Theory of cavity-assisted microwave cooling of polar molecules.

# Theory of cavity-assisted microwave cooling of polar molecules.

## Abstract

We analyze cavity-assisted cooling schemes for polar molecules in the microwave domain, where molecules are excited on a rotational transition and energy is dissipated via strong interactions with a lossy stripline cavity, as recently proposed by A. André et al., Nature Physics 2, 636 (2006). We identify the dominant cooling and heating mechanisms in this setup and study cooling rates and final temperatures in various parameter regimes. In particular we analyze the effects of a finite environment temperature on the cooling efficiency, and find minimal temperature and optimized cooling rate in the strong drive regime. Further we discuss the trade-off between efficiency of cavity cooling and robustness with respect to ubiquitous imperfections in a realistic experimental setup, such as anharmonicity of the trapping potential.

###### pacs:
37.10.Mn, 85.25.-j, 37.10.Vz

## 1 Introduction

Cold molecules are nowadays very actively explored [1]. Of particular interest are cold heteronuclear molecules prepared in their electronic and vibrational ground state, which in view of their electric dipole moments can provide a strong dipole-dipole interaction, and a strong coupling of rotational excitations to microwave fields and cavities. The field of potential applications of cold samples of polar molecules ranges over ultracold chemistry [2] and precision measurements [3, 4], quantum computation [5, 6] and the simulation of exotic condensed matter models [7, 8, 9]. A necessary prerequisite for these developments is the ability to cool molecular ensembles and single trapped molecules to very low temperatures.

Several techniques for cooling, and preparation of ultracold molecular ensembles have been investigated [10] both experimentally and theoretically, including cavity-assisted laser cooling [11, 12, 13], buffer gas cooling, and Stark deceleration of polar molecules including trapping in magnetic and electric traps [14, 15, 16, 17, 18], evaporative [19] and sympathetic cooling schemes [20], and preparation of ground state molecules by photoassociation of cold atoms [21, 22, 23, 24]. Recently a technique for trapping and cooling of polar molecules on a chip has been suggested, which relies on strong coupling of rotational states of the molecule to a superconducting stripline cavity [25]. When trapped at a distance m above the cavity electrodes, the vacuum Rabi frequency associated with the exchange of a rotational excitation and a single microwave photon is in the order of kHz. In combination with a comparable photon loss rate this coupling serves as an efficient dissipation channel for (the in free space essentially stable) rotational states, which is exploited for cooling the motion of a trapped molecule. Cooling forces in this setup arise from the strong gradients of microwave fields in the near-field regime, in analogy to cavity-assisted laser cooling techniques discussed for atoms in the optical regime [11, 26, 27, 28, 29, 30, 31]. Microwave cooling does not rely on specific electronic or collisional properties [2, 19, 20, 32, 33, 34] of the molecules and therefore should be applicable for a large set of di- and also multiatomic polar molecules. As molecules are prepared in the motional ground state of a chip-based trap, this cooling technique is in particular interesting for applications in the context of on-chip (hybrid) quantum information processing [25, 35].

In this work we will present a detailed theoretical study of cavity assisted cooling of a single trapped polar molecule in the microwave domain. The goal of our theoretical analysis is to identify the physical mechanisms behind different cooling schemes, and to discuss the cooling rates and final temperatures in various parameter regimes including weak and strong driving fields and a finite temperature of the environment. The main conclusions of this analysis are as follows. Depending on the detuning between the relevant rotational transition frequency and the microwave fields, cavity cooling is dominated either by cavity assisted sideband cooling (CASC), where cooling forces arise from the gradient of a classical driving field, or “” cooling where forces are due to gradients in the coupling between the molecule and a single microwave photon. Zero environmental temperature in principle allows ground state cooling in the resolved sideband limit for the respective processes. At finite temperature , thermal cavity photons at frequency introduce an additional heating mechanism, and the minimal temperature is given by with the trap frequency. We find that this limit can be reached in the strong driving regime, where also cooling rates are optimized. Our analysis of cavity cooling in the presence of imperfections such as anharmonicity of the trapping potential shows that by switching between the CASC and the cooling regime, cooling efficiency is traded for an increased robustness of the cooling scheme. This may in particular be important for ongoing experiments.

The paper is structured as follows. In section 2 we introduce the model for a trapped polar molecule coupled to a stripline cavity, and present a brief overview of the basic physical ideas and mechanisms underlying the cavity assisted microwave cooling. In section 3 we proceed with a technical derivation of the cooling and heating rates in the bad cavity limit. Based on these results we present in section 4 a more detailed discussion of cooling rates and final temperatures as a function of different system parameters. Finally, in section 5 we summarize the main results and conclusions of this work.

