1. Introduction

Optical cavity QED, where a single atom strongly interacts with a single or a few field modes of a high-Q cavity, has been at the heart of theoretical and experimental quantum optics research during the last decades. Starting from demonstrating basic consequences as vacuum Rabi splitting [1] up to the realization of a single photon source [2], the Jaynes–Cummings model and its generalizations are certainly among the most studied systems in physics. For a practical implementation, one has to ensure the unperturbed strong interaction of a single atom with a tiny optical resonator.

The most straightforward idea is to put an atom between two closely spaced, very highly reflecting mirrors. Even for rather slowa toms, the atomic motion limits the interaction time and leads to a time dependent coupling. In the optical domain, light forces induced by the cavity field further perturb the atom and introduce noise into the system. One idea to overcome this problem is to capture and trap an atom at the desired position to allowf or long interaction times and lown oise. As the cavity volume has to be kept as small as possible, it is very hard to put an extra trap between the mirrors. Fortunately, it turns out that for suitable operating conditions the cavity field itself could be used to keep the atom fixed at the position of maximum atom-field coupling and provide for a cooling force to counteract the heating processes [3, 4]. The existence of the trapping potential and cooling force have been experimentally verified in the meantime [5, 6].

In practice, one still has some difficulties to load this trap, by capturing atoms moving within the cavity field, at the desired locations. The central difficulty in this respect is the fact that the potential (field intensity distribution) is rather steep with periodicity λ/2 in the longitudinal direction (along the cavity axis), while it is rather flat, of the order of the mode waist size w ₀, transversally. Hence there is much less cavity-induced friction force and confinement in the radial direction, and the atoms cannot be trapped radially or escape after a short time [7]. A strong improvement of such capturing has recently been reported by an external feedback mechanism which turns up the pump intensity when an atom is detected in the field [8]. However, the interaction time is still limited and the required feedback electronics complicates the setup. In addition, one does not get damping of the angular momentum as the field intensity has no angular dependence.

In this work, we propose an alternative method to facilitate and enhance the transverse capture rate of the atoms by making use of several instead of a single cavity mode. As these modes possess different radial field amplitudes, the relative phase of these modes determines the total field intensity distribution and hence the shape of the optical potential. This has two desired consequences: on the one hand, one gets higher local radial field gradients [9], and, on the other hand, one can have spatially dependent stimulated photon scattering between the modes. The second effect of coherent mode coupling has already been predicted to lead to faster cooling of atoms in a ring cavity, as compared to a standing wave setup [10]. Here, in particular, the use of modes with different angular symmetry will lead to a breaking of this symmetry and a damping of the angular motion of the atom. This effect turns out to be particularly manifest if the atom itself is coherently driven from the side and acts as an effective, spatially dependent cavity pump. Note that the possibility of accurate atom tracking in a cavity field by use of a degenerate mode family has been recently proposed [11].

This paper is organized as follows. After presenting the basic ingredients of the model in Sect. II, we calculate and plot the steady-state field distribution for a spatially fixed atom in Sect. III. In Sect. IV, we then numerically demonstrate the atom capture and trapping for selected parameters.

 

Fig. 1. Schematic representation of the system composed of an laser-driven atom strongly coupled to the field of a coherently pumped cavity.

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2. Semiclassical description of an atom coupled to a multimode field

Let us consider a two-level atom with transition frequency ω_a strongly coupled to M modes with nearly degenerate frequencies ω_n ≈ ω_c of a high-finesse cavity (e.g. in quasi-confocal geometry). The atom is transversally injected into the cavity field. For simplicity, we assume longitudinally very cold atoms (in practice this is automatically guaranteed by spatial filtering of the atoms by a cavity entrance slit), so that we are able to restrict our study to the radial dimensions, i.e. perpendicular to the cavity axis, only. In general, we assume two coherent laser fields pumping the atom-field system, which are set to have the same frequency ω_p . One field part is injected into the cavity through one of the mirrors, yielding an effective pump strengths η_n for the nth mode. The second part is directly driving the atom in the form of a broad standing wave, transverse to the cavity axis. Hence, we can safely reduce its spatial dependence to simple plain standing wave with effective pump strength η_t (x) = ζh(x) = ζ cos(k_p y). A schematic sketch of the system is depicted in Fig. 1. Both the atom and the cavity field are coupled to external reservoirs, which gives rise to spontaneous emission (γ) and cavity decay (κ).

