Superior dark-state cooling via nonreciprocal couplings in trapped atoms

Cooling the trapped atoms toward their motional ground states engages a series of improving laser cooling techniques from Doppler to Sisyphus and subrecoil cooling schemes [1]. Upon approaching the motional ground state of atoms, the linewidth or the spontaneous decay rate of the transition puts a limit on the temperature that these cooling schemes can achieve. Their ultimate performance depends on the balance between cooling and heating mechanisms, where the fluctuations of spontaneously emitted and rescattered photons give rise to the constraint that forbids further cooling. In one of the subrecoil cooling platforms, the resolved sideband cooling [2–6] in the Lamb–Dicke (LD) regime suppresses the carrier transition and specifically excites the transition with one phononic quanta less. Effectively, it moves the atoms toward the zero-phonon state via spontaneous emissions, and essentially, the steady-state phonon occupation is determined by a squared ratio of the spontaneous emission rate over the trapping frequency, which can be much smaller than one.

Alternatively, the dark-state sideband cooling [7–22] utilizes Λ-type three-level structure of atoms with two laser fields operating on two different hyperfine ground states and one common excited state. This leads to an effective dark-state picture owing to quantum interference between two ground states and forms an asymmetric absorption profile under the two-photon resonance condition with a large detuning. This profile allows a narrow transition in the resolved sideband to cool down the atoms by removing one phonon, similar to the sideband cooling scheme with a two-level structure. In contrast to the two-level structure, the effective laser drivings and decay rates in dark-state sideband cooling can be tunable, which makes it a flexible and widely-used cooling scheme in ultracold atoms.

Recently, a novel scheme of using nonreciprocal couplings in optomechanical systems [23, 24] manifests motional refrigeration, which gives insights in heat transfer by utilizing unequal decay channels. This coupling can be facilitated as well in a nanophotonics platform of atom–waveguide interface [25–27] which allows quantum state engineering via mediating collective nonreciprocal couplings between atoms [27–47]. In this quantum interface, a strong coupling regime in light–matter interaction can be reached by coupling atoms with evanescent fields at the waveguide surface, where the directionality of light exchange processes can be further manipulated and tailored by external magnetic fields [34]. This gives rise to exotic phenomena of collective radiation behaviors [38–40] and a new topological waveguide-QED platform that can host photonic bound states [48].

In this work, we theoretically investigate the dark-state sideband cooling scheme in trapped atoms with nonreciprocal couplings, as shown in figure 1. The nonreciprocal couplings can be facilitated in an atom–waveguide interface or via a photonic quantum link in free space [49], which leads to collective spin–phonon correlations [18, 50] and distinct heat exchanges. This cooperative mechanism of cooling has been considered as well in two ions in free space [51], cavity-mediated atoms [52, 53], and one-dimensional chain of optically bound cold atoms [54]. Here we present a superior dark-state cooling compared to the conventional single atom results by utilizing the nonreciprocal couplings between two atoms. We denote the first atom as the target atom and the second as residual or spectator one, and place the first atom left to the second atom without loss of generality. We explore various tunable parameter regimes which are allowed in dark-state cooling and identify the parameter region that gives better cooling performance without compromising its cooling rate. We further obtain an analytical form of the phonon occupation for the target atom under the asymmetric driving conditions. Our results demonstrate a distinct heat transfer within the atoms, which leads to a superior dark-state sideband cooling assisted by collective spin–exchange interactions. This opens new avenues in surpassing the cooling obstacle [18, 55, 56], which is crucial in scalable quantum computation [55, 57–59], quantum simulations [60, 61], and preparations of large ensemble of ultracold atoms [21, 62] or molecules [63]. This paper is organized as follows. We present the theoretical model of dark-state cooling with nonreciprocal couplings in section 2. We then identify the parameter regimes that demonstrate superior cooling behaviors and their cooling dynamics in section 3. In section 4, we further show the analytical prediction of the steady-state phonon occupation of the target atom and confirm its validity under dark-state mapping. Finally we discuss possible experimental realizations and conclude in section 5.

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We start with a conventional EIT cooling scheme [7] with nonreciprocal couplings between constituent trapped atoms, as shown in figure 1. Each atom involves Λ-type three-level atomic structure, where two hyperfine ground states |g⟩_j and |r⟩_j for jth atom couple to their common excited state |e⟩_j with laser fields of Rabi frequencies  Ω_g,j and  Ω_r,j , respectively. The intrinsic decay rates from the excited state are γ_g and γ_r . With the nonreciprocal couplings introduced as γ_L and γ_R, the dynamics of a density matrix ρ for N atoms with mass m can be expressed as  (ℏ = 1)

$$\begin{equation}\frac{\mathrm{d}\rho }{\mathrm{d}t}=-i[{H}_{\text{LD}}+{H}_{\mathrm{L}}+{H}_{\mathrm{R}},\rho ]+{\mathcal{L}}_{\mathrm{L}}[\rho ]+{\mathcal{L}}_{\mathrm{R}}[\rho ],\end{equation} \tag{ 1 }$$

