Theory of cavity-assisted microwave cooling of polar molecules

Margareta Wallquist^1,2,5, Peter Rabl^3,4, Mikhail D Lukin^3,4 and Peter Zoller^1,2

¹ Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria
² Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, 6020 Innsbruck, Austria
³ Institute for Theoretical Atomic, Molecular and Optical Physics, Cambridge, MA 02138, USA
⁴ Physics Department, Harvard University, Cambridge, MA 02138, USA

⁵ Author to whom any correspondence should be addressed.

E-mail: margareta.wallquist@uibk.ac.at

Received 18 March 2008
Published 4 June 2008

Contents

• 1. Introduction
• 2. Overview on cavity-assisted microwave cooling of a trapped polar molecule
  • 2.1. Cavity quantum electrodynamics (QED) with a single trapped polar molecule coupled to a microwave cavity
  • 2.2. Cavity cooling of polar molecules in the Lamb-Dicke regime
  • 2.3. Cavity cooling of polar molecules in the bad cavity regime
    • 2.3.1. CASC/Doppler cooling
    • 2.3.2. ∇g-cooling
    • 2.3.3. Cooling by quantum interference
• 3. Theory of cavity cooling in the bad cavity limit
  • 3.1. Adiabatic elimination of the cavity mode
  • 3.2. Effective molecule dynamics in the Lamb-Dicke regime
  • 3.3. Adiabatic elimination of the internal degrees of freedom
• 4. Discussion
  • 4.1. Cavity-assisted sideband cooling
  • 4.2. ∇g cooling
  • 4.3. Combining cavity-assisted microwave cooling and ∇g cooling
    • 4.3.1. Weak saturation regime
    • 4.3.2. Strong driving regime
    • 4.3.3. Fully saturated regime
• 5. Summary and conclusion
• Acknowledgments
• Appendix A. Molecular master equation in the Lamb-Dicke limit
• Appendix B. Bloch equation
• Appendix C. Evaluation of S_&b.Omega;(&b.omega;)
• Appendix D. Evaluation of S_g(&b.omega;)
• References

1. Introduction

Nowadays cold molecules are being very actively explored (see e.g. [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]–[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^Note6  both experimentally and theoretically, including cavity-assisted laser cooling [11]–[13], buffer gas cooling, and Stark deceleration of polar molecules including trapping in magnetic and electric traps [14]–[18], evaporative [19] and sympathetic cooling schemes [20], and preparation of ground state molecules by photoassociation of cold atoms [21]–[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 d~1 μm above the cavity electrodes, the vacuum Rabi frequency g associated with the exchange of a rotational excitation and a single microwave photon is in the order of 100 kHz. In combination with a comparable photon loss rate 2κ~g this coupling serves as an efficient dissipation channel for (in the 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]–[31]. Microwave cooling does not rely on specific electronic or collisional properties [2, 19, 20], [32]–[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 `∇g' 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 T, thermal cavity photons at frequency ω_c introduce an additional heating mechanism, and the minimal temperature T_f is given by T_f = (ν/ω_c)T  T with ν 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 ∇g 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 (BCL). 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 quantum electrodynamics (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 H_S,

is given in terms of the Hamiltonian H_c for the microwave cavity, the Hamiltonian H_m 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, H_int describes the dipole interaction between rotational states of the molecule and the electric field of the cavity. Apart from the coherent evolution governed by H_S, 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 H_S which are relevant for the discussion in this paper while more details can be found in [25, 38].

Figure 1. (a) Schematic plot of the setup considered in this paper for cavity-assisted cooling of polar molecules. A more realistic trap design can be found elsewhere [25]. (b) Level diagram for the cavity–molecule system restricted to the lowest states |g|0c, |e|0c and |g|1c. Dashed lines indicate different quantized motional levels. See text for more details.

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, E_rot(N) = BN(N + 1), with B the rotational constant and N the angular momentum quantum number, allows us to identify the two specific rotational states |g and |e with a transition frequency typically in the tens of GHz regime; for CaBr ω_eg/2π≊11 GHz. 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 |e via a microwave stripline cavity. When the frequency of the relevant cavity mode ω_c is near resonance with ω_eg, the dynamics of the molecule is restricted to the two states {|g, |e} and the molecule is well approximated by a two-level system (TLS).

The microwave cavity sketched in figure 1 is a (quasi-) one-dimensional (1D) super-conducting stripline with length L~1 cm, and transverse dimension of the order of d~1 μm. Allowed wavevectors k_n = nπ/L correspond to a mode spacing in the GHz regime. Restricted to a single resonator mode, the cavity Hamiltonian is with c (c^†) the annihilation (creation) operators for photons of frequency ω_c. 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 L along the cavity axis, the spatial dependence of in the transverse direction or at the cavity endcaps is determined by the typical electrode spacing d. 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 ω_c/2π~10 GHz, quality factors in the range of Q~10³–10⁶ [39]–[41] translate into a photon loss rate 2κ 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 |g and |e are coupled to the quantized electric field of the cavity. For the moment, we disregard the motional degrees of freedom, and further introduce a classical microwave field E_mw cos ω_mt which drives transitions between |g and |e with Rabi frequency Ω = μE_mw. We can describe the combined cavity–molecule system by a Jaynes–Cummings-type model in a frame rotating with the microwave frequency ω_m,

where Δ_c = ω_c–ω_eg and Δ = ω_m–ω_eg 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 ω_c≈ω_eg, equation (2) describes coherent oscillations between an excitation in the TLS, |e|n_cc, and an additional photon in the cavity, |g|n_c + 1_c, on a timescale  [42, 43]. For a molecule trapped at a distance h~d~1–0.1 μm above the cavity and a molecular dipole moment of several Debye (μ = 4.36 D for CaBr) the vacuum Rabi frequency g is of the order of 2π×50–500 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 |e. For κ > g, the corresponding characteristic decay rate is given by

