A Master Equation

Experimental and theoretical comparison of the cooling performance between standard and double-bright EIT cooling

Abstract

We introduce and demonstrate double-bright electromagnetically induced transparency (D-EIT) cooling as a novel scalable approach to standard EIT cooling. By involving an additional ground state, two bright states can be shifted individually into resonance for cooling of motional modes of different frequency. This allows three-dimensional ground state cooling of a Ca ion trapped in a linear Paul-trap with a single cooling pulse. Measured cooling rates and steady-state mean motional quantum numbers for this D-EIT cooling are compared with those of standard EIT cooling as well as concatenated standard EIT cooling pulses for multi-mode cooling. The failure of theoretical descriptions within the Lamb-Dicke regime to reproduce the experimental results is explained by the development of a time-dependent rate theory to simulate the cooling for both, standard EIT and D-EIT cooling. Limitations of the different cooling techniques and possible extensions to multi-ion crystals are discussed.

pacs:
42.50.-p, 03.75.Be, 37.10.De, 37.10.Mn, 06.30.Ft

Current adrdress: ]National Research Council Canada, Ottawa, Ontario, K1A 0R6, Canada Current adrdress: ]Geo++ GmbH, 30827 Garbsen

I Introduction

Electromagnetically induced transparency (EIT) cooling Morigi et al. (2000); Eschner et al. (2003) has become a standard technique for fast ground-state cooling (GSC) of trapped ions and atoms. It has been demonstrated for cooling of a single ion Roos et al. (2000), for cavity-assisted EIT cooling of a neutral atom in a dipole trap Kampschulte et al. (2014), for sympathetic cooling of ion chains Lin et al. (2013); Lechner et al. (2016), and a quantum-gas microscope Haller et al. (2015). Due to the narrow cooling resonance, standard EIT cooling restricts GSC to a narrow range of nearby vibrational modes in one single cooling pulse Lin et al. (2013); Lechner et al. (2016). The larger the spectral-mode spread, the less efficient the cooling, potentially leading to heating of some modes. However, many quantum optics experiments such as quantum simulations of Ising models Porras and Cirac (2004), ion-strings Jurcevic et al. (2014); Richerme et al. (2014); Kiethe et al. (2017), and quantum logic spectroscopy Wan et al. (2014); Wolf et al. (2016); Rosenband et al. (2008); Schmidt et al. (2005); Hempel et al. (2013); Chou et al. (2017) would benefit from a multi-mode GSC technique. EIT cooling is a natural extension of dark state cooling of atoms in free spaceAspect et al. (1988). The analogy for trapped ions was first analyzed theoretically in Lindberg and Javanainen (1986) and the main advantages over regular sideband cooling where put forward in Morigi et al. (2000) and verified experimentally in Roos et al. (2000); Schmidt-Kaler et al. (2001). The advantage of the combination of EIT cooling with ground state driving was analyzed in Cerrillo et al. (2010).

Recently, a novel scalable approach to standard EIT cooling, referred to as double-bright EIT (D-EIT) cooling, has been demonstrated Scharnhorst et al. (2017). Here, we provide a detailed theoretical model for D-EIT cooling and compare its performance to single EIT cooling theoretically and experimentally. The D-EIT scheme includes a third ground state that is coupled via an additional laser to an excited state. A similar scheme had been proposed by Evers and Keitel Evers and Keitel (2004) for a model system to suppress higher-order heating processes. In real atomic systems heating processes due off-resonant scattering typically dominate, so that the suppression of higher-order heating is of little benefit. Instead, we use the second freely tunable EIT resonance in the D-EIT level scheme to enable simultaneous cooling of spectrally separated modes. At the same time, the scheme protects the dark state from decoherence due to off-resonant scattering and removes limitations due to other non-ground state cooled modes (spectator modes). We analytically characterize the performance of D-EIT cooling within second-order Lamb-Dicke perturbation theory in terms of final temperatures and cooling rates. Beyond this limit, this characterization becomes unsuitable: on the one hand, the vibrational degrees of freedom are not strictly thermal and, on the other hand, the decay process does not follow a single exponential. Instead, the final Fock state occupation is used and a generalized cooling rate is defined with help of a generalized fluctuation-dissipation theorem. This generalized rate evidences that the initial cooling stages are slower, and we ascribe this phenomenon to a depletion of the dark state induced by decoherence effects associated to the motional excitation and the higher coupling strength.

We experimentally demonstrate multi-mode ground-state cooling of all degrees of freedom of a single Ca ion via D-EIT cooling within a single cooling pulse. The measured results on cooling rate and final temperature are compared to standard EIT cooling and sequential standard EIT cooling pulses for multi-mode cooling. All experimental results are supported by full density matrix calculations.

The article is organized as follows: the experimental setup is introduced in Sec. II including the experimental sequence and the measurement method for the mean motional quantum number and cooling rate. The cooling schemes of single EIT and D-EIT are presented in Sec. III. Sec. IV discusses the theoretical description of the implemented cooling schemes, especially giving a description of cooling beyond the Lamb-Dicke perturbative limit. The experimental results are presented and discussed in Sec. V. Sec. VI summarizes and concludes the findings.

Ii Experimental setup

Figure 1: (color online) (a) Schematic of the beam setup with respect to the trap and magnetic field . (b) Energy levels of Ca. (c) Relevant levels for single EIT cooling. (d) Relevant levels for D-EIT. In (c), (d) the pump beam(s) are drawn with thicker lines. Figure taken from Scharnhorst et al. (2017).
Figure 2: (color online) Evolution of in radial and axial direction, measured after a delay time after D-EIT cooling, during which all lasers have been switched off. The background heating rate is derived from a linear fit of as \SI[parse-numbers=false](45.0±2.8)\per\second in the axial direction and \SI[parse-numbers=false](28.2±3.4)\per\second in the radial direction.