where H_LD for the EIT cooling in the LD regime (in the first order of LD parameter η_g(r)) reads

$$\begin{align}\hfill {H}_{\text{LD}}& =-\sum\limits _{j=1}^{N}{{\Delta}}_{j}\vert {e\rangle }_{j}\langle e\vert +\nu \sum\limits _{j=1}^{N}{a}_{j}^{{\dagger}}{a}_{j}+\sum\limits _{j=1}^{N}\left(\frac{{{\Omega}}_{g,j}}{2}\vert {e\rangle }_{j}\langle g\vert +\frac{{{\Omega}}_{r,j}}{2}\vert {e\rangle }_{j}\langle r\vert +\mathrm{H}.\mathrm{c}.\right)\hfill \\ \hfill & \quad +i{\eta }_{g}\,\cos ({\psi }_{g})\sum\limits _{j=1}^{N}\left[\frac{{{\Omega}}_{g,j}}{2}\vert {e\rangle }_{j}\langle g\vert \left({a}_{j}+{a}_{j}^{{\dagger}}\right)-\mathrm{H}.\mathrm{c}.\right]+i{\eta }_{r}\,\cos ({\psi }_{r})\sum\limits _{j=1}^{N}\left[\frac{{{\Omega}}_{r,j}}{2}\vert {e\rangle }_{j}\langle r\vert \left({a}_{j}+{a}_{j}^{{\dagger}}\right)-\mathrm{H}.\mathrm{c}.\right],\hfill \end{align} \tag{ 2 }$$

with the common laser detuning Δ_j ≡ ω_g,j − ω_eg = ω_r,j − ω_er as required in EIT cooling scheme, i.e. the same difference between the jth-site laser central frequencies (ω_g(r),j ) and the atomic transition frequencies (ω_eg and ω_er ). Projection angles of the laser fields to the motional direction are denoted as ψ_g(r). The harmonic trap frequency is ν with a creation (annihilation) operator ${a}_{j}^{{\dagger}}$ (a_j ) in the quantized phononic states |n⟩, and LD parameters are ${\eta }_{g(r)}={k}_{g(r)}/\sqrt{2m\nu }$ with k_g(r) ≡ ω_g(r),j /c. The coherent and dissipative nonreciprocal couplings in the zeroth order of η_g,r are [35]

$$\begin{equation}{H}_{\mathrm{L}(\mathrm{R})}=-i\sum\limits _{m=g,r}\frac{{\gamma }_{m,\mathrm{L}(\mathrm{R})}}{2}\sum\limits _{j< ( > )l}^{N}\left({\mathrm{e}}^{\mathrm{i}{k}_{s}\vert {x}_{j}-{x}_{l}\vert }\vert {e\rangle }_{j}\langle m\vert \otimes \vert {m\rangle }_{l}\langle e\vert -\mathrm{H}.\mathrm{c}.\right),\end{equation} \tag{ 3 }$$

and

$$\begin{align}\hfill {\mathcal{L}}_{\mathrm{L}(\mathrm{R})}[\rho ]& =-\sum\limits _{m=g,r}\frac{{\gamma }_{m,\mathrm{L}(\mathrm{R})}}{2}\sum\limits _{j,l=1}^{N}{\mathrm{e}}^{\mp \mathrm{i}{k}_{s}({x}_{j}-{x}_{l})}\left(\vert {e\rangle }_{j}\langle m\vert \otimes \vert {m\rangle }_{l}\langle e\vert \rho +\rho \vert {e\rangle }_{j}\langle m\vert \otimes \vert {m\rangle }_{l}\langle e\vert -2\vert {m\rangle }_{l}\langle e\vert \rho \vert {e\rangle }_{j}\langle m\vert \right),\hfill \end{align} \tag{ 4 }$$

respectively, where k_s denotes the wave vector in the guided mode that mediates nonreciprocal couplings γ_L(R) ≡ γ_g,L(R) + γ_r,L(R), and ξ ≡ k_s |x_j+1 − x_j | quantifies the light-induced dipole–dipole interactions between the relative positions of trap centers x_j and x_l .

Throughout the paper, we define spin–exchange interactions specifically denoted to two lower states (|g⟩ and |r⟩) and excitation–exchange in general for the effect of nonreciprocal couplings. We also note that in actual atom–waveguide interfaces, a small but finite nonguided coupling γ_ng can not be avoided entirely, which can be quantified and included in equation (4), for example of the probe field transition, as −γ_ng /2(|e⟩_j ⟨e|ρ + ρ|e⟩_j ⟨e| − 2|g⟩_j ⟨e|ρ|e⟩_j ⟨g|) for all individual atoms. We will elaborate more on its role in experimental realizations in section 5.