Although the validity of this expression is limited to γ  g, we find a significant increase of the rotational decay rate from Γ_eg~ mHz in free space to a cavity enhanced decay of γ/2π~10–100 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 ν/2π~1 MHz are experimentally feasible [25]^Note7 . While in general the trapping potentials will be different for the two states |g and |e, there exist the 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 a (a^†) the usual annihilation (creation) operators. Coupling between the 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 upon 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 H_S (1),

The bare energy levels of H_S and the different couplings between them are sketched in figure 1(b). We emphasize at this point that the validity of H_m given in equations (4) is based on the 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 x-, y- and z-directions independently. This justifies the use of the 1D 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 T~0.02–4 K the thermal occupation number of the cavity mode, , is typically of the order of N~0–10. In addition to the vacuum contribution in equation (3), a finite N results in cavity-assisted decay of rate γ(N + 1) and a cavity-assisted excitation rate γN. 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 d~1 μm, 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 η≡x₀∇Ω/Ω|_x = 0 and η_c≡x₀∇g/g|_x = 0 in analogy to the Lamb–Dicke parameter in optical cooling schemes; e.g. in the case of trapped ions, η = 2πx₀/λ with λ the laser wavelength [36]. As the linear extension of the molecule trap is approximately limited by d we find that cooling in this setup naturally occurs in the so-called Lamb–Dicke limit η, η_c≈x₀/d  1, where the spatial extension of the molecule wavepacket is small compared to variations of the microwave fields; x₀≊6 nm for CaBr in a ν/2π = 1 MHz trap. While typically η and η_c 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. H_S≊H₀ + H₁. Here, H₀ 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

such that is the mean dipole force exerted on the molecule. The small parameters η and η_c allow us to treat H₁ as a weak perturbation for the decoupled dynamics of the system. For |H₁|  Γ_I, with Γ_I = min{γ, κ} 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 μ(t). The resulting equation for the mean occupation number n(t) = Tr{a^†aμ(t)} (see also [29, 36, 45]) is of the form

For a positive damping rate W > 0, equation (6) describes relaxation of the mean occupation number to a final steady state value n₀. The relevant quantities characterizing the cooling dynamics, W and n₀, are given by

where the cooling and heating rates A_– = S(ν) and A ₊  = S(–ν) are determined by the spectrum of the force operator,

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 S(ν)–S(–ν) is maximized (large cooling rate) and the ratio S(–ν)/S(ν) 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 S(ω).

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 BCL, g  κ, 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 S(ω) given in equation (8) into three contributions,

The labels on the first two terms refer to their origin; the spectrum S_Ω is associated with the force from the drive field gradient ∇Ω and exhibits sharp resonances of width ~γ, whereas S_g is associated with the force due to the cavity–TLS coupling gradient, ∇g, and exhibits a broader resonance of width 2κ. 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, S_I, describes destructive and constructive interference between the different cooling processes. For illustration, the spectrum S(ω) and its decomposition according to equation (9) are shown in figure 2 for the case of zero temperature, (a) for a weakly (Ω  |Δ|) and (b) for a strongly (Ω~|Δ|) driven molecule.

Figure 2. Example for the force spectrum S(ω) for zero temperature (N = 0) and for (a) Ω  |Δ| and b) Ω~|Δ|. In both the figures, the contribution of SΩ(ω) and Sg(ω) are indicated by the dashed and dotted lines, respectively.

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

2.3.1. CASC/Doppler cooling We first consider the limit η_c→0, where the cavity field exerts no force on the molecule but still provides an effective decay channel for the excited state |e. In this limit, the force spectrum reduces to the first term in equation (9), S(ω)≡S_Ω(ω). For zero temperature and Ω  γ the spectrum exhibits a single Lorentzian peak of width γ centered at a frequency ω = –Δ (see figure 3). We identify S_Ω(ω) 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 γ, we can choose the detuning Δ = –ν such that S(ν)  S(–ν). Under those conditions sideband cooling leads to final occupation numbers close to the quantum ground state, n₀ = (γ/4ν)².

Figure 3. CASC. (a) Spectrum SΩ(ω) in the weak driving regime Ω  |Δ| for N = 0 (solid line) and N = 0.5 (dashed line). The parameters chosen for this plot are (all in units of ν): Ω = 0.1, Δ = –1, Δc = 0, κ = 5 and g = 0.5. (b) Resonant processes contributing in the sideband resolved regime. Dashed lines indicate processes at finite environment temperature.

For nonzero temperature the cavity acts as a finite temperature reservoir for the molecule, which has a corresponding equilibrium excited state population ρ_ee⁰ = N/(2N + 1). In the spectrum, another resonance appears at ω = –ν with a height ~ρ_ee⁰. As seen from figure 3(b), this resonance corresponds to transitions from the excited state to the ground state which for Δ = –ν 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 n₀ in the trap is equal to the thermal cavity occupation number N. This equilibrium condition translates into a final temperature T_f = (ν/ω_c)T  T. In section 4, we show that for a weak driving field Ω  γ_N = (2N + 1)γ and for γ_N  ν the correct interpolation between the zero temperature limit n₀ = (γ/4ν)² and the finite temperature result n₀≊N of sideband cooling is given by

While in the sideband resolved regime cooling is most efficient we may instead consider the so-called Doppler limit γ_N  ν. In this regime, cooling rates are optimized by choosing the detuning Δ = –γ_N/2 and the minimum final occupation number is

In contrast to the sideband resolved regime, we find that when the linewidth γ_N exceeds the trapping frequency ν the scaling of the final occupation number with temperature is less favorable (~N²) and cooling becomes inefficient for large N.

