Microwave Rabi resonances beyond the small-signal regime

TL;DR

This paper investigates microwave-atom interactions beyond the small-signal regime, revealing higher-harmonic responses in Rabi resonances that enhance understanding and measurement techniques for quantum systems.

Contribution

It demonstrates experimental observation of higher-harmonic Rabi responses beyond the small-signal approximation in cold rubidium atoms, supported by numerical simulations.

Findings

Observation of higher-harmonic population responses

Validation of numerical simulations with experimental data

Enhanced understanding of microwave-atom coupling

Abstract

The coupling between microwave fields and atoms (or atom-like systems) is inherently weaker than for optical fields, making microwave signal manipulation for applications like quantum information processing technically challenging. In order to better understand this coupling and to develop tools for measuring it, we explore the microwave coupling to atoms using the "atomic candle" technique, and push it beyond the bounds of the small-signal regime in order to deliver a larger signal. In familiar two-level systems, responses beyond the usual Rabi oscillations can arise when a single-tone drive is phase-modulated, causing the steady-state populations to oscillate at integer harmonics of the modulation frequency. Resonant behavior of the first two harmonics for frequencies near the Rabi frequency, known as $\alpha$ and $\beta$ Rabi resonances, is widely used for microwave-field magnetometry and as a power standard known as the atomic candle. Here, we explore Rabi resonances beyond the small-signal approximation and report upon experimental observations of higher-harmonic population response for microwave hyperfine transitions in cold $^{87}\rm Rb$ atoms, which we compare to numerical simulations.

Figures

Equations

$$\dot{u}$$

$$\dot{v}$$

$$\dot{w}$$

$$\theta (t)={\theta }_0+m\sin ({\omega }_{\mathrm{m}}t+{\varphi }_{\mathrm{m}}).$$

$${\rho }_{22}(t)={\rho }_{22,0}+{\rho }_{22,\mathrm{m}}(t),$$

$$v(t)=v_0+v_{\mathrm{m}}(t),$$

$$u(t)=u_0+u_{\mathrm{m}}(t),$$

$${\ddot{\rho }}_{22,\mathrm{m}}+{\Gamma }_1{\dot{\rho }}_{22,\mathrm{m}}+$$

$$2{\Omega }_{\mathrm{R}}{\Gamma }_2{v}_{\mathrm{m}}+2{\Omega }_{\mathrm{R}}(\delta +\dot{\theta })u_{\mathrm{m}}+u_0{\Omega }_{\mathrm{R}}\dot{\theta },$$

$${\rho }_{22,m}(t)=P_1\sin ({\omega }_{\mathrm{m}}t+{\varphi }_1)+P_2\sin (2{\omega }_{\mathrm{m}}t+{\varphi }_2),$$

$$P_1$$

$$P_2$$

$${\rho }_{11}(t)=\sum _n{P}_n\sin (n{\omega }_{m}t+{\varphi }_n),$$

$$a_n=\frac{1}{N}\sum _{k=1}^N{\rho }_{11}(t_k)\sin n{\omega }_{\mathrm{m}}{t}_k,$$

$$b_n=\frac{1}{N}\sum _{k=1}^N{\rho }_{11}(t_k)\cos n{\omega }_{\mathrm{m}}{t}_k,$$

$$f(t)$$

$$+C_2{e}^{-{\Gamma }_{2}t}{\sin }^2\left(\frac{{\Omega }_R}{2}t+\varphi \right)+\sum _{n=1}^6{P}_n\sin (n{\omega }_{m}t+{\varphi }_n),$$

Full text

I Introduction

The control and manipulation of microwave signals with atoms is undergoing a renaissance in the scientific world, moving beyond traditional applications such as providing the definition of the second Bandi2012a ; Pellaton2012 ; Petremand2012 ; Bandi2014 , towards precision metrology in direct-current (DC) Sun2018a and alternating current (AC) Horsley2016 magnetometers, and novel applications for quantum information processing, such as microwave-to-optical transduction of qubits Hafezi2012 ; Kiffner2016 ; Adwaith2019 . These two examples highlight two different ways atoms are used: first, as a measurement tool to detect the fields; and second, as the interaction medium through which the fields are manipulated. Both cases demand an intimate understanding of the microwave-to-atom coupling, and a way to measure it precisely and with a large signal-to-noise ratio.

