the quantized energy-levels in superconducting qubits are similar, and lie in the range from $1\,000\,e\,a_{0}$ to $10\,000\,e\,a_{0}$, where $e$ is the elementary charge, and $a_{0}$ is the Bohr radius [1]. Interfaces between Rydberg atoms and superconducting qubits, that exploit these strong electric dipole transitions, for example, by resonant coupling to a common mode of a superconducting coplanar waveguide (CPW) microwave resonator, therefore offer a scalable approach to networking neutral atom [2] and superconducting [3] quantum processors. Since strong optical transitions between low-lying energy levels are employed in the process of laser photoexcitation of the atoms to the Rydberg states in this hybrid system, while superconducting qubits only exhibit transitions at microwave frequencies, Rydberg-atom—superconducting-circuit interfaces also open opportunities to use telecom-band optical photons — which can propagate with low loss over long distances in optical fibers — to prepare or manipulate the quantum states of superconducting qubits in an interaction mediated by the atoms [4, 5, 6]. This type of scheme, that exploits optical excitation to the hybridized quantum states of a Rydberg atom strongly coupled to one or more superconducting qubits through a common mode of a CPW resonator, would, for example, enable the realization of long-range, fiber coupled networks of superconducting processors.

To efficiently interface Rydberg atoms with superconducting circuits, so that the optical photons used for Rydberg state excitation may be exploited to manipulate the photon number state of a superconducting CPW resonator [7], or in the longer term the quantum state of a superconducting qubit, resonant coupling is best achieved using strong single-photon microwave transitions between Rydberg states. In experiments reported up to now with helium (He) Rydberg atoms, coherent coupling to the third harmonic mode of a $\lambda/4$ superconducting CPW resonator at $2\pi\times 19.556$ GHz was achieved using single-color, and two-color two-photon transitions between the 1s55s 3S1 and 1s56s 3S1 Rydberg levels [8, 9, 10]. This work was performed with stray dc electric fields above the superconducting chip surface that approached zero [11]. However, the use of a two-photon transition meant that the effective electric dipole transition moment, available for coupling to a single photon in the resonator, for example, in the case of the two-color two-photon scheme, was only $\sim 50\,e\,a_{0}$.

Experiments have also been reported in which rubidium (Rb) Rydberg atoms were coherently coupled to a superconducting CPW resonator operating in a similar frequency range [12]. However, the propensity for Rb to adsorb on the cryogenically cooled surfaces of the superconducting chip used in this work resulted in stray electric fields which were challenging to compensate [13]. Consequently, it was necessary to apply a comparatively strong dc offset electric field of $\sim 3.6$ V/cm, at the location of the atoms above the CPW resonator, to dominate over these stray fields and allow the isolation of a pair of Rydberg levels with a comparatively small differential static electric dipole polarizability, which were suitable for coupling to the resonator. The trade off in using a strong offset electric field to overcome the effects of stray fields emanating from the superconducting chip surface was that the resulting single-photon electric dipole transition moments were reduced to $\sim 30\,e\,a_{0}$.

With good control over stray electric fields close to the surface of a superconducting chip on which a CPW resonator is fabricated – through the choice of atomic species to minimize surface adsorption and effects of stray electric fields from adsorbates, and the use of a $\lambda/4$ resonator geometry in which the center conductor was open at one end but grounded at the other to avoid charge build-up [11] – we demonstrate here the coupling of the single-photon transition between the $1\mathrm{s}50\mathrm{s}\,^{3}\mathrm{S}_{1}$ ($|50\mathrm{s}\rangle$) and $1\mathrm{s}50\mathrm{p}\,^{3}\mathrm{P}_{J}$ ($|50\mathrm{p}\rangle$) Rydberg levels in He to the $2\pi\times$11.752\text{,}\mathrm{G}\mathrm{H}\mathrm{z} second harmonic mode of a superconducting CPW resonator operated at 3.66 K. To achieve this while maintaining tunability at the interface without strongly polarizing the atoms and increasing their sensitivity to stray dc electric fields, a strong microwave dressing field was applied to ac Stark shift the $|50\mathrm{p}\rangle$ state. This field was detuned from the $1\mathrm{s}50\mathrm{p}\,^{3}\mathrm{P}_{J}\rightarrow 1\mathrm{s}50\mathrm{d}\,^{3}\mathrm{D}_{J}$ ($|50\mathrm{p}\rangle\rightarrow|50\mathrm{d}\rangle$) transition and operated at an amplitude of $\sim 100$ mV/cm to tune the $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition, which occurs at $\omega_{\mathrm{50s,50p}}=2\pi\times$12.149\,181\text{\,}\mathrm{G}\mathrm{H}\mathrm{z} in zero field, into resonance with the resonator mode. In the presence of this strong microwave dressing field, the single-photon electric dipole transition moment for the $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition remained large, at $d\sim 1500\,e\,a_{0}$. Consequently, the microwave power in the resonator in the experiments could be reduced by more than three orders of magnitude from that used previously [10], moving this hybrid quantum system an important step closer to operation in the single-photon strong-coupling regime.