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 Ω we benefit not only from a higher damping rate W, but CASC also gets more robust with respect to imperfections in the trapping potential.

2.3.2. ∇g-cooling We now consider the opposite limit η→0, 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 g→0 such that the spectrum of force fluctuations reduces to the second term in equation (9), S(ω)≡S_g(ω).

For simplicity, we first look at the case of zero temperature N = 0. As shown in figure 2, S_g(ω) has a single resonance at ω = Δ_c–Δ with a width equal to the photon loss rate 2κ. The deviation from the naively expected value ω = Δ_c 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 ∇g-cooling, namely the resolved sideband regime κ  ν (SB) and the Doppler regime κ  ν (D) with the corresponding limits for the steady state occupation number,

Therefore, at zero temperature the final occupation number produced by ∇g-cooling may be discussed in analogy to the cavity-assisted cooling, with the decay rate γ replaced by the photon decay rate 2κ. The sideband resolved regime, which has been discussed in the optical regime, e.g. in [11], seems to be accessible in the present setup.

For finite temperature, ∇g-cooling becomes more involved as can be seen from the spectrum S_g(ω) plotted in figure 4. In the resolved sideband regime, S_g(ω) now consists of four distinct resonances at ω = ±(Δ_c–Δ) and ω = ±Δ_c. The resonance at ω = Δ_c–Δ corresponds to the process discussed above for zero temperature, and the resonance at ω = –(Δ_c–Δ) corresponds to the analogue process with the molecule initially in the excited state. The two resonances at the cavity detuning ω = ±Δ_c describe thermally activated processes which do not involve the external driving field Ω. 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 ω = ±Δ_c are of equal height and therefore result in a purely diffusive (W = 0) dynamics. In other words, due to the large frequency mismatch ω_c  ν 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 κ  ν, we minimize re-thermalization rates by choosing either |Δ_c|  |Δ_c–Δ| or |Δ_c| = 0. In section 4, we show that ∇g-cooling at finite temperatures results in a steady state occupation number,

where n_th~8(N + 1)ρ_ee⁰(κ/Ω)² for both choices on |Δ_c|. Thus in contrast to cavity-assisted sideband cooling, we find that for ∇g-cooling at finite temperature the final occupation number depends crucially on the driving field Ω. A similar result is obtained for the Doppler regime,

with β_th~16(N + 1)ρ_ee⁰(κ/Ω)² for the same choices on |Δ_c|. From equations (13) and (14), we conclude that to employ the mechanism of ∇g-cooling efficiently even at finite temperature we must go beyond the weak driving limit Ω→0 and consider either Ω~κ or Ω~|Δ|. In the regime Ω~κ  |Δ|, |Δ_c|, 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 Ω~|Δ| such that the driven excited state population ρ^Ω_ee = Ω²/[(4Δ² + 2Ω² + γ²)] becomes comparable to ρ_ee⁰ = N/(2N + 1) and stimulated cooling transitions eventually dominate over thermally activated heating processes. However, in this regime equations (13) and (14) are no longer valid and we postpone the discussion of ∇g cooling in the strongly driven regime Ω  |Δ| to section 4, where we find a significant improvement also for a low driving power Ω > γ.

2.3.3. 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 S(ω) plotted in figure 2 clearly differs from the naively expected sum of S_Ω(ω) and S_g(ω). In equation (9), we have introduced the term S_I(ω) 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 S_Ω(ω). In (b), the molecule is excited on the carrier transition and changes its motional state in the decay process, described by S_g(ω). 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 (see equation (32), in section 3.2 below). In a semiclassical picture, the oscillating motion of the molecule in the trap therefore translates into modulations of γ and δ at the frequency ν and results in additional sidebands in the excitation process. Note that interference between the paths (b) and (c) also occurs for η→0, and in general we cannot write S(ω)≡S_g(ω) even in this limit. However, in section 4, we will argue that in the parameter regimes relevant for ∇g 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 ρ_ee⁰.

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 (S(–ν)→0) by interference, while S(ν) is still finite, which in the ideal case leads to ground state cooling, n₀ = 0. 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 ~ρ^Ω_ee dominate over thermally activated processes ~ρ_ee⁰. 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 ∇g-cooling. In [26, 28, 29] more details on interference effects in cavity-assisted cooling of atoms can be found for the weak excitation regime (N = 0, Ω→0).

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 (Ω→0 and N = 0), analytic expressions for S(ω) and the resulting cooling rates have been derived in the previous works [26, 29] on a related system of trapped atoms inside optical cavities. However, for nonzero 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 BCL g  κ and allows us to study cavity cooling for a wide range of the system parameters.