A single Ca ion is produced via ablation loading using a frequency-doubled pulsed Nd:YAG laser and ionized in a two-photon process Sheridan et al. (2011); Wübbena (2011); Gulde (2003). The Ca ion is confined in a linear Innsbruck-style Paul trap as described in Doležal et al. (2015); Wübbena (2014) with \SI2.5mm endcap-to-ion distance and \SI0.8mm radio-frequency (RF) electrode-to-ion distance, and hollow endcap electrodes providing axial laser access. The radially confining RF quadrupole field is generated by applying \SI28.209MHz to both pairs of diagonally opposite RF electrodes while the relative RF phase between those two pairs is near . This symmetric driving of the RF electrodes minimizes axial micromotion, since in this configuration the ground RF potential coincides with the symmetry axis of the trap and the ground of the 3D DC quadrupole field generated by the endcap electrodes. With a loaded Q-factor of 270 and \SI5W of power fed to the helical resonator, the radial vibrational mode frequencies are the measured to be \SI2.552MHz and \SI2.540MHz for a single Ca ion, which implies a voltage between the RF trap electrodes of around \SI600V. The symmetry in the drive of the quadrupole field is deliberately slightly broken in order to lift the degeneracy of the radial modes. This represents a compromise between efficient radial cooling, which benefits from non-degenerate modes, and axial micromotion compensation, which benefits from symmetry. Radial micromotion is compensated via two pairs of DC electrodes running parallel to the RF electrodes. Axial confinement is provided by the two DC endcap electrodes. By applying a differential voltage between them, the ion can be placed at the point of minimal axial micromotion along the symmetry axis of the trap. The axial vibrational mode frequency is \SI904.6kHz with DC voltages of \SI568.5\volt and \SI460\volt applied to the endcap electrodes. Micromotion is compensated to an overall residual second order fractional Doppler shift of less than via the sideband technique Berkeland et al. (1998); Keller et al. (2015), corresponding to a modulation index for a single Ca ion. As shown in Fig. 2, the background heating rate of the system is \SI[parse-numbers=false](45.0±2.8)\per\second in the axial and \SI[parse-numbers=false](28.2±3.4)\per\second in the radial direction. These values are negligible compared to the cooling rates discussed in this paper. Three pairs of field coils create a constant magnetic field oriented perpendicular to the axial direction with a field strength of \SI416\microT (see Fig. 1). An additional three pairs of field coils are used for active magnetic field stabilization. The overall field is measured in three orthogonal directions with a flux gate magnetic field probe mag () and stabilized by feeding back on the current drivers of the coil. Along the quantization axis ( direction), we achieve a suppression of \SI10dB for noise frequencies up to \SI50Hz and a residual rms field of \SI3.3nT in the frequency band from Hz to kHz.

Figure 3: (color online) Fluorescence plot of dark resonances between \SI866nm laser and \SI397nm beam, where the latter is red-detuned by \SI21.34MHz from unperturbed resonance. Dashed lines indicate theoretically expected frequencies of dark resonances for the given polarizations and magnetic field.

Fig. 1(b) shows the relevant energy levels of Ca, connected by the available lasers top () labeled with their respective wavelength. As shown in Fig. 1(a), two \SI397nm beams are available. One of them is purely radial and polarized, the other one is directed along the axial direction and is polarized. The \SI866nm laser is also purely axial and polarized. The \SI729nm beam has a projection onto the axial and radial directions. The \SI729nm, \SI866nm, and \SI397nm lasers are phase-locked to the same ultra-stable, cavity-stabilized reference using a high-bandwidth transfer-lock technique described in Scharnhorst et al. (2015). This ensures phase stability among the lasers, demonstrated by dark resonances between the \SI866nm and \SI397nm laser shown in Fig. 3.

ii.1 Experimental sequence

Every beam is switched and tuned in frequency individually via acousto-optical modulators (AOM). The timing and frequency of the optical pulses generated by the AOMs are controlled by an FPGA-based control-system Schindler (2008); Wübbena (2014). All measurements use the same experimental pulse order: For each data point the ion is first Doppler cooled for \SI1\milli\second via both \SI397nm lasers connecting the to levels while simultaneously applying the \SI866nm beam to act as a repumper. This prepares the ion close to the Doppler cooling temperature limit, corresponding to in axial and in radial direction assuming the natural linewidth of \SI20.7MHz of the S to P transition. After this first cooling step, EIT cooling is applied. This is followed by a low-power \SI2\micro\second optical pumping pulse of \SI397nm and \SI866nm light preparing the population into the state. The polarization purity of the \SI397nm is better than , which implies a measurement offset due to imperfect pumping and heating during the pump pulse of . This offset is negligible for our purposes. Information about the motional state of the ion is probed by mapping it onto the internal electronic states. For this purpose, we apply laser radiation at \SI729nm detuned to the red (blue) of the optical carrier by the oscillation frequency of the ion in the trap, driving red sidebands (RSB) (blue sidebands (BSB)) from to for a Rabi time . These pulses not only change the electronic state, but also decrease (increase) the motional quantum state. The final electronic states are discriminated by probing the S to P transition with \SI397nm light for \SI250\micro\second. During this time the ion’s fluorescence is detected through imaging optics and a photo-multiplier tube. For each setting the sequence is repeated 250 times and from the resulting excitation statistics the population in the logic state is inferred via the threshold technique Roos et al. (1999); Hemmerling et al. (2012). Typically, around \SI100\per\milli\second scattering events are detected if the ion is in the electronic ground state, and around \SI0.1\per\milli\second if the ion is in the state. Every event with a scattering rate less than \SI6.5\per\milli\second is attributed to the ion being in the state (dark state). The excitation probability is estimated as a fraction of trials in which the dark state was observed.

ii.2 Measurement of and cooling rates

Figure 4: (color online) Typical dependency of on EIT cooling time, here for the axial direction after D-EIT cooling at . The solid line shows the exponential fit. The horizontal dashed line indicates, where the exponential fit approaches the fitted to within . is calculated over all after the fit crosses below the dashed line.