In the following, an overview of the experimental apparatus is provided in Section II. In Section III measurements made to characterize the superconducting CPW resonator, and stray dc electric fields at the atom-resonator interface are presented. This is followed in Section IV by a description of the microwave dressing scheme employed to ac Stark shift the $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition into resonance with the resonator, and an overview of the Floquet calculations performed to support the interpretation of the experimental data. In Section V the results of the experiments are presented before the outcomes of this work are discussed and conclusions drawn in Sections VI and VII, respectively.

II Experiment

A schematic overview of the core cryogenic components of the apparatus used in the experiments is shown in Figure 1. A pulsed supersonic beam of He atoms, traveling at a mean longitudinal speed of $2000\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$, was generated using a pulsed valve operated at a repetition rate of $25\text{\,}\mathrm{H}\mathrm{z}$. Atoms in this beam were prepared in the long-lived ($7870\text{\,}\mathrm{s}$ lifetime [14]) metastable 1s2s 3S1 level in a dc electric discharge at the exit of the valve [15]. The discharge was operated with a sharp metal anode (+230 V) positioned $\sim$1 mm from the valve opening, and seeded with electrons from a heated tungsten filament. Approximately 10 cm downstream from the valve, the beam was collimated by a 3-mm-diameter skimmer. Charged particles produced in the discharge were then removed by an electrostatic filter. The resulting neutral atom beam subsequently entered the cryogenic region of the apparatus through 4- and 3-mm-diameter apertures in heat shields maintained at $\sim$30 K and $<$4 K, respectively.

Inside the cold sub-4 K core of the apparatus, the atomic beam was intersected at right angles by continuous-wave ultra-violet (UV) and infrared (IR) laser beams. These copropagating laser beams were focused to \sim$$100\text{\,}\mathrm{\SIUnitSymbolMicro m} full-width-at-half-maximum (FWHM) waists between a pair of parallel copper electrodes, and operated at wavelengths of 388.975 and 787.297 nm, respectively, to drive the resonance-enhanced two-color two-photon 1s2s 3S1 $\rightarrow$ 1s3p 3P2 $\rightarrow$ 1s50s 3S1 excitation scheme [16]. Short pulses of excited Rydberg atoms were generated by detuning the IR laser $\sim$ 220 MHz below the field-free 1s3p 3P2 $\rightarrow$ 1s50s 3S1 transition frequency, and using a pulsed electric field of $\sim 650$ mV/cm to Stark shift the Rydberg state into resonance for select periods of time. These Rydberg excitation electric field pulses were applied for $3\text{\,}\mathrm{\SIUnitSymbolMicro s}$, resulting in the photoexcitation of 6-mm-long bunches of Rydberg atoms.

After excitation, the atoms traveled 25 mm to a $10\leavevmode\nobreak\ \mathrm{mm}\times 10\leavevmode\nobreak\ \mathrm{mm}$ niobium nitride (NbN) superconducting chip (100-nm-thick NbN film on a silicon substrate), containing an L-shaped $\lambda$/4 CPW resonator (length 6.335 mm; center-conductor width $20\text{\,}\mathrm{\SIUnitSymbolMicro m}$; gap width $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$), with one end grounded and the other open end capacitively coupled to a coplanar microwave waveguide [8]. The CPW was connected to a microwave source outside the vacuum chamber with the coaxial cables used to achieve this thermalised through attenuators on the 30 K and 4 K stages of the cryocooler. The total attenuation between the source and the CPW on the superconducting chip was -22.5 dB.

As shown in Figure 1, the longer ($\sim 5$ mm) straight section of the resonator was aligned with the propagation axis of the atomic beam. The Rydberg atoms interacted primarily with the evanescent microwave field above this part of the resonator at a typical atom-surface distance of \sim$$300\text{\,}\mathrm{\SIUnitSymbolMicro m} [11]. A split copper electrode was located $\sim 10$ mm above the superconducting chip in the apparatus (not shown in Figure 1) to allow compensation of stray electric fields at the location of the atoms above the resonator.

The resonance frequency of the resonator was coarsely controlled by setting the temperature, $T_{\mathrm{res}}$, of the NbN chip. In the experiments described here, the chip was operated at nominal temperatures of either $T_{\mathrm{res}}=3.65$ or $5.00$ K, which resulted in resonance frequencies in the second harmonic mode of $\omega_{2}\simeq 2\pi\times 11.752$ GHz, or $2\pi\times 11.710$ GHz, respectively. To couple the single-photon electric-dipole allowed $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition ($\omega_{50\text{s},50\text{p}}=2\pi\times 12.149181$ GHz in the absence of external fields) to the second harmonic mode of the resonator, a strong microwave dressing field, $\omega_{\text{dress}}$ = 2$\pi\times$ 3.350 GHz, detuned by $\sim$2$\pi$ x 100 MHz below the field-free $|50\mathrm{p}\rangle$ $\rightarrow$ $|50\mathrm{d}\rangle$ transition frequency ($|50\mathrm{d}\rangle\equiv$ 1s50d 3DJ) was generated above the superconducting chip. This field emanated from a microwave antenna located $\sim 13$ mm above the chip surface as shown in Figure 1, and it caused in an ac stark shift the $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition toward lower frequencies – as required to tune it into resonance with $\omega_{2}$. In each cycle of the experiment, the dressing field was applied as a 1-$\text{\,}\mathrm{\SIUnitSymbolMicro s}$-duration pulse, $12\text{\,}\mathrm{\SIUnitSymbolMicro s}$ after Rydberg state photoexcitation.