In terms of the eigenstates with only atomic spin degrees of freedom in equation (2), we can further diagonalize the EIT Hamiltonian [64] in a single-particle limit, where three eigenstates are

$$\begin{equation}\vert +\rangle =\sin \,\phi \vert e\rangle -\cos \,\phi \vert b\rangle ,\vert -\rangle =\cos \,\phi \vert e\rangle +\sin \,\phi \vert b\rangle ,\vert d\rangle =\cos \,\theta \vert g\rangle -\sin \,\theta \vert r\rangle ,\end{equation} \tag{ 5 }$$

with |b⟩ ≡ sin θ|g⟩ + cos θ|r⟩, and the corresponding eigenenergies are

$$\begin{equation}{\omega }_{+}=\left(-{\Delta}+\sqrt{{{\Omega}}_{g}^{2}+{{\Omega}}_{r}^{2}+{{\Delta}}^{2}}\right)/2,\qquad {\omega }_{-}=\left(-{\Delta}-\sqrt{{{\Omega}}_{g}^{2}+{{\Omega}}_{r}^{2}+{{\Delta}}^{2}}\right)/2,\enspace {\omega }_{d}=0.\end{equation} \tag{ 6 }$$

The angles θ and 2ϕ are defined as tan⁻¹(Ω_g /Ω_r ) and ${\tan }^{-1}(-\sqrt{{{\Omega}}_{g}^{2}+{{\Omega}}_{r}^{2}}/{\Delta})$, respectively. Under the EIT cooling condition of ω₊ = ν [6, 7], a red sideband transition between |d, n⟩ ↔|+, n − 1⟩ becomes resonant, and we can safely ignore the influences of off-resonant state |−⟩ and blue sideband transition between |d, n⟩ ↔|+, n + 1⟩.

The effective Hamiltonian within the subspace |d⟩ and |+⟩ can then be obtained from equation (2), which reads in an interaction picture [6, 22],

$$\begin{equation}{H}_{\text{LD}}^{\text{eff}}=i{\eta }_{\text{eff}}\sum\limits _{j=1}^{N}\frac{{{\Omega}}_{\text{eff},j}}{2}\left[\vert {d\rangle }_{j}\langle +\vert \left({a}_{j}^{{\dagger}}+{a}_{j}\right)-\mathrm{H}.\mathrm{c}.\right],\end{equation} \tag{ 7 }$$

with η_eff ≡ η_g  cos(ψ_g ) − η_r  cos(ψ_r ), ${{\Omega}}_{\text{eff},j}\equiv {{\Omega}}_{g,j}{{\Omega}}_{r,j}\,\sin \,{\phi }_{j}/\sqrt{{{\Omega}}_{g,j}^{2}+{{\Omega}}_{r,j}^{2}}$, $\tan (2{\phi }_{j})\equiv -\sqrt{{{\Omega}}_{g,j}^{2}+{{\Omega}}_{r,j}^{2}}/{{\Delta}}_{j}$, and ${{\Delta}}_{j}=-\nu +({{\Omega}}_{g,j}^{2}+{{\Omega}}_{r,j}^{2})/(4\nu )$. The associated nonreciprocal coupling terms as in equations (3) and (4) reduce to

$$\begin{equation}{H}_{\mathrm{L}(\mathrm{R})}^{\text{eff}}=-i\sum\limits _{j< ( > )l}^{N}\frac{{\gamma }_{\text{eff},\mathrm{L}(\mathrm{R})}(j,l)}{2}\left({\mathrm{e}}^{\mathrm{i}{k}_{s}\vert {x}_{j}-{x}_{l}\vert }\vert {+\rangle }_{j}\langle d\vert \otimes \vert {d\rangle }_{l}\langle +\vert -\mathrm{H}.\mathrm{c}.\right),\end{equation} \tag{ 8 }$$

$$\begin{align}\hfill {\mathcal{L}}_{\mathrm{L}(\mathrm{R})}^{\text{eff}}[\rho ]& =-\sum\limits _{j\ne l,l=1}^{N}\frac{{\gamma }_{\text{eff},\mathrm{L}(\mathrm{R})}(j,l)}{2}{\mathrm{e}}^{\mp \mathrm{i}{k}_{s}({x}_{j}-{x}_{l})}\left(\vert {+\rangle }_{j}\langle d\vert \otimes \vert {d\rangle }_{l}\langle +\vert \rho +\rho \vert {+\rangle }_{j}\langle d\vert \otimes \vert {d\rangle }_{l}\langle +\vert -2\vert {d\rangle }_{l}\langle +\vert \rho \vert {+\rangle }_{j}\langle d\vert \right)\hfill \\ \hfill & \quad -\sum\limits _{j=1}^{N}\frac{{\gamma }_{\text{eff},j}}{2}\left(\vert {+\rangle }_{j}\langle +\vert \rho +\rho \vert {+\rangle }_{j}\langle +\vert -2\vert {d\rangle }_{j}\langle +\vert \rho \vert {+\rangle }_{j}\langle d\vert \right),\hfill \end{align} \tag{ 9 }$$