Very generally, an oscillating electromagnetic field with a frequency near the resonance of a two-level system periodically drives population transfer between levels, in a process commonly known as Rabi oscillation. In the presence of relaxation between these levels (due to one or more of many possible decoherence mechanisms), the steady-state populations tend towards a constant value. In contrast, these steady-state populations oscillate when the phase of the oscillating field is modulated at a frequency $\omega_{\rm m}$, and the amplitude of the population oscillations is enhanced when $\omega_{\rm m}$ is equal to the Rabi frequency, $\Omega_{\rm R}$. This phenomenon, known as “Rabi resonance” Cappeller1985 ; Camparo1998 ; Camparo1998a , can be used to find the the system’s Rabi frequency and, correspondingly, the power of the oscillating field at the position of the atom(s). In analogy to standard candles in astronomy, the Rabi-flopping atoms respond to AC and DC fields in exactly the same way in all laboratories, making them “atomic candles” Camparo1998a ; Coffer2002 for electromagnetic power standards. This technique has also attracted significant interest for its applications for AC magnetometry inside MW cavities Sun2017a ; Sun2018a , MW wave guides Kinoshita2009 ; Kinoshita2011 ; Kinoshita2013 , and in free space Swan-Wood2001 ; Sun2018a ; Liu2018 ; Shi2018 ; Kinoshita2017 . Applying static magnetic fields Sun2017a and using multispecies vapor cells Sun2018a adds frequency tunability and expands the operational bandwidth of this technique.

In previous work, the Rabi resonance was analyzed and measured only in the small-signal regime, where a sufficiently small modulation depth permits an approximate analytic solution to the Maxwell-Bloch equations for the populations. These population oscillations, which have frequency components at $\omega_{\rm m}$ and $2\omega_{\rm m}$, can be described analytically, including an amplitude that depends on $\omega_{\rm m}$. Here, we explore the atomic candle technique beyond the small-signal approximation both numerically and experimentally. Using large-deviation phase modulation on the $\rm MW$ field, we observe higher-order spectral features whose characteristics and scaling behave differently than the first and second harmonics. In this work, we focus our attention on the component oscillating at $4\omega_{\rm m}$, namely, the fourth harmonic, and discuss how it may be used to extend the atomic candle technique.

II Theory: Rabi oscillations with a phase-modulated driving field

As we explore higher order Rabi resonances theoretically, we will consider the two-level system used in our experiments [Fig. 1], which is comprised of two Zeeman states in different hyperfine levels of an alkali metal, which are described by the total angular momentum $F$ and projections $m_{F}$. For this two-level system, separated by energy $\hbar\omega_{0}$, the levels can be coupled by a non-zero matrix element for magnetic dipole transitions, $V_{12}=\mathinner{\langle{2}|}\hat{\boldsymbol{\mu}}\cdot\mathbf{B}_{0}\mathinner{|{1}\rangle}$, where $\hat{\boldsymbol{\mu}}$ is the magnetic dipole operator, and $\mathinner{|{1}\rangle}$ and $\mathinner{|{2}\rangle}$ represent the two levels. Here, the AC driving field $\mathbf{B}(t)={B}_{0}\mathbf{e}_{B0}\cos\omega t$ is described by the peak field $\mathbf{B}_{0}={B}_{0}\mathbf{e}_{B0}$, with AC-field amplitude ${B}_{0}$ and direction $\mathbf{e}_{B0}$. When the AC field is resonant ($\omega=\omega_{0}$), the population oscillates at the Rabi frequency $\Omega_{\rm R}=V_{12}/\hbar$. When the detuning $\delta=\omega-\omega_{0}$ is non-zero, the oscillations occur at the generalized Rabi frequency is $\Omega=\sqrt{\Omega_{\rm R}^{2}+\delta^{2}}$.

In the presence of decoherence or damping, the oscillatory behavior of the two-level system damps out and the system equilibrates to a steady population ratio (e.g., equal populations for a strong resonant AC field and small damping). These dynamics are well-described by the Maxwell-Bloch equations Noh2010 .