After passing over the superconducting chip the atoms moved into the detection region of the apparatus. There, 16 mm beyond the chip and $25\text{\,}\mathrm{\SIUnitSymbolMicro s}$ after initial laser photoexcitation, the final populations of the $|50\mathrm{s}\rangle$ and $|50\mathrm{p}\rangle$ Rydberg levels were determined by state-selective pulsed electric field ionization. This was implemented by applying a slowly-rising negative potential to one of the electrodes in this region to ionize the atoms. The resulting electrons were then accelerated out of the cryogenic part of the apparatus to a microchannel plate detector operated at room temperature.

III Superconducting CPW resonator and stray electric field characterization

To characterize the CPW resonator, microwave spectra were recorded by transmission through the co-planar waveguide which was capacitively coupled to it on the superconducting chip. These spectra, obtained for $T_{\mathrm{res}}=3.66$ K and 5.00 K, and covering microwave frequencies close to resonance with the second harmonic mode of the resonator are displayed in Figure 2(a) and (b), respectively. In these data a notch-type resonance is observed. This resonance lineshape is a consequence of interference between the microwave field transmitted directly through the waveguide, and that scattered from the resonator. To determine the resonator resonance frequencies, loaded quality factors and insertion loss from each spectrum, Fano functions were fit to the data using least-square methods. For the case in which $T_{\mathrm{res}}=3.66$ K, this yielded a resonance frequency of $\omega_{2}=2\pi\times$11.751,98\text{,}\mathrm{G}\mathrm{H}\mathrm{z}, and a loaded quality factor, determined from the spectral FWHM, of $Q=3960$. The insertion loss from the waveguide on resonance was $L_{\mathrm{ins}}$ = $26.67\text{\,}\mathrm{d}\mathrm{B}$. To perform experiments with the resonator off resonance from the atomic transition frequency, the chip was also operated at the higher temperature of $T_{\mathrm{res}}$ = $5.00\text{\,}\mathrm{K}$ [Figure [2](https://arxiv.org/html/2508.20292v1#S2.F2)(b)]. In this case the resonance frequency shifted to a lower value of $\omega_{2}=2\pi\times$11.709\,69\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}, the loaded quality factor reduced to $Q=1640$, and the insertion loss was $L_{\mathrm{ins}}$ = $32.86\text{\,}\mathrm{d}\mathrm{B}$. The temperature-dependent shift of the resonator resonance frequency of $-2\pi\times$42.29\text{,}\mathrm{M}\mathrm{H}\mathrm{z} between the data in Figure [2](https://arxiv.org/html/2508.20292v1#S2.F2)(a) and (b), is larger than the resonance width when $T_{\mathrm{res}}=3.66$ K or $5.00$ K, which were $\kappa=2\pi\times$2.97\text{\,}\mathrm{M}\mathrm{H}\mathrm{z} and $2\pi\times$7.17\text{,}\mathrm{M}\mathrm{H}\mathrm{z}$$, respectively. The additional use of the phase of the detected microwave field in the circle-fit method [17] to determine the resonator characteristics from the data in Figure 2 yielded values in agreement with those of the best fit Fano functions to within 10%.

Stray electric fields in the region where the atoms interacted with the microwave field in the superconducting resonator were minimized in the experiments by applying pulsed offset electric potentials to a split electrode above the superconducting chip. The resulting compensation field was optimized, using the atoms as in situ quantum sensors, by driving the antenna at frequencies close to the single-color two-photon $|50\mathrm{s}\rangle\rightarrow|51\mathrm{s}\rangle$ ($|51\mathrm{s}\rangle\equiv 1\mathrm{s}51\mathrm{s}\,^{3}\mathrm{S}_{1}$) transition frequency at $\omega_{\mathrm{50s,51s}}/2=2\pi\times$26.001,859\text{,}\mathrm{G}\mathrm{H}\mathrm{z}, and monitoring population transfer to the $|51\mathrm{s}\rangle$ state. An example spectrum recorded in this way, for a compensation potential of $+$1.2\text{\,}\mathrm{V} is shown in Figure 3. When recording this data, the resonator and microwave dressing fields were inactive. The microwave pulses generated using the antenna had a duration of $1\text{\,}\mathrm{\SIUnitSymbolMicro s}$, and were applied at the time in the experimental sequence when the atoms were located above the resonator. From this spectrum, a frequency shift of $-2\pi\times$0.419\text{,}\mathrm{M}\mathrm{H}\mathrm{z}$$ from the field-free transition frequency of $\omega_{\mathrm{50s,51}}/2$ is observed. Comparison of this shift with the calculated dc Stark shift of the two-photon transition, allowed a residual uncanceled electric field of $F_{\mathrm{u}}$ $\sim$ $104\text{\,}\mathrm{mV}\text{\,}{\mathrm{cm}}^{-1}$ at the location of the atoms to be determined.

