Fast thermometry for trapped ions using dark resonances

For a quantitative determination of the ion temperature from the measured fluorescence spectra, a model for the theoretical lineshape is required. Therefore we describe the dynamics of the system by a Lindblad master equation, the time evolution of the density matrix. We consider the Hamiltonian of the system

$$\begin{eqnarray}&&\mathcal{H}={{\mathcal{H}}_{{\rm dip}}}+{{\mathcal{H}}_{{\rm ion}}}\end{eqnarray} \tag{ 1 }$$

which is composed of ${{\mathcal{H}}_{{\rm dip}}}$, the ion-light dipole interaction, and ${{\mathcal{H}}_{{\rm ion}}}$, describing internal states of the ion including the detunings of the two light fields ${{\Delta }_{{\rm a}}}$ and ${{\Delta }_{{\rm b}}}$ [30–32]. The motional state of the ion, not included in the Hamiltonian, will be introduced later as a spectral Doppler shift of the detunings ${{\Delta }_{{\rm a},{\rm b}}}$.

The Lindblad master equation for the density matrix $\hat{\rho }$ reads [33]

$$\begin{eqnarray}&&\frac{{\rm d}\hat{\rho }}{{\rm d}t}=-\frac{{\rm i}}{\hbar }\left[ \mathcal{H},\hat{\rho } \right]+\mathcal{L}(\hat{\rho }).\end{eqnarray} \tag{ 2 }$$

The spontaneous decay of the population of the excited levels is introduced by the dissipative Lindblad operator

$$\begin{eqnarray}&&\mathcal{L}(\hat{\rho })=\mathop{\sum }\limits_{i}\left[ {{\mathcal{C}}_{i}}\hat{\rho }\mathcal{C}_{i}^{\dagger }-\frac{1}{2}\left( \mathcal{C}_{i}^{\dagger }{{\mathcal{C}}_{i}}\hat{\rho }+\hat{\rho }\mathcal{C}_{i}^{\dagger }{{\mathcal{C}}_{i}} \right) \right]\end{eqnarray} \tag{ 3 }$$

where the transition operators ${{\mathcal{C}}_{i}}=\sqrt{{{\Gamma }_{pq}}}|q\rangle \langle p|$ describe the dissipation along the dipole transitions $p\to q$. They depend on the decay rates according to Fermi's golden rule [31, 34]

$$\begin{eqnarray}&&{{\Gamma }_{{\rm ab}}}=\frac{8{{\pi }^{2}}}{3{{\epsilon }_{0}}\hbar {{\lambda }_{pq}}}|\langle p|\vec{D}|q\rangle {{|}^{2}}\end{eqnarray} \tag{ 4 }$$

with the atomic dipole operator $\vec{D}$ and the transition wavelength ${{\lambda }_{pq}}$.

The spectral widths of the light fields ${{\Gamma }_{{\rm a}}}$ and ${{\Gamma }_{{\rm b}}}$ are included as an additional broadening of the corresponding atomic level $|q\rangle$. Phase fluctuations between the atomic state and the light field lead to a decay of the dark state and are modeled by transition operators ${{C}_{{\rm a},{\rm b}}}=\sqrt{{{\Gamma }_{{\rm a},{\rm b}}}}|q\rangle \langle q|$ [30, 34] (see appendix equations 16 and 17).

To obtain the resulting spectrum of the fluorescence light the steady-state solution ${\rm d}\hat{\rho }/{\rm d}t=0$ is required. In the following we will solve this equation numerically. We transform equation (2) into a matrix equation, in which the density matrix $\hat{\rho }$ with dimension $N\times N$ is represented by a vector with N² entries, where N is the number of atomic levels involved. Consequently, the master equation with the Lindblad operator L_ik in matrix form is described by

$$\begin{eqnarray}&&\frac{{\rm d}{{\rho }_{i}}}{{\rm d}t}=\mathop{\sum }\limits_{k=1}^{{{N}^{2}}}{{L}_{ik}}{{\rho }_{k}}=0\quad {\rm with}\quad i\in \left\{ 1,\ldots ,{{N}^{2}} \right\}.\end{eqnarray} \tag{ 5 }$$

One of these equations (for simplicity i = 1) is replaced by the normalization condition ${\rm Tr}(\hat{\rho })=1$. Finally, to obtain the steady-state solution ${{\rho }^{0}}$ we solve equation (5) by numerical matrix inversion. The fluorescence rate $\mathcal{F}$ is proportional to the population in the excited state(s). To obtain a fluorescence spectrum including the dark resonances, as shown in figure 1(b), $\mathcal{F}$ has to be calculated for every detuning Δ_i of the corresponding light field within the range of interest.