For a single trapped particle a mean motional quantum number can be assigned to each of its motional degrees of freedom (two radial modes, one axial mode). In case of a thermal distribution, the corresponds to a temperature. The temporal evolution after applying GSC for time is obtained by sideband thermometry Turchette et al. (2000). For this, the excitation of RSB and BSB at a given Rabi time are measured as a function of cooling time. The temperature is then computed as

from the sideband excitations. A typical temperature measurement with corresponding fit is shown in Fig. 4. We determine the steady-state mean motional quantum number by averaging over several measurements after the equilibrium is reached. The cooling time at which equilibrium is achieved is defined as the time at which the exponential fit of the form , with being the cooling rate, approaches the fitted to within .

For temperatures , the measurement SNR of is improved by first fitting the sideband excitations individually with an exponential function and then deriving the from the fitted , . A further improvement of the SNR of is achieved by probing the sidebands at a time corresponding to the -time of a Doppler-cooled Rabi flop. This shorter -time and the additional fitting step for the sidebands has been chosen to determine the cooling rate .

For comparison with the simulations, Rabi frequencies of the cooling beams (\SI397nm and ) have been calibrated in the following way. A part of the power of each beam is picked off before the beams enter the vacuum chamber and is monitored with a photo diode (PD) for active intensity stabilization by feeding back on the AOM RF drive power using a digital sample & hold stabilization circuit. The induced Stark-shift on the state is deduced by taking the difference between the carrier resonance frequency of the logic transition with and without one of the cooling beams being on during the logic pulse. Doing so for different cooling laser powers, a linear dependency between Stark-shift and cooling laser voltage on the PD is obtained. The Rabi frequency can then be calculated using the relationship . The \SI866nm Rabi frequency is a free parameter in the simulation which is chosen to fit the experimental results.

Figure 5: (color online) Cross section through the center of the ion trap, showing the orientation of radial modes with respect to RF electrodes, schematically represented as gray shaded areas.

As shown in Fig. 1(a) all cooling beams run in the same plane to which the \SI729nm logic laser has an angle . The two radial modes are orthogonal with respect to each other. They are rotated around trap axis such that the modes have a and angle with respect to the plane of the cooling lasers, see Fig. 5. The rotation angle has been deduced from the different sideband Rabi-frequencies for the two radial modes for the same logic laser intensity. In the radial direction the temperature is obtained from the mode that has a lower projection onto the plane spanned by the cooling beams. The other radial mode has a higher projection onto this plane, resulting in more efficient cooling of this mode. The \SI12kHz spectral spacing between the two radial modes is small compared to all cooling resonance linewidths.

Iii Cooling schemes

Figure 6: (color online) Simulated scattering rate during D-EIT cooling versus detuning from the virtual level . Vertical straight lines indicate motional frequencies. The inset shows single EIT spectrum in the respective direction optimized for axial cooling. D-EIT has a bright state at both radial and axial frequency, while for single EIT there is only one bright state. The position of the radial (axial) bright state is dominated by the Rabi frequency of the \SI397nm (\SI866nm ) beam. Figure taken from Scharnhorst et al. (2017).

iii.1 Single EIT cooling

EIT cooling is a technique based on a -like level scheme with two ground states and one shared excited state Morigi et al. (2000). Fig. 1(c) shows the relevant levels and beams for single EIT cooling in Ca. Here, the ground states are the levels and the excited state is chosen to be the state. If lasers connecting the respective ground are brought into two-photon resonance near the excited state, a dark resonance is generated, as shown for a red-detuned resonance in Fig. 3 and for a blue detuned resonance in the simulated scattering spectrum in the inset of Fig. 6. Each dark resonance is associated with a bright state that is theoretically described by a Fano-profile Janik et al. (1985). Its spectral position (shift) with respect to the dark resonance depends on the Rabi frequency of the two lasers forming the dark resonance. Typically, one of the two lasers is much stronger (pump) than the other one (probe) such that the shift of the bright state is dominated by the pump beam. In our setup the \SI397nm beam is used as pump and the \SI397nm serves as probe beam. Pump and probe are brought into resonance on a virtual energy level which is blue detuned by with respect to the unperturbed level. That choice of probe beam reduces the effect of off-resonant scattering on the to transition Roos et al. (2000). Our beam geometry allows cooling of the axial or radial modes by shifting the bright state to the respective mode frequency. Due to the large mode spacing, a single EIT pulse can cool either the axial or the two radial modes, but not all three at once. All modes can be cooled using single EIT by applying two sequential pulses, which is hereafter referred to as “single EIT with precooling” or “3D EIT cooling”.

iii.2 Double-bright EIT cooling

In D-EIT cooling, the scheme of single EIT cooling is extended to a double -shaped level scheme, as shown in Fig. 1(d). This is accomplished by introducing a \SI866nm beam as an additional pump connecting the to the virtual level. It contributes to the dark resonance suppression of carrier scattering and creates a second bright state together with the \SI397nm beam coupling to , hence the name double-bright EIT. For our beam geometry, it is formed by purely axial beams and the Stark-shift-induced bright state by the \SI866nm pump beam is brought into resonance with the axial motional sideband. The bright state formed by the \SI397nm and beams is shifted into resonance with the radial sidebands. The shift of the two bright states can be chosen independently by adjusting the Rabi frequency of the respective pump beam accordingly. This way, both radial and axial direction can be cooled simultaneously, despite their frequency splitting of more than \SI1.5\mega\hertz. Furthermore, the off-resonant coupling of the \SI397nm transition to the level is made dark via coupling to the D-manifold, which reduces off-resonant scattering.

Iv Theory

iv.1 Lamb-Dicke Theory

In order to theoretically model the performance of both cooling schemes and provide predictions for and , we firstly apply a commonly used approach involving adiabatic elimination of the electronic degrees of freedom in the Lamb-Dicke limit Cirac et al. (1992). In this limit, two competing processes dictate the cooling rate and the steady-state temperature, which are the diffusion due to spontaneous emission and the coordinated effect of sideband excitation and dissipation. The first one is known as carrier scattering and contributes with a diffusion term of the form

where the sum is over all excited states and all ground states , denotes the steady state population of the excited state, is the effective Lamb-Dicke parameter associated with the to transition and its scattering rate. The second one is quantified in terms of the steady-state, two-time correlation function in frequency domain

of the operator corresponding to the electronic part of the first order term of the Lamb-Dicke expansion of the Hamiltonian. Here we make use of the averaging with respect to the zeroth-order steady-state electronic density matrix . Its evaluation at the oscillator frequency provides the heating () and cooling () rates

where stands for the real part. These enter the rate equation governing the population dynamics in the Fock space of the oscillator

(1)

corresponding to a matrix of the form

(2)

The predicted steady state probability distribution

(3)

corresponds to a thermal state of temperature , average occupation and ground state occupation . Additionally, the approach to the steady state is governed by the cooling rate .

