Doppler cooling thermometry of a multi-level ion in the presence of micromotion

TL;DR

This paper presents an analytical model for Doppler cooling thermometry of multi-level ions in RF traps, accounting for micromotion effects, and validates it with experiments on different energy distributions.

Contribution

The novel model extends Doppler thermometry to multi-level ions and includes micromotion effects, improving initial energy estimation accuracy.

Findings

Model accurately describes fluorescence during Doppler cooling.

Experimental validation with various energy distributions.

Effective initial energy extraction from fluorescence data.

Abstract

We study the time-dependent fluorescence of an initially hot, multi-level, single atomic ion trapped in a radio-frequency Paul trap during Doppler cooling. We have developed an analytical model that describes the fluorescence dynamics during Doppler cooling which is used to extract the initial energy of the ion. While previous models of Doppler cooling thermometry were limited to atoms with a two-level energy structure and neglected the effect of the trap oscillating electric fields, our model applies to atoms with multi-level energy structure and takes into account the influence of micromotion on the cooling dynamics. This thermometry applies to any initial energy distribution. We experimentally test our model with an ion prepared in a coherent, thermal and Tsallis energy distributions.

Figures

Equations

$$\Delta E_i=\hslash \left(k_i^{inc}-k_i^{scatt}\right)v_i+\frac{1}{3}\frac{{\left(\hslash \left|k\right|\right)}^2}{2m}{\left({\hat{k}}^{inc}-{\hat{k}}^{scatt}\right)}^2.$$

$$\Delta E_i=\hslash k_i^{inc}{v}_i+\frac{2}{3}\frac{{\left(\hslash \left|k\right|\right)}^2}{2m}.$$

$$P_{1\text{D}}^i\left({\delta }_M^i,{\delta }^i\right)=\frac{1}{\pi \sqrt{({\delta }_M^i{)}^2-({\delta }^i{)}^2}}.$$

$$\gamma ={\int }_{-{\delta }_{max}}^{{\delta }_{max}}\gamma \left(\mathbf{par},{\delta }_D\right)\cdot P_{3\text{D}}\left({\delta }_{max},{\delta }_D\right)d{\delta }_D.$$

$$\frac{\mathrm{d}{E}_i}{\mathrm{d}t}\left({\delta }_M^{x,y,z}\right)=-\hslash \iiint {\delta }^i\cdot \gamma \left(\mathbf{par},{\delta }_D={\delta }^x+{\delta }^y+{\delta }^z\right)\prod _{i=x,y,z}{P}_{1\text{D}}\left({\delta }_M^i,{\delta }^i\right)d{\delta }^{x}d{\delta }^{y}d{\delta }^z+\frac{2}{3}\frac{{\left(\hslash k\right)}^2}{2m}\gamma .$$

$$I\left(x\right)={\int }_0^{\frac{2\pi }{\omega }}C\exp \left(-\frac{{\left(x-A\cos (\omega t)\right)}^2}{2{\sigma }^2}\right)\text{d}t.PSP$$

$$x_i(t)=\left[A_i^{emm}+A_i^{sec}\cos \left({\omega }_{i}t\right)\right]\left[1+\frac{q_i}{2}\cos ({\Omega }_{\text{rf}}t)\right].MATHIEU$$

$$\begin{gathered} \mathbf{E}\left(t\right)={\mathbf{E}}_0{e}^{i\left(\mathbf{k}\cdot \left[{\mathbf{x}}_0(t)+\frac{q}{2}{\mathbf{A}}^{\mathbf{emm}}\cos \left({\Omega }_{\text{rf}}t\right)\right]-{\omega }_{laser}t\right)}= \\ {\mathbf{E}}_0{e}^{\left[i\mathbf{k}\cdot {\mathbf{x}}_0(t)\right]}\sum _n{J}_n\left({\beta }^{emm}\right)e^{i\left(n\left({\Omega }_{\text{rf}}t+\pi /2\right)-{\omega }_{laser}t\right).} \end{gathered}$$

$$\begin{gathered} {\mathbf{E}}_0{e}^{\left[i\mathbf{k}\cdot {\mathbf{x}}_0(t)\right]}= \\ {\mathbf{E}}_0\prod _i{e}^{i{k}_i\left(A_i^{sec}\cos \left({\omega }_{i}t\right)+A_i^{sec}\frac{q_i}{2}\cos \left({\omega }_{i}t\right)\cos \left({\Omega }_{\text{rf}}t\right)\right)}= \\ {\mathbf{E}}_0\prod _i{e}^{i{k}_i\left(A_i^{sec}\cos \left({\omega }_{i}t\right)+A_i^{sec}\frac{q_i}{4}\left(\cos \left(\left({\Omega }_{\text{rf}}+{\omega }_i\right)t\right)+\cos \left(\left({\Omega }_{\text{rf}}-{\omega }_i\right)t\right)\right)\right)}. \end{gathered}$$