## 2 Overview on cavity-assisted microwave cooling of a trapped polar molecule

The goal of this work is to present a theory for ground-state cooling of trapped polar molecules, in analogy to the ideas and theoretical tools employed for cooling of trapped ions in the optical domain [36, 37]. In this section we start with a brief review of the system involving a polar molecule coupled to a stripline cavity, as proposed in [25]. Further, the general relation between cooling/heating rates and the spectrum of fluctuations of the dipole forces which act on the molecule allows us to identify different cooling mechanisms in this setup. A qualitative discussion of the individual mechanisms is presented together with analytical results for a weakly driven molecule. The goal of this overview is to provide a physical understanding of the main cooling and heating processes which serves as a guideline through the more technical derivation of the force spectrum in section 3 and the extended discussion in section 4.

### 2.1 Cavity QED with a single trapped polar molecule coupled to a microwave cavity

We consider the setup shown in figure 1, where a single polar molecule is trapped close to a superconducting stripline cavity. The system Hamiltonian ,

 HS=Hc+Hm+Hint, (1)

is given in terms of the Hamiltonian for the microwave cavity, the Hamiltonian describing the internal degrees of freedom of the molecule including a classical microwave driving field, as well as its motion (external degree of freedom) in the trap. Finally describes the dipole interaction between rotational states of the molecule and the electric field of the cavity. Apart from the coherent evolution governed by , coupling of microwave photons to the electromagnetic environment introduces a dissipative decay channel for cavity photons. In section 3 we incorporate dissipation in terms of a master equation description of our system. In the following we summarize the properties of which are relevant for the discussion in this paper while more details can be found in [25, 38].

With initial temperatures of the molecule in the order of 1-100 mK, electronic and vibrational degrees of freedom are frozen out and the molecule is well described by its rotational degrees of freedom only. The anharmonicity of the rotor spectrum, , with the rotational constant and the angular momentum quantum number, allows us to identify two specific rotational states and with a transition frequency typically in the tens of GHz regime. With spontaneous emission rates of low lying rotational states in the mHz regime for zero temperature, the idea is to create an effective decay channel for the excited state via a microwave stripline cavity. When the frequency of the relevant cavity mode is near resonance with , the dynamics of the molecule is restricted to the two states and the molecule is well approximated by a two-level system (TLS).

The microwave cavity sketched in figure 1 is a (quasi-) one-dimensional superconducting stripline with length cm, and transverse dimension of order m. Allowed wavevectors correspond to a mode spacing in the GHz regime. Restricted to a single resonator mode, the cavity Hamiltonian is with () the annihilation (creation) operators for photons of frequency . The electric field generated by the cavity is given by the operator with the field strength associated with a single microwave photon. While varies on the scale of along the cavity axis, the spacial dependence of in the transverse direction or at the cavity end-caps is determined by the typical electrode spacing . For strong cooling forces we are in particular interested in those strong gradients in the near field regime. The microwave photons dissipate through the coupling of the stripline to its surrounding leads. At frequencies of GHz, quality factors in the range of  [39, 40, 41] translate into a photon loss rate in the range of MHz to kHz.

Molecules interact with microwave photons via the dipole interaction , where is the electric dipole operator of the molecule. The rotational states and are coupled to the quantized electric field of the cavity. Disregarding, for the moment, the motional degrees of freedom we can describe the combined cavity-molecule system, including a classical microwave field of frequency and Rabi-frequency which drives transitions between and , by a Jaynes-Cummings type model in a frame rotating with the microwave frequency ,

 HJC=ℏ(Δc−Δ)c†c−ℏΔ2σz+ℏg(σ+c+σ−c†)+ℏΩ2σx, (2)

where and are the detunings of the cavity field and the classical driving field from the rotational transition frequency, respectively, and we have adopted the usual Pauli operator notation for the TLS. Under resonance conditions equation (2) describes coherent oscillations between an excitation in the TLS, , and an additional photon in the cavity, , on a timescale  [42, 43]. For a molecule trapped at a distance m above the cavity and a molecular dipole moment of the order of several Debye the vacuum Rabi frequency is of the order of kHz [25]. In combination with this coherent energy transfer between molecule and cavity, the photon decay opens an effective decay channel for the rotationally excited state . For the corresponding characteristic decay rate is given by

 γ=2g2κκ2+Δ2c. (3)

Although the validity of this expression is limited to , we find a significant increase of the rotational decay rate from mHz in free space to a cavity enhanced decay of kHz. This also implies that in principle cooling rates of the same order can be achieved in this system.