Now, we take the finite temperature of the ion into account, assuming a thermal state at temperature T. For conditions with phonon numbers $n\gg 1$, the motional state of the ion can be introduced in a semi-classical way. We consider the two light fields to have an angle α with respect to each other. A velocity $\vec{v}$ of the ion results in a Doppler shift of the two frequencies ${{\omega }_{{\rm a}}}$ and ${{\omega }_{{\rm b}}}$ and thus shifts their detuning by $\delta {{\Delta }_{i}}(v,\alpha )={{\vec{k}}_{i}}\cdot \vec{v}$, where ${{\vec{k}}_{i}}$ are the corresponding wave-vectors. Through the inclusion of the Doppler shifted detunings in the calculation of the spectrum we obtain a fluorescence rate $\mathcal{F}(\alpha ,\vec{v})$ depending on α and $\vec{v}$. We obtain a temperature-dependent fluorescence rate $\tilde{\mathcal{F}}(T)$

$$\begin{eqnarray}&&\tilde{\mathcal{F}}(T)=\int {{P}_{T}}(\vec{v})\cdot {{\mathcal{F}}_{\alpha }}(\vec{v})\ {\rm d}\vec{v},\end{eqnarray} \tag{ 6 }$$

by integrating over all velocity classes in the Maxwell-Boltzmann distribution

$$\begin{eqnarray}&&{{P}_{T}}(\vec{v})={{\left( \frac{m}{2\pi {{k}_{{\rm B}}}T} \right)}^{3/2}}{{{\rm e}}^{\left( -m{{{\vec{v}}}^{2}}/2{{k}_{{\rm B}}}T \right)}}\end{eqnarray} \tag{ 7 }$$

with atomic mass m and Boltzmann-constant ${{k}_{{\rm B}}}$. For a numerical solution of equation (6) we discretize $\vec{v}$ into distinct velocity classes along the directions of the laser beams. The line shape of the dark resonances as a function of the temperature T is shown in figure 2(b). To determine the ion temperature the measured fluorescence data is fitted by $\tilde{\mathcal{F}}(T)$ using a Markov chain Monte Carlo parameter estimation [35].

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For ⁴⁰Ca⁺ ions with an electronic ground-state configuration of [Ar]$\;4{{\;}^{2}}{{S}_{1/2}}$, having one single s electron above a closed shell, the energetic structure can be approximated by a three-level system with fine structure levels $^{2}{{S}_{1/2}}=|1\rangle$, $^{2}{{D}_{3/2}}=|3\rangle$ and $^{2}{{P}_{1/2}}=|2\rangle$; see figure 2(a). The ${{S}_{1/2}}\leftrightarrow {{P}_{1/2}}$ dipole transition is used for laser cooling and is driven by a laser near $397\;$ nm. In order to reach a closed cycling transition all the population which decays into the metastable state ${{D}_{3/2}}$ with a lifetime of $1.2\;$ s is pumped back to the ${{P}_{1/2}}$ state by a laser near $866\;$ nm. In our experimental setting both laser frequencies are red detuned with respect to the corresponding atomic transition ${{\Delta }_{397,866}}\lt 0$. To obtain a resonance fluorescence spectrum, scattered photons near $397\;$ nm are recorded as a function of the detuning Δ₈₆₆, while Δ₃₉₇ is kept constant. Note, that scanning Δ₃₉₇ is unfavorable for thermometry, as its detuning has a strong influence on the cooling rate. As a magnetic field is applied to define the quantization axis, the degeneracy of the electronic states is lifted, leading to an eight-level system [36, 37]; see figure 2(a). Depending on the polarizations of the laser beams up to 12 dark resonance lines can be observed. In our experiment the polarizations of both beams are chosen such that only ${{\sigma }^{+}}$ and ${{\sigma }^{-}}$ transitions are addressed, leading to the appearance of four dark resonances (figure 2). Regarding the ${{P}_{1/2}}\leftrightarrow {{D}_{3/2}}$ transition this polarization is avoiding dark states in the ${{D}_{3/2}}$ manifold. For the ${{S}_{1/2}}\leftrightarrow {{P}_{1/2}}$ transition a ${{\sigma }^{+}}+{{\sigma }^{-}}$ polarization is convenient, as it leads to an excitation of the transitions which are maximally apart and therefore most easily resolved.