$${⟨\frac{\text{d}N}{\text{d}t}⟩=\int P\left(E\right)\frac{\text{d}N}{\text{d}t}|}_{E_0=E}.$$

$$\begin{array}{rl}& \frac{\text{d}\hat{\rho }}{\text{d}t}=\mathcal{L}\left[\hat{\rho }\right],\\ & \mathcal{L}=-\frac{i}{\hslash }\left[\hat{H},\hat{\rho }\right]+\hat{D}.\end{array}$$

$$\hat{D}=-\frac{1}{2}\sum _i{\hat{C}}_i^{†}{\hat{C}}_i\rho +\rho {\hat{C}}_i^{†}{\hat{C}}_i-2{\hat{C}}_i^{†}\rho {\hat{C}}_i.$$

$$\begin{array}{rl}& {\hat{C}}_1=\sqrt{\frac{2}{3}{\Gamma }_{P\to S}}|1⟩⟨4|,\\ & {\hat{C}}_2=\sqrt{\frac{2}{3}{\Gamma }_{P\to S}}|2⟩⟨3|,\\ & {\hat{C}}_3=\sqrt{\frac{{\Gamma }_{P\to S}}{3}}\left(|1⟩⟨3|-|2⟩⟨4|\right),\end{array}\begin{array}{rl}& {\hat{C}}_4=\sqrt{\frac{{\Gamma }_{P\to D}}{6}}\left(\sqrt{3}|5⟩⟨3|+|6⟩⟨4|\right),\\ & {\hat{C}}_5=\sqrt{\frac{{\Gamma }_{P\to D}}{6}}\left(|7⟩⟨3|+\sqrt{3}|8⟩⟨4|\right),\\ & {\hat{C}}_6=\sqrt{\frac{{\Gamma }_{P\to D}}{3}}\left(|6⟩⟨3|+|7⟩⟨4|\right).\end{array}$$

$$\begin{array}{rl}& {\hat{C}}_7=\sqrt{2{\Gamma }_{422}}\left(|1⟩⟨1|+|2⟩⟨2|\right),\\ & {\hat{C}}_8=\sqrt{2{\Gamma }_{1092}}\left(|5⟩⟨5|+|6⟩⟨6|+|7⟩⟨7|+|8⟩⟨8|\right).\end{array}$$

$${\hat{H}}_{atom}=\sum _i^8\hslash {\omega }_i|i⟩⟨i|=\sum _{i=1}^2\hslash {\omega }_S|i⟩⟨i|+\sum _{i=3}^4\hslash {\omega }_P|i⟩⟨i|+\sum _{i=5}^8\hslash {\omega }_D|i⟩⟨i|.$$

$$\begin{array}{rl}& {\hat{H}}_{1,3}=+\frac{1}{\sqrt{3}}{\Omega }_{422}\cos (\alpha )e^{i{\omega }_{422}t},\\ & {\hat{H}}_{2,3}=-\frac{1}{\sqrt{3}}{\Omega }_{422}\sin (\alpha )e^{i{\omega }_{422}t},\\ & {\hat{H}}_{5,3}=-\frac{1}{2}{\Omega }_{1092}\sin (\beta )e^{i{\omega }_{1092}t},\\ & {\hat{H}}_{6,3}=-\frac{1}{\sqrt{3}}{\Omega }_{1092}\cos (\beta )e^{i{\omega }_{1092}t},\\ & {\hat{H}}_{7,3}=+\frac{1}{2\sqrt{3}}{\Omega }_{1092}\sin (\beta )e^{i{\omega }_{1092}t},\\ & {\hat{H}}_{8,3}=0,\end{array}\begin{array}{rl}& {\hat{H}}_{1,4}=-\frac{1}{\sqrt{3}}{\Omega }_{422}\sin (\alpha )e^{i{\omega }_{422}t},\\ & {\hat{H}}_{2,4}=+\frac{1}{\sqrt{3}}{\Omega }_{422}\cos (\alpha )e^{i{\omega }_{422}t}\\ & {\hat{H}}_{5,4}=0,\\ & {\hat{H}}_{6,4}=-\frac{1}{2\sqrt{3}}{\Omega }_{1092}\sin (\beta )e^{i{\omega }_{1092}t},\\ & {\hat{H}}_{7,4}=-\frac{1}{\sqrt{3}}{\Omega }_{1092}\cos (\beta )e^{i{\omega }_{1092}t},\\ & {\hat{H}}_{8,4}=+\frac{1}{2}{\Omega }_{1092}\sin (\beta )e^{i{\omega }_{1092}t}.\end{array}$$