Trapping of a molecule close to the surface of a chip requires a sufficiently strong trapping potential; electrostatic traps with frequencies of the order of MHz are experimentally feasible [25] 1. While in general the trapping potentials will be different for the two states and , there exist so-called ‘sweet spots’ for the electric bias fields, where this difference vanishes [25]. Let us consider a 1D model in which case the external degrees of freedom are described by a harmonic oscillator Hamiltonian , with () the usual annihilation (creation) operators. Coupling between internal and external degrees of freedom arises from the spatial dependence of the Rabi frequencies and , as compared to the case of laser cooling which we already remarked in the introduction. Here is the position operator of the molecule with respect to the center of the trap. Including motional degrees of freedom into equation (2), we end up with the explicit form for the system Hamiltonian (1),

 Hc = ℏ(Δc−Δ)c†c, Hm = ℏνa†a−ℏΔ2σz+ℏΩ(^x)2σx, Hint = ℏg(^x)(σ+c+σ−c†). (4)

The bare energy levels of and the different couplings between them are sketched in figure 1 b). We emphasize at this point that the validity of given in equation (4) is based on two crucial assumptions: i) the trapping potential is harmonic and ii) the trapping potential is state independent. Assumption i) is in general fulfilled near the trap center, i.e., in the final stage of the cooling process and as mentioned above, assumption ii) is fulfilled for certain conditions on the trapping fields. As neither of those assumptions will be strictly satisfied in a realistic experiment, we will investigate small deviations from these conditions in section 4. Note that as long as condition i) is approximately fulfilled we can treat the motion of the molecule along the , and directions independently. This justifies the use of the one dimensional model for the motion of the molecule.

While for all cooling schemes in the optical domain the temperature of the electromagnetic environment can safely be set to zero, this assumption is in general not valid in the microwave regime. In a cryogenic environment with a temperature the thermal occupation number of the cavity mode, , is typically of the order of . In addition to the vacuum contribution in equation (3), a finite results in cavity-assisted decay of rate and a cavity-assisted excitation rate . As we discuss in more detail below, thermally activated processes are one of the limiting factors in achieving cold temperature.

### 2.2 Cavity cooling of polar molecules in the Lamb-Dicke regime

When trapped close to the surface of the chip, cooling forces on the molecule arise from variations of the microwave fields on the typical length scale m, which is only determined by the electrode geometry. For a harmonic trapping potential we compare this scale with the harmonic oscillator length and introduce the dimensionless parameters and in analogy to the Lamb-Dicke parameter in optical cooling schemes; e.g. in the case of trapped ions, with the laser wavelength [36]. As the linear extension of the molecule trap is approximately limited by we find that cooling in this setup naturally occurs in the so-called Lamb-Dicke limit , where the spatial extension of the molecule wavepacket is small compared to variations of the microwave fields. While typically and will be equal or at least of the same order, in the following discussion we treat them as two independent parameters.

In the Lamb-Dicke limit we can expand the Hamiltonian (4) to first order in , i.e. . Here is the Hamiltonian describing the free evolution of the internal and motional degrees of freedom of the molecule. The coupling between the internal and motional degrees of freedom is of the form

 H1=^x^F,^F=ℏ∇Ω2σx+ℏ∇g(σ+c+σ−c†), (5)

such that is the mean dipole force exerted on the molecule. The small parameters and allow us to treat as a weak perturbation for the decoupled dynamics of the system. For , with the characteristic relaxation rate of the cavity-molecule system, we can adiabatically eliminate the internal degrees of freedom (see section 3) and derive an effective equation of motion for the reduced external density operator . The resulting equation for the mean occupation number (see also [29, 36, 45]) is of the form

 ⟨˙n⟩=−W[⟨n⟩−⟨n⟩0]. (6)

For a positive damping rate , equation (6) describes relaxation of the mean occupation number to a final steady state value . The relevant quantities characterizing the cooling dynamics, and , are given by

 W=A−−A+,⟨n⟩0=A+/(A−−A+), (7)

where the cooling and heating rates and are determined by the spectrum of the force operator,

 S(ω)=2x20ℏ2Re∫∞0dτ⟨^F(τ)^F(0)⟩0e+iωt. (8)

Equations (6)-(8) provide the general relation between the cooling dynamics of the external degrees of freedom and the fluctuation spectrum of the internal operator . Efficient cooling schemes correspond to the case where both the difference is maximized (large cooling rate) and the ratio is minimized (low final temperatures). Throughout this paper we make extensive use of this relation to discuss different cooling schemes in terms of the properties of .