The corresponding matrix form of $\mathcal{H}$ as well as the coupling operators ${{\mathcal{C}}_{i}}$ are given in the appendix. The Zeeman splitting leads to a 64-dimensional Lindblad operator L_ik in equation (5) and thus to a time consuming matrix decomposition. To reduce the computational complexity we approximate the effect of temperature on the dark resonances by a relative Doppler broadening ${{\Gamma }_{{\rm D}}}(T,\alpha )$. It can be deduced from the variance of the relative Doppler shift $\delta \Delta =\delta {{\Delta }_{397}}-\delta {{\Delta }_{866}}$ and reads

$$\begin{eqnarray}&&{{\Gamma }_{{\rm D}}}(T,\alpha )=\left| {{{\vec{k}}}_{397}}-{{{\vec{k}}}_{866}} \right|\sqrt{\frac{{{k}_{{\rm B}}}T}{2\;m}}=\sqrt{k_{397}^{2}+k_{866}^{2}-2\;{{k}_{397}}{{k}_{866}}{\rm cos} (\alpha )}\;\sqrt{\frac{{{k}_{{\rm B}}}T}{2\;m}}.\end{eqnarray} \tag{ 8 }$$

Regarding the shape of the dark resonances, the laser linewidths and the Doppler broadening act in a similar way. Therefore, ${{\Gamma }_{{\rm D}}}$ is treated as an additional contribution to the broadening due to the spectral widths of the lasers. During the following applications both methods, using equation (6) and equation (8), have been compared for different spectra and lead to consistent results for the temperatures within their uncertainties. The thermal broadening ${{\Gamma }_{{\rm D}}}(T,\alpha )$ in equation (8) shows a strong dependence on the relative orientation α of the k-vectors of the two lasers. The strongest influence of temperature, and thus the highest sensitivity even at low temperatures, is achieved for counter-propagating laser beams ($\alpha =\pi$), where Δ₃₉₇ and Δ₈₆₆ are Doppler shifted with opposite signs. Even for moderate temperatures of a few millikelvin the contrast of the dark resonances becomes too small to assign a reliable temperature. In this range, however, the best choice is a collinear beam configuration ($\alpha =0$), which has reduced sensitivity, but is well suited for a wide range of temperatures up to several tens of millikelvin, as shown in figure 2(b).

The modification of the photon scattering rate on the laser cooling transition in the vicinity of dark resonances affects the equilibrium ion temperature. In the following this effect is analyzed in a semi-classical picture, where the Doppler shift reduces the light scattering of selected velocity classes.

For Doppler cooling, during which the $866$ nm laser may be far detuned from dark resonances, the cooling rate is dominated by the spontaneous emission lifetime $\tau =7\;$ ns for the decay of ${{P}_{1/2}}$ into ${{S}_{1/2}}$ [38–40]. The period of the secular motion is much larger than the lifetime, $\omega \ll 1/\tau$ (weak binding regime), and hence the ion does not move considerably during the scattering process [41, 42]. The steady state of the internal dynamics is typically reached after ${{10}^{-7}}$ s and therefore, in our case, much faster than any secular motion. In the vicinity of a dark resonance the velocity dependent Doppler cooling force is affected by the modified scattering rate. To analyze its influence we introduce the effective Doppler shift $\delta {{\Delta }^{{\rm DR}}}$ of the relative frequency detuning of the laser frequencies with respect to a dark resonance ${{\Delta }^{{\rm DR}}}={{\Delta }_{866}}-{{\Delta }_{397}}$. Including $\delta {{\Delta }^{{\rm DR}}}$ the condition for a dark resonance reads