$$\hat{U}=\sum _{i=1}^2{e}^{-i{\omega }_{422}t}|i⟩⟨i|+\sum _{i=5}^8{e}^{-i{\omega }_{1092}t}|i⟩⟨i|.$$

$${\hat{H}}^{\prime }=\hat{U}\hat{H}{\hat{U}}^{†}-i\hslash \hat{U}\frac{\text{d}{\hat{U}}^{†}}{\text{d}t},$$

$${\rho }^{\prime }=\hat{U}\rho {\hat{U}}^{†}.$$

$${\Delta }_{422}={\omega }_{422}-({\omega }_P-{\omega }_S),$$

$${\Delta }_{1092}={\omega }_{1092}-({\omega }_P-{\omega }_D).$$

$${\rho }^{\prime }=({\rho }_{11},{\rho }_{12},...,{\rho }_{87},{\rho }_{88})$$

$$\frac{\text{d}{\rho }_{rs}}{\text{d}t}=\sum _{kj}{L}_{rs,kj}{\rho }_{kj}.$$

$$L_{rs,kj}=-\frac{i}{\hslash }\left({\mathcal{H}}_{rk}{\delta }_{js}-{\mathcal{H}}_{js}^{†}{\delta }_{rk}\right)+\sum _m{\left(C_m\right)}_{rk}{\left(C_m^{†}\right)}_{js},$$

$$\frac{\text{d}{x}_i}{\text{d}t}=v_i$$

$$\frac{\text{d}{v}_i}{\text{d}t}=-x_i(a_i+2q_i\cos ({\Omega }_{\text{rf}}t)){\Omega }_{\text{rf}}^2/4+\hslash {\rho }_P\left({\Gamma }_{P\to S}{k}_{i,422}+{\Gamma }_{P\to D}{k}_{i,1092}\right)/m.$$

$$\frac{\text{d}{\rho }_{rs}}{\text{d}t}=\sum _{kj}{L}_{rs,kj}{\rho }_{kj}.$$

$$\begin{array}{rl}& {\delta }_{422}={\omega }_{422}-({\omega }_P-{\omega }_S)\\ & {\delta }_{1092}={\omega }_{1092}-({\omega }_P-{\omega }_D)\end{array}$$

$$\begin{array}{rl}& {\delta }_{422}={\delta }_{422}^0+k_{422}{v}_x^{inh}\left\{\cos (\left(\Omega +{\omega }_x\right)t)+\cos (\left(\Omega -{\omega }_x\right)t)\right\}\\ & {\delta }_{1092}={\delta }_{1092}^0+k_{1092}{v}_x^{inh}\left\{\cos (\left(\Omega +{\omega }_x\right)t)+\cos (\left(\Omega -{\omega }_x\right)t)\right\}\end{array}$$

$$\begin{array}{rl}& \Delta L_{422}=L\left(1,0,par\right)-L\left(0,0,par\right)\\ & \Delta L_{1092}=L\left(0,1,par\right)-L\left(0,0,par\right)\end{array}$$

$$L=L_0+2\Delta L\left\{\cos (\Omega +{\omega }_x)t+\cos (\Omega -{\omega }_x)t\right\}$$

Full text

1. I Introdution

2. II Doppler cooling thermometry

3. II.1 The eight-level system

4. II.2 Weak binding limit

5. II.3 Harmonic trap

6. II.3.1 Validity of the weak binding approximation

7. II.3.2 Experimental verification of the model.

8. II.4 Paul trap

9. II.4.1 Micromotion sidebands

10. II.4.2 Ion thermometry

11. III Conclusion

I Introdution

Doppler cooling of atoms was proposed by Hansch and Schawlow and independently by Wineland and Dehmelt in 1975 Hänsch and Schawlow (1975); Wineland and Dehmelt (1975). Here, laser light is tuned such that, due to the Doppler effect, an atom moving against the laser propagation direction has a higher probability of scattering photons than an atom moving along the laser direction. Photon scattering thus cools the atom until it reaches the Doppler temperature limit, typically in the mK range. The ability to cool trapped ions to mK temperatures is a requirement for many applications, such as quantum information Leibfried et al. (2003), optical clocks Margolis (2009) and quantum simulations Blatt and Roos (2012). Doppler cooling is also a pre-requisite for sideband cooling in which the ion is cooled to its motional ground-state.