In a frame rotating with the pump frequency ω_p , the quantum master equation for this system is given by

where the Hamiltonian in the rotating-wave approximation and the damping terms read

Here Δ_a = ω_p - ω_a and Δ_c = ω_p - ω_c are the atomic and cavity-field detunings respectively. Further σ ⁺ and σ ^- denote the atomic raising and lowering operators, respectively, $a_n^{†}$ and a_n the field creation and annihilation operators for the nth mode. The coupling between the atom and the nth mode is given by $g_n\left(x\right)=d\sqrt{\mathit{ħ}{\omega }_c/2∊{\mathcal{V}}_n}{f}_n\left(x\right)$ where d is the atomic dipole moment and V_n the effective mode volume. The second term in Eq. (2b) contains the momentum recoil due to spontaneous emission.

Although we assume rather cold atoms, their temperature is still well above the recoil limit. Hence, we can refrain from a quantum treatment of the center of mass (CM) motion. Following the lines of the phase space method presented in [12], we derive a systematic semiclassical model. This simply gives a set of coupled differential equations for the particle momentum and position as well as for the field amplitudes, which can be solved numerically.

As a first step to achieve this task, we adiabatically eliminate the internal atomic degrees of freedom by setting

This can be done in the low-saturation regime, where the atomic operators σ^± evolve on a fast timescale due to a large detuning Δ_a or a large damping rate γ.

We then use Wigner function representation of the quantum master equation, which yields coupled partial differential equations for the combined atom-field Wigner function W(x, p, α ₁...α_M , ${\alpha }_1^{*}$...${\alpha }_M^{*}$). This is truncated at second-order leading to a Fokker–Planck-equation (FPE), valid for not too weak fields and not too cold atoms. Still central quantum properties of the system are kept in this way. Its implications for several special cases have already been discussed before [12]. Here, we are interested in the initial capture process governed by the frictional forces which are represented by the firstorder derivatives in the FPE. Keeping only these terms, an equivalent set of ordinary differential equations of motion can be read off:

where U ₀ = g ²Δ_a/(${\mathrm{\Delta }}_a^2$ + γ²) is the light shift per photon and Γ₀ = g ²γ/(${\mathrm{\Delta }}_a^2$ + γ²) the photon scattering rate. As a shortcut in Eq.(4), we have introduced the field amplitude at the position of the atom,

The first term in Eq. (4b) is the dipole force due to photon redistribution between the cavity modes by the atom. The second one is a radiative force emerging from scattering photons out of the cavity, whereas the last three terms arise from the direct coupling of the atom to the external laser field. This interaction yields an additional cavity pump with amplitude η _eff = U ₀ ζ/g, which describes photon scattering into the cavity via the atom. Further, there appears another radiative force characterized by the scattering rate γ_eff = Γ₀ ζ/g. The last term is simply the free-space dipole force acting on an atom in a standing wave. It is clear that these last three terms vanish if only the cavity is driven directly. Similarly, Eq. (4c) points up the effect of the laser pumping the atom on the cavity field. Two additional terms appear describing the coherent dynamics (η _eff) and the decay (γ_eff) of the cavity field. Since these effects are only provided by the presence of the atom, they strongly depend on the atomic position.