where

$$\begin{equation}{\gamma }_{\text{eff},\mathrm{L}(\mathrm{R})}(\,j,l)\equiv \sin \,{\phi }_{j}\,\sin \,{\phi }_{l}\left[{\gamma }_{g,\mathrm{L}(\mathrm{R})}\,\cos \,{\theta }_{j}\,\cos \,{\theta }_{l}+{\gamma }_{r,\mathrm{L}(\mathrm{R})}\,\sin \,{\theta }_{j}\,\sin \,{\theta }_{l}\right],\end{equation} \tag{ 10 }$$

$$\begin{equation}{\gamma }_{\text{eff},j}\equiv {\sin }^{2}\,{\phi }_{j}\left({\gamma }_{g}\,{\cos }^{2}\,{\theta }_{j}+{\gamma }_{r}\,{\sin }^{2}\,{\theta }_{j}\right),\end{equation} \tag{ 11 }$$

with γ_g = γ_g,L + γ_g,R, γ_r = γ_r,L + γ_r,R, and tan θ_j = Ω_g,j /Ω_r,j . γ_g,r are total decay rates to the states |g⟩ and |r⟩, respectively. We note that ${\gamma }_{\text{eff},j}{\gamma }_{\text{eff},l}\ne {[{\gamma }_{\text{eff},\text{L}}(\,j,l)+{\gamma }_{\text{eff},\text{R}}(\,j,l)]}^{2}$ in general, unless θ_j = θ_l , which would otherwise be assured in quantum systems with collective and pairwise dipole–dipole interactions [35]. This inequality arises owing to the mapping to the reduced Hilbert space, where different ϕ_j determined by external laser fields and θ_j by laser Rabi frequencies can further allow extra and tunable degrees of freedom for cooling mechanism. We note that to have different θ_j requires a spatially inhomogeneous field condition.

The above dark-state mapping sustains when the state |−⟩ can be detuned significantly from the subspace |+, n − 1⟩ and |d, n⟩, which leads to the requirement of  Δ_j ≫ Ω_r,j ≫ ν for the jth atom based on the condition of |ω₋| ≫ ν, the EIT cooling condition ω₊ = ν, and the usual assumption of a weak probe field  Ω_g,j ≪ Ω_r,j . The resultant effective dark-state sideband cooling should also be legitimate by fulfilling the sideband cooling condition of ν ≫ η_effΩ_eff,j and γ_eff,j . This can be guaranteed when ϕ_j and θ_j are made small enough, which can easily be achieved respectively by choosing large enough laser detuning  Δ_j ≫ Ω_r,j , Ω_g,j and control fields  Ω_r,j ≫ Ω_g,j . In the setting of EIT cooling with nonreciprocal couplings, it is these tunable excitation parameters and the controllable directionality of excitation–exchange couplings that give rise to distinct cooling behaviors we explore in the following section.

Here we consider a setting of two atoms with  Λ-type three-level configurations with nonreciprocal couplings, which represents the building block for a large-scale atomic array. In this basic unit under EIT cooling with collective excitation–exchange interactions, we numerically obtain the steady-state phonon occupations tr$[\rho (t\to \infty ){a}_{j}^{{\dagger}}{a}_{j}]={\langle {n}_{j}\rangle }_{\text{ss}}$ from equation (1), under the Hilbert space |α, n⟩_j with α ∈ {g, r, e} and n ∈ {0, 1, 2}. This truncation of phonon numbers is valid when the composite two-atom system is close to their motional ground state, that is when ${\langle {n}_{j}\rangle }_{\text{ss}}\ll 1$.

As a comparison to the single atom result $({\langle {n}_{j}\rangle }_{\text{ss}}^{\text{s}})$ without nonreciprocal spin exchange interactions, we calculate the ratio ${\tilde{n}}_{j}\equiv {\langle {n}_{j}\rangle }_{\text{ss}}/{\langle {n}_{j}\rangle }_{\text{ss}}^{\text{s}}$ to account for the superior or inferior cooling regime when ${\tilde{n}}_{j}$ is less or more than 1. In figure 2, we explore various parameter regimes of Ω_g,j , Ω_r,j , γ_g,j , ξ, γ_R and look for superior cooling regions in the allowable parameters under EIT cooling conditions. An essential parameter γ_R is defined as γ_R = γ_g,R + γ_r,R with γ_L = γ_g,L + γ_r,L, a total decay rate γ_R + γ_L = γ, γ_g = γ_g,L + γ_g,R, and γ_r = γ_r,L + γ_r,R. In all plots, we investigate an asymmetric setup of driving fields in two atoms, where we focus on the cooling behavior of the target atom $({\tilde{n}}_{1})$, or otherwise we always observe heating effect. In figure 2(a), we find a superior cooling regime for the target atom when γ_R/γ > 0.5 with a minimum at γ_R/γ ≈ 0.75 for a smaller probe-to-control field ratio Ω_g,j /Ω_r,j . This presents the essential effect of nonreciprocal couplings, which leads to surpassing cooling behaviors owing to distinct heat transfer from collective excitation–exchange interactions. The heating effect from symmetric EIT settings can be seen in figure 2(b), where ${\tilde{n}}_{1(2)}$ at γ_R/γ < (>)0.5 becomes larger than the respective single atom cases when Ω_g(r),2/Ω_g(r),1 = 1. We note that at the unidirectional coupling when γ_R/γ = 1(0), the ${\tilde{n}}_{1(2)}$ reaches the single atom limit as if the atom experiences one-way decay channel without the back action from the other atom. The asymmetric setting is favored for superior cooling and evidenced in figure 2(b) when Ω_g(r),2/Ω_g(r),1 < 0.5.