Recently, the dynamics of recorded fluorescence during Doppler cooling of an initially hot ion was proposed and demonstrated as a tool for the evaluation of the ion energy Epstein et al. (2007); Wesenberg et al. (2007). This method is widely used for thermometry of ions Häffner et al. (2008); Allcock et al. (2010); Blakestad et al. (2009); Steiner et al. (2013); Zipkes et al. (2010a); Huber et al. (2008); Walther et al. (2012); Zipkes et al. (2010b); Abah et al. (2012); Daniilidis et al. (2011). The Doppler cooling thermometry used in these experiments assumes a simple two-level cooling transition. However, in many of these experiments, the cooling transition involved multiple levels in a $\Lambda$ transition structure. The use of a two-level model produces inaccurate results.

While for ions with a simple two-level energy structure, both theory of Doppler cooling Cirac et al. (1992) and its use for thermometry are well established Wesenberg et al. (2007), an accurate analytic model for cooling dynamics involving multiple levels in a $\Lambda$ transition structure does not exist. Such complex energy level structure has benefits in reaching sub Doppler-limit temperatures or even ground-state cooling as in EIT cooling schemes Lechner et al. (2016). Recently, a new thermometry method which uses a dark-resonance in multi-level ions was proposed Roßnagel et al. (2015). However, such dark resonance thermometry is only applicable up to tens of mK.

A further complication to the modeling of Doppler cooling and thermometry in Paul traps is the distortion of the absorption spectrum by micromotion. Micromotion is a fast, ion motion driven by the time-dependent trapping radio-frequency (rf) field. Excess micromotion can be eliminated by positioning the ion in the position where rf fields nulls Berkeland et al. (1998). Inherent micromotion is always present due to the finite amplitude of ion’s motion. Previous works on Doppler cooling theory treated the effect of micromotion by including the sidebands in the absorption spectra in the low-saturation limit Wesenberg et al. (2007), or by restricting to the well-resolved micromotion sidebands regime Cirac et al. (1994).

In this paper, we present an analytical Doppler cooling model applicable to multi-level ions, trapped in a radio-frequency Paul trap. We consider the effect of both excess and inherent micromotion outside the low-saturation limit. We also do not restrict our treatment to resolved micromotion sidebands. Our model predicts the time dependence of both energy and fluorescence during Doppler cooling. Using this model we obtain the initial energy distribution of the ion from an experimental time-resolved Doppler cooling fluorescence signal. We test our model using a single $\mathrm{Sr}^{+}$ ion in a linear Paul trap. The energy levels of $\mathrm{Sr}^{+}$ form a typical $\Lambda$ system with an eight-level manifold due to the Zeeman splitting under a constant magnetic field (Figure 1). Although our method applies to any level structure, without loss of generality, throughout this paper we will deal specifically with the level structure of $\mathrm{Sr}^{+}$, shown in Figure 1.

II Doppler cooling thermometry

Doppler cooling thermometry is based on the fact that the ions’ kinetic energy influences the fluorescence rate due to the Doppler effect. To determine the initial energy of an ion, we monitor its fluorescence during Doppler cooling. The ion interacts with a red detuned, linearly polarized, laser light. As the ion scatters photons, its energy decreases towards the Doppler cooling limit and its fluorescence rate increases. Once the ion cools to an energy where Doppler shifts are too small to induce an appreciable change to the scattering rate, Doppler cooling thermometry loses its sensitivity. In our case, the transition on which cooling occurs is $\gamma=2\pi\times 20.4\ MHz$ and the Doppler cooling signal loses sensitivity below $\approx 10\ mK$. The effect of the energy quantization of the ions motional levels is only important close to the trap ground-state and can be safely ignored in our analysis.

II.1 The eight-level system

For a two-level system, the steady state scattering rate can be expressed analytically Wesenberg et al. (2007), however, for more than two-levels, there is no closed analytic formula. We numerically calculate the steady-state scattering rate of ${}^{88}\textrm{Sr}^{+}$ by solving the coupled Bloch equations for all involved levels with given laser couplings and decay channels (see SM II.). Here the levels we consider are the $5S_{1/2}$ ground-state and the $4D_{3/2}$ meta-stable state, with a lifetime of $390\ \mathrm{msec}$. These levels are connected via allowed dipole transitions to the $5P_{1/2}$ excited state at transition wavelengths of $422\ \mathrm{nm}$ and $1092\ \mathrm{nm}$ respectively. The lifetime of the $5P_{1/2}$ level is $8\ \mathrm{nsec}$. Including all Zeeman states of these levels we get 64 coupled Bloch equations (see Figure 1). We calculate the scattering rate, $\gamma=\Gamma\rho\left(P_{1/2}\right)$, from the steady state population of the excited $5P_{1/2}$ states. To experimentally calibrate our laser couplings, we measure the scattering rate as a function of the $422\ \mathrm{nm}$ laser detuning (Figure 2). From a fit to the spectrum we find the Rabi frequencies and exact detuning of both lasers. As seen, the experimental data agrees well with the numerical solution.