### 2.3 Cavity cooling of polar molecules in the bad cavity regime

While the results for cooling in the Lamb-Dicke limit summarized in equations (6)-(8) are valid for arbitrary cavity-molecule interactions, we concentrate in the remainder of this paper on the bad cavity limit (BCL), , which we expect to be the most relevant regime in experiments. For additional discussions on cavity cooling of atoms in the good cavity limit the reader is referred to [29].

The BCL allows us to eliminate the cavity degrees of freedom and to study and interpret cavity-cooling in terms of the resulting effective dynamics for the molecule. However, based on the underlying coherent interaction and the finite bandwidth of the cavity mode, the effective dynamics of the molecule differs significantly from the usual case of a two level atom coupled to the free radiation field. The adiabatic elimination of the cavity mode, which is discussed in more detail in section 3, results in a decomposition of the general form of given in equation (8) into three contributions,

 S(ω)=SΩ(ω)+Sg(ω)+SI(ω). (9)

The labels on the first two terms refer to their origin; the spectrum is associated with the force from the drive field gradient and exhibits sharp resonances of width , whereas is associated with the force due to the cavity-TLS coupling gradient, , and exhibits a broader resonance of width . As explained in more detail in the following discussion, we can with each of these forces associate a different cooling mechanism in our system. The third contribution to the spectrum, , describes destructive and constructive interference between the different cooling processes. For illustration, the spectrum and its decomposition according to equation (9) are shown in figure 2 for the case of zero temperature and a) for a weakly () and b) for a strongly () driven molecule.

The results presented in this overview section apply to a weakly driven molecule, and generalizations to arbitrary are derived in section 4.

#### Cavity-assisted sideband/Doppler cooling.

We first consider the limit where the cavity field exerts no force on the molecule but still provides an effective decay channel for the excited state . In this limit the force spectrum reduces to the first term in equation (9), . For zero temperature and the spectrum exhibits a single Lorentzian peak of width centered at a frequency (see figure 3). We identify with the excitation spectrum of a decaying TLS and recover a situation which is familiar from laser cooling of trapped ions [36, 37]. In the so-called resolved sideband limit, i.e. when the trap frequency is large compared to the effective decay rate , we can choose the detuning such that . Under those conditions sideband cooling leads to final occupation numbers close to the quantum ground state, .

For non-zero temperature the cavity acts as a finite temperature reservoir for the molecule, which has a corresponding equilibrium excited state population . In the spectrum another resonance appears at with a height . As seen from figure 3 b) this resonance corresponds to transitions from the excited state to the ground state which for preferably increase the motional quantum number and lead to heating. In equilibrium heating and cooling rates balance each other, meaning that the steady state occupation number in the trap is equal to the thermal cavity occupation number . This equilibrium condition translates into a final temperature .

In section 4 we show that for a weak driving field and for the correct interpolation between the zero temperature limit and the finite temperature result of sideband cooling is given by

 ⟨n⟩0=N+(2N+1)(γN4ν)2. (10)

While in the sideband resolved regime cooling is most efficient we may instead consider the so-called Doppler limit . In this regime cooling rates are optimized by choosing the detuning and the minimum final occupation number is,

 ⟨n⟩0=(2N+1)γN4ν. (11)

In contrast to the sideband resolved regime, we find that when the linewidth exceeds the trapping frequency the scaling of the final occupation number with temperature is less favorable and cooling becomes inefficient for large .

Equations (10) and (11) have been derived for a weakly driven molecule and generalize the usual limits for sideband and Doppler cooling of trapped ions to the case of a finite temperature reservoir. In section 4 we study the limit of a strongly driven molecule (see figure 2 b)), where we show that by increasing we benefit not only from a higher damping rate , but cavity assisted sideband cooling also gets more robust with respect to imperfections in the trapping potential.

#### ∇g-cooling.

We now consider the opposite limit where cooling forces arise from gradients of the cavity-molecule coupling only. In addition, to avoid interference effects discussed below, for the moment we assume such that the spectrum of force fluctuations reduces to the second term in equation (9), .

For simplicity we first look at the case of zero temperature . As shown in figure 2, has a single resonance at with a width equal to the photon loss rate . The deviation from the naively expected value can be understood from the level diagram shown in figure 4 b). With the molecule initially in the ground state, the energy extracted from the trap has to match the energy to fully excite the TLS and the additional energy to excite the cavity (). In analogy to the discussion given above we distinguish two regimes of -cooling, namely the resolved sideband regime (SB) and the Doppler regime (D) with the corresponding limits for the steady state occupation number,

 (SB):⟨n⟩0=(κ2ν)2,(D):⟨n⟩0=κ2ν. (12)

Therefore, at zero temperature the final occupation number produced by -cooling may be discussed in analogy to the cavity-assisted cooling, with the decay rate replaced by the photon decay rate . The sideband resolved regime, which has been discussed in the optical regime e.g. in reference [11], seems to be accessible in the present setup.