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Furthermore in figure 2(c), we explore the effect of decay channels in the probe field and control field transitions. As γ_g increases along with a finite nonreciprocal coupling γ_R, a superior cooling region emerges and allows an impressive performance of almost twofold improvement $({\tilde{n}}_{1}\approx 0.5)$ compared to the case without excitation–exchange interactions. Finally in figure 2(d), we find the optimal operations of EIT cooling at ξ = 2π or π, which also reflects the optimal interparticle distances that permit superior EIT cooling. For ${\tilde{n}}_{2}$ of the residual atom in all lower panels, we find no significant cooling or even heating behaviors under asymmetric EIT driving conditions. This shows the essential role of the residual atom that hosts excitation–exchange couplings to remove extra heat from the target atom. We note that throughout the paper, we consider the same ratio of nonreciprocal couplings between the left- and the right-propagating channels in either γ_g or γ_r for simplicity.

Next we numerically simulate the time dynamics of two-atom system in figure 3. To genuinely simulate the time evolutions in equation (1), we postulate the system in a thermal state initially [6, 65],

$$\begin{equation}\rho (t=0)=\prod\limits _{j=1}^{N}\sum\limits _{n=0}^{\infty }\frac{{n}_{0}^{n}}{{({n}_{0}+1)}^{n+1}}\vert g,{n\rangle }_{j}\langle g,n\vert ,\end{equation} \tag{ 12 }$$

where n₀ is an average phonon number for both atoms. We assume n₀ ≲ 1 and make a sufficient truncation on the finite motional states to guarantee the convergence in numerical simulations. As shown in figure 3, we select three contrasted regimes for heating, cooling, and neutral time dynamics. The characteristic times to reach the steady-states are shown shorter in cooling dynamics in figure 3(b). This presents an enhancement in cooling rate compared to the EIT cooling scheme in a single atom, in addition to the outperforming cooling behaviors in the steady-state phonon occupations. By contrast, the time dynamics in the heating regime shows a longer time behavior than the single atom case, while the neutral regime has a similar characteristic timescale. We attribute the enhanced cooling rate or prolonged time dynamics in heating to the collective spin–phonon couplings between the atoms.

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In equations (7)–(9), we have demonstrated the mapping from EIT cooling with three-level structures to the effective dark-state sideband cooling scheme. Within n ∈ {0, 1}, the mapping has reduced the Hilbert space dimensions in two atoms from 36 to 16 with 16² coupled linear equations. We follow the wisdom in sideband cooling with chiral couplings by considering asymmetric driving conditions when η_effΩ_eff,2 ≪ η_effΩ_eff,1 [66]. In this way, we are able to perform a partial trace on the motional degree of freedom (a₂) and focus on the target atom behavior ${\langle {n}_{1}\rangle }_{\text{ss}}$, which further diminishes the dimension of the Hilbert space to 8. The dynamics of this two-atom system can then be determined by the reduced density matrix ${\text{tr}}_{{a}_{2}}(\rho )$. This neglect of a₂ is legitimate since its effect becomes a higher order correction compared to the motions of the target atom owing to a relatively small driving field in the second atom.

The steady-state solutions for the density matrix elements of ${\text{tr}}_{{a}_{2}}(\rho )={\rho }_{{\alpha }_{1}{n}_{1}{\alpha }_{2}{\beta }_{1}{m}_{1}{\beta }_{2}}=\langle {\alpha }_{1},{n}_{1};{\alpha }_{2}\vert {\text{tr}}_{{a}_{2}}\rho \vert {\beta }_{1},{m}_{1};{\beta }_{2}\rangle$ can further be simplified by taking advantage of the fact that ρ_d0dd0d is $\mathcal{O}(1)$ when ${\gamma }_{\text{eff}}^{2}/{\nu }^{2}$ and ${\eta }_{\text{eff}}^{2}{{\Omega}}_{\text{eff},j}^{2}/{\nu }^{2}$ are much smaller than one. Here we use α₁₍₂₎ and β₁₍₂₎ to denote the first (second) atomic internal levels |d⟩ and |+⟩ in the dark-state mapping, and n₁ and m₁ for phonon occupation numbers of the first atom. The essential focus of this section is that (a) with the assistance of dark-state mapping, we are able to reduce three-level EIT cooling scheme to an effective two-level sideband cooling scheme in two atoms, (b) we can simplify the coupled equations of all density matrix elements by tracing over the second atom's motional degrees of freedom and keeping the leading order terms, and obtain some analytical forms, and (c) we later prove that a dark-state mapping is legitimate in figure 4, where we directly compare a full numerical simulations of three-level EIT cooling scheme in two atoms with equation (15).