With all laser parameters determined, we use the calculated spectrum to estimate the ion scattering rate at any given velocity. Here the Doppler shifts from both transitions are related through $\delta_{D}=\delta_{422}=\frac{1092}{422}\delta_{1092}$, where, $\delta_{422}=\mathbf{k_{422}\cdot v}$ and $\delta_{1092}=\mathbf{k_{1092}\cdot v}$ and we assume that both lasers are co-linear. The green solid line in Figure 3 shows the calculated Doppler shifted spectrum of the ion.

II.2 Weak binding limit

Even with the fluorescence spectrum in hand, solving the energy-time-dependent problem of the scattering rate of a trapped ion in a laser field is a computationally intensive task. One has to solve the optical Bloch equations for an eight-level system which are also coupled with motion due to Doppler shifts to get the cooling rate. To develop a computationally efficient model which also gives a physical insight into the dynamics of Doppler cooling, several approximations have to be made. The first approximation is that of weak binding, which assumes that level populations, and therefore the fluorescence rate, reach steady-state faster than the ion is moving in the trap. This approximation was also used in the case of a two-level ion Wesenberg et al. (2007); Epstein et al. (2007).

In the case of two-level atoms Wesenberg et al. (2007); Epstein et al. (2007) where the two levels are connected with a strong dipole transition (natural linewidth of several 10’s MHz), the time scale over which populations reach steady state is given by the lifetime of the excited state and is independent of the laser intensity or detuning Noh and Jhe (2010). For a multi-level atom, Doppler cooling involves additional re-pump transitions; the 1092 nm transition in our case. Typically, these transitions have natural linewidth about an order of magnitude smaller. The time scale over which populations reach steady state will depend on the intensity of the re-pump laser and will typically be an order of magnitude slower.

II.3 Harmonic trap

In this section, we concentrate on the effect of multi-level structure on Doppler cooling and ignore the effect of both inherent and excess micromotion. In linear Paul traps, the electric field in the axial direction is static. The simplified model of this section is appropriate for the determination of the ions’ energy along the axial mode. The effects of micromotion and the extension of our model to motion in the radial directions are discussed in the following sections.

Similarly to Wesenberg et al. (2007), we consider an event in which the atom scatters a single photon. From conservation of energy and momentum, we can show Wineland and Itano (1979) that the energy of the ion in each mode $(i=x,y,z)$ per single scattering event will change by:

[TABLE]

Here, $v_{i}$ is the initial velocity of the ion, $k_{i}^{inc}$ is the incident photon $k$-vector and $k_{i}^{scatt}$ is the scattered photon $k$-vector projected on the $i$-th axis. In the non-relativistic limit, the magnitude of the k-vector changes by order of $\sim v/c$ which is small and can be neglected. Since the re-scattered photon is isotropically emitted Letokhov et al. (1976), we can integrate $k_{i}^{scat}$ over all scattering angles and Equation 1 simplifies to,

[TABLE]

The cooling rate is now given by $\frac{\mathrm{d}E_{i}}{\mathrm{d}t}=\gamma\left(\mathbf{par},\delta_{D}\right)\Delta E_{i}$, where $\gamma$ is the instantaneous scattering rate which we calculated by solving the eight-level optical Bloch equations. The scattering rate depends naturally on the Doppler shift, $\delta_{D}$, and also on $\mathbf{par}$ which is a vector of experimental parameters such as laser frequencies and intensities, magnetic field and more (see SM I.). These parameters are energy independent and therefore do not change during the cooling process.

In the weak binding regime, the internal states of the ion reach steady-state much faster than the typical time scale determined by the trap. The steady state population and the instantaneous scattering rate depend on the laser parameters and the instantaneous velocity of the ion. The velocity probability distribution along a single dimension for a harmonic oscillator which also determines the Doppler shift distribution is given by,

[TABLE]

Here, $\delta_{M}^{i}$ is the maximal Doppler shift for a given ion energy, $E_{i}$, in a given trap direction $(i=x,y,z)$, $\delta_{M}^{i}=\frac{k_{i}}{2\pi}\sqrt{\frac{2E_{i}}{m}}$ and $k_{i}$ is the projection of the laser k-vector on a given trap axis.