For finite temperature, -cooling becomes more involved as can be seen from the spectrum plotted in figure 4. In the resolved sideband regime, now consists of four distinct resonances at and . The resonance at corresponds to the process discussed above for zero temperature, and the resonance at corresponds to the analogue process with the molecule initially in the excited state. The two resonances at the cavity detuning describe thermally activated processes which do not involve the external driving field . We interpret the corresponding heating and cooling rates as a cavity mediated thermalization process of the molecular motion, which otherwise would be highly decoupled from the electromagnetic environment. In the weak driving limit we find that the resonances at are of equal height and therefore result in a purely diffusive () dynamics. In other words, due to the large frequency mismatch implicitly assumed in the derivation of our model, any finite thermal occupation of the cavity mode translates into an effective infinite temperature reservoir for the molecular motion.

In the resolved sideband limit we minimize re-thermalization rates by choosing either or . In section 4 we show that -cooling at finite temperatures results in a steady state occupation number,

 ⟨n⟩0=N+(N+1)(κ2ν)2+⟨n⟩th, (13)

where for both choices on . Thus in contrast to cavity-assisted sideband cooling, we find that for -cooling at finite temperature the final occupation number depends crucially on the driving field . A similar result is obtained for the Doppler regime,

 ⟨n⟩0=κ2ν[(2N+1)+βth], (14)

with for the same choices on . From equations (13) and (14) we conclude that to employ the mechanism of -cooling efficiently even at finite temperature we must go beyond the weak driving limit and consider either or . In the regime , the cavity is far detuned from the rotational transition frequency and cavity mediated thermalization is suppressed, but it requires enough power to scatter photons from the driving field off-resonantly into the cavity. Alternatively we may consider the regime such that the driven excited state population becomes comparable to and stimulated cooling transitions eventually dominate over thermally activated heating processes. However, in this regime equation (13) and (14) are no longer valid and we postpone the discussion of cooling in the strongly driven regime to section 4 where we find a significant improvement also for a low driving power .

#### Cooling by quantum interference.

As a consequence of interference effects between forces exerted on the molecule by the cavity and the classical microwave field, the spectrum plotted in figure 2 clearly differs from the naively expected sum of and . In equation (9) we have introduced the term to account for this difference.

In figure 5 we have sketched the three different paths which contribute to the interference effects. In a) the molecule is excited on the red sideband followed by an effective decay via the cavity on the carrier transition. This process is described by . In b) the molecule is excited on the carrier transition and changes its motional state in the decay process, described by . The third path sketched in figure 5 c) might not be obvious in the first place and has the following interpretation. The position dependent coupling of the TLS to the cavity mode, , in general leads to a position dependent decay constant as well as a position dependent Stark shift . In a semiclassical picture, the oscillating motion of the molecule in the trap therefore translates into modulations of and at the frequency and results in additional sidebands in the excitation process. Note that interference between path b) and path c) also occurs for and in general we cannot write even in this limit. However, in section 4 we will argue that in the parameter regimes relevant for cooling those interference effects only slightly modify the cooling process. For finite temperature a fourth process, the decay of a thermally excited molecule as sketched in figure 5 d), adds incoherently to the processes a)-c). Again, this process is proportional to the thermal equilibrium excited state population .

Cooling by quantum interference was discussed and demonstrated in the context of electromagnetically induced transparency [46, 47] and has been proposed for atoms in optical cavities [26, 28]. In all those schemes cooling relies on a reduction or ideally a cancelation of blue sideband transitions () by interference, while is still finite, which in the ideal case leads to ground state cooling, . From figure 2 we see that in the present system such an exact cancelation of the heating rate appears in the weak driving regime, but is no longer present in the strongly driven system. At finite temperature thermally activated processes reduce the contrast of the interference features, and from the discussion of the individual paths given above we conclude that cooling by quantum interference is only efficient as long as driven cooling and heating processes dominate over thermally activated processes . Due to the sensitivity to temperature as well as other system parameters, we do not discuss cooling by quantum interference in much detail in the present work, but in section 4 we point out several consequences of interference effects for the combination of CASC and -cooling. In references [26, 28, 29] more details on interference effects in cavity-assisted cooling of atoms can be found for the weak excitation regime (, ).