IV Dressing scheme

The comparatively large quantum defects of the triplet $|50\mathrm{s}\rangle$ ($\delta_{\mathrm{50s}}\simeq 0.2966720$ [18]) and $|50\mathrm{p}\rangle$ ($\delta_{\mathrm{50p}}=0.0683528$ [18]) states in He means that they are energetically well-isolated from other higher-$\ell$ states with the same value of $n$. The single-photon electric-dipole allowed microwave transition between these states therefore remains strong in weak dc fields. For example, the transition moment is $d\simeq 2029\,e\,a_{0}$ in zero-field and remains $>2020\,e\,a_{0}$ in dc fields up to $200$ mV/cm. The corresponding field-free transition frequency $\omega_{\mathrm{50s,50p}}$ lies close to $\omega_{2}$, with a detuning of $\Delta_{\mathrm{res}}=\omega_{\mathrm{50s,50p}}-\omega_{2}=-2\pi\times$397.2\text{,}\mathrm{M}\mathrm{H}\mathrm{z} when $T_{\mathrm{res}}=$3.66\text{\,}\mathrm{K}. Therefore, to resonantly couple the atoms to the second harmonic mode of the resonator using the $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition it must be tuned to lower frequency. Because the differential static electric dipole polarizability of this transition is greater than zero, the dc Stark effect could, in principle, be used to induce the required frequency shift by applying an offset field of $F_{\mathrm{dc}}=704$ mV/cm. However, in such comparatively large dc fields, the $|50\mathrm{p}\rangle$ state is strongly polarized and increasingly sensitive to electric field noise, as it approaches the $\ell$-mixed hydrogenic Rydberg-Stark states. Consequently, tuning the frequency of the $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition using a dc electric field in the region close to the superconducting chip surface is anticipated to lead to decoherence and spectral broadening.

A more effective approach to tuning the $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition into resonance with the $\omega_{2}$ when $T_{\mathrm{res}}=3.66$ K, is to use an off-resonant microwave dressing field to induce an ac Stark shift of, for example, the $|50\mathrm{p}\rangle$ state toward lower frequency. This can be achieved by choosing the dressing field frequency to lie below the frequency of the $|50\mathrm{p}\rangle\rightarrow|50\mathrm{d}\rangle$ transition, with a detuning $\Delta_{\mathrm{dress}}=\omega_{\mathrm{dress}}-\omega_{\mathrm{50s,\,50d}}$. In the presence of this microwave dressing field, the $|50\mathrm{p}\rangle$ state is ac stark shifted to lower energy, while the $|50\mathrm{d}\rangle$ state is shifted to higher energy. This is seen on the left side of Figure 4. In the experiments described here, the detuning of this dressing field from the field-free $|50\mathrm{p}\rangle\rightarrow|50\mathrm{d}\rangle$ transition frequency was chosen to be $\Delta_{\mathrm{dress}}=-2\pi\times$102.77\text{,}\mathrm{M}\mathrm{H}\mathrm{z}$$.

In the region above the cryogenically cooled superconducting chip surface where the atoms interacted with the resonator, in addition to the dressing field, a residual uncanceled dc electric field $F_{\mathrm{u}}=104$ mV/cm was also present, as determined from the Stark shift in the spectrum in Figure 3. This dc field introduces a $-2\pi\times$7.927\text{,}\mathrm{M}\mathrm{H}\mathrm{z}$$ Stark shift of the $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition frequency in the absence of the dressing field. To account for this in the calculations, the combined ac and dc Stark shifts of the $|50\mathrm{s}\rangle$ and $|50\mathrm{p}\rangle$ states were treated. This was achieved by considering the Hamiltonian of the Rydberg atom within the electric dipole approximation, such that,

[TABLE]

where $H_{0}$ is the field-free Hamiltonian, and the dc field $\vec{F}_{\mathrm{dc}}=[0,0,F_{\mathrm{dc}}]$ and microwave dressing field $\vec{F}_{\mathrm{dress}}=[0,0,F_{\mathrm{dress}}\cos(\omega_{\mathrm{dress}}t)]$ act in the $z$ dimension. Since this Hamiltonian is periodic in time, it can be transformed into a time-independent Floquet Hamiltonian [19]. Following this transformation, the matrix representation of the Floquet Hamiltonian can be constructed in the atomic basis expanded by the number of Floquet modes, $|n\ell m_{\ell}q\rangle$, where $\ell$ and $m_{\ell}$ are the orbital angular momentum and azimuthal quantum numbers of the Rydberg electron, respectively, and $q$ is the Floquet mode order, such that,

[TABLE]