Since the total Doppler shift, $\delta_{D}=\delta^{x}+\delta^{y}+\delta^{z}$, for a 3D harmonic oscillator is a scalar quantity, it is convenient to write a single distribution of the Doppler shift in 3D. Following the derivation of Wesenberg et al. (2007) we convolve the $P_{1\textrm{D}}^{i}$ for all axis and obtain a 3D Doppler shift distribution $P_{3\textrm{D}}=P_{1\textrm{D}}^{x}\ast P_{1\textrm{D}}^{y}\ast P_{1\textrm{D}}^{z}$. The dashed line in Figure 3 shows the convolved 3D Doppler shift distribution for an ion with a 1$\mathrm{\ K\cdot k_{B}}$ energy in each direction.

The scattering rate $\gamma$ of a hot ion is obtained by integrating the function $\gamma\left(\mathbf{par},\delta_{D}\right)$ multiplied by $P_{3D}$ over the Doppler shift,

[TABLE]

Here, $\delta_{max}=\delta_{M}^{x}+\delta_{M}^{y}+\delta_{M}^{z}$ is determined by the energy of the ion. The average energy change per scattering event is $\hbar\mathbf{k_{422}}\cdot\mathbf{v}=\hbar\delta_{D}$. We use this fact and rewrite Equation 2 as,

[TABLE]

Equation 4 and Equation 1 together determine the time-dependent fluorescence rate of the ion during Doppler cooling. While Equation 1 determines the average change in energy per unit of time, Equation 4 is used to calculate the scattering rate as a function of energy.

Equation 1 takes into account only the momentum transfer due to the cooling light because the scattering rate contains information only on the 422 nm scattering. The repump light also imparts momentum on the ion, however, this effect is smaller by a factor of $\frac{\lvert\mathbf{k_{1092}}\rvert}{\lvert\mathbf{k_{422}}\rvert}\frac{\mathrm{\Gamma_{1092}}}{\mathrm{\Gamma_{422}}}\approx 0.02$. We can include this effect by scaling the $\gamma\rightarrow\gamma\cdot\frac{\mathrm{\Gamma_{1092}}}{\mathrm{\Gamma_{422}}}$ and $\delta^{i}\rightarrow\delta^{i}\cdot\frac{\lvert\mathbf{k_{1092}}\rvert}{\lvert\mathbf{k_{422}}\rvert}$ in Equation 1 and summing both contributions.

II.3.1 Validity of the weak binding approximation

The timescale, $\tau$, in which the excited state reaches steady state depends greatly on the laser parameters. From a numerical solution of the eight-level Bloch equations, we observe that by saturating the $D_{3/2}\rightarrow P_{1/2}$ transition we can reduce $\tau$ and hence regain the weak-binding limit (inset of Figure 4). The timescale also depends on laser detuning and polarization. It is important to verify the validity of the weak binding approximation before using the Doppler cooling model.

To validate our model, we compare it with the result of an optical Bloch simulation (OBS) in which we do not assume steady state Meir et al. (2017a). Instead, we obtain the instantaneous scattering rate by propagating in time the 64 coupled dynamical Bloch equations of the 8-level system together with the equations of motion of the ion in the presence of a scattering force (see SM III.). The OBS is computationally very demanding and time-consuming which makes it impractical as a regular thermometry tool. Furthermore, the OBS does not provide any physical insight into the mechanisms that affect Doppler cooling. However, it does produce an exact result and can be used in regimes where the assumption of our model are no longer valid.

Figure 4 compares the photon scattering rate dynamics calculated by our model to the OBS result using different repump parameters. As expected, for weak repump intensities the weak-binding limit does not hold, and our model does not reproduce the OBS result, while for strong repump we see good agreement between the OBS and our model.

II.3.2 Experimental verification of the model.

To experimentally validate our model, we compare two different thermometry methods on a single ${}^{88}\mathrm{Sr}^{+}$ ion. We initialize the ion in a classical coherent state using an oscillating rf electric field drive, on-resonance with the axial trap frequency ($\omega_{ax}=417.5\ kHz$), with a constant rf drive power. We initialize the ion in different energies by changing the rf drive pulse length.

The first thermometry we use is the Doppler cooling thermometry. At the end of the ion’s initialization, we turn on the cooling and repump lasers and monitor the ion’s fluorescence for 10 ms with 30 $\mu$s binning. We repeat this process 200 times. In Figure 5 we show the experimental results together with a single parameter fit (initial energy) to the Doppler cooling model. The good agreement between the model and the experimental time-dependent fluorescence signal suggests that the ion was prepared in a well-defined energy and that our model is valid.