$$\begin{eqnarray}&&\Delta _{\delta }^{{\rm DR}}={{\Delta }^{{\rm DR}}}-\delta {{\Delta }^{{\rm DR}}}={{\Delta }_{866}}-{{\Delta }_{397}}-\left( {{\nu }_{866}}-{{\nu }_{397}} \right)\frac{\vec{v}\cdot \vec{e}}{c}=0,\end{eqnarray} \tag{ 9 }$$

where ${{\nu }_{i}}$ are the absolute frequencies of the lasers, $\vec{e}$ is their direction and $\vec{v}$ the velocity vector of the ion. The relative Doppler shift $\delta {{\Delta }^{{\rm DR}}}$ has the opposite sign of the two individual Doppler shifts, because ${{\nu }_{866}}\lt {{\nu }_{397}}$. If the ion moves in the same direction as the laser beams $\vec{v}\uparrow \uparrow {{\vec{k}}_{i}}$, photon scattering leads to an amplification of the motion of the ion. This heating process limits the achievable temperature during Doppler cooling [42]. However, if the frequency of the $866$ nm laser is red detuned with respect to the dark resonance (${{\Delta }^{{\rm DR}}}\lt 0$), those velocity classes with $\vec{v}\uparrow \uparrow {{\vec{k}}_{i}}$ are Doppler shifted into resonance $\Delta _{\delta }^{{\rm DR}}\approx 0$. Photon scattering then is suppressed and the resulting excitation of the ion motion is reduced. As a consequence, for ${{\Delta }^{{\rm DR}}}\lt 0$ a lower temperature can be achieved, similarly to the velocity dependent coherent population trapping (VSCPT) schemes for samples of neutral atoms [43].

In a complementary analysis we show that scattering on motional sidebands of the coherent two photon Λ-transition ${{S}_{1/2}}\leftrightarrow {{D}_{3/2}}$ is enhanced near dark resonances. In figure 3 we present the result of a numerical analysis of the populations of the individual ${{S}_{1/2}}$ and ${{D}_{3/2}}$ sublevels in the vicinity of dark resonances. On resonance with the ${{P}_{1/2}}\leftrightarrow {{D}_{3/2}}$ transition, far detuned from any dark resonances with ${{\Delta }^{{\rm DR}}}\ll 0$, the decay from ${{P}_{1/2}}$ to ${{S}_{1/2}}$ is favored over the decay from ${{P}_{1/2}}$ to ${{D}_{3/2}}$ and therefore the population is accumulated in the ${{S}_{1/2}}$ states [44, 45]. However, this situation is inverted if ${{\Delta }^{{\rm DR}}}\approx 0$. As can be seen from figure 3 the population is then higher in the ${{D}_{3/2}}$ manifold than in the ${{S}_{1/2}}$ states. For this reason the direction of the Λ-transition from ${{D}_{3/2}}$ to ${{S}_{1/2}}$ is favored over the reverse direction. As a consequence, for a negative detuning ${{\Delta }^{{\rm DR}}}\lt 0$ the effective transition from ${{D}_{3/2}}$ to ${{S}_{1/2}}$ removes phonons from the motional mode, while phonons are added for ${{\Delta }^{{\rm DR}}}\gt 0$ [18]. The cooling cycle is closed by an excitation into the ${{P}_{1/2}}$ states and subsequent spontaneous decay. In the steady-state solution each ${{S}_{1/2}}$ sublevel is effectively coupled to all resonances because a strong ${{S}_{1/2}}\leftrightarrow {{P}_{1/2}}$ scattering ensures a constant redistribution of the population within the ${{S}_{1/2}}$ manifold. Therefore, the population of both ${{S}_{1/2}}$ levels shows four resonances. The different shapes of the population of the two ${{S}_{1/2}}$ sublevels is are due to the detuning of the $397\;$ nm laser, which leads to different scattering rates.