In this expression, the first term on the right hand side represents the field-free energy of the Rydberg states, i.e., $\langle n^{\prime}\ell^{\prime}m^{\prime}_{\ell}|H_{0}|n\ell m_{\ell}\rangle=E_{n\ell}=-hcR_{\mathrm{He}}/(n-\delta_{n\ell})^{2}$. The second and third terms account for the interaction of the atom with the time-independent dc, and time-dependent microwave dressing fields, respectively. The final term represents the energy offset of the Floquet sidebands, which lie at integer multiples of the dressing field photon energy $q\hbar\omega_{\mathrm{dress}}$. The values of these matrix elements were determined by numerical integration using the Numerov method [20] and the quantum defects of the corresponding triplet Rydberg states in He [18]. The combined effects of the dc electric field, and microwave dressing field on the Rydberg energy-level structure were obtained from the eigenvalues of the Hamiltonian matrix. The corresponding eigenvectors were then used to determine the electric dipole moments for transitions between Rydberg states in the presence of the fields. Convergence was achieved in the eigenvalues, for the particular states of interest here with $n=50$ over the range of field strengths considered, using a basis set with $49\leq n\leq 51$, $|q_{\mathrm{dress}}|\leq 3$, and all allowed values of $\ell$.

An overview of the effect of dc electric fields of up to 150 mV/cm on the ac stark shifted $n=50$ Rydberg states in the presence of the strong microwave dressing field is shown in the right side of Figure 4. The uncanceled dc field of $F_{\mathrm{u}}=104$ mV/cm encountered in the experiments is indicated by the dashed gray vertical line in this panel of the figure. The importance of accounting for this residual uncanceled field, to accurately calculate the effect of the microwave dressing field can be seen from the dc Stark shift of the microwave dressed $|50\mathrm{p}\rangle$ state in the figure.

A more detailed view of the calculated frequency shifts of the $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition can be seen in Figure 5. These data show that, in the presence of the residual dc electric field of $F_{\mathrm{u}}=104$ mV/cm, a microwave dressing field amplitude of $367$ mV/cm is required to shift the $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition into resonance with $\omega_{2}$ when $T_{\mathrm{res}}=$3.66\text{,}\mathrm{K}$$ (dashed horizontal line). Under these conditions the electric dipole moment of the $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition reduces from its field-free value $>2000\,e\,a_{0}$ indicated by the color scale on the right side of the figure. However, it remains significant with a value of $d\sim 1500\,ea_{0}$ at the point where the lower curve, calculated in the presence of the microwave dressing field, crosses the dashed horizontal line.

V Results

To resonantly couple the single-photon $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition to the $\omega_{2}=2\pi\times 11.752$ GHz second harmonic mode of the superconducting CPW resonator when operated at $T_{\mathrm{res}}=3.66$ K, it was necessary to determine the optimal output power from the microwave source, $P_{\mathrm{dress}}$, at which to drive the microwave antenna above the superconducting chip to generate the required $\omega_{\mathrm{dress}}=2\pi\times$3.350\text{,}\mathrm{G}\mathrm{H}\mathrm{z} microwave dressing field amplitude of $367$ mV/cm at the location of the atoms above the resonator. This was found by injecting a 1-$\text{\,}\mathrm{\SIUnitSymbolMicro s}$-duration pulse of microwave radiation into the resonator at its resonance frequency, $\omega_{2}=2\pi\times$11.751\,96\text{\,}\mathrm{G}\mathrm{H}\mathrm{z} when the atoms were located above it and monitoring the population of the $|50\mathrm{s}\rangle$ in the detection region of the apparatus. The peak power of this flat-top microwave pulse as it propagated through the CPW on the superconducting chip was $P_{\mathrm{inc}}=-50.5$ dBm. Since the microwave radiation at this frequency was far off resonance from the $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition frequency in the absence of the dressing field, it alone had no observable effect on the $|50\mathrm{s}\rangle$ population. With this weak microwave field in the resonator, a set of measurements were then made with the simultaneous application of the pulsed dressing field while continuing to monitor the $|50\mathrm{s}\rangle$ population. The resulting data are presented in Figure 6(a). From this figure it is seen that as $P_{\mathrm{dress}}$ is increased from -5 dBm to $\sim+7$ dBm no significant change occurs to the population of the $|50\mathrm{s}\rangle$. However, as $P_{\mathrm{dress}}$ approaches 12 dBm, the $|50\mathrm{s}\rangle$ population reduces by $\sim 0.2$ before returning again to 1 when $P_{\mathrm{dress}}$ reaches 16 dBm. This resonant depletion of the $|50\mathrm{s}\rangle$ population occurs when the $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition is ac Stark shifted into resonance with $\omega_{2}$, and the resulting resonant coupling to the microwave field in the resonator causes a transfer of population to the $|50\mathrm{p}\rangle$ state.