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 ρ(t) is determined by the following master equation,

The system Hamiltonian H_S 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

with . Starting from the full master equation (15) we proceed in two steps. First, we make use of the small parameter g/κ 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 η and η_c, 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

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

and the molecular degrees of freedom,

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

In the decoupled limit g→0 the cavity field evolves independently of the molecule. On a timescale κ^–1 the system relaxes into the state ρ(t)≈ρ⁰_c⊗ρ_m(t), with ρ_c⁰ the cavity equilibrium density operator defined by , and the molecular operator ρ_m(t) evolving under the action of . For a finite coupling g  κ deviations from the factorized form of ρ(t) are small, but the coupling term modifies the dynamics of ρ_m(t). To proceed we adopt a projection operator technique along the lines of [45] and define the projector and its orthogonal complement . Inserting the decomposition into equation (15) we obtain the two coupled equations

As the population in the subspace is damped with a rate κ 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 g. For times t  κ^–1 we end up with an effective master equation for the molecule density operator which is given by

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, g  κ.

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])

Here, we introduced the abbreviations and

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 T (25) depends on the bare molecule Hamiltonian H_m. Only in the infinite bandwidth limit, where κ is large compared to Δ, Ω and ν, will equation (25) reduce to the simple form T = S/(κ–iΔ_c).

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 η, η_c  1 where the 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.

Under the same approximation the operator T gets the form

with Σ_–(ν), Σ ₊ (ν) = Σ_–(–ν) effective TLS operators defined in (A.3). Up to second order in the Lamb–Dicke parameters we obtain the master equation

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

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

Explicit expressions for and are listed in appendix A. Let us for the moment consider the effective dynamics of the TLS (30), with the corresponding Bloch equation for the set of operators {σ_x, σ_y, σ_z} of the form

Here, and the remaining parameters are defined in appendix B. In the discussion in section 4, we will mainly focus on the limit Ω  κ, where the parameters δ_{x, y}, γ_{x, y}, Ω_{x, y} and Γ_{x, y} are small and equation (31) reduces to the normal Bloch equation for a TLS coupled to a thermal bath. Further, the Stark shift and the effective decay rate reduce to the familiar expressions

For the purpose of readability, we will use the simplification in the following. Finally, note that when the driving strength Ω is comparable to the cavity linewidth κ, unless |Δ|, |Δ_c|  κ there will be modifications of δ and γ. In addition, the non-standard terms in the Bloch equation (31) introduce new features in the TLS dynamics which modify the force spectrum S(ω) 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, ηΩ, η_cg²/κ  γ_N, 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 ,

with ρ_I⁰ the internal equilibrium state defined by . The second-order perturbation theory in the Lamb–Dicke parameters η, η_c  1 results in a master equation for the external operator , which describes the evolution of the population of the individual trap levels. We obtain

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

and recover the cooling equation (6) given in section 2 with W = (A_––A ₊ ) and n₀ = A ₊ /W. The heating and cooling coefficients A_± which follow from an evaluation of equation (34) can be divided into three contributions,

where the physical motivation for the decomposition (36) has been presented in section 2. The first contribution to the force spectrum, S_Ω(ω), follows from the first term of equation (34) in the limit η_c→0 and is defined as

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

with effective TLS operators Σ_±(ν) given by (A.5). Finally, remaining contributions from the first term in equation (34) are found by making the ansatz,

The result is summarized by S_I(ω) defined as

with the action of the superoperators given by

In conclusion, we have expressed heating and cooling rates A_± 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 ηΩ  γ_N, η_c  1, but at this stage no further assumptions have been made. However, in the following, we mainly concentrate on the case Ω  κ or on the regime Ω~κ, |Δ_c|, |Δ|  Ω where the Bloch equation reduces to the standard form (see appendix B). Using the quantum regression theorem and the steady state solution of equation (31) we outline the derivation of analytic expressions for S_Ω(ω) and S_g(ω) in appendices C and D. Analytic expressions for S_I(ω) 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 the cavity-assisted cooling of polar molecules both in the weak and strong driving regimes, and base the discussion on the result for S(ω) derived in section 3. For a physical understanding, we again find it useful to discuss the CASC and ∇g 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 CASC where the corresponding cooling rate W_Ω = S_Ω(ν)–S_Ω(–ν) and steady state occupation n_{0, Ω} = S_Ω(–ν)/W_Ω depend on the spectrum S_Ω(ω) defined in equation (37). Following the calculations outlined in appendix C we obtain for the spectrum in the weak driving limit, Ω  γ_N,

For a fixed ratio ν/γ_N we optimize n_{0, Ω} with respect to the detuning Δ, where obviously Δ must be negative to promote cooling. For the two limits of interest, the resolved sideband limit (SB) ν  γ_N and the Doppler limit (D) ν  γ_N, we find Δ = –ν and Δ = –γ_N/2, 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,

Although in the weak driving limit the final occupation number is independent of Ω, the cooling rate scales as W_Ω~Ω² and under the validity of equation (43) it is bounded by W_Ω < η²γ. 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 Ω, but focus on CASC in the sideband resolved limit γ_N/ν  1. As shown in appendix C, the spectrum now exhibits a three-peak structure and reads,

with . For sideband cooling, we are only interested in the peaks at nonzero frequency. The relative heights of the resonances, α_±, and the widths and which are proportional to γ_N are defined in appendix C and are functions of Ω, Δ and N. For W_Ω to be positive, we require a red-detuned microwave field Δ < 0 and cooling is optimized for . Under those conditions we obtain

with and

with . The functions g() and g₁() are plotted in figure 6 and depend on the angle defined as . We find that the cooling rate increases with Ω up to Ω~|Δ| but then goes to zero for Ω  |Δ|. It has a maximum value of g(₀)≊0.2 at , and we want to point out that this holds independently of the temperature N. The final steady-state occupation number increases monotonically with increasing Ω; g₁(₀)≊0.02 and for Ω  |Δ|.