The second order terms in the FPE would imply additional fluctuation terms in Eqs. (4) which are required to get quantitative answers for the final temperatures and trapping times as it has been previously discussed [13]. In the noisy atomic trajectories, however, the effect of the friction force is strongly masked. Since we want to focus on the multi-mode effect underlying the damped atomic center-of-mass motion, it is desirable to neglect the fluctuation terms. This approximation does not lead us to wrong conclusions as long as the friction is concerned. On the other hand, it must be kept in mind that the diffusion, besides inducing noise, also limits the time the atom is trapped within the resonator. The trapping time τ can be estimated by comparing the energy spread due to momentum diffusion, ΔE = (Δp)²/2m ≈ D _recτ/2m, with the depth of the potential ħU ₀|ε|². Here we use only D _rec, the momentum diffusion coefficient due to recoil heating, because the other contribution arising from the fluctuations of the dipole force is of the same order of magnitude in the longitudinal and much less in the transverse directions. This consideration leads to the condition

where ω _rec is the recoil frequency of the atom. With sufficiently large detunings (Δ_A/γ > 50), τ can attain values of the order of milliseconds for a Rubidium atom, which is in agreement with the numerical simulations of the full dynamics [12].

For the validity of Eqs. (4) the atomic excited level has to be weakly populated which is equivalent to keeping the saturation parameter

small. The transverse intensity distribution is given by

Here one has to distinguish carefully between the atomic position x and the spatial coordinates x and y. In order to avoid confusion, we will hold this notation in the following.

Let us recall at this point that in the vicinity of the center of a confocal cavity, the mode functions are given by:

Here n and m indicate the Hermite polynomials and w ₀ is the spot size. As mentioned above, we consider only very cold atoms in the z-direction as they are spatially filtered from the source to the cavity entrance. Hence, we can set cos(kz) ≈ 1 in Eq. (9) and consider the transverse motion only. Of course, on a longer time scale when momentum diffusion heats up the z-motion, this would not be valid and some spatial averaging on the z-motion should be included. However, we expect our results not to be qualitatively changed by this assumption.

 

Fig. 2. Steady-state field intensity for a fixed laser-driven atom where Δ_c = 2U ₀ = -20 μs ^-1, corresponding to Δ_a = -1440 μs ^-1 and g = 120 μs ^-1. The other parameters are (κ, γ) = (10, 20) μs ^-1 and the pumping strength ζ is chosen such that s = 0.03. The arrow indicates the position of the atom.

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3. Steady-state intensity distributions

As can be seen from Eqs. (4), the atom in the cavity acts as a moving refractive index and source term for the cavity field modes. In order to get a first insight into this complicated dynamics, we look at the case of a very slow atom, so that the fields can attain their steady values according to the momentary position. In this limit the mode amplitudes α_n can be easily calculated by setting $\dot{\alpha }$_n = 0 in Eq. (4c) and solving the linear system

with the matrix

and the vector

Let us now look at the solution for some typical cases. If only the atom is coherently illuminated, photons can only be created via scattering by the atom. Due to the spatial dependence of the atom-field coupling, the atom scatters photons preferentially into modes with f_n (x) large. The relative phases are locked and, accordingly, the cavity field exhibits a localized peak at the position of the atom. This behavior is illustrated in Fig. 2, where we took the first four sets of modes into account, i.e. n + m ≤ 3 and M = 10. If Δ_c is of the order of U ₀ the atom shifts the cavity field into resonance and the total intensity becomes maximum while it decreases for a large detuning Δ_a.

Note that although it is hardly visible in the figure, in general, there is a slight spatial shift between the field maximum and the atomic position. Hence we have a nonzero field gradient at the atomic position, which implies a remaining force pushing the atom towards the cavity center.

 

Fig. 3. Steady-state field intensity for a fixed atom in a driven cavity, where (a) Δ_c = 2U ₀ = -20 μs ^-1 and (b) Δ_a = 0, Δ_c = -20 μs ^-1. Only η ₀₀ is different from zero and chosen such that s = 0.03. The other parameters are the same as in Fig. 2. The arrows indicate the positions of the atom.

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In a second generic case, we look at an atom sitting in the cavity, where the TEM _00- mode is driven through the mirrors. Interestingly, a similar effect occurs as above. The atom raises the local field drastically for Δ_c ≃ U ₀ and creates a maximum close to its position. This can be seen in Fig. 3a, where we considered the first ten modes and the driving laser only coupled to the ground mode (i.e. η_ij = 0, ∀i, j ≠ 0). However, a big drop appears at the atomic position if |Δ_a| ≃ |Δ_c|. The atom pushes the field maximum away and the intensity decreases. Fig. 3b shows the sharp intensity decline at the atomic position. In order to get similar values for the saturation parameter and thus for the photon number as in the red detuned case, the pumping strength has to be chosen significantly higher here. The other parameters are left the same. In the case if |Δ_a| ≫ |Δ_c|, the effect of the atom becomes small. The rise (for equal signs of the detunings) and the drop (opposite signs) of the cavity field is does not play any important role anymore.