From the results of the calculations shown in Figure 5, the dressing field amplitude required to most strongly couple the atoms to the resonator is the lowest field for which the ac Stark shift of the $|50\mathrm{p}\rangle$ state gives rise to a $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition frequency equal to $\omega_{2}$. Consequently, the resonant depletion of the $|50\mathrm{s}\rangle$ population close to $P_{\mathrm{dress}}=12$ dBm in the data in Figure 6(a) indicates the value of $P_{\mathrm{dress}}$ at which this optimal dressing field is generated at the location of the atoms. The dashed black curve overlaid on the experimental data represents a Gaussian function fit to the experimental data. The fitting procedure implemented involved first converting the logarithmic power scale of $P_{\mathrm{dress}}$ to a linear scale representing the dressing field amplitude. This allowed the Gaussian function to be fit to the data using least-squares methods, before transforming back to the logarithmic scale for direct comparison with the experimental results. In the field-amplitude domain the free parameter of the Gaussian function were limited to determining the dimensionless ratio, $\sigma_{\mathrm{F}}/F_{\mathrm{dress}}$, which was found to be $\simeq 0.2$. Comparison of this with the results of the Floquet calculations indicate that this corresponds to a spectral FWHM in the frequency domain of $\simeq$55\text{,}\mathrm{M}\mathrm{H}\mathrm{z}$$ when the peak is $F_{\mathrm{dress}}=367$ mV/cm. Since this is larger than the Fourier transform limit of the 1-$\text{\,}\mathrm{\SIUnitSymbolMicro s}$-duration pulsed field injected into the resonator, this suggests that the resonance widths encountered in the experiments when the $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition is ac Stark shifted into resonance with $\omega_{2}$ are dominated by inhomogeneities in the microwave dressing field amplitude.

For values of $P_{\mathrm{dress}}\gtrsim 17$ dBm in Figure 6(a) further depletion of the $|50\mathrm{s}\rangle$ population is observed. This feature in the data, which appears at between 3 and 10 times the microwave source power required to tune the $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition into resonance with $\omega_{2}$ results from the complex multilevel character of the Rydberg states and the effect of the higher-$\ell$ Rydberg states on the ac Stark shift of the $|50\mathrm{p}\rangle-|50\mathrm{d}\rangle$ transition in the corresponding fields.

To confirm that the microwave dressed Rydberg atoms were resonantly coupled to the microwave field in the CPW resonator when $P_{\mathrm{dress}}=12$ dBm, and not, for example, the evanescent field of the CPW or microwave connectors on the PCB, a second similar measurement was made, but with the temperature of the superconducting chip increased to $T_{\mathrm{res}}=5.00$ K. This resulted in a $-2\pi\times 42.29$ MHz frequency shift of the resonator resonance. In this situation the $2\pi\times$11.751,96\text{,}\mathrm{G}\mathrm{H}\mathrm{z}$$ microwave field for which $P_{\mathrm{inc}}=-50.5$ dBm did not build up in the resonator upon pulsed injection through the CPW. Consequently, as can be seen from the corresponding data in Figure 6(b), no resonant depletion of the $|50\mathrm{s}\rangle$ signal is observed when $P_{\mathrm{dress}}=12$ dBm.

With $T_{\mathrm{res}}=3.66$ K, and $P_{\mathrm{dress}}$ set to $12$ dBm, so that the ac Stark shift caused by the microwave dressing field ensured that the $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition was resonant with $\omega_{2}$, a further set of measurements were made of the dependence of the depletion of the $|50\mathrm{s}\rangle$ population on the value of $P_{\mathrm{inc}}$, i.e., the power of the microwave field in the CPW. These are shown in Figure 6(c). From these data (filled points) it can be seen that as $P_{\mathrm{inc}}$ is reduced from -43.5 dBm to -53.5 dBm the depletion of the $|50\mathrm{s}\rangle$ population reduces from $\sim 0.35$ to $\sim 0.1$. This depletion depends linearly on the value of $P_{\mathrm{inc}}$ when displayed on the logarithmic dBm scale, as seen from the linear function indicated by the dashed black line which was fit to the data using least squares methods. Extrapolation of this fit function to lower values of $P_{\mathrm{inc}}$, approaching -60 dBm, indicates that for the $1\text{\,}\mathrm{\SIUnitSymbolMicro s}$ interaction time of the atoms with the resonator field in the experiments, microwave fields in the CPW with powers down to -58 dBm can be detected using the atoms at frequencies resonant with $\omega_{2}$ and the $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ Rydberg-Rydberg transition frequency. This is the value of $P_{\mathrm{inc}}$ above which the depletion of the $|50\mathrm{s}\rangle$ state is greater than one standard deviation in the measured population as indicated by the shaded gray region around the continuous line representing unit population.

To obtain frequency-domain information on the resonant coupling of the atoms to the resonator field in the presence of the strong microwave dressing field, spectra were recorded by varying the frequency of the microwave field in the CPW while monitoring the $|50\mathrm{s}\rangle$ population. These measurements were made for $T_{\mathrm{res}}=$3.66\text{,}\mathrm{K}, so that $\omega_{2}=2\pi\times$11.751\,96\text{\,}\mathrm{G}\mathrm{H}\mathrm{z} [as in Figure 6(a)], and with $P_{\mathrm{inc}}=-43.5$ dBm with the results displayed in Figure 7. Under these conditions, when the microwave dressing field is off – so that the $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition frequency is off resonance from the resonator – no depletion of the $|50\mathrm{s}\rangle$ population is observed as the frequency of the microwave field injected into it is adjusted [Figure 7(a)]. With the dressing field on and set to $P_{\mathrm{dress}}=12$ dBm, such that the ac Stark shifted $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition frequency is resonant with $\omega_{2}$, resonant depletion of the $|50\mathrm{s}\rangle$ is observed at the frequency $\omega_{2}$ as seen in Figure 7(b).