Figure 6. Dimensionless cooling rate g() and final occupation number g1() for CASC in the resolved sideband limit, as functions of sin = Ω/ν. The vertical line marks the optimal drive strength Ω = 0.65ν which maximizes the cooling rate.

We conclude that driving moderately, Ω~|Δ|~ν/2, 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, ην~ηΩ < γ_N, and thus the expected upper limit W≤γ still holds. Nevertheless, this bound is a factor of 1/η² higher than in the weak driving case.

4.2. ∇g cooling

Let us now move on to discuss the mechanism of ∇g cooling, where the cooling rate W_g = S_g(ν)–S_g(–ν) and the final occupation number n_{0, g} = S_g(–ν)/W_g are given in terms of the spectrum S_g(ω) defined in equation (38). In appendix D, we show that for finite temperature N≠0 and arbitrary Ω  κ or Ω~κ  |Δ|, |Δ_c| the structure of S_g(ω) (D.4) is quite involved and generally exhibits six resonances centered at frequencies ω = ±(Δ_c–Δ) and . Note that we assume , which holds naturally when we later combine ∇g cooling with CASC. As the height of two of the resonances, at , where s = sgn(Δ), scales as ~(Ω/Δ)⁴ we can for weak driving Ω  |Δ| neglect those contributions and obtain the spectrum

We note that only the driven terms in equation (47) contribute to the cooling rate W_g, which for optimized detuning Δ_g = Δ_c–Δ = κ in the Doppler limit and Δ_c–Δ = ν in the sideband limit, respectively, are given by

The last term in (47) is symmetric in frequency and thus corresponds to a purely diffusive process which is present only when the thermal excited state population ρ_ee⁰ is nonzero. For finite temperature there is thus a competition between the driven cooling processes proportional to and thermal diffusive processes. In the resulting expressions for the final occupation number n_{0, g}, which are given in equations (13) and (14), corrections due to the additional diffusion process appear in the quantities β_th and n_th which, in general, are given by

As already mentioned in section 2, to reduce β_th and n_th there are two different strategies. We can either stay in the far-detuned regime |Δ_c|~|Δ|  κ, Ω and increase the driving power Ω~κ, or alternatively, use a resonant driving field |Δ|→0. In the first case, we are still in the far-detuned regime, Ω  |Δ|, and equation (47) is still valid, even for κ~Ω. For a resonantly driven molecule we, however, need to reevaluate S_g(ω), as we will do in the following.

We now consider the strong driving regime Ω  |Δ|, γ_N where the rotational transition of the molecule is saturated; ρ_ee→1/2. In this limit, the general spectrum (D.4) is of the form,

We find that for a saturated molecule at moderate driving strength |Δ|, γ_N < Ω < κ, where the second condition is already implicitly assumed in the derivation of equation (50), the three peaks overlap almost completely and thus to a very good approximation we can consider all resonances to be centered around Δ_c. We then obtain the two-peak structure

in analogy to the spectrum S_Ω(ω) (42) in the weakly driven regime. In the fully saturated regime, the final temperatures achievable by ∇g cooling are given by

and exhibit a similar dependence as in the case of CASC but with γ replaced by 2κ. However, for a finite environment temperature n_{0, g} grows in both regimes only linearly with N. The cooling rates are independent of N and approach the optimal values of

In conclusion, we find that the apparent inefficiency of ∇g-cooling at finite temperatures can be overcome either in the regime where the cavity is far-detuned from the rotational transition and the driving strength Ω is comparable to the cavity linewidth κ, or in the fully saturated regime Δ = 0, Ω  γ. In the latter case, the cooling rates are optimized and the resulting final temperatures grow only linearly with N.

4.3. Combining cavity-assisted microwave cooling and ∇g cooling

Until now, we have only considered the idealized limits η→0 and η_c→0 where either the mechanism of CASC/Doppler cooling or ∇g-cooling dominate the cooling process. In this section, we focus on the more realistic case where both Lamb–Dicke parameters are of the same order and discuss the combination of the cooling mechanisms, being aware that in general W≠W_Ω + W_g due to interference effects. In view of the tight confinement required to trap the molecule close to the surface of the chip, the following discussion is restricted to the resolved sideband regime γ_N < ν.

From our analysis so far, we conclude that under ideal conditions cavity cooling of polar molecules is dominated by the mechanism of CASC, which provides the highest cooling rates and results in the lowest final temperature both in the weak and strong driving regime. However, although CASC is the more effective of the two mechanisms, we expect it also to be more sensitive to imperfections in the system, which in a chip based setup might be harder to avoid than in, e.g. a large ion trap. Therefore, in the following we are particularly interested in the robustness of cavity cooling under non-ideal trap conditions. In order to discuss different types of imperfections like anharmonicity or a state-dependent trapping potential within a general and simple model, we proceed as follows. We assume that the system parameters are optimized with respect to the expected trapping frequency ν which differs from the real trapping frequency ν_r = ν + δν by a shift δν, and then we study the dependence of the cooling parameters W(δν) and n₀(δν). For an anharmonic trap we can then interpret W_n = W(δν_n) as a level-dependent cooling rate, assuming a frequency spacing of the trap levels ν_n = ν + δν_n with δν₀ = 0. Small deviations from a state-independent trapping potential (e.g. assuming ν is the trap level spacing as seen by state |g and ν + δν as seen by state |e) translate into a level-dependent detuning Δ_n→Δ–nδν. Since for sideband cooling, which is most sensitive to imperfections, a shift of the transition frequency and a shift of the trapping frequency affect cooling the same way we can interpret W_n = W(nδν) as a level-dependent cooling rate as well.