 

Fig. 4. Steady-state field intensity for two laser-driven atoms simultaneously inside a cavity where Δ_c = 2U ₀ = -20 μs ^-1. The other parameters are the same as in Fig. 2. The arrows indicate the positions of the atoms.

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Let us finally take a look at the case, where two atoms are simultaneously present in the field. Obviously, if only a single mode is involved, the overall field amplitude and phase are the only available observables and in general we cannot distinguish between one or two atoms, if f_n (x) = f_n (x₁) + f_n (x₂). This is completely different if several modes are involved as the whole field intensity distribution is changed in this case. We demonstrate this at an example in Fig. 4, where we plot the steady-state field intensity distributions for two atoms simultaneously in the field and parameters analogous to Fig. 2. Clearly, two peaks are visible at the positions of the two atoms. From Fig. 4 and the fact that the field intensity is directly proportional to the optical potential for the atoms, it is also clear, that one gets a strongly cavity-enhanced atom-atom interaction which was observed in Ref. [14].

 

Fig. 5. Trajectory of a Rubidium atom with initial velocity v = 12 cm/s for (a) the ground mode and (b) the first six modes. In both cases, the ground mode is pumped. The blue curves shows asymmetric oscillations for 0 ≤ t ≤ 2 ms, the red circles a few rotations after t = 5 ms. The other parameters are the same as in Fig. 2.

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Note that by choosing the detuning and amplitude of the pump field, the atom-atom interaction can be largely tailored in this case.

4. Dynamic capturing and trapping of an atom

Let us now consider the full dynamics of Eqs. (4). Obviously, an analytical solution in this case is rather hard to find and we will only present numerical calculations of the atomic motion for a set of representative parameter values by directly integrating Eqs. (4). As the atom is simultaneously coupled to many modes, one expects larger local field intensities, stronger field gradients, and thus a higher capture probability than in the single-mode case, where capturing an atom crossing the mode is very unlikely.

As a first example, we consider the case of an atom in the field of a single driven TEM mode function f ₀₀(x, y). This is the standard setup of most previous theoretical and experimental treatments and we will use it as a reference here. The atom is put at a random position inside the cavity with a small initial velocity and we assume that the interaction starts at a given initial time t = 0. In Fig. 5a we have plotted the trajectory for a Rubidium atom with initial velocity v = 12 cm/s. The blue curve shows how the atom moves for the first two milliseconds (0 < t < 2 ms) showing elliptic orbits around the cavity axis avoiding the region of maximal coupling due to the angular momentum barrier. As the field is angularly symmetric, the dynamic cooling will only influence the radial motion which is damped in this case. This is shown by the red circle, which represents the atomic trajectory for some time after 5 ms. In this steady-state, r is fixed and hence the cavity field remains constant and constitutes a conservative potential. Orbiting in such a large radius state, diffusion is very likely to kick the atom out of the potential.

 

Fig. 6. Trajectory of a driven Rubidium atom initial at rest for (a) the ground mode and (b) the first six modes. The atom is trapped in y-direction within about a quarter of a wavelength already for 0 < t < 1 ms (blue curve). The atomic motion after t = 5 ms, which is indicated by the red curve, is much more damped in the multimode case.

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In contrast to converging to a stable circle, the atom is captured and confined to a much smaller spatial region, when we include more modes in the dynamics. This is illustrated in Fig. 5b, where we took the six lowest index modes into account, according to n + m ≤ 2. We even decreased the pump strength by 30 percent in this case to keep the saturation of the atom low. Again, the blue curve represents the atomic motion for 0 < t < 2 ms, which is more extended and irregular now. However, the atom is slowed down and captured to a position very close to the cavity axis with velocity v = 0.3 cm/s after about 5 ms. Its trajectory reduces to the red point shown in the center.