A phase-modulated oscillating field changes these dynamics. Here, we consider the case where a time-dependent phase $\theta(t)$ modulates the AC field: $\mathbf{B}(t)={B}_{0}\mathbf{e}_{B0}\cos\left[\omega t+\theta(t)\right]$. In this case, the standard Maxwell-Bloch equations can be modified to include additional time-dependent terms:

[TABLE]

where $u$ and $v$ are the in- and out-of-phase coherences [$u=2\rm Re(\rho_{12})$ and $v=2\rm Im(\rho_{12})$] and $w=\rho_{22}-\rho_{11}$ is the population difference between levels, and where $\rho$ is the usual $2\times 2$ density matrix of the two-level system ($\rho_{11}$ is the ground-state population and $\rho_{22}$ is the excited-state population). In the loss terms, $\Gamma_{1}$ and $\Gamma_{2}$ represent longitudinal and transverse damping rates, respectively. We consider the case where the AC field is modulated about a constant phase offset $\theta_{0}$ with frequency $\omega_{\rm m}$, modulation depth $m$, and offset phase $\phi_{\rm m}$:

[TABLE]

We explore the numerical solutions of Eqs. 1-3 to find the time-dependent coherences and population dynamics under a variety of parameters. In particular, we are interested in the excited state population $\rho_{22}=(w+1)/2$ in the steady-state regime ($\Gamma_{1}t\gg 1$), which is found to oscillate, even long after the damping times associated with $\Gamma_{1}$ and $\Gamma_{2}$ [FIG. 2].

Below, we have a closer look at the time dependence of the steady-state populations of the ground and excited states, which is directly related to a measurable quantity: the absorption of light from one of these levels to an optically excitable third level.

II.1 Small-modulation approximation

Treating the modulation term as a perturbation allows us to assume that in the steady-state regime the parameters describing the atomic state evolve around their unmodulated steady-state values $u_{0},~{}v_{0},~{}\rho_{22,0}$:

[TABLE]

where $\rho_{22,\rm m},v_{\rm m},u_{\rm m}$ describe the evolution due to the phase modulation.

FIG. 2 shows that in the case of a weak modulation, the steady-state ground-state population $\rho_{11}(t)$ oscillates around its unmodulated value. Indeed, it can be shown Coffer2002 that in this case the excited-state population evolution is described by

[TABLE]

which is a damped harmonic oscillator equation, where the driving is due to coherence between the levels ($v_{\rm m}$ and $u_{\rm m}$) and phase modulation $\dot{\theta}$.

For small modulations ($m<\sqrt{2\Gamma_{1}/\omega_{\rm m}}$ ) and low decoherence ($\Gamma_{2}\ll\omega_{\rm m}$), known as the small-signal approximation, the steady-state solution for the excited state population has been found analytically Coffer2002 . The population dynamics in this regime are represented by the first two harmonics of $\omega_{\rm m}$, and the excited-state dynamics with respect to its unperturbed value may be expressed as

[TABLE]

with amplitudes

[TABLE]

The main feature of these solutions is that the amplitudes $P_{1}(\omega_{\rm m},\Omega_{\rm R})$ and $P_{2}(\omega_{\rm m},\Omega_{\rm R})$ peak when the frequency of the corresponding harmonic is resonant with the Rabi frequency, $\omega_{\rm m}=\Omega_{\rm R}$ and $2\omega_{\rm m}=\Omega_{\rm R}$, respectively, which are known as the $\alpha$ and $\beta$ Rabi resonances. Experimentally, scanning the modulation frequency $\omega_{\rm m}$ in search of this peak provides a tool for measuring $\Omega_{\rm R}$ and, thus, the power of the driving field, since $\Omega_{\rm R}\propto B_{0}^{2}$ Sun2018a . In addition, the amplitude of the first harmonic disappears at zero-carrier-detuning, $\delta=0$ ($P_{1}\propto\delta$). By tuning the carrier frequency $\omega$ to field-sensitive transitions Sun2018a , this dependence provides a means by which to measure static magnetic fields.