We compare the energies that we obtained from the model with an alternative method for measuring the ions’ energy when it is in a classical coherent state. In this method, we image the ion on a CCD and extract its energy from the shape of the intensity profile. For a 1D classical coherent state, the intensity profile is given by:

[TABLE]

Here, $\sigma$ is the width of our imaging system point-spread-function and $A$ is the ion’s oscillation amplitude with frequency $\omega/2\pi$ from which we determine the energy of the ion: $E=\frac{1}{2}mA^{2}\omega^{2}$. We used on-resonance light such that Doppler shifts only reduce the scattering rate. In the analysis, we neglect the Doppler shift effect on the intensity profile since we are interested only in the oscillation amplitude. To prevent any mechanical effects of laser light on the energy of the ion, we image the ion with low laser intensity ($\Omega_{422}\approx 0.1$ MHz) during 10 ms CCD exposures and repeat the measurement 5000 times to improve the signal-to-noise ratio. The experimental images are also shown in Figure 5.

A comparison between the energy extracted from the Doppler cooling fit and the CCD images for different energies are shown in Figure 5. The initial energies of the ion obtained by the two methods agree well.

II.4 Paul trap

In the preceding sections, we discussed the dynamics of Doppler cooling in a true harmonic potential while neglecting the oscillating fields of the ion’s Paul trap. In linear Paul traps only one axis is static. In the remaining axes, the motion of the ion is a superposition of a slow frequency ($\omega/2\pi$) and fast frequency ($\Omega_{rf}/2\pi$) motion,

[TABLE]

Here, $|q_{i}|<1$ is the Mathieu parameter which nulls for the static axis and has a finite value for the other two axes. $A_{i}^{sec}$ is the harmonic motion amplitude which is now superimposed with a fast modulation. We refer to this motion as inherent micromotion since it is proportional to the ions harmonic amplitude. The displacement of the ion from the rf null due to stray electric fields is $A_{i}^{emm}$. This displacement results in excess-micromotion with an amplitude $q_{i}A_{i}^{emm}/2$. While excess-micromotion is an artifact in Paul traps and can be reduced to negligible values Keller et al. (2015) the inherent part of micromotion is an intrinsic aspect of the ion’s motion.

Because the micromotion velocity typically changes faster than the time in which the internal states of the atom reach steady-state, the absorption and emission of a photon can not be localized in phase space, and the weak binding approximation breaks. The analysis of the cooling process, therefore, requires a more involved approach.

II.4.1 Micromotion sidebands

Micromotion affects Doppler cooling by modifying the spectrum. To evaluate the effect of micromotion on the scattering rate we move to the micromotion reference frame. In this frame, the ion still undergoes a secular motion, but there is no micromotion. The electric field in the new reference frame is modulated at the rf frequency, $\Omega_{\textrm{rf}}$.

In the case of excess micromotion, the ion senses additional sidebands with frequencies $\omega_{laser}\pm n\Omega_{\textrm{rf}}$, where $n$ is an integer number. We express the electric field in the presence of excess micromotion as,

[TABLE]

Here, $\mathbf{x_{0}}(t)$ is the secular and inherent part of the ion’s motion and $\beta^{emm}=\frac{1}{2}\sum_{i=x,y}q_{i}k_{i}A_{i}^{emm}$ is the excess-micromotion modulation index. To first order in $\beta^{emm}$, excess-micromotion adds two sidebands to the ion spectrum at $\omega_{laser}\pm(\Omega_{\textrm{rf}})$.

The inherent micromotion part of $\mathbf{x_{0}}(t)$ modulates the electric field twice, once in the harmonic trap frequency $\omega_{i}$ and second in the rf frequency $\Omega_{\textrm{rf}}$. The electric field in the presence of inherent micromotion can be written as:

[TABLE]

An expansion to Bessel functions can be performed here as well (see SM V.). To first order in the inherent micromotion modulation index, $\beta^{imm}_{i}=\frac{1}{4}q_{i}k_{i}A_{i}^{sec}$, this modulation adds four more additional sidebands at $\pm(\Omega_{\textrm{rf}}\pm\omega_{i})$ for each of the radial modes, $i=x(y)$. The modified spectrum is obtained by recalculating the spectrum with amplitudes of sidebands which are obtained from the Bessel series expansion as we did in Equation 8. The relative intensities of the inherent micromotion sidebands in first order can be expressed as a square of electric field components at frequency $\omega_{laser}\pm\Omega_{\textrm{rf}}\pm\omega_{x(y)}$ (for more details see SM V. Oberst (1999)). The scattering rate is calculated by using the modified spectrum in Equation 4.

II.4.2 Ion thermometry

In the previous section, we have shown how micromotion affects the scattering rate by introducing motional sidebands. Here we will examine the effect of these sidebands on Doppler cooling fluorescence signal.