## 3 Theory of Cavity Cooling in the bad cavity limit

In the previous overview section we have presented for the case of a weakly driven molecule a physical picture of the individual cooling mechanisms which contribute to the overall cavity-assisted cooling process. For the weak driving regime and for zero temperature, analytic expressions for and the resulting cooling rates have been derived in previous work [26, 29] on a related system of trapped atoms inside optical cavities. However, for non-zero temperature we have identified thermal cavity-mediated heating processes which degrade the cooling efficiency, and thus in the present setup there is a clear necessity to extend the discussion of cavity cooling schemes to the regime of a strongly driven molecule. In this section we present a derivation of heating and cooling rates which is valid in the bad cavity limit and allows us to study cavity cooling for a wide range of system parameter.

The starting point for the derivation is the combined cavity-molecule system discussed in section 2, for which the evolution of the total density operator is determined by the following master equation,

 ˙ρ = L(ρ) = −i[HS,ρ]+Lκ(ρ). (15)

The system Hamiltonian is defined in equations (1) and (4), and the coupling of the cavity field to a thermal reservoir is taken into account by the term

 Lκ(ρ)=κ(N+1) D[c](ρ)+κN D[c†](ρ), (16)

with . Starting from the full master equation (15) we proceed in two steps. First, we make use of the small parameter to eliminate the dynamics of the cavity mode and derive an effective master equation for the internal (TLS) and external degrees of freedom of the molecule. In a second step, the weak coupling between the TLS and the external degrees of freedom, represented by the small parameters , also allow us to eliminate the TLS and derive an effective master equation describing only the motion of the molecule in the trap.

### 3.1 Adiabatic elimination of the cavity mode

For the elimination of the cavity degrees of freedom we decompose the total Liouville operator defined in equation (15) into three contributions

 L=Lc+Lm+Lg. (17)

The first two terms describe the uncoupled dynamics of the cavity mode,

 Lc(ρ)=−i[(Δc−Δ)c†c,ρ]+Lκ(ρ), (18)

and the molecular degrees of freedom,

 Lm(ρ)=−i[Hm,ρ], (19)

whereas the interaction between the cavity mode and the molecule is described by the last term in equation (17),

 Lg(ρ)=−i[g(^x)(σ+c+σ−c†),ρ]. (20)

In the decoupled limit the cavity field evolves independently of the molecule. On a timescale the system relaxes into the state , with the cavity equilibrium density operator defined by , and the molecular operator evolving under the action of . For a finite coupling deviations from the factorized form of are small, but the coupling term modifies the dynamics of . To proceed we adopt a projection operator technique along the lines of reference [45] and define the projector and its orthogonal complement . Inserting the decomposition into equation (15) we obtain the two coupled equations

 P˙ρ(t) = PLmPρ(t)+PLgQρ(t), (21) Q˙ρ(t) = Q(Lc+Lm+Lg)Qρ(t)+QLgPρ(t). (22)

As the population in the subspace is damped with a rate which is fast compared to the coupling term , we formally integrate equation (22), insert the result into equation (21) and expand the final expression up to second order in . For times we end up with an effective master equation for the molecule density operator which is given by

 ˙ρm(t)=−i[Hm,ρm(t)]+∫∞0dτTrc{LgQe(Lc+Lm)τQLge−Lmτ(ρm(t)⊗ρ0c)}. (23)

Strictly speaking the effective master equation (23) is valid in the limit , but we expect it to hold to a good approximation also for a slightly relaxed condition, .

The second term in equation (23) describes the effect of the cavity on the molecular dynamics. We evaluate this term by inserting the definitions of , and given in equations (18)-(20) and obtain an effective master equation on the form (see also [45])

 ˙ρm=−i[Hm,ρm]+[(N+1)(TρmS†−S†Tρm)+N(T†ρmS−ST†ρm)+H.c.]. (24)

Here we introduced the abbreviations and

 T=∫∞0dτe−i(Δc−Δ)τe−κτS(−τ),S(t)=eiHmtSe−iHmt. (25)

In summary we have derived an effective master equation (24) for the molecule, where the first term describes the bare evolution, and the second term describes the influence of the cavity on the dynamics. We want to point out that the latter term in general has a complicated form, since the operator (25) depends on the bare molecule Hamiltonian . Only in the infinite bandwidth limit, where is large compared to and , will equation (25) reduce to the simple form .