The dynamics gets even more complicated, if the atom is coherently driven instead of the cavity mode. To be consistent, we have to include the pump laser in the optical potential, which turns out to have a big effect on the atomic motion. As mentioned above, the atom is assumed to be the only source of cavity photons, by scattering. Hence, only red detuning Δ_a < 0, which ensures that the atom is attracted to high field intensities, is a reasonable choice here. Further, the intracavity intensity is directly related to the saturation and attains sufficiently strong values only for large g and large Δ_a. The latter condition imposes a high pump strength ζ. The atom is also confined in the standing wave of the pump field which acts as an extra dipole trap to guide the atoms into the cavity. It is clear that the number of modes close to resonance determines the total intracavity photon number as well. Again, in the various parameter sets, we keep the saturation constant, which requires a much larger decrease of ζ than before, if one takes more modes into account.

Except for the strong extra effect of the external laser in the y-direction the trajectories showa qualitatively similar behavior as above. Figs. 6a and b correspond to the case of only the ground mode and the lowest six modes considered, respectively. As mentioned above, we increased the pumping strength ζ while the other parameters are the ones as in Fig. 2. Clearly, the x as well as the y-motion are much more strongly damped in the multi-mode case. This plot also demonstrates that cavity cooling and trapping can be applied in an efficient way for atoms trapped by an external potential. Here it is the dipole potential created by the pump laser, which, at least conceptually, further simplifies the setup.

In general, the two generic cases of cavity pump or atomic pump will not be the optimum for trapping and cooling, but some combined action will give the best results.

 

Fig. 7. Influence of an incoming atom (blue curve) on an initial trapped one (green curve) in the atom driving case for Δ_c = -30 μs ^-1 and U ₀ = -3.33 μs ^-1. The atom initially at rest starts to oscillate in phase with the incoming one (a) and after a few milliseconds the kinetic energy is periodically exchanged between the two atoms (b).

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We will, however, not try to optimize the setup here, as the best choice strongly depends on the kind of atoms, mirrors and laser parameters. This clearly goes beyond the scope of this work.

As a final case, we will again consider the dynamics of two driven atoms simultaneously in the field and study their cavity-mediated interaction. Fig. 7a shows the effect of an atom entering the cavity with velocity v = 12 cm/s (blue curve) where a second atom is already trapped and is at rest (green curve). We use the case of atom driving where both atoms are trapped and confined in the potential wells of the transverse pump laser from the beginning. The interesting part of the motion, hence, will take place along the x-direction, which we will consider in the following. The incoming atom shifts the field maximum in the resonator created by the already present atom towards its position. This causes a field gradient at the position of the trapped atom and an effective attractive force between the two particles, making them approach each other. At the point where the atoms get closest, they have the maximum relative velocity and also the local field reaches its maximum value. When the atoms move apart again, the field maximum is now behind both of the moving atoms, leading to a deceleration. As the field is now stronger than in the phase where they approach each other, the relative attractive force is enhanced when they separate. Hence, in addition to the cavity induced damping, motional energy is transferred from the fast to the slow atom. This strongly enhances the capturing probability for the second atom. After a certain time the kinetic energy is evenly distributed among the atoms and the roles of the two atoms interchange. We get a periodic energy exchange between the two atoms. This can be seen in Fig. 7b where the green curve corresponds to the atom initial at rest. Note that the oscillations have nowthe same amplitudes and are damped simultaneously.

5. Conclusions

We have demonstrated that using a multimode configuration and tailoring the pump and resonator geometry, one can strongly enhance the friction forces and trapping potentials for atoms entering in the field of a high-Q cavity. This increases the probability of the atoms to be captured. Apart from the field intensity, also the relative mode phases and hence the field shape is now a dynamic quantity. Monitoring the fields would allow highly precise tracking of the motion of even several particles simultaneously. In addition by properly choosing the parameters, one has a new handle of the effective atom-atom interaction. Besides a buildup of correlations as it happens for two-level atoms, this could be used for controlled entanglement in the case of atoms with more complex internal structure.