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The experiments are performed with ⁴⁰Ca⁺ ions, trapped in a linear Paul trap with four radial electrodes of diameter $0.8\;$ mm and a distance to the trap axis of $1.5\;$ mm. Each two diagonally opposing electrodes are driven by the same phase of a radio frequency at ${{\omega }_{{\rm RF}}}/2\pi =21.45\;$ MHz, but with a shift of π with respect to the other two electrodes. A peak to peak voltage of about $900\;$ V leads to radial frequencies ${{\omega }_{x}}/2\pi$ and ${{\omega }_{y}}/2\pi$ of $468\;$ kHz and $472\;$ kHz, respectively. Applying a constant voltage of $9\;$ V to the end-caps at a distance of $8\;$ mm from each other results in an axial trap frequency of ${{\omega }_{{\rm ax}}}/2\pi =170\;$kHz. The ⁴⁰Ca⁺ ions are loaded into the trap by ionizing neutral calcium atoms effusing from a thermal oven using a resonant two step photoionization process. The ions are Doppler cooled on the ${{S}_{1/2}}\leftrightarrow {{P}_{1/2}}$ transition. A static magnetic field of $4.7\times {{10}^{-4}}$ T is applied at 45° to the trap axis, which lifts the degeneracy of the fine structure levels. The laser system consists of two extended-cavity diode lasers. One amplified (tapered amplifier) and frequency doubled (second harmonic generation) diode laser provides laser light at $397\;$ nm (Toptica TA-SHG 110) and a second diode laser at $866\;$ nm (Toptica DL100). A sketch of the full experimental setup is shown in figure 4. Both of the lasers are locked to an external reference cavity using the Pound-Drever-Hall technique. From the shape of the in-loop Pound-Drever-Hall error signal we have deduced the full linewidths at half maximum of the two lasers as $450\pm 30\;$ kHz for the $397\;$ nm laser and $490\pm 30\;$ kHz for the $866\;$ nm laser. Acousto-optical modulators (AOMs) are installed in the beam paths of both lasers, using a double pass configuration to switch the beams on and off, as well as to allow for frequency shifts. A field-programmable gate array (FPGA) controls an rf-switch to change between two different rf-drive frequencies of the AOM in the branch of the $866\;$ nm laser and thus allows for fast experimental sequences with nanosecond precision. For the spectral scans a constant intensity of the $866\;$ nm laser beam over a range of $40\;$ MHz is required. To avoid power fluctuations due to a frequency dependent diffraction efficiency of the AOM (Brimrose TEF-270-100), the beam intensity is measured on a photodiode (PD) as a function of the AOM drive frequency, and its driving power is corrected accordingly.

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The two laser beams are oriented parallel to one another and perpendicular to the magnetic field (beam path A in figure 4). They have an angle of 45° to the axis of the trap and 60° to the radial directions, and thus all vibrational modes are addressed simultaneously. The polarization of both laser beams is chosen by waveplates to be perpendicular to the magnetic field as well, leading to a superposition of ${{\sigma }^{+}}$ and ${{\sigma }^{-}}$ polarization with respect to the quantization axis. The intensities of both laser beams are $35\;\mu {\rm W}$ at a beam waist of $160\;\mu {\rm m}$ for the $397\;$ nm laser and $80\;\mu {\rm W}$ at a beam waist of $380\;\mu {\rm m}$. A flip-mirror permits us to change the beam path of the $397\;$ nm laser to propagate parallel to the direction of the magnetic field (beam path B; dashed line in figure 4). The polarization of this beam path is again chosen by waveplates such that both ${{\sigma }^{+}}$ and ${{\sigma }^{-}}$ transitions are driven. For beam path B the orientations of the two laser beams are perpendicular to one another and the Doppler shift does not cancel to first order anymore. Consequently, this configuration leads to a higher sensitivity on the ion movement and therefore is more sensitive to temperature differences.