### 3.2 Effective molecule dynamics in the Lamb-Dicke regime

To proceed with the derivation, we now consider the master equation (24) in the Lamb-Dicke limit where internal and external degrees of freedom interact weakly. We expand the position dependent Rabi frequencies and up to first order in the Lamb-Dicke parameters, i.e.,

 Ω(^x)≃Ω[1+η(a+a†)]andS≃gσ−[1+ηc(a+a†)]. (26)

Under the same approximation the operator gets the form

 T≃g[Σ−+ηcΣ−(ν)a+ηcΣ−(−ν)a†], (27)

with effective TLS operators defined in (59). Up to second order in the Lamb-Dicke parameters we obtain the master equation

 ˙ρm(t) ≃ (L0+L1+L2) ρm(t), (28)

where the subscripts indicate the respective order in the Lamb-Dicke parameters . The zeroth order term is given by

 L0=LI+LE, (29)

where describes the uncoupled external dynamics of the molecule, and describes the effective dynamics of the uncoupled TLS,

 LIρm = −i[HI,ρm] + g2(N+1)[Σ−ρmσ+−σ+Σ−ρm+H.c.] (30) + g2N[Σ+ρmσ−−σ−Σ+ρm+H.c.].

Explicit expressions for and are listed in A. Let us for the moment consider the effective dynamics of the TLS (30), with the corresponding Bloch equations for the set of operators of the form

 Unknown environment '%' (31)

Here and the remaining parameters are defined in B. In the discussion in section 4 we will mainly focus on the limit where the parameters and are small and equation (31) reduces to the normal Bloch equations for a two level system coupled to a thermal bath. Further the stark shift and the effective decay rate reduce to the familiar expressions

 ~δ→δ=(2N+1)g2Δcκ2+Δ2c,~γ→γ=2g2κκ2+Δ2c. (32)

For the purpose of readability, we will use the simplification in the following. Finally note that when the driving strength is comparable to the cavity linewidth , unless there will be modifications of and . In addition, the non-standard terms in the Bloch equations (31) introduce new features in the TLS dynamics which modify the force spectrum and in the end the cooling dynamics. A detailed discussion of the effective dynamics of a strongly driven TLS inside a cavity is given in [48].

### 3.3 Adiabatic elimination of the internal degrees of freedom

Starting from equation (28) we further eliminate the internal degrees of freedom in a similar fashion as the above elimination of the cavity mode, following the procedure in [45]. First we notice that the zeroth order term in the expansion (28) describes the uncoupled dynamics of the internal and external degrees of freedom, whereas the higher order terms couple the external and internal dynamics and therefore introduce forces acting on the molecule motion. Provided the coupling is weak compared to the energy scales defining the TLS dynamics, , the state of the TLS will deviate only slightly from the equilibrium state due to the interaction, and the dynamics of the TLS can be eliminated. We define a projection operator ,

 Pμ(ρm)=ρ0I⊗∑n|n⟩⟨n|TrI{⟨n|ρm|n⟩}, (33)

with the internal equilibrium state defined by . Second order perturbation theory in the Lamb-Dicke parameters results in a master equation for the external operator , which describes the evolution of the population of the individual trap levels. We obtain

 Extra open brace or missing close brace (34)

Inserting the definitions of given in A we end up with a master equation on the form

 ˙μ(t)=A−D[a](μ(t))+A+D[a†](μ(t)), (35)

and recover the cooling equation (6) given in section 2 with and . The heating and cooling coefficients which follow from an evaluation of equation (34) can be divided into three contributions,

 A±=SΩ(∓ν)+Sg(∓ν)+SI(∓ν), (36)

where the physical motivation for the decomposition (36) has been presented in section 2. The first contribution to the force spectrum, , follows from the first term of equation (34) in the limit and is defined as

 SΩ(ω)=η2Ω22Re∫∞0dτTrI{σxeL0τ(σxρ0I)}eiωτ. (37)

The second term, , follows from the second term in equation (34) and is given by

 Sg(ω)=2η2cg2 Re{(N+1)⟨σ+Σ−(ω)⟩I+N⟨σ−Σ+(ω)⟩I}, (38)

with effective TLS operators given by (61). Finally, remaining contributions from the first term in equation (34) are found by making the Ansatz,

 L1(ρ0I⊗μ)=K+(ρ0I)⊗aμ+K−(ρ0I)⊗a†μ+K†−(ρ0I)⊗μa+K†+(ρ0I)⊗μa†. (39)

The result is summarized by defined as

 SI(±ν)=2Re∫∞0dτTrI{K†±(eL0τK±(ρ0I))}e±iντ−SΩ(±ν), (40)

with the action of the superoperators given by

 K±(ρ)=−iηΩ2σxρ + ηcg2(N+1)[σ−ρΣ++Σ−(±ν)ρσ+−σ+Σ−(±ν)ρ−σ+Σ−ρ] (41) + ηcg2N[σ+ρΣ−+Σ+(±ν)ρσ−−σ−Σ+(±ν)ρ−σ−Σ+ρ].