4.3.1. Weak saturation regime We first consider the weak saturation regime Ω  |Δ| and assume the relation γ < ν < κ for the damping parameters. We choose Δ = –ν to maximize excitations of the molecule on the red sideband and choose Δ_c = κ–ν with the intention to optimize ∇g-cooling in the Doppler limit. In that case, the spectrum S(ω) is similar to the one plotted in figure 2(a) in section 2. As an example, we plot in figure 7 the cooling rate W(δν) and the final occupation number n₀(δν) for a specific set of parameters. We find that in the weak saturation regime the cooling rate is dominated by the mechanism of CASC and under non-ideal conditions δν≠0 it scales as

The correction due to the gradients of lead to an increase of W(δν) for δν < 0, and a decrease for δν > 0. The asymmetry shows that corrections arise from interference effects in the excitation of the molecule. As expected from the small value of ρ_ee^Ω~Ω²/Δ²  1, the pure mechanism of ∇g-cooling plays essentially no role for the cooling rate of a weakly driven molecule. However, as discussed above, the mechanical coupling to the cavity introduces an additional source of diffusion, which becomes important when the thermally activated excited state population ρ_ee⁰ exceeds ρ_ee^Ω. In contrast to pure CASC, where n₀(δν) is highly insensitive to δν (since the heating rate scales like the cooling rate, A ₊ (δν)~1/[1 + (2δν/γ_N)²]), the additional diffusion for η_c > 0 scales differently with δν and will affect the final occupation number. In the limit κ  ν, we obtain

with

Figure 7. Cavity cooling in the weak saturation regime for the parameters (all in units of ν) Ω = 0.1, Δ = -1, κ = 4, Δc = 3, g = 0.2 and η = ηc. (a) Cooling rate W(δν) in units of W0 = η2Ω2/γ. The dashed (dashed–dotted) line indicates the corresponding result for ηc = 0 (η = 0). (b) Steady-state occupation number n0(δν) for N = 0.1 (blue), N = 0.5 (red) and N = 1 (green). Dashed lines for ηc = 0.

In summary, we find that for a weakly driven molecule the cavity-assisted microwave cooling is dominated by CASC and under non-ideal conditions |δν| > γ_N the total cooling rate decreases as W≊W_Ω~(γ_N/δν)². For finite temperature the additional diffusion caused by thermal ∇g processes affects the final occupation number and n₀≈N is reached only for δν  γ and (γ_N/Ω)²ρ_ee⁰  1.

4.3.2. Strong driving regime We now consider the case of a strongly driven molecule Ω~|Δ|, where according to our previous discussion the cooling rate for CASC, W_Ω, is (close to) optimized. In figure 8(a), we plot the cooling rate W(δν) with a fixed (Δ < 0) but different values of Ω. Apart from the improvement compared to weaker drive under ideal conditions, we note that W is also less sensitive to deviations δν for Ω~|Δ|. We attribute this feature partially to power broadening of the resonances, and also to a stronger contribution from interference effects. The asymmetry of the latter improves the robustness of cavity cooling for δν < 0 but has a negative effect for δν > 0. In figure 8(b), we also plot the final steady-state occupation number for N = 0.5 and different values of Ω. We see that by increasing Ω up to Ω₀ the improved robustness of the cooling rate translates into a corresponding robustness of the final temperature with respect to deviations from ideal harmonic trapping conditions. In contrast to W_Ω, the thermal diffusion rates do not increase with Ω, and thus their contribution to the final occupation number is small; the limit n₀ = N can be reached for (γ²_N/ν²)(N + 1)ρ_ee⁰ < 1. When we increase Ω beyond the optimal value of Ω₀≊0.65ν both the efficiency and the robustness of cavity cooling decrease again.

Figure 8. Cavity cooling in the strong driving regime for the parameters (all in units of ν) (Δ <  0), κ = 4, Δc = 3, g = 0.2 and η = ηc. (a) Cooling rate W(δν) in units of W0 = η2γ for N = 0 and different values of Ω = 0.1 (solid), 0.65 (dashed), 0.95 (dotted). (b) Steady-state occupation number n0(δν) for N = 0.5.

4.3.3. Fully saturated regime We finally consider the fully saturated regime Ω  γ_N, |Δ| where the effect of CASC vanishes while ∇g cooling is most effective. When we assume in addition that , excitations on the motional sidebands do not play a relevant role any longer and we recover the case of pure ∇g cooling as discussed above in section 4.2. According to that discussion, we conclude that cavity cooling in this limit has the same efficiency and robustness as CASC in the weak driving limit, but with the relevant scale γ_N replaced by 2κ. As γ_N  κ, we trade cooling efficiency for robustness by switching between CASC and ∇g cooling.