Single EIT Cooling

The traditional concept for EIT cooling arising in the context of a Raman laser configuration Morigi et al. (2000) is such that, within the regime of validity of the Lamb-Dicke perturbative expansion, the presence of a dark state eliminates the carrier scattering and additionally simplifies the mathematical derivation of the absorption spectrum . The dark state arising in our configuration is . It involves a superposition of the two ground states with , and the Rabi frequencies of the 397 nm and polarized lasers respectively, whereas the cooling operator may be defined , where is an unnormalized state involving the Lamb-Dicke parameters of the two lasers along the relevant axis as shown in Fig.1(a).

In this case, the real part of the absorption spectrum as derived by means of the quantum regression theorem Gardiner and Zoller (2004) becomes

(4)

where and is the decay rate of the state. Eq.(4) vanishes for and contains two maxima at , corresponding to the frequencies of the dressed excited state and bright state respectively. By means of a large blue detuning, the maximum of the dressed bright state can be adjusted to the relevant oscillator frequencies so that and an optimal cooling rate

(5)

and a final occupation number

may be achieved, where corresponds to the heating contribution of the spurious 397 coupling. This contribution is approximately of the form in the limit of highly asymmetric laser intensities used in the implementation of EIT in . The dependence on clarifies the necessity of this limit to minimize the spurious coupling between and that depletes the dark state . Although the condition maximizes Eq.(5) for constant , it implies , so that the probe laser establishes the order of magnitude of the rate. Since , the pump laser controls the position of the dressed state and, by the relations above, . Therefore, this configuration fundamentally limits the rates that can be achieved for a given detuning. The additional laser coupling proposed in double bright EIT circumvents this limitation and allows EIT cooling with higher laser intensities.

Double-bright EIT cooling

In our system, judicious choice of the detunings allows for six possible implementations of a double--Raman configurations. The choice involving the level generates the following dark state

with and the Rabi frequency associated to the 866 nm laser. The cooling operator has the form

where only the ground states with non-vanishing overlap with are considered. In particular, with the unnormalized state

As in the case of single EIT, the presence of the dark state facilitates the analytical calculation of the scattering spectrum, which can be expressed in the form of a matrix inversion (see App. B)

where stands for imaginary part. It contains the denominator of a cooling spectrum exhibiting two EIT features (analogous to double EIT cooling as described in Evers and Keitel (2004))

and, what is more interesting, the denominator of a cooling spectrum with three EIT features

This triple EIT structure has never been described in the literature and is a novel aspect of our cooling scheme. Given the determinant of the matrix

the final form of the scattering spectrum becomes

The values of the Lamb-Dicke parameters control whether triple or double EIT cooling takes place, or a combination of both. The radial degrees of freedom are easiest to analyze since, due to the beam geometry [see Fig.1(a)], and a modified triple EIT term survives

On the other hand, the axial degree of freedom is characterized by , and

with .

The benefit of such a rich spectral structure is highlighted when several vibrational modes need to be addressed. In this case, the position of several maxima can be distributed along the spectrum in such a way that all modes enjoy the condition . As with single EIT, this is achieved by adjusting the Stark shift of the bright states to the appropriate frequency so that optimal rates of up to

may be accessed. Stark shifts are in general proportional to the square of the relevant Rabi frequency and inversely proportional to the detuning of the shifting beam. Since Stark shifts need to be kept fixed in this cooling configuration, cooling rates are roughly proportional to the detuning of the laser configuration . For the final occupation

may be achieved. In summary, D-EIT cooling allows cooling of two individually adjustable mode ranges. Furthermore, as a consequence of the absence of spurious carrier heating in D-EIT, smaller steady-state temperatures are achieved, while at the same time saturating the possible cooling rate. These results strictly hold only within the Lamb-Dicke approximation, and need to be modified to account for experimentally observed effects beyond the Lamb-Dicke limit. Additionally, it is also possible to adjust to take advantage of the double and triple EIT structures appearing in the scattering spectrum, such that the phenomenon described in Evers and Keitel (2004) can be exploited to reach vanishing occupation within second-order perturbation theory.

iv.2 Beyond the Lamb-Dicke Limit

The double configuration of D-EIT avoids the detrimental coupling present in single EIT cooling, such that the Lamb-Dicke theory predicts significantly higher rates with sufficient laser power. These rates can become so large that the Lamb-Dicke approximation is no longer fulfilled, and strong correlations between the motional and the electronic degrees of freedom set in that effectively limit the cooling capability. A realistic prediction of the behavior in this limit requires a more advanced theory and, in practical terms, numerical simulations of the full electronic and vibrational master equation. In this section, we provide an interpretation of this limitation in terms of linear response theory, which we first show to be equivalent to the Lamb-Dicke limit, and then express deviations therefrom in terms of generalized fluctuation-dissipation theorems. In general, this treatment shows that the definition of cooling rate is ambiguous, and a time dependent cooling rate is a more appropriate description. This is a direct consequence of the build-up of non-negligible correlations between the vibrational and electronic degrees of freedom that seem to effectively reduce the decay rate of higher-lying Fock states. Therefore, the first stages of cooling take longer, and the observed timescales for cooling remain larger than the Lamb-Dicke prediction. Additionally, the resulting vibrational steady state does not exhibit a thermal distribution, so it is not appropriate to assign a temperature to the measured .