Similarly, to what we did in section C.2, we compare our results to those of OBS. Extension of OBS from harmonic trap to Paul trap is straightforward (see SM III.). In Figure 6 we compare the time evolution of fluorescence under Doppler cooling of an ion with initial energy $E=1\ \textrm{K}\cdot\textrm{k}_{\textrm{B}}$ in the $y$ direction calculated using OBS, and our model with and without including first-order micromotion sidebands. As seen, incorporating micromotion sidebands is necessary for our model to better match with the OBS result. With the inclusion of higher-order sidebands, agreement is expected to improve. The need to include higher order sidebands increases with higher initial energies.

As an application of our model we determine the energy distribution of an ion in two different cases. Generally, the ion is not found in a specific energy state but rather in a statistical distribution of energies, $P\left(E\right)$. The averaged scattering rate is obtained from our model by weighting the fluorescence curves according to the distribution,

[TABLE]

Here, we test our model with two different distributions (Figure 7). The Tsallis distribution, $P(E)\propto E^{2}/(1+E/n\text{k}_{\text{B}}T)^{n}$, which evolves after the ion collides multiple times with ultracold atoms DeVoe (2009); Zipkes et al. (2011); Chen et al. (2014). In our example, the energy scale, $\mathrm{k_{B}}\cdot T$, is determined by the intense excess micromotion, $E_{emm}=250\ \mathrm{mK\cdot k_{B}}$, induced in this experiment. The power-law, $n=4$, of the distribution is determined from a molecular-dynamics simulation Meir et al. (2016, 2017b). The second distribution we study is thermal, $P(E)\propto E^{2}e^{-E/\text{k}_{\text{B}}T}$. This distribution is produced by applying white voltage noise to one of the trap electrodes. We determine the heating rate due to this white noise using carrier Rabi spectroscopy on a narrow linewidth transition to be $305\ \mathrm{mK\cdot k_{B}/sec}$ (SM IV.).

We use our model to extract the distribution parameters from the experimental fluorescence curves. For the ion with a Tsallis energy distribution (red in Figure 7) the cooling rate is slower due to the broader energy distribution. We extract the ion’s ”temperature” Meir et al. (2016), $T_{\text{ion}}=Tn/(n-2)$ from a fit to our model. The temperature, $T_{\text{model}}=155\pm 13\ \mathrm{mK}$ agrees with the molecular dynamics simulation results, $T_{\text{sim}}=0.62*E_{emm}+7=162\ \mathrm{mK}$. The scaling is slightly different than in the referenced article due to different trap parameters used in this experiment Meir et al. (2016). For the ion with a Maxwell-Boltzmann thermal distribution, the cooling rate is faster due to the relatively narrower energy distribution. We extract the ions’ temperature $T_{\text{model}}=1080\pm 50\ \mathrm{mK}$ using our model. This temperature agrees with the heating rate extrapolation of $T_{\text{extrap}}=915\pm 45\ \mathrm{mK}$ using Rabi thermometry Leibfried et al. (2003).

III Conclusion

In this paper, we developed a model of Doppler cooling of a single multi-level ion trapped in a rf Paul trap. Our analysis includes the effects of both excess and inherent micromotion. Our model is a simple tool for understanding Doppler cooling dynamics. This model can be used for ion thermometry in the range of 10’s mK to 10’s K. Below 10 mK Doppler shifts are too small to result in sufficient changes to the scattering rate. Above 10’s K more and more micromotion sidebands have to be included which complicates the calculation significantly. With a good experimental signal-to-noise ratio, our model can also be used to distinguish between different energy distributions. We have bench-marked our method using coherent, thermal and non-thermal Tsallis energy distributions for energies between 0.5 to 3 $\mathrm{K\cdot k_{B}}$, and obtained a good agreement with alternative measurement methods and simulations. Doppler thermometry is a practical method because it requires only the same lasers used for Doppler cooling. It is an important tool for studying non-linear dynamics in ion crystals Lin and Duan (2011),transport of ions Blakestad et al. (2009); Walther et al. (2012); Huber et al. (2008) or atom-ion collisions Meir et al. (2016); Zipkes et al. (2010b, a). It can also serve as an important method for future thermodynamics experiments in ion traps Dawkins et al. (2016); Abah et al. (2012).

This work was supported by the Crown Photonics Center, ICore-Israeli excellence center circle of light, the Israeli Science Foundation, the U.S.-Israel Binational Science Foundation, and the European Research Council (consolidator grant 616919-Ionology). We thank Asaf Miron for helpful conversations and for remarks on the manuscript.