The arguments presented in the previous paragraph are valid under the assumption ν  Ω  γ_N, |Δ|, but these conditions might not always be satisfied in a real experiment. In particular, if the trapping potential is slightly state-dependent, the detuning Δ depends, in a semiclassical picture, on the position of the molecule in the trap. In figure 9, we plot the total force spectrum S(ω) for the case Δ = 0 and for Δ = ±Ω. Under resonance conditions (note that our definition of Δ includes the cavity induced Stark shift δ) the spectrum is to a good approximation given by S_g(ω), assuming γ_N  ν. For Δ comparable to Ω, we find that S(±ν) is reduced since the excited state population drops below its maximal value of 1/2. In addition, for red detuning, Δ < 0, destructive interference for and constructive interference for degrade ∇g cooling, whereas a blue-detuned driving field, Δ > 0, has just the opposite effect. We conclude that cavity cooling in the saturated regime can be efficiently applied for a range of parameters fulfilling the condition ν > Ω > γ_N, Δ≥0. Outside this parameter regime, excitations of the molecule on the red and blue motional sidebands as well as destructive interference effects play an important role and ∇g cooling no longer provides a valid description of the full cooling (or heating) dynamics.

Figure 9. Spectrum S(ω) for ∇g cooling in the saturated regime. The three different lines show S(ω) in units of S0 = ηc2 g2/κ for Δ = 0 (solid), Δ = Ω (dashed) and Δ = -Ω (dotted). The remaining parameters for this plot have been chosen as (all in units of ν): g = 0.3, κ = 3, Δc = 3, Ω = 0.2 and η = ηc. Multiple resonance in the frequency range are due to excitations of dressed states of the strongly driven TLS.

5. Summary and conclusion

In this paper, we have studied cavity-assisted cooling methods for a single polar molecule, which are based on a strong coupling between the two rotational states of the molecule and an on-chip microwave resonator. We have shown that in the Lamb–Dicke limit the cooling dynamics in this system can be understood in terms of the two main cooling mechanisms, namely CASC and ∇g cooling, interference effects between those two mechanisms and a cavity mediated thermalization channel introducing additional heating at finite temperature. Our main conclusions are as follows.

Cavity-cooling is most efficient by employing CASC, meaning that the molecule is resonantly excited on the red motional sideband followed by cavity-enhanced decay of rate γ. In the sideband resolved regime, γ_N = γ(2N + 1)  ν, the lower bound for the final occupation number n₀ is then given by N, the thermal equilibrium occupation number of the cavity mode, and translates into a final temperature of T_f = (ν/ω_c)T  T. The cooling rate, which scales as ~Ω² for low excitation power is optimized for a value Ω₀≊0.65ν where cooling rates close to the maximum value of W~γ can be achieved. Including imperfections like the anharmonicity of the trapping potential characterized in our model by a level-dependent shift of the trapping frequency δν_n, the cooling rate decreases as W_n~(γ_N/δν_n)² and eventually re-thermalization processes become dominant. Our analysis shows that deviations from the ideal case n₀≊N are expected when either the frequency shift exceeds the excitation linewidth |δν_n| > γ_N or when the sideband cooling rate drops below the re-thermalization rate for an insufficient driving strength Ω. Together with the cooling rate also the robustness of CASC is optimized for Ω~Ω₀.

In the far-off-resonant regime |Δ|, |Δ_c|  ν, Ω as well as under resonance conditions Δ≈0 cavity cooling is dominated by the mechanism of ∇g-cooling. In the resonant case, ∇g cooling is optimized in the fully saturated regime Ω  γ_N, |Δ|, where the limit n₀≊N can be reached when the cavity linewidth 2κ is smaller than the trapping frequency, while the result for Doppler cooling n₀ = (2N + 1)κ/(2ν) applies in the opposite limit. The maximal cooling rate under resonance conditions is W = η_c² g²/κ while in the off-resonant case the molecule is only virtually excited and the cooling rate is reduced by a factor of Ω²/Δ². In this case, to overcome thermally activated heating processes the additional condition Ω > κ is required to reach a final occupation number close to N. While in general it might be harder to reach the sideband resolved limit κ  ν, ∇g cooling is by far less sensitive to imperfections as the condition δν_n < γ_N is replaced by δν_n < 2κ.

In this work, we have analyzed cooling rates and final occupation numbers for a fixed set of parameters to obtain a physical understanding of the characteristics and limitations of cavity cooling in different parameter regimes. From this analysis more efficient cooling strategies can be derived by considering the possibilities to switch between different parameter regimes during the cooling process. For example, at the initial stage where the anharmonicity of the trapping potential may be most severe, ∇g cooling in the Doppler regime offers a very robust cooling method. In a second stage, starting with the precooled molecule localized near the center of the trap, sideband cooling can then efficiently cool the molecule to the ground state. For the design of the optimal cooling strategies, we should emphasize that the present setup not only allows us to change Δ(t) and Ω(t) in time, but also offers different possibilities to control γ(t), Δ_c(t) and κ(t) during the cooling process. For example, using a tunable cavity [50]–[52] the cavity frequency ω_c(t) and therefore also the effective decay rate γ(t) can be controlled in a time-dependent way. In addition, when the cavity is coupled off-resonantly to another dissipative solid state device, e.g. a Cooper pair box, photon states will hybridize with the lossy TLS and a substantial modulation of the effective photon loss rate can be achieved by changing the detuning between the two systems.

Table 1. Limits for the final occupation number n0 and the cooling rate W for the different parameter regimes studied in this paper. Typical values are ν/2π = 1 MHz, γ/2π = 10–100 kHz, N = 0–10, η, ηc = 10–2, κ/2π = 0.01–10 MHz and g/2π = 50–500 kHz. For ∇g cooling in the weak driving regime, optimized values for nth, βth~N(κ/Ω)2 are given in section 2.