The centroid of the resonance in Figure 7(b), determined by fitting a Gaussian function (continuous blue curve) to the experimental data using least squares methods, lies at $2\pi\times$11.7517\text{,}\mathrm{G}\mathrm{H}\mathrm{z} (continuous vertical gray line). This is in good agreement with the value of $\omega_{2}$ obtained from the microwave transmission measurements on the superconducting chip when $T_{\mathrm{res}}=$3.66\text{\,}\mathrm{K} in Section III. The measured FWHM of the resonance, $\Delta\omega_{\mathrm{FWHM}}\simeq 2\pi\times$7.1\text{,}\mathrm{M}\mathrm{H}\mathrm{z}, is greater than the Fourier transform limit of the $1\text{\,}\mathrm{\SIUnitSymbolMicro s}$ duration microwave pulse injected into the resonator and exceeds the FWHM of, or decay rate $\kappa=2\pi\times$2.97\text{\,}\mathrm{M}\mathrm{H}\mathrm{z} from the resonator at this temperature. The corresponding spectral broadening is attributed to a combination of (1) the inhomogeneity in the microwave dressing field in the region above the superconducting chip, which causes variations in the magnitude of the ac Stark shift of the $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition frequency across the distribution of Rydberg atoms, and (2) the inhomogeneity in the dc field, to which this transition is sensitive because of its larger polarizability, for example, when compared to the single-color two-photon $|50\mathrm{s}\rangle\rightarrow|51\mathrm{s}\rangle$ transition in Figure 3.

The resonant behavior of the atom-resonator interaction in the presence of the microwave dressing field indicates that the calculated microwave dressing field amplitude of $367$ mV/cm required to ac Stark shift the $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition into resonance with the resonator when operated at $\omega_{2}=2\pi\times$11.751,96\text{,}\mathrm{G}\mathrm{H}\mathrm{z} is achieved when $P_{\mathrm{dress}}=12$ dBm. However, since the change in this ac Stark shift with field amplitude around this mean field is $\sim 1$ MHz/(mV/cm), a homogeneity of the dressing field of better than $3\times 10^{-3}$ over the distribution of Rydberg atoms as they move over the resonator is required to achieve a spectral width of the transition that approaches the Fourier-transform limit of the interaction time. As seen from the diagram of the superconducting chip-holder and surrounding electrode structures in Figure [8](https://arxiv.org/html/2508.20292v1#S5.F8), these components which strongly affect the spatial distribution of the microwave dressing field emanating from the antenna, have size scales on the order of 10 mm. Consequently, for the spatially extended (6-mm-long in the $z$ dimension by $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wide in the $x$ and $y$ dimensions) distribution of atoms moving $\sim300\text{,}\mathrm{\SIUnitSymbolMicro m}$above the superconducting chip surface, even the$\sim 1$% homogeneity in the microwave dressing field amplitude required to ensure that any spectral broadening caused by a distribution of ac Stark shifts remains below the resonator FWHM is challenging to achieve and it is this which represents the dominant contribution to the spectral broadening in the measurements.

VI Discussion

The inhomogeneity of the microwave dressing field amplitude across the spatial distribution of atom positions that contributes to the broadening of the resonance data in Figure 7(b) also results in a dephasing of the atom-resonator interaction. This dephasing is a result of the corresponding distribution of frequency detunings from the resonator resonance frequency, and to a lesser extent the distribution of resonant Rabi frequencies. The use of a fast moving beam of Rydberg atoms in the experiments reported here offers the advantage of easily spatially separating the Rydberg state laser photoexciation region, from the region in which the atoms interact with the superconducting circuit, and the detection region in the apparatus. However, it has the disadvantage of yielding comparatively long (typically 2 – 6 mm long) bunches of excited Rydberg atoms that move up to 2 mm within a $1\text{\,}\mathrm{\SIUnitSymbolMicro s}$ atom-resonator interaction time. In the future, narrower spectral features and reduced dephasing can be achieved by laser slowing and cooling the atoms [21, 22] and ultimately positioning them precisely above the CPW resonator using, for example, optical tweezers. With cold atoms localized in this way, to a precision on the order of $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$, together with a resonator fabricated with a resonance frequency specifically chosen to lie close to the field-free $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition frequency to reduce the required AC Stark shift, the inhomogeneity of the microwave dressing field will have a significantly reduced effect on the measurements.