The inset of FIG. 2 compares the spectra of the steady-state $\rho_{11}$, simulated for $\delta=0$ and $\delta=0.1\omega_{m}$. As is expected, the first harmonic vanishes at zero-carrier-detuning. In the simulation, the magnitudes of the decoherence rates are comparable with the modulation frequency, which violates the small-signal approximation and leads to additional oscillations at third and fourth harmonics, even though their amplitudes are a few orders of magnitude smaller than $P_{1}$ and $P_{2}$.

II.2 Large modulation: Numerical analysis

A general solution of the phase-modulated oscillatory dynamics of a two-level system, with arbitrary phase- and amplitude-modulation, was obtained in Ref. Alekseev1992 .

Still, there is no simple analytical expression for the case where the phase is modulated according to Eq. 4. Numerical simulations of the large-modulation condition [FIG. 3] show that the steady-state ground-state population has a periodic solution, but, unlike in the small-modulation case, its average is shifted from the unmodulated value and its spectrum consists of many harmonics at multiples of $\omega_{\rm m}$:

[TABLE]

where $n$ can be any positive integer and $\phi_{n}$ is the phase of the response. As might be expected from small-signal theory, the relative height of the spectral consituents $P_{n}$ depends on $m,\omega_{\rm m},\Omega_{\rm R},\Gamma_{1,2}$ and $\delta$. In the case when $\delta=0$, all odd components disappear, which is consistent with the analytical small signal solutions (Eq. 5). However, compared to the small signal solution, the large-modulation dynamics include additional spectral peaks, and thus more (correlated) information from which to extract the field calibration, providing more options for extracting the final results.

Next, we evaluate the response of the excited-state population when the carrier frequency is equal to the frequency of the transition between the ground and excited states (i.e., $\delta=0$), which is the regime of practical interest. Here, the ground state population’s response to phase modulation includes only the even harmonics of $\omega_{\rm m}$ and we study, in particular, the second and fourth harmonics for different values of the modulation index $m$. We are interested only in the steady-state response, where $\Gamma_{1}t_{1}\gg 1$ and any “normal” Rabi oscillations would have damped out.

To extract values of the spectral amplitudes $P_{n}$ of the steady-state ground-state population signal $\rho_{11}(t)$ from our numerical simulations, we consider both the in- and out-of-phase responses of the signal at frequency $n\omega_{\rm m}$:

[TABLE]

where the $t_{k}$ represent the evenly-spaced discrete time points, and the index $k$ runs over all time indices. This approach to the analysis provides the amplitudes of the components at each harmonic of the modulation frequency $\omega_{\rm m}$, and the overall overall amplitude is given by $P_{n}=2\sqrt{a_{n}^{2}+b_{n}^{2}}$.

This analysis relies on the fact that $a_{n}$ and $b_{n}$ approach zero as $N$ approaches infinity for any frequency other than the reference frequency (i.e., $\omega\neq n\omega_{\rm m}$). In the case when $N$ is finite, the contribution from these unwanted frequencies can be significant, so we filter them out in the frequency domain using Fast Fourier transform and its inverse. To reduce the contribution of the numerical artifacts due to finite duration of the analyzed signal, we apply a Hamming window to $\rho_{11}(t)$ before calculating $a_{n}$ and $b_{n}$.

II.2.1 Rabi resonance: low decoherence rate

Figure 4(a) shows the simulated amplitude response for the second and fourth harmonics as a function of the Rabi frequency with fixed $\omega_{\rm m}$ and $\Gamma_{1},\Gamma_{2}\ll\omega_{\rm m}$. In agreement with Eq. 7, the amplitude $P_{2}$ of the second harmonic peaks when $\Omega_{\rm R}=2\omega_{\rm m}$. The fourth harmonic has two resonances: at $\Omega_{\rm R}=2\omega_{\rm m}$ and $\Omega_{\rm R}=4\omega_{\rm m}$. The latter peak is similar to the frequency response of a damped harmonic oscillator whose natural frequency is $\Omega_{\rm R}$ and is driven harmonically at $4\omega_{\rm m}$. In contrast, the resonant behavior of the second and fourth harmonics in the case of large phase modulation deviates from this small signal approximation, as seen in FIG. 4(b). Both harmonics’ resonance peaks shift toward smaller values of $\Omega_{\rm R}/\omega_{\rm m}$, but the shift of $P_{2}$’s peak (i.e., $\Delta_{2}$) is much more significant than that of $P_{4}$ (i.e., $\Delta_{4}$). Additionally, its shape changes drastically, including an increase in the linewidth. The value of this shift as a function of $m$ is shown in FIG. 4(c), which indicates that the fourth-harmonic Rabi resonance is more stable against variation of the modulation depth.