Linear response theory

In the Lamb-Dicke limit, laser cooling can be understood as a thermalization process ensuing from contact of the oscillator with a thermal bath, and linear response theory suffices to characterize this phenomenon. From this perspective, it is possible to associate the rate at which the cooling process takes place with the conductance of the thermal environment. The fluctuation-dissipation theorem Kubo (1966), both in its quantum and classical versions, relates the thermal conductance of the oscillator to the fluctuations around equilibrium

(6)

where is the thermal current departing the oscillator and is the variance of energy fluctuations around equilibrium, with the Hamiltonian of the oscillator and the inverse temperature in units of the Boltzmann constant . This establishes a dynamical picture whereby small temperature deviations from equilibrium generate a current well approximated by Fourier’s law . The heat capacitance of the oscillator allows us to formulate a linear response equation

(7)

which, by using for an oscillator of frequency and number operator , can be solved to find the dynamics of the occupation of the oscillator in its approach to the steady state

(8)

where and correspond to the initial and final occupation numbers respectively.

In the following, we make use of the fluctuation-dissipation theorem to show , corresponding to the Lamb-Dicke cooling rate . The variance of the energy fluctuations at equilibrium can be computed with help of the two-time measurement probability of observing the oscillator at level when it is at equilibrium and at level in a subsequent measurement performed at a later time

The probability can be decomposed into the probability for the oscillator to be in state initially and the probability to perform a transition between states and in the time . The latter is also the dynamical map Rivas and Huelga (2011), which maps the initial probability distribution to the one at time , . Given Eq.(1), the dynamical map is easily computed as , with from Eq.(2). Therefore,

(9)

restricts the sum over to values , so that

Therefore, given the heat capacity of the oscillator in its steady state the rate is confirmed to coincide with that of the linear response prediction

(10)

Non-linear response theory

The linear response prediction only applies as long as the assumptions of weak coupling and adiabatic separation of the timescales is satisfied. This is expressed in our case by the condition for all laser couplings , which sets an upper bound to the value of the Rabi frequencies for which the Lamb-Dicke theory remains valid. In practice, only the Lamb Dicke parameters and Rabi frequencies that appear in the expression of the cooling rate play a role in the strength of the dissipative dynamics and are therefore subject to the upper bound. In consequence, for an EIT cooling rate of the form , this implies

(11)

which in our setting translates to rates much smaller than for the axial degree of freedom and for the radial degrees of freedom.

The behavior beyond these limits can only be predicted by considering the full master equation for both the vibrational and electronic degrees of freedom. In this situation, the reduced density matrix of the vibrational degrees of freedom does not necessarily fit a thermal distribution as in Eq.(3) and the average occupation at the steady state , although well defined, is not associated to a single temperature value. This is clear evidence of the breakdown of the linear response limit, and the definition of a thermodynamically grounded cooling rate as in Eq.(10) becomes troublesome.

In practical terms, the occupation number does not experience an exponential decay as in Eq.(8), but rather a decay whose rate depends on the initial oscillator temperature and that varies as the steady state is approached. This can be explored by introducing the concept of an instantaneous rate . For an initial thermal state of the oscillator, one may express the derivative as the quotient inv ().

(12)

where is a generalized, time-dependent capacitance of the oscillator and is the corresponding generalized conductance. Whereas the fluctuation dissipation theorem of the form Eq.(6) cannot be applied in this case, it can be corrected by means of an additional term as explained in Cerrillo et al. (2016) so that

The first term on the right hand side is analogous to Eq.(9), so

with the thermal distribution of temperature . The second term corrects the fluctuation dissipation theorem and, as shown in Cerrillo et al. (2016), corresponds to a computation of the variance of the transferred energy

Expression (12) reduces to Eq.(7) in the linear response limit since the fraction becomes constant and independent of the initial temperature . Although the solution of Eq.(12) does not have a simple exponential decay form, the quotient does provide us with some information about the timescale of the cooling process.

The dynamical map can be well approximated with help of the transfer tensors Cerrillo and Cao (2014); Rosenbach et al. (2016), which can be extracted from short timescale simulations of the global master equations and contracted to provide values for the quotient to very long times.

To theoretically evaluate the D-EIT cooling process outside the Lamb-Dicke regime, we performed numerical full density matrix master equation simulations, taking into account all 8 relevant electronic and the lowest 17 motional states. An instance of this is presented in Fig. 7, where is shown as a function of the elapsed cooling time and initial mean population . falls as increases, but it recovers as a function of time as lower thermal states become involved in the dynamics. This effect is connected with the observation of an increase of the P-state population when leaving the Lamb-Dicke regime. A possible explanation for this observation is that the strong laser-induced spin-motional coupling prevents establishing an electronic dark state that has no admixture of the excited P-state. Increasing P population results in an enhanced scattering on carrier and sidebands, limiting the cooling rate and achievable . The effect is more pronounced the higher Fock states are involved, but even at an a \SI20% reduction in the cooling rate can be observed (see Fig. 7). This behavior is characteristic of strong coupling dissipative theory Cerrillo et al. (2016). A full understanding of this behavior requires the development of an analytical model for cooling in the regime beyond the LD approximation, which is subject to future theoretical investigations.

Figure 7: Time dependent rate in units of phonon for axial single EIT at as a function of the initial occupation number . The Lamb-Dicke prediction for the rate corresponds to phonon, whereas the measured value is phonon. Figure taken from Scharnhorst et al. (2017).

iv.3 Simulated data

The simulated data is obtained by full density matrix theory as described in App. A. For each data point the experimentally obtained and are used in the simulation. In D-EIT cooling the Rabi frequencies of the pump beams are chosen such that the corresponding bright states coincide with the desired mode frequencies. In EIT cooling the simulated Rabi frequency of the \SI866nm beam is set to have no effect on the cooling other than clearing out the D fast enough not to limit the cooling rate. The given simulated are derived via sideband thermometry as in the experimental case. From these the cooling rates are derived to ensure comparability with the experimental data. The sideband thermometry leads to smaller than would be obtained by averaging over all Fock states. However, that difference is not significant within the error bars of the measured . As discussed in this section, the cooling rates depend on the initial temperature of the cooled mode and change with elapsed cooling time, which is why it is not possible to define a single cooling rate for a given EIT cooling scheme. In order to be able to compare a simulated cooling rate to the experimental data, the simulated rate is given at .