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To proceed, we can neglect those ${\rho }_{{\alpha }_{1}{n}_{1}{\alpha }_{2}{\beta }_{1}{m}_{1}{\beta }_{2}}$ whose leading terms are higher than the second order. Setting e^iξ = 1 and keeping the terms of $\mathcal{O}(1)$, $\mathcal{O}({\gamma }_{\text{eff},1(2)})$, $\mathcal{O}({\gamma }_{\text{eff},\text{L}(\text{R})})$, $\mathcal{O}({\eta }_{\text{eff}}{{\Omega}}_{\text{eff},1})$, $\mathcal{O}({\gamma }_{\text{eff},1(2,\text{L},\text{R})}{\eta }_{\text{eff}}{{\Omega}}_{\text{eff},1})$, we obtain

$$\begin{equation}\begin{aligned}\hfill 0\,=\,& {\eta }_{\text{eff}}{{\Omega}}_{\text{eff},1}({\rho }_{+0dd1d}-{\rho }_{d1d+0d})+2i{\gamma }_{\text{eff},1}{\rho }_{+0d+0d}+2i{\gamma }_{\text{eff},\mathrm{L}}(1,2)({\rho }_{d0++0d}+{\rho }_{+0dd0+}),\hfill \\ \hfill 0\,=\,& -{\eta }_{\text{eff}}{{\Omega}}_{\text{eff},1}{\rho }_{d0+d1d}-2i{\gamma }_{\text{eff},\mathrm{L}}(1,2){\rho }_{d0+d0+}-2i{\gamma }_{\text{eff},\mathrm{R}}(1,2){\rho }_{+0d+0d}-i({\gamma }_{\text{eff},1}+{\gamma }_{\text{eff},2}){\rho }_{d0++0d},\hfill \\ \hfill 0\,=\,& {\eta }_{\text{eff}}{{\Omega}}_{\text{eff},1}({\rho }_{+0d+0d}-{\rho }_{d1dd1d})-2i{\gamma }_{\text{eff},\mathrm{L}}(1,2){\rho }_{d1dd0+}-i{\gamma }_{\text{eff},1}{\rho }_{d1d+0d},\hfill \\ \hfill 0\,=\,& -2i{\gamma }_{\text{eff},\mathrm{R}}(1,2)({\rho }_{+0dd0+}+{\rho }_{d0++0d})-2i{\gamma }_{\text{eff},2}{\rho }_{d0+d0+},\hfill \\ \hfill 0\,=\,& {\eta }_{\text{eff}}{{\Omega}}_{\text{eff},1}{\rho }_{+0dd0+}-2i{\gamma }_{\text{eff},\mathrm{R}}(1,2){\rho }_{d1d+0d}-i{\gamma }_{\text{eff},2}{\rho }_{d1dd0+},\hfill \\ \hfill 0\,=\,& {\eta }_{\text{eff}}{{\Omega}}_{\text{eff},1}({\rho }_{+0dd1d}-{\rho }_{d1d+0d})+i{\gamma }_{\text{eff},1}\frac{{\eta }_{\text{eff}}^{2}{{\Omega}}_{\text{eff},1}^{2}}{8{\nu }^{2}}{\rho }_{d0dd0d},\hfill \end{aligned}\end{equation} \tag{ 13 }$$

along with three other relationships,

$$\begin{align}\hfill & {\rho }_{+1d+1d}=\frac{{\eta }_{\text{eff}}^{2}{{\Omega}}_{\text{eff},1}^{2}}{16{\nu }^{2}}{\rho }_{d0dd0d},\hfill \\ \hfill & {\rho }_{d1+d0d}=-\frac{i{\gamma }_{\text{eff},\mathrm{R}}(1,2){\eta }_{\text{eff}}{{\Omega}}_{\text{eff},1}}{8{\nu }^{2}}{\rho }_{d0dd0d},\hfill \\ \hfill & {\rho }_{+1dd0d}=-\frac{4\nu -i{\gamma }_{\text{eff},2}}{16{\nu }^{2}}{\eta }_{\text{eff}}{{\Omega}}_{\text{eff},1}{\rho }_{d0dd0d}.\hfill \end{align} \tag{ 14 }$$