Furthermore, the height $P_{4,\rm r}$ of the fourth-harmonic Rabi-resonance peak corresponding to $\Omega_{R}=4\omega_{\rm m}$ scales as $m^{4}$, compared to quadratic ($m^{2}$) dependence of the second harmonic’s peak value $P_{2,\rm r}$ [FIG. 5] (note that the quadratic behavior also fails at large $m$, indicating that Eq. 7 is no longer valid). Therefore, in the large-phase-modulation regime, $P_{4}$’s peak is comparable to $P_{2}$’s, and thus it becomes advantageous to use a fourth-harmonic atomic candle (4HAC), where the Rabi resonance of the fourth harmonic is used.

II.2.2 Rabi resonance: high decoherence rate

According to Eq. 7, the width of $P_{2}$-peak should increase with increasing decoherence rates $\Gamma_{1,2}$. FIG. 6 shows that this applies to both $P_{4}$-peaks as well, and when the decoherence rate is comparable to $\omega_{\rm m}$, these two peaks overlap. Even at small $m$, the $P_{2}$ peak’s shape is different from a Lorentzian (as in FIG. 4), which indicates that Eq. 7 is no longer valid. For stronger modulation, line-broadening and asymmetry make both features unviable for practical applications.

III Experiment

III.1 Experimental setup

To verify the findings of our numerical calculations, we tested the fourth-harmonic atomic candle (4HAC) technique using our cold-atom apparatus [FIG. 1].

In our experiment we use a cloud of laser-cooled 87Rb atoms, which have undergone standard magneto-optical trapping, followed by an “optical molasses” step giving a 1-mm-wide cloud with typical atom number of $10^{8}$ and temperature of $70\ \mu$K. The atoms are optically pumped into $F=1$ ground state by switching off the repumping light 1 ms before the cooling and trapping beams are turned off and the atoms begin to expand in time-of-flight. Next, the cloud is illuminated by a travelling $\rm MW$ field emitted from an open-ended rectangular waveguide, whose phase is modulated periodically with frequency $\omega_{\rm m}$ using built-in functions of the $\rm MW$ source (SRS SG384). The near-resonant $\rm MW$ field (with carrier frequency $\omega/2\pi\approx 6.8~{}{\ \text{GHz}}$) transfers the population between the ground states $F=1$ and $F=2$. This transfer is observed by monitoring the absorption of of a weak probe laser beam resonant with $F=2\rightarrow F^{\prime}=2$ transition [FIG. 1], which is proportional to the population of state $F=2$. Since the best signal is produced when both the optical and MW fields are on resonance with the corresponding transitions, this technique is known as the double resonance (DR) Bandi2012a . The transmission signal is measured by a high-sensitivity photodetector and recorded with an oscilloscope. During the laser trapping and cooling stages, the $\rm MW$ source is detuned by $\delta/2\pi=100$ MHz from the hyperfine splitting, but during the DR interrogation it is quickly brought back to resonance by mixing with an external 100 MHz radio-frequency signal. The $\rm MW$ signal is amplified by a separate amplifier, providing about $2$ W of power to the waveguide.

To measure the response of the atoms to the modulated microwave fields, and to test the 4HAC technique, we systematically varied the carrier detuning $\delta$ to find the resonant condition. Next, once resonance was found ($\delta=0$), the phase-modulation frequency was varied to find the Rabi-resonance condition $\delta_{\rm m}=0$. The time-dependent transmission of the resonant optical beam served as a measure of the population in the $F=1$ ground state, and a frequency analysis of the transmission dynamics was performed numerically following the data acquisition.