V Results

In this section, the experimental results are presented and supported by full density matrix simulations as described in Sec. IV.3 and App. A. The scope of this work is the demonstration of D-EIT cooling of all three modes of a single Ca ion in one cooling pulse and the comparison of its performance with the standard single EIT cooling technique. We first describe the implementation of single EIT cooling of the axial mode or both radial modes and provide resulting cooling rates and achievable . Special emphasis is put on detrimental effects of single EIT cooling on the spectator modes for which the cooling pulse has not been optimized. Secondly, we demonstrate cooling of all three modes of a single Ca ion by applying two consecutive single EIT cooling pulses, one to cool the axial mode and one to cool both close-by radial modes. Besides the cooling performance, the effect of the second cooling pulse on the initially cooled mode is investigated. This 3D single EIT cooling illustrates the possibilities and limitations of multi-mode cooling via standard EIT cooling. Finally, the results of D-EIT cooling are presented and compared to the single EIT cooling schemes and to the simulations.

In the following, represents the non steady-state mean motional quantum number and the steady-state mean motional quantum number, while denotes the cooling rate. The index refers to either the axial (a) or the radial modes (r).

v.1 Single EIT cooling

Figure 8: (color online) False-color map of the residual axial RSB excitation, depending on the Rabi frequencies of the pump and probe cooling beams after axial single EIT cooling. Areas of low RSB excitation correspond to pump/probe Rabi frequency combinations leading to low . The horizontal line indicates the pump Rabi frequency along which the measurements shown in Fig. 9 were taken. A constant combined Stark-shift of the bright state corresponding to the axial trap frequency due to the pump and probe lasers is given by red dashed line.
Figure 9: (color online) Dependency of and on the probe Rabi frequency after axial single EIT cooling at , while the pump Rabi frequency is kept fixed at \SI14.6\mega\hertz. The solid lines are quadratic polynomials fitted to the data points to guide the eye.

Fig. 9 illustrates the dependence of and on the probe Rabi frequency after axial single EIT cooling at and a fixed pump Rabi frequency. Low probe Rabi frequencies result in low and small . With increasing Rabi frequency the rate , but also increases due to off-resonant scattering via the unused excited state , as predicted by Eq. 5. Due to this trade-off, there is no single probe beam Rabi frequency where both parameters are optimized.

In order to account for the trade-off between and and to retrieve consistent results for single EIT cooling of a particular mode, the Rabi frequencies of the pump and probe beam are experimentally determined in a two-step process. In the first step, to find the Rabi frequencies of the cooling lasers that minimize the steady-state mode occupation , we use a \SI500\micro\second cooling pulse long enough to reach steady-state for all reasonable cooling parameters. To maximize the cooling rate in the second step, we minimize after a \SI100\micro\second pulse too short to reach steady-state. As a consequence, the combination of Rabi frequencies for pump and probe leading to the smallest RSB excitation is shifted towards higher probe Rabi frequencies, maximizing the cooling rate. The combined optimum of the two optimization steps was obtained from the overlapped Rabi frequency maps of the two steps, resulting in a compromise between a fast cooling rate and a low corresponding .

Fig. 8 shows an example of a map of the optimization of pump and probe Rabi frequencies for single axial EIT cooling with . The colors represent the residual excitation on the axial RSB depending on the pump and probe Rabi frequencies after a fixed EIT cooling time of \SI300\micro\second, where blue indicates areas of low . Fig. 9 has been obtained by scanning the probe Rabi frequency along the red solid line indicated in Fig. 8 for a fixed pump power of \SI14.6\mega\hertz.

Figure 10: (color online) of the radial (black circles) and the axial (blue crosses) modes during single EIT cooling of the axial mode at , MHz, MHz, of the radial mode (black circles) and axial mode (blue crosses). The dashed and dotted lines represent exponential decay functions using the simulation results. Both curves employ the same cooling rate of \SI439\per\second, confirming that the time to reach is limited by radial cooling rate. Radial mode: , initial temperature . Axial mode: , initial temperature of .

Each cooling pulse affects all three modes, not only the one for which it had been optimized. For each cooling pulse setting a cooling rate and a steady-state occupation can be assigned to every mode according to the resulting scattering spectrum as shown in the inset of Fig. 6. Generally, the bright resonance of single EIT cooling allows optimized cooling around one frequency, while modes further away experience smaller and higher . The mode frequency range within which single EIT cooling to low is possible, is defined by the width of the bright state Morigi (2003). E.g. for MHz, cooling of modes within a frequency range of MHz to has been demonstrated in Lechner et al. (2016).

In the case of radial EIT cooling, the pump Rabi frequency required to shift the bright state into resonance with the radial mode frequency makes it broad enough to also provide fast cooling on the axial mode. Although the axial mode is located at the slope of the bright state scattering spectrum, because of the larger Lamb-Dicke factor for the axial mode (). Conversely, for axial EIT cooling the bright resonance is narrower than for radial EIT cooling since the pump Rabi frequency necessary to shift the bright resonance onto the axial mode is smaller. Including the ratio of the Lamb-Dicke factors, is then more than one order of magnitude smaller than .

Lamb-Dicke and master equation predictions do not take into account the perturbation that the presence of an additional, vibrationally hot degree of freedom (spectator modes Wineland et al. (1998); Lin et al. (2013)) generates in the cooling process. In general, one may regard any spectator modes as a source of dephasing that harms the cooling process. Therefore, the efficiency of cooling depends on the motional excitation of the spectator modes.