In conclusion, we have expressed heating and cooling rates in terms of two-point correlation functions and steady state expectation values of a TLS whose evolution is described in terms of the effective Bloch equations (31). The results are valid for and , but at this stage no further assumptions have been made. However, in the following we mainly concentrate on the case or on the regime , where the Bloch equations reduce to the standard form (see B). Using the quantum regression theorem and the steady state solution of equation (31) we outline the derivation of analytic expressions for and in C and D. Analytic expressions for could in principle be derived along the same lines but in general do not have simple enough form to provide further insight. Therefore we will rather use numerical calculations in combination with previously derived results for the weak excitation limit [29] as a basis for the discussion of the interference terms.

## 4 Discussion

In this section we discuss cavity-assisted cooling of polar molecules both in the weak and strong driving regime, and base the discussion on the result for derived in section 3. For a physical understanding we again find it useful to discuss cavity assisted sideband cooling and cooling independently in a first step. In a second step we will then study the general case where both mechanisms as well as interference effects are taken into account. Analytical results for cooling rates and final occupation numbers for the different parameter regimes are summarized at the end of this paper in table 1.

### 4.1 Cavity assisted sideband cooling

We start the discussion with the mechanism of cavity assisted sideband cooling (CASC) where the corresponding cooling rate and steady state occupation depend on the spectrum defined in equation (37). Following the calculations outlined in C we obtain for the spectrum in the weak driving limit, ,

 SΩ(ω)=η2Ω24[ρ0ggγN(ω+Δ)2+γ2N/4+ρ0eeγN(ω−Δ)2+γ2N/4]. (42)

For a fixed ratio we optimize with respect to the detuning , where obviously must be negative to promote cooling. For the two limits of interest, the resolved sideband limit (SB) and the Doppler limit (D) , we find and respectively. The final occupation number is given by the respective expressions (10) and (11) in section 2, and the corresponding cooling rates are given by,

 (SB):WΩ=η2Ω2(2N+1)γN,(D):WΩ=η2(2ν/γN)Ω2(2N+1)γN. (43)

Although in the weak driving limit the final occupation number is independent of , the cooling rate scales as and under the validity of equation (43) it is bounded be . In competition with the cavity-mediated thermalization process of the molecule, discussed in section 2, it is therefore necessary to study sideband cooling beyond the weak driving regime.

Let us now consider an arbitrary , but focus on CASC in the sideband resolved limit . As shown in C, the spectrum now exhibits a three-peak structure and reads,

 SΩ(ω)≃η2Ω24[α+¯γ(ω−¯Δ)2+¯γ2/4+α0¯γ0ω2+¯γ20/4+α−¯γ(ω+¯Δ)2+¯γ2/4], (44)

with . For sideband cooling we are only interested in the peaks at nonzero frequency. The relative heights of the resonances, , and the widths which are proportional to are defined in C and are functions of , and . For to be positive we require a red detuned microwave field and cooling is optimized for . Under those conditions we obtain

 WΩ = η2Ω24(α+−α−¯γ) = γ(ηνγN)2g(φ), (45)

with , and

 ⟨n⟩0,Ω = α−α+−α−+O(γ2N/ν2) ≃ N+(2N+1)g1(φ), (46)

with . The functions and are plotted in figure 6 and depend on the angle defined as . We find that the cooling rate increases with up to but then goes to zero for . It has a maximum value of at , and we want to point out that this holds independently of the temperature . The final steady state occupation number increases monotonically with increasing ; and for .

We conclude that driving moderately, , increases the cooling rate tremendously while only marginally increasing the final occupation number. We point out that the validity of the adiabatic elimination constraints the parameters, , and thus the expected upper limit still holds. Nevertheless, this bound is by a factor higher than in the weak driving case.

### 4.2 ∇g cooling

Let us now move on to discuss the mechanism of cooling, where the cooling rate and the final occupation number are given in terms of the spectrum defined in equation (38). In D we show that for finite temperature and arbitrary or the structure of (76) is quite involved and generally exhibits 6 resonances centered at frequencies and . Note that we assume , which holds naturally when we later combine cooling with CASC. As the height of two of the resonances, at where , scales as we can for weak driving neglect those contributions and obtain the spectrum

 Sg(ω)=η2cg22[ κ(N+1)(Ω/Δ)2κ2+(ω−Δg)2 + κN(Ω/Δ)2κ2+(ω+Δg)2 + ∑s=±4κ(N+1)ρ0eeκ2+(ω+sΔc)2 ]. (47)

We note that only the driven terms in equation (47) contribute to the cooling rate , which for optimized detuning in the Doppler limit and in the sideband limit, respectively, are given by

 (SB):W=η2cg