`CASC'	Sideband limit (ν  γN)	Doppler limit (ν  γN)
Weak driving regime	n0 = N + (2N + 1)(γN/4ν)2	n0 = (2N + 1)γN/4ν
	W = η2Ω2/((2N + 1)γN)	W = 2η2Ω2ν/((2N + 1)γN2)
Strong driving regime		—
	W≊0.2(ην/γN)2γ	—
`∇gcooling'	Sideband limit (ν    κ)	Doppler limit (ν    κ)
Weak driving regime	n0 = N + (N + 1)(κ/2ν)2 + nth	n0 = [(2N + 1) + βth](κ/2ν)
	W = ηc2 g2/(2κ)(Ω/Δ)2	W = ηc2 g2ν/(2κ2)(Ω/Δ)2
Saturated regime	n0 = N + (2N + 1)(κ/2ν)2	n0 = (2N + 1)(κ/2ν)
	W = ηc2g2/κ	W = ηc2g2ν/κ2

Acknowledgments

We acknowledge discussions with D DeMille, J Doyle, J Schmiedmayer and R Schoelkopf and thank them for stimulating input for this work. M W acknowledges support from the Marie Curie Research Training Network ConQuest and EuroSQIP. Work at Harvard was supported by the Packard Foundation and P R acknowledges support by the NSF through a grant for the Institute for Theoretical Atomic, Molecular and Optical Physics at Harvard University and Smithsonian Astrophysical Observatory.

Appendix A. Molecular master equation in the Lamb–Dicke limit

In this section, we present a few details of the Lamb–Dicke expansion (28) of the molecular master equation (24) which was derived in section 3. The first- and second-order terms in the Lamb–Dicke parameters η, η_c  1 read,

and

The operators Σ_±(ν) are defined by,

and depend on the Heisenberg operators σ_±(t) = e^iH_Itσ_±e^–iH_It with H_I = –(Δ/2)σ_z + (Ω/2)σ_x the bare TLS Hamiltonian. For the evaluation of the integral (A.3), we follow [26] and write the exponential as

Using the algebra of Pauli matrices we arrive at the general form

with the complex coefficients B_i(ω) given by

with Δ_g = Δ_c–Δ.

Appendix B. Bloch equation

For later convenience, we write the Bloch equation (31) corresponding to the effective TLS dynamics (30) in the form

with and

The parameters Δ and Ω describe the evolution of a driven free TLS. The interaction of the TLS with the cavity field introduces energy shifts and δ_y,

with the complex frequency-dependent coefficients B_i(ω) presented in appendix A. Further, the effective decay via the cavity is described by the rates ,

Γ_x and Γ_y,

Finally, there are additional effective Rabi frequencies Ω_x and Ω_y,

From the definition of the B_i (A.6) we find that the non-standard terms γ_{x, y}, δ_{x, y}, Ω_{x, y} and Γ_{x, y} vanish in the limit Ω  κ or else for Ω~κ when the additional condition |Δ|, |Δ_c|, |Δ_c–Δ|  Ω is fulfilled.

Appendix C. Evaluation of S_&b.Omega;(&b.omega;)

In this section, we calculate the excitation spectrum defined in equation (37),

This expression can be evaluated using the quantum regression theorem [49], which relates the two-point correlation function to the dynamics of the TLS represented by the Bloch equation (B.1). After performing the integral we obtain

where the function in the numerator is given by,

Here, we also used the relation [iω1 + A]^–1 = Ad(iω1 + A)/Det(iω1 + A), where Ad(x) is the adjugate matrix of x. By writing the determinant as

with _i the eigenvalues of A, we clearly see the three-resonance structure familiar from the excitation spectrum of a driven TLS.

Simple analytic expressions for S_Ω(ω) can be obtained in certain limits. In the weak driving regime Ω  γ_N, we can set Ω = 0 in the evaluation of h(iω) and _i, and obtain the expression given in equation (42). For general Ω but in the limit , the spectrum consists of three well-separated peaks, with the center frequencies and the widths given by the imaginary and real parts of the _i. For κ such that the Bloch equation (B.1) acquires a standard form (see appendix B) we find

with . Using a partial fraction decomposition of equation (C.1) and keeping only the lowest relevant orders of we end up with the simple structure for S_Ω(ω) given in equation (44). Since cooling and heating rates are determined by the peaks at nonzero frequencies, , we can approximately write the spectrum S_Ω(ω) as

In section 4, we use this expression to determine the cooling and heating rates in the resolved sideband limit for arbitrary values of Ω.

Appendix D. Evaluation of S_g(&b.omega;)

Let us now evaluate the ∇g spectrum (38), using the expression (A.5) for the generalized TLS operators Σ_±(±ν). We write the resulting spectrum on the form,

where the complex functions B_i are defined in (A.6), and the TLS steady-state expectation values ρ_ij = i|ρ_I⁰|j are derived from the Bloch equation (31) under the assumption Ω  κ or Ω, κ  |Δ|, |Δ_c|. For simplicity, we also assume , which is compatible with the constraints for CASC. For transparency, we separate the thermal and the driven population,

Regarding the off-diagonal elements ρ_eg/ge, it can be shown that only the real part contributes to the spectrum in the limit ,

Without further approximation, we obtain a spectrum with six resonances,

with Δ_g≡Δ_c–Δ. The two limits of interest are the resolved sideband limit κ  ν and the Doppler limit ν  κ.

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Notes

Note6  For a review on experimental progress is the field of cold molecules, see [10] and references cited.

Note7  In principle our derivation applies equally well to magnetic traps, as used in current atom chip experiments (see for example [44] and references therein), in the case of a molecule with an unpaired electron spin. Note, however, that the trap fields must be compatible with the superconducting setup.