The fluorescence light at $397\;$ nm is collected by an objective lens system with a numerical aperture of 0.26. The image of the trapped ions is projected onto a CCD camera. A pickup mirror reflects $30\%$ of the collected photons onto a photomultiplier tube (PMT). The camera image is used for trapping, controlling and preparing the ions. During the experimental sequences we use the PMT to be able to count single photons, which enables us to make measurements with exposure times of a few microseconds. The PMT events are counted by a $40\;$ MHz micro-controller (MC), which is triggered by the FPGA to guarantee precise timing with respect to the laser pulses.

To obtain a spectrum of resonance fluorescence the frequency of the $866$ nm beam is scanned via the drive frequency of the corresponding AOM. We can chose two different modes of operation. First, a wide scan of the spectrum including all dark resonances is recorded, where the frequency is tuned continuously over the regions of interest and scattered photons near 397 nm are recorded. A fit of the data by the calculated fluorescence rate $\mathcal{F}({{\Delta }_{866}})$ allows us to determine Rabi frequencies ${{\Omega }_{397}}$ and ${{\Omega }_{866}}$, the detuning of the $397$ nm laser ${{\Delta }_{397}}$ and the magnitude of the magnetic field B.

To greatly reduce the influence of scattered photons on the temperature of the ion we use an alternative mode of data acquisition, which consists of two steps. (i)For an equilibration time ${{t}_{{\rm e}}}\geqslant 2\;$ ms the frequency of the $866\;$ nm beam is set at some fixed detuning ${{\Delta }_{{\rm e}}}$, resulting in a steady state of the ion temperature. (ii)Then, for a much shorter measurement time of typically ${{t}_{{\rm m}}}=20\;\mu {\rm s}$, the drive frequency of the AOM is switched to the detuning ${{\Delta }_{{\rm m}}}$. Only during ${{t}_{{\rm m}}}$ are scattered fluorescence photons recorded on the PMT. The sequence of steps (i) and (ii) is repeated multiple times to detect in total about 10⁴ photons to reduce the statistical error. Then, ${{\Delta }_{{\rm m}}}$ is stepped to the next value to obtain the full shape of one dark resonance to determine the temperature T. All other parameters, such as B-field or Rabi frequencies, had been determined from the scan over the full spectrum. Unless noted otherwise, the collinear beam configuration (beam path A) is employed to make use of a maximum range of temperatures which are accessible.

The multi-level system of ⁴⁰Ca⁺ ions allows us to tailor the line shape and vary the resulting temperature. Here, we demonstrate how the temperature can be controlled and measured in the vicinity of the dark resonance at $\Delta _{0}^{{\rm DR}}$ corresponding to the $|{{D}_{3/2}},-3/2\rangle$ state; see figure 5. We apply the measurement procedure where the frequency of the $866$ nm laser is switched, as described above. If ${{\Delta }_{{\rm e}}}$ is close to the ${{P}_{1/2}}\leftrightarrow {{D}_{3/2}}$ transition and far from any dark resonances during the equilibration ${{t}_{{\rm e}}}$ in step (i), we measure a temperature of $3.1(4)$ mK. However, if the frequency of the $866$ nm laser is close to a dark resonance but red detuned ${{\Delta }_{{\rm e}}}\lt \Delta _{0}^{{\rm DR}}$ during ${{t}_{{\rm e}}}$, we see that the temperature drops by a factor of four and measure $0.7(4)$ mK at ${{\Delta }_{{\rm e}}}=\Delta _{0}^{{\rm DR}}-1.2\;$ MHz. In contrast, we observe strong heating effects if ${{\Delta }_{{\rm e}}}$ is blue detuned to the dark resonance ${{\Delta }_{{\rm e}}}\gt \Delta _{0}^{{\rm DR}}$. At ${{\Delta }_{{\rm e}}}=\Delta _{0}^{{\rm DR}}+2\;$MHz the ion is excited so much that the dark resonance feature, recorded during step (ii), vanishes completely. We can only give a lower bound of $T\gt 80\;$ mK.