III.2 Results and analysis

Using large modulation depths, phase-modulated MW signals applied to our cold atomic samples revealed multi-harmonic nature of the steady oscillations beyond the small-signal regime. FIG. 7 shows the first $1.2$ ms of the measured transmission signals (total duration is $5$ ms) and derived spectra for three different conditions in our atomic candle experiments. First, we demonstrate “pure” Rabi oscillations [FIG. 7(a)] by applying a resonant but unmodulated $\rm MW$ field. The spectrum here is featureless. Next, when $\rm MW$-field is strongly modulated but detuned from the hyperfine transition ($m=2\pi$, $\delta\neq 0$), the population and thus the transmission signal have additional modulations [FIG. 7(b)], and the spectrum contains peaks at integer values of $\omega_{\rm m}$. Finally, when the carrier frequency is resonant ($\delta=0$), the odd harmonics of the frequency of the phase modulation are suppressed in the transmission signal [FIG. 7(c)]. We observed that the amplitudes of the spectral components slightly varied from shot to shot under constant parameters of the MW field setup, and during some of the shots we do not see the total elimination of the odd harmonics. This might be due to noise in the response of the radio-frequency mixer, and for later analysis of the amplitudes we take average values over a few trials.

Because our experiment allows us to record only a very limited number of oscillations in the time-domain, spectral analysis using numerical FFTs does not yield a consistent set of results. Instead, we retrieve the spectral information by using the same numerical routine as in Section II.2. For additional validation we fit the time-series of the transmission signal $f(t)$ to a model

[TABLE]

where the first term accounts for the loss of atoms due to interaction with the probe; the second term accounts for longitudinal damping; the third describes damped Rabi oscillations, and the final terms correspond population dynamics at the harmonics of the modulation frequency. In this model, $A,~{}B,~{}C_{1,2},~{}P_{n},~{}\phi,~{}\phi_{n}$ are all fit parameters, where: $A$ represents the initial density of atoms at the position of the probe; $B$ is the rate of hole burning due to momentum kicks from the probe; $C_{1}$ and $C_{2}$ the amplitude of the transmission oscillations, which are proportional to the amplitude oscillations; $\phi$ and $\phi_{\rm n}$ represent the initial phase of the carrier and modulation frequencies at the start of the transmission measurement; and $P_{n}$ are the steady-state amplitudes of the population oscillation components that we are looking for in this atomic candle measurement.

To investigate practicability of the 4HAC, we analyzed the amplitudes of the second and fourth harmonics, $P_{2}$ and $P_{4}$ as a function of the modulation frequency $\omega_{\rm m}$ at a fixed $\rm MW$ power. (For these analyses, we use only the tail of the data, which corresponds to the steady-state regime.) FIG. 8 shows that the amplitudes of both harmonics peak as $\omega_{\rm m}$ is scanned through $\Omega_{\rm R}/2=2\pi\times(3.8\pm 0.3)$ kHz in a manner similar to FIG. 6(b), where $\Omega_{\rm R}$ was estimated from an analysis of a set of unmodulated signals. The uncertainty in $\Omega_{\rm R}$ is largely of the same origin as the uncertainties in the amplitudes (as was discussed before), since when the carrier-wave is not exactly on resonance with the transition, the oscillation response is at a higher frequency. By fitting a set of transmission data corresponding to unmodulated driving field, we estimate the decoherence rates in our experiment to be $\Gamma_{1}=(2.0\pm 0.1)\times 10^{4}$ s*-1* and $\Gamma_{2}=(0.97\pm 0.04)\times 10^{4}$ s*-1*. Even though in this regime Rabi resonances are not practically useful, the qualitative behavior supports our model in numerical analysis.

III.3 Experimental considerations

In Coffer2002 it was shown that increasing the probe’s power leads to a broadening of the line shapes of the atomic-candle Rabi resonances. The absorption and subsequent spontaneous emission of a photon by the atom from the upper ground state contributes to the longitudinal relaxation, and thus $\Gamma_{1}$ is proportional to the absorption rate, which in turn is proportional to the probe’s intensity.