Solving the above equations leads to the solutions of density matrix elements expressed in terms of ρ_d0dd0d . Finally we obtain the steady-state phonon occupation for the target atom as (ρ_d0dd0d ≈ 1)

$$\begin{align}\hfill {\langle {n}_{1}\rangle }_{\text{st}}& ={\rho }_{+1++1+}+{\rho }_{+1d+1d}+{\rho }_{d1+d1+}+{\rho }_{d1dd1d},\hfill \\ \hfill & \approx \frac{{\gamma }_{\text{eff},1}^{2}}{16{\nu }^{2}}+\frac{{\eta }_{\text{eff}}^{2}{{\Omega}}_{\text{eff},1}^{2}}{8{\nu }^{2}}-\frac{{\gamma }_{\text{eff},1}}{{\gamma }_{\text{eff},2}}\frac{{\gamma }_{\text{eff},\mathrm{R}}(1,2){\gamma }_{\text{eff},\mathrm{L}}(1,2)}{4{\nu }^{2}}\hfill \\ \hfill & \quad +\frac{{({\gamma }_{\text{eff},1}+{\gamma }_{\text{eff},2})}^{2}/({\gamma }_{\text{eff},1}{\gamma }_{\text{eff},2})[{\gamma }_{\text{eff},\mathrm{R}}(1,2){\gamma }_{\text{eff},\mathrm{L}}(1,2)]}{({\gamma }_{\text{eff},1}+{\gamma }_{\text{eff},2})/{\gamma }_{\text{eff},1}\left[{\gamma }_{\text{eff},1}{\gamma }_{\text{eff},2}/4-{\gamma }_{\text{eff},\mathrm{R}}(1,2){\gamma }_{\text{eff},\mathrm{L}}(1,2)\right]+{\eta }_{\text{eff}}^{2}{{\Omega}}_{\text{eff},1}^{2}/4}\frac{{\eta }_{\text{eff}}^{2}{{\Omega}}_{\text{eff},1}^{2}}{16{\nu }^{2}},\hfill \end{align} \tag{ 15 }$$

where the approximate form assumes ν ≫ γ_eff,L(R)(1, 2), γ_eff,1(2), and the first two terms are exactly the steady-state phonon occupation for a single atom under the sideband cooling [6]. The remaining terms are the modifications originating from the nonreciprocal couplings. At unidirectional coupling γ_eff,L(R)(1, 2) = 0, equation (15) again reduces to the single atom result, which is also shown in figures 2(a) and (b) when γ_R/γ = 1 or 0.

In the LD regime as η_eff → 0, we can estimate equation (15) as

$$\begin{equation}{\langle {n}_{1}\rangle }_{\text{ss}}{\vert }_{{\eta }_{\text{eff}}\to 0}\approx \frac{{\gamma }_{\text{eff},1}^{2}}{16{\nu }^{2}}-\frac{{\gamma }_{\text{eff},1}}{{\gamma }_{\text{eff},2}}\frac{{\gamma }_{\text{eff},\mathrm{R}}(1,2){\gamma }_{\text{eff},\mathrm{L}}(1,2)}{4{\nu }^{2}},\end{equation} \tag{ 16 }$$

where the phonon occupation of the target atom is no longer limited by the effective spontaneous emission rate as in the single atom result of ${\gamma }_{\text{eff},1}^{2}/(16{\nu }^{2})$. If we assume θ₁ = θ₂ ≈ 0 and γ_g ≫ γ_r , by comparing to the single atom result, we obtain

$$\begin{equation}{\tilde{n}}_{1}{\vert }_{{\eta }_{\text{eff}}\to 0}\approx 1-\frac{4{\gamma }_{g,\mathrm{L}}{\gamma }_{g,\mathrm{R}}}{{\gamma }_{g}^{2}}.\end{equation} \tag{ 17 }$$

The above shows the minimum of ${\tilde{n}}_{1}{\vert }_{{\eta }_{\text{eff}}\to 0}=0$ when γ_g,L(R) = 1/2. This coincides with the predictions of the target ion under chiral-coupling-assisted sideband cooling [66] when an extreme LD regime as η → 0 is applied. Furthermore in this extreme regime as in an infinite mass limit, a reciprocal coupling emerges to host the heat removal in the target atom from the mechanism of collective spin–exchange interaction. The values of γ_g,R at the minimum of ${\tilde{n}}_{1}$ would shift to nonreciprocal coupling regimes of γ_g,R ≶ 0.5 when finite corrections of η_eff in equation (15) are retrieved, where two minimums arise from a quartic dependence of γ_g,R in equation (15). These optimal conditions can be obtained [66] with a perturbation of η_eff,

$$\begin{equation}{\gamma }_{g,\mathrm{R}}^{\text{opt}}=\frac{\gamma }{2}\pm \frac{\sqrt{{\eta }_{\text{eff}}{{\Omega}}_{\text{eff},1}\gamma \sqrt{2}}}{2},\end{equation} \tag{ 18 }$$

where the optimal condition γ_g,R = 1/2 migrates toward nonreciprocal coupling regime as a square root function of η_effΩ_eff,1.