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Because different frequencies along the dark resonance lead to different equilibrium temperatures, we investigate experimentally the influence of the measurement duration ${{t}_{{\rm m}}}$ in step (ii) on the resulting temperature. Therefore, we vary ${{t}_{{\rm m}}}$ between a few microseconds and $1\;$ ms. In the latter case, ${{t}_{{\rm m}}}$ is sufficiently long to let the thermal state equilibrate to a temperature determined by ${{\Delta }_{{\rm m}}}$ during the measurement. For short times of ${{t}_{{\rm m}}}$, however, the temperature is governed by ${{\Delta }_{{\rm e}}}$ and not significantly influenced by the measurement. To characterize the influence of the measurement we have tuned ${{\Delta }_{{\rm e}}}$ to two different values at $\Delta _{0}^{{\rm DR}}-1.2\;$MHz and $\Delta _{0}^{{\rm DR}}+8.5\;$ MHz (triangular markers in figure 5(a)), corresponding to temperatures of $0.7(4)$ mK and $3.1(4)$ mK, respectively. The equilibration process is shown in figure 5(b), where the measured temperature difference $\Delta T$ is plotted as a function of ${{t}_{{\rm m}}}$. An exponential fit of the temperature differences by $\Delta T({{t}_{{\rm m}}})=a\;{\rm exp} (-{{t}_{{\rm m}}}/{{\tau }_{{\rm m}}})$ reveals a decay time of ${{\tau }_{{\rm m}}}=0.38(7)$ ms, corresponding to a rate of about 2.6 kHz. Indeed, for measurement times ${{t}_{{\rm m}}}\leqslant 20\;\mu {\rm s}$ the modification of the measured temperature of $\Delta T\leqslant 0.1\;$ mK is negligible as compared to the concurrent errors. At low temperatures the dominant error results from the systematic uncertainty of the spectral linewidth Γof the laser sources. An uncertainty in the knowledge of the laser linewidth of $30\;$ kHz results in a temperature uncertainty of $0.4\;$ mK.

For many applications, it might be important to extend the range of control also to thermal states with higher temperature. To this end we apply electrical noise to the trap electrodes, which are used for compensating micromotion. This leads to a random displacement of the trap potential, while the trap frequency is constant. Combined with the dissipative process of laser cooling it is applied for ${{t}_{{\rm e}}}\geqslant 2\;$ms, such that a steady state with adjustable temperature T is achieved, where T is controlled by the amplitude of the electric field noise as $T\propto E_{{\rm rms}}^{2}$. This environment mimics an effective thermal heat bath for the trapped ion [46, 47]. We make the measurements with a linear crystal of five ions and could vary its temperature over more than an order of magnitude. Figure 6 shows the spectrum at different noise amplitudes. We fit temperatures of $T=3.1(5)$ mK, $9(1)$ mK and $46(4)$ mK to the data. Note that, for these scans, we have used the continuous acquisition scheme. At high noise amplitudes, where the dark resonances are broadened, we find an excellent fit to the model curves. For the data without excitation we observe deviations from the fit curve near ${{\Delta }_{866}}=-6\;$ MHz, which are explained by cooling/heating in the vicinity of the dark resonance, an artifact absent in the pulsed acquisition mode.

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To capture the time evolution of temperatures we apply an electrical noise pulse of $2\;$ ms while laser cooling the ions with a fixed detuning Δ₈₆₆ to reach an initial steady-state temperature ${{T}_{{\rm i}}}$. Then we switch the noise off and observe the thermal recooling process. Therefore we record fluorescence spectra by the pulsed measurement technique with different time delays between 50 and $650\;\mu {\rm s}$ after the noise pulse; see figure 7(a). Fitting the measured temperatures with $T\propto {\rm exp} (-\Delta t/{{\tau }_{{\rm c}}})$ reveals a time constant of ${{\tau }_{{\rm c}}}=87(3)\mu {\rm s}$. Similarly, we observe the inverse heating process: starting with a laser cooled ion we measure the re-equilibration process after switching on the noise pulse; see figure 7(b). An exponential fit results in a heating time of ${{\tau }_{{\rm h}}}=257(40)\mu {\rm s}$. In the language of reservoir engineering [46, 47], such dynamics may be interpreted as thermalization of the ion with an effective heat bath. Once the noise pulse is switched the ion is out of equilibrium and subsequent re-equilibration can be observed.

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