A nearby hot mode results in decoherence, indistinguishable from laser noise, via higher order scattering processes, which leads to an imperfect dark state and allows for weak carrier scattering Wineland et al. (1998). As a consequence, the cooling rate of the cooled mode decreases while its increases compared to the case of a perfect dark state or cold spectator mode Lin et al. (2013). In our setup, this cross-talk becomes relevant in axial EIT cooling, where is much slower than . This behavior is illustrated in Fig. 10, which shows and during axial EIT cooling at . After \SI500\micro\second the axial mode is cooled to , while the radial mode is still close to its initial temperature. Hence, the radial mode is hot enough to undermine the dark state and thus limits the . For longer cooling times, decreases further at the rate , since the dark state becomes purer with decreasing . This is supported by the dashed and dotted exponential decay functions for the radial and axial , plotted together with the experimental data in Fig. 10, both using the same simulated s, and the individual initial , and during axial EIT cooling.

The results of EIT cooling optimized individually for the axial and radial mode are shown in Fig. 11 together with simulated values for and . Typical results of single EIT cooling are summarized in Tab. 1. It is impractical to wait for the axial mode to reach its final . Therefore, in the case of axial EIT cooling, is measured after a time where it would have reached its equilibrium value if there was no cross-talk from hot radial modes. As a consequence, there is an offset of 0.1 between the simulated and the measured . In accordance with the ratio of the Lamb-Dicke factors between the radial and axial mode , , as can be seen from the top panels in Fig. 11. For radial EIT cooling is well described by the full master equation theory. The simulated rates agree with the measured ones for . For higher blue detunings the simulated radial rates are larger than the measured ones. This may be explained by cross talk from the axial mode, analogous to the cross-talk of the radial modes in the case of axial EIT cooling described above and shown in Fig. 10. Although this effect cannot be captured by single-oscillator simulations, the Lamb-Dicke prediction for within the radial single EIT cooling pulse is on the order of \SI8e3\per\second, which is similar to the measured . This serves as an indication that cross-talk from the axial mode is indeed limiting the , since departure of the Lamb-Dicke limit is ruled out by the coincidence between master equation simulations and Lamb-Dicke predictions.

v.2 3D single EIT cooling

Simultaneous cooling of very different mode frequencies is not possible within a single EIT cooling pulse due to the limited width of the bright state (see Fig. 6, Lechner et al. (2016); Lin et al. (2013)). Instead, several EIT pulses have to be applied sequentially, each of them optimized to cool around one particular frequency, in our case the axial or radial mode(s). In the following, the first cooling pulse is termed precooling pulse. It is optimized to cool one of the modes (precooled mode) with high cooling rate and aims to reach a low . During that pulse the off-resonant mode(s) are cooled as well, but at a slower rate and to a higher , as demonstrated in Fig. 10. To keep the overall cooling time short, the precooling pulse is terminated once the precooled mode reached its , although the other mode(s) might not have reached their steady state, yet. After that, a second pulse optimized for cooling the other mode(s) is applied.

Figure 11: (color online) Cooling rates and equilibrium for EIT and D-EIT as a function of the blue detuning . (a) Radial modes. (b) Axial mode. The top panels display the cooling rates and the bottom panels the . The data for D-EIT cooling is taken from Scharnhorst et al. (2017). The solid and dashed lines are quartic polynomials fitted to the data points to guide the eye.
Figure 12: (color online) Temperature of the precooled mode during the second cooling pulse of 3D single EIT cooling. For long enough cooling times of the second cooling pulse, both precooled modes will approximate .

A cooling pulse tailored for a certain mode may detrimentally affect other modes. As one can conclude from the analysis of typical cooling spectra , away from a bright resonance the ratio is not favorable and steady-state occupation numbers above are expected. Therefore, it is generally the case that a concatenation of pulses may result in less effective cooling.

The of the precooled mode associated with the scattering spectrum of the second cooling pulse is higher than the at the end of the precooling pulse and it will be heated up, as depicted in Fig. 12. It shows the increase of of the axial and radial precooled mode caused during the second cooling pulse. This effective heating is smaller for the precooled radial mode than for the precooled axial mode. Thus, the ordering of the two cooling pulses leads to different for both modes after the second cooling pulse, see also Table 1.

Typical results of 3D EIT cooling are summarized in Tab. 1. The listed of the precooled mode depends on the cooling time of the second mode. Since the precooling pulse cools the second mode below Doppler temperature, the time for the second mode to reach is reduced. Axial EIT with radial precooling is the most beneficial implementation of 3D single EIT cooling in our setup. All three modes of a single Ca ion are cooled close to the ground state within \SI640\micro\second combined cooling time.

v.3 D-EIT cooling

Fig. 11 displays and compares the results of single EIT and D-EIT cooling. For each method the and cooling rate for the radial and axial modes is shown as a function of the blue detuning of the virtual from the magnetic field-free to transition. The maximum detuning is limited by the tuning range of the \SI866nm AOM in case of D-EIT and by the \SI397nm AOM in case of single EIT cooling. This limitation is different for both methods due to different center frequencies and tuning ranges of the respective AOMs.

As the detuning increases, the steady-state occupation falls while the cooling rate increases. Increasing detuning yields both a colder steady state and faster cooling. The optimal cooling parameters were found experimentally by adjusting the three involved Rabi frequencies to obtain ac Stark shifts of the bright states matching the two mode frequencies, while maximizing the cooling rate and minimizing the steady-state . Therefore, larger detunings allow higher Rabi frequencies and thus stronger cooling rates. As the detuning is increased, the bright state scattering resonances narrow while the residual blue sideband excitations are suppressed. This results in an improved ratio between cooling and heating scattering events and thus a lower final .

D-EIT achieves cooling rates and steady-state energies similar to those obtained from single EIT cooling, but does so for all modes of a Ca ion’s motion simultaneously, while single EIT cooling had to be optimized for GSC of each mode separately. The simulated as well as the radial cooling rates agree well with the measured values. In the axial direction the simulated cooling rates are higher than the measured ones. This difference increases with and may be explained by the limited maximum number of Fock states that can be numerically simulated. The truncation of high Fock states in the simulation with non-negligible occupation leads to an overestimation of . We have verified, that in D-EIT decreases with increasing number of simulated Fock states. This effect is more pronounced in the axial direction where the lower trap frequency results in population of larger Fock states compared to the radial direction for the same initial temperature. Furthermore, we suspect that high-frequency relative phase noise between the \SI866\nano\meter and \SI397\nano\meter lasers results in a deterioration of the dark state, further restricting the experimentally observed cooling rate at high Rabi frequencies.