In figure 4, we show the superior dark-state sideband cooling under nonreciprocal couplings, where two lowest minimums can be identified as superior cooling compared to the single atom results at γ_R = 1. We compare the exact results from EIT cooling scheme with three-level atomic structures and the predictions from the effective dark-state sideband cooling, which are well described by an analytical form in equation (15) under asymmetric driving conditions. This confirms the validity of dark-state mapping when large enough two-photon detunings Δ_j are applied. The symmetry of the curves in figure 4 results from the asymmetric driving condition, where the residual atom behaves as an entity to allow spin–exchange interactions only. In this scenario, the target atom exchanges spin excitations with the residual one at finite directional coupling strengths. An interference from nonreciprocal couplings can build up owing to the collective nature of spin–exchange interactions, which leads to a multiplication factor of γ_eff,R γ_eff,L in equation (16) or γ_g,R γ_g,L in equation (17), and therefrom the symmetric profiles in figure 4.

A superior cooling performance in the constituent atoms emerges owing to the presence of nonreciprocal couplings. With this extra degree of freedom, the composite system is allowed to access new parameter regimes with potentially better cooling behaviors. These unequal coupling channels provide a directional heat transfer from the target atom to the residual one under an asymmetric driving condition. This distinct heat removal can be attributed to an effective heat sink of the residual atom, where an asymmetric driving strengthens an imbalance of the heat distribution in the composite systems. Due to the collective nature of excitation–exchange interactions, a finite directional decay channel allows a superior cooling region that surpasses the single atom results. For experimental realizations, we propose to apply a side excitation scheme in an atom–waveguide system [34], where a quantum-state-controlled directional coupling strength can be realized and tunable in a strong coupling regime of β ≡ (γ_L + γ_R)/(γ_L + γ_R + γ_ng ) [33, 38] greater than 90% with a nonguided mode quantified by γ_ng . In this system, γ_L(R)/γ can be tailored between 0 and 1 [34], which we have theoretically explored here. The nonguided mode coupling would further provide an extra dissipation channel that can turn the heating mechanism at the reciprocal coupling regime (γ_R = γ_L) to cooling instead, indicating a reduction in the spin–spin correlation that is otherwise more significant in the heating regime [66].

For experimental consideration of EIT cooling schemes, we have several options in the level transitions of rubidium atoms [21], where various hyperfine states or field polarizations can be utilized. In addition to the tunable strengths of the probe and control fields with a large enough detuning, the essential requirement for superior cooling relies on the ratio of decay rates as shown in figure 2(c). We take ⁸⁵Rb atoms as an example: the probe to control field transitions with polarizations σ⁺ to σ⁻ can be chosen as $\vert F=2,{m}_{F}=1\rangle \to \vert {F}^{\prime }=3,{m}_{F}^{\prime }=2\rangle \to \vert F=3,{m}_{F}=3\rangle$ or $\vert F=2,{m}_{F}=0\rangle \to \vert {F}^{\prime }=3,{m}_{F}^{\prime }=1\rangle \to \vert F=2,{m}_{F}=2\rangle$ for D₂ transition and $\vert F=2,{m}_{F}=1\rangle \to \vert {F}^{\prime }=3,{m}_{F}^{\prime }=2\rangle \to \vert F=3,{m}_{F}=3\rangle$ or $\vert F=3,{m}_{F}=-3\rangle \to \vert {F}^{\prime }=2,{m}_{F}^{\prime }=-2\rangle \to \vert F=3,{m}_{F}=-1\rangle$ for D₁ transition, where the branching ratios γ_g :γ_r can be 32:15, 6:1, 10:3, and 15:1, respectively. This ratio can be tuned by using different polarizations, for example σ⁺ to σ⁺ transition, or different atomic species of ¹³³Cs [62]. For ⁸⁵Rb atoms, we have γ ≈ 2π × 6 MHz and ν ≈ 2π × 0.3 MHz, which can be satisfied in typical EIT settings and optical traps with a weak confinement.

In summary, we have shown theoretically that a superior EIT cooling behavior can be feasible when nonreciprocal couplings between constituent atoms are introduced. This provides a way to get around the cooling bottleneck [18, 55, 56] that an atom-based quantum computer or other applications of quantum technology may encounter. The conventional EIT cooling in a single atom is limited by the steady-state phonon occupation ∼(γ/Δ)² [22], while the corresponding dark-state sideband cooling is by $\sim {({\gamma }_{\text{eff}}/\nu )}^{2}$. By utilizing nonreciprocal couplings between constituent atoms, an intriguing dark-state cooling scheme in Λ-type three-level structure manifests a distinct heat transfer to further cool the target atom surpassing its single atom limit. We explore the allowable system parameters especially under asymmetric driving conditions and identify the regions for superior cooling performance with an enhanced cooling rate. Our results unravel the mechanism of collective spin–exchange interactions between trapped atoms, which enable new possibilities in driving the atoms toward their motional ground states. Future work will consider a multiatom platform, where more complex spin–phonon correlations may arise to further enhance the performance of the EIT cooling scheme.

We acknowledge support from the Ministry of Science and Technology (MOST), Taiwan, under Grant No. MOST-109-2112-M-001-035-MY3. We are also grateful for support from TG 1.2 and TG 3.2 of NCTS.

The data that support the findings of this study are available upon reasonable request from the authors.