In our experiment we can observe a similar effect, where by varying the power of the probe beam we can obtain qualitatively different oscillation patterns. FIG. 9 shows that increased probe’s power leads to a higher decoherence rate and changing it by a factor of $10$ from $0.54\ \mu$W to $0.06\ \mu$W brings the system from highly-damped [FIG. 9(a)] to underdamped [FIG. 9(b)] oscillations. This is in a good qualitative agreement with our simulations, where the damped case corresponds to FIG. 3, and a simulation of the underdamped case is shown in [FIG. 9(c)]. Since our simulation model is based on the atom as a two-level system and does not describe the absorption of the probe, it does not capture the monotonically increasing transmission over time that is experimentally observed in the case of the higher probe power. The fact that the average transmission level does not change significantly in the case of the smaller probe intensity indicates that the nonzero steady-state slope of the transmission signal is due to the hole-burning in the cloud rather than thermal expansion.

For our steady-state analysis, we want to work in a regime where the Rabi oscillations are damped. Since in our case the interrogation time is limited, this can be achieved by using a higher probe power, corresponding to a strongly-damped regime. In addition, higher probe power gives a better signal to noise ratio.

The main limitation in our experiment was the short interrogation time, limited by thermal expansion of the atomic cloud and interaction with the probe beam, which was “burning” a hole in the cloud. In addition, we observed slight discrepancies between several successive measurement of the DR signal at constant $\rm MW$ field parameters, which are probably due to fluctuations in the turn-on time of the field. These should not be an issue in the real-world configurations, such as vapor cells, where interaction with the buffer gas keeps the atoms in place, or in cold atoms whose interrogation time is increased by adding a trapping potential, e.g., an optical dipole trap. Another issue is that, due to the geometry of our experiment, it is hard to select a specific Zeeman transition by applying a bias magnetic field with a particular direction with respect to the polarization of the $\rm MW$ field. Again, in vapor-cell applications this problem usually is not present.

IV Concluding remarks

In summary, using double-resonance measurements in cold atoms, we have shown that the steady-state populations of two-level atoms interacting with a phase-modulated microwave field oscillate at multiples of the modulation frequency. We have applied the Rabi resonance technique to cold atoms, and made the first observations of higher-order harmonics in any system. In the case when the carrier wave is exactly on resonance, we observe that only the even harmonics are present, which is confirmed by numerical solution of modified Maxwell-Bloch equations. Finally, we find that it is experimentally advantageous to use large optical probe powers in these measurements to damp “regular” Rabi oscillations and observe only the response to the modulation. These observations shed new light on the dynamics of a two-level system, which is currently a workhorse of many practical quantum information applications.

In addition, we have shown that the amplitude of the oscillations at $4\omega_{\rm m}$ has a resonance when $\omega_{\rm m}$ is varied. Our simulations show that at weak modulation the fourth harmonic has two resonance peaks: when $\Omega_{\rm R}=2\omega_{\rm m}$ and $\Omega_{\rm R}=4\omega_{\rm m}$. The quartic dependence of the height of latter peak and its smaller linewidth compared to that of the Rabi $\beta$-resonance, making a possible alternative for atomic-candle applications in experiments with a weak absorption signal, e.g., in cold atoms.

By exploring in-depth the interactions between atoms and phase-modulated microwave fields, we have provided the foundations for a high signal-to-noise tool for measuring microwave fields, and thus the coupling to atomic systems. With the increasing use of 3D microwave cavities Reagor2013 ; Souris2017a , including with atomic systems Adwaith2019 , atomic candle techniques can play an important role in calibrating the microwave field strengths for accurate measures of the coupling strength between the cavity field and, for instance, a microwave qubit Grezes2014 ; Lachance-Quirion2017 ; Reed2017 . These calibrations will be especially important for techniques that rely on precise timing, such as pulse-area-based quantum memories Lvovsky2009 ; Grezes2016 ; Saglamyurek2018 and quantum transduction protocols Hafezi2012 ; Mcgee2013 ; Andrews2015 ; Kiffner2016 .

Acknowledgements.

This work was generously supported by the University of Alberta; the Faculty of Science; the Natural Sciences and Engineering Research Council (NSERC); Alberta Innovates; the Canada Foundation for Innovation (CFI), the Canadian Institute for Advanced Research (CIFAR), and the Canada Research Chairs (CRC) Program. We thank Taras Hrushevskyi, Erhan Saglamyurek and Benjamin Smith for their help with the experimental setup, and thank the Davis lab for the loan of their photodetector.