EIT Method (s)
Single EIT ax. 0.10 7 720
Single EIT ax. w/ rad. prec. 0.04 0.20 6401
Single EIT rad. 1.3 0.10 490
Single EIT rad. w/ ax. prec. 1 0.10 700
D-EIT 0.11 0.14 670
Table 1: Comparison of experimental values for single EIT, single EIT with precooling and D-EIT cooling from initial Doppler cooling temperatures, each optimized for minimum at . In case of axial single EIT cooling the meta-stable is given.

Table 1 compares the final and the time needed to reach the final temperature between D-EIT cooling and the results of single EIT cooling in axial and radial direction with and without precooling. These numbers were obtained from measurements at , which is the maximum blue detuning that usable for both single and D-EIT cooling in our apparatus. In our system the cooling performance of D-EIT is comparable to that of precooled axial single EIT cooling.

v.4 Comparison between Lamb-Dicke analytical theory, master equation simulation and experiment

Figure 13: (color online) Axial cooling rates for D-EIT as a function of the blue detuning . Compared are the measured rates with the simulated values obtained by the full master equation approach with limited Fock states and the Lamb-Dicke approximation. The dash-dotted line indicates the limit for the validity of the Lamb-Dicke approximation, see Eq. 11, the background gradient represents the gradual failure of the Lamb-Dicke approximation.

Fig. 13 illustrates the breakdown of the simple analytical model in the Lamb-Dicke approximation. As detuning increases, the cooling rates predicted by Eq. 5 grow an order of magnitude faster than those observed in the experiment. Some of the discrepancy is due to the correlations between electronic and vibrational degrees of freedom captured by the master equation simulations. The simulations, accordingly, come closer to the experimentally observed behavior. It is not yet clear whether the remaining discrepancy is due to numerical limitations or to other features not captured by our model such as relative phase noise between the cooling lasers.

Vi Conclusion and outlook

In conclusion, we developed a detailed theoretical model for double-bright EIT (DEIT) cooling Scharnhorst et al. (2017) as a novel scalable approach to standard EIT cooling. We experimentally demonstrate D-EIT GSC of all three motional degrees of freedom of a single Ca ion within one cooling pulse and compared the steady-state mean motional quantum numbers and cooling rates to those of single EIT cooling and 3D EIT cooling theoretically and experimentally. We give a thermodynamic interpretation of laser cooling by using an analogy between linear response theory and the Lamb-Dicke regime. Within the Lamb-Dicke regime, a description of the system in form of an analytical theory for both single EIT and D-EIT cooling is given. A theory for time-dependent rates has been developed to describe the system beyond the Lamb-Dicke regime and it has been compared to the measured data. We demonstrate how single EIT cooling addresses mostly only one mode and can heat other motional degrees of freedom that are outside the cooling bandwidth. Furthermore, the detrimental effect of spectator modes on the of the cooled mode is observed. In the Ca level structure, cooling rates for single EIT cooling are limited by off-resonant scattering via the unused magnetic sublevel of the state. The chosen level scheme of D-EIT cooling protects the dark state from such decoherence processes as well as limitations of cooling rates and due to motional spectator modes. Thus, for systems with smaller Lamb-Dicke factors, it can reach higher cooling rates and lower and outperform single EIT cooling.

The D-EIT technique is attractive for e.g. multi-ion, multi-species experiments in which quantum control of several spectrally separated motional modes is required. Examples include ion strings to study many-body physics Schneider et al. (2012); Kiethe et al. (2017), quantum simulations Porras and Cirac (2004); Blatt and Roos (2012), and quantum information processing Kienzler et al. (2016). The technique will be particularly useful for quantum logic spectroscopy Wan et al. (2014); Wolf et al. (2016); Rosenband et al. (2008); Schmidt et al. (2005); Hempel et al. (2013); Chou et al. (2017), and quantum logic clocks Rosenband et al. (2008); Ludlow et al. (2015) for which two ions of a different species need to be cooled to suppress systematic frequency shifts. In our experiment, a single Ca ion is used to sympathetically cool a single Al clock ion and serves as the logic ion in quantum logic spectroscopy Schmidt et al. (2005). Due to their mass ratio and the high radial symmetry of our trap, it is possible to choose a set of experimental parameters such that all six motional modes of the two ion crystal bunch at two frequencies. By rotating the trap by in the horizontal (xz) plane with respect to the k-vectors of the cooling beams, D-EIT will enable ground state cooling of all six modes of the two ion-crystal in one single cooling pulse. This helps to reduce the dead-time, and thus reduces the instability of the clock. In ion traps with large heating rates, near-resonant D-EIT cooling would establish a steady-state below the Doppler cooling limit with low incident power levels to minimize ac-Stark shifts on the clock transition Rosenband et al. (2008). Furthermore, the D-EIT cooling scheme can provide even triple EIT cooling if the magnetic field strength is chosen such that the sub-level is red detuned from the virtual excited state by the frequency of the radial mode and if the \SI397\nano\meter is polarized. That way, the D-EIT scheme additionally provides a second dark state at the position of the radial motional blue sideband, suppressing all second order radial heating processes. In principle, it is possible to generate additional bright states for additional GSC around other frequencies by extending the D-EIT scheme through coupling more ground states from the manifold to the virtual excited states.

Acknowledgements.
The authors would like to thank Nicolas Spethmann for helpful comments on the manuscript. We acknowledge support from the DFG through CRC 1128 (geo-Q), project A03 and CRC 1227 (DQ-mat), project B03, and the state of Lower Saxony, Hannover, Germany.

Appendix A Master Equation

The simulations have been performed with a master equation including the 8 electronic levels (, ) and one vibrational degree of freedom represented by the annihilation operator . We assume the presence of 397 nm laser in and polarizations and 866 nm in and polarizations. The Hamiltonian in units is

where , and correspond to the detunings of the laser frequencies with respect to the unperturbed transitions of the 397 nm laser in and