In previous work, with an apparatus similar to that in Figure 8 but without the microwave antenna present between the split electrode components labeled E3 and E4, stray electric fields at the position of the atoms above the CPW resonator were compensated to $\sim 25$ mV/cm. The introduction of the antenna to generate strong microwave dressing fields above the superconducting chip surface in the experiments described here leads to an inhomogeneity in the dc field generated by electrodes E3 and E4. This dc field inhomogeneity is the cause of the larger residual uncanceled electric field of $F_{\mathrm{u}}=104$ mV/cm in the current experiments. For this reason, this residual uncanceled field is not randomly oriented. Its orientation, at the location of the atoms \sim$$300\text{\,}\mathrm{\SIUnitSymbolMicro m} above the chip surface, was determined from finite element calculations to be in the $y$ dimension – as indicated by the central red vector crossing the distribution of atoms in Figure 8. Similar finite element calculations of the frequency-dependent polarizations of the microwave dressing field and the evanescent microwave field from the CPW resonator indicate that at the position of the atoms both of these fields are also linearly polarized in the $y$ dimension, i.e., parallel to $\vec{F}_{\mathrm{u}}$. This observation is the reason the Floquet calculations described in Section IV were restricted to the situation in which the dc and time-varying microwave fields were chosen to be polarized parallel to each other. Closer to the CPW resonator, when the atom-surface distance is on the same order of magnitude as the features in the superconducting circuit, i.e., $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ it will become important to treat more completely the possibility of a non-zero angle between the polarizations of the dressing field and the resonator field. This can be achieved by expanding the Floquet calculations to include a time-varying component polarized in the perpendicular direction.

From the data in Figure 6(c), when $P_{\mathrm{inc}}=$-53.5\text{,}\mathrm{d}\mathrm{B}\mathrm{m}, the depletion of the population of the $|50\mathrm{s}\rangle$ state after the $1\text{\,}\mathrm{\SIUnitSymbolMicro s}$ interaction time of the atoms with the resonator is $\sim 0.1$. This indicates that the resonant Rabi frequency associated with the atom-resonator interaction under these conditions is approximately $2\pi\times 100$ kHz. The photon occupation number of the resonator when $P_{\mathrm{inc}}=$-53.5\text{\,}\mathrm{d}\mathrm{B}\mathrm{m} can be estimated from the the circulating power in the resonator $P_{\mathrm{circ}}$, which can be expressed as [23]

[TABLE]

where $m=2$ and $Q=3960$ are the mode number and quality factor of the resonator, $P_{\mathrm{inc}}$ is expressed in units of $\mathrm{W}$ and the insertion loss is $L_{\mathrm{ins}}=$26.96\text{,}\mathrm{d}\mathrm{B} (see Section [III](https://arxiv.org/html/2508.20292v1#S3)). Thus $P_{\mathrm{circ}}$ is estimated to be $\sim0.1\text{,}\mathrm{\SIUnitSymbolMicro W}$which corresponds to a photon occupation number in the second harmonic mode of$N\sim 10^{6}$. Since the Rabi frequency associated with the resonant interaction of the atoms with the resonator is dependent on$\sqrt{N}$, the single photon Rabi frequency in the current experiments can be estimated to be$2\pi\times 100$ Hz.

In the future this single photon Rabi frequency can be increased by improving the homogeneity of the dc electric fields generated above the CPW resonator. This will be achieved by biasing the antenna to the same offset potential as the split electrodes E3 and E4. A further increase will be achieved by fabricating a resonator with a resonance frequency that lies closer to the field-free $|50\mathrm{s}\rangle\rightarrow|50\mathrm{p}\rangle$ transition frequency, reducing the AC Stark shift required to tune the transition into resonance with it. This will mean that the electric dipole transition moment between these states with the dressing field applied will increase and approach its field-free value.

However, the most significant increase in single-photon Rabi frequency will be seen by positioning the atoms closer to the resonator and the surface of the superconducting chip. In the current experiments the atoms are located \sim$$300\text{\,}\mathrm{\SIUnitSymbolMicro m} above the chip surface when they interact with the resonator. If this distance is reduced to between $30$ and $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$, single-photon Rabi frequencies on the order of $2\pi\times 1$ MHz are expected to be achievable because of the corresponding increase in the amplitude of the evanescent field close to the resonator.

VII Conclusion

The results presented here represent an important step toward reaching the single-photon strong-coupling regime in hybrid microwave cavity QED with Rydberg atoms coupled to CPW resonators in superconducting circuits. They demonstrate control of the Rydberg-Rydberg transition frequency at this hybrid interface using a strong off-resonant microwave dressing field, and represent the first demonstration of coupling a strong ($d\sim 1500\,\mathrm{ea}_{0}$) single-photon microwave transition between Rydberg states to a superconducting CPW resonator. Since this transition dipole moment is $\sim 30$ times larger than in previous related experiments, the circulating power in the resonator could be reduced by almost a factor of 1000 over that employed previously. Although the experiments were performed with a microwave field injected into the resonator to ensure an observable population transfer between the Rydberg states in the $1\text{\,}\mathrm{\SIUnitSymbolMicro s}$ interaction time of the atoms with the resonator, consideration of the circulating power in the resonator and corresponding photon occupation number allowed the single-photon Rabi frequency to be estimated to be $2\pi\times 0.1$ kHz. With future upgrades to the apparatus to better localize the atoms, and move them closer to the surface of the superconducting chip, limitations resulting from the homogeneity of the microwave dressing field will be overcome to allow this type of microwave dressed atom-circuit interface to be exploited for optical manipulation of the microwave-photon number state in the resonator, or ultimately the quantum state of a superconducting qubit in an interaction mediated by the atoms.