Realising a quantum absorption refrigerator with an atom-cavity system

The three-body quantum absorption refrigerator comprises three subsystems with Hamiltonian

$$\begin{eqnarray}&&H=\displaystyle \sum _{j=A,B,C}{H}_{j},\end{eqnarray} \tag{ 1 }$$

where the operators H_j act non-trivially on subsystem j only. Subsystem A is the body to be cooled. We assume that it has an equally spaced energy spectrum with level splitting E_A, i.e.,

$$\begin{eqnarray}&&{H}_{A}={E}_{A}\displaystyle \sum _{n=0}^{D-1}n| n\rangle \langle n| ,\end{eqnarray} \tag{ 2 }$$

where D is the local Hilbert space dimension (possibly infinite). This general form may describe a qubit, a spin or a harmonic oscillator. We also assume that H_B and H_C each possess at least one pair of eigenstates differing in energy by E_j, such that

$$\begin{eqnarray}&&{E}_{A}+{E}_{B}={E}_{C}.\end{eqnarray} \tag{ 3 }$$

The form of H_B and H_C is otherwise arbitrary. The subsystems are coupled together by a three-body interaction,

$$\begin{eqnarray}&&V=g({L}_{A}{L}_{B}{L}_{C}^{\dagger }+{L}_{A}^{\dagger }{L}_{B}^{\dagger }{L}_{C}),\end{eqnarray} \tag{ 4 }$$

where g is the interaction energy and L_j is a lowering operator connecting pairs of Hamiltonian eigenstates separated by an energy E_j, i.e.,

$$\begin{eqnarray}&&[H,{L}_{j}]=-{E}_{j}{L}_{j},\end{eqnarray} \tag{ 5 }$$

while ${L}_{j}^{\dagger }$ is the corresponding raising operator. The condition (3) ensures that $[H,V]=0$, so that the interaction (4) enacts resonant transitions between degenerate energy eigenstates of H.

Cooling is achieved by coupling subsystem B to a hot thermal bath at temperature ${T}_{h}\gt {T}_{r}$, while subsystems A and C remain coupled to the environment at temperature T_r. Energy exchange between the subsystems then allows heat to naturally flow from the hot reservoir to the colder environment. However, due to the specific form of the interaction (4), subsystem C can only absorb a quantum of energy from the hot body B by simultaneously absorbing energy from A, thus leading to cooling. The heat flow through the refrigerator is illustrated in figure 1.

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The concepts of virtual qubits and virtual temperatures provide a convenient and intuitive way to analyse autonomous thermal machines [38]. A virtual qubit is a pair of states in the composite Hilbert space of subsystems B and C which directly couples to the target subsystem A. By choosing the parameters of the system appropriately, the virtual qubit can be placed at an effective virtual temperature that may be lower than T_r. The operation of the refrigerator can then be understood as a simple thermalisation process between A and the virtual qubit.

To make this notion explicit, we observe that the interaction Hamiltonian (4) can be written as

$$\begin{eqnarray}&&V=g({L}_{A}{L}_{v}^{\dagger }+{L}_{A}^{\dagger }{L}_{v}),\end{eqnarray} \tag{ 6 }$$

where ${L}_{v}={L}_{B}^{\dagger }{L}_{C}$. Equations (3) and (5) together imply that

$$\begin{eqnarray}&&[H,{L}_{v}]=-{E}_{A}{L}_{v}.\end{eqnarray} \tag{ 7 }$$

This means that L_v is a lowering operator connecting pairs of states differing by an energy E_A in the composite Hilbert space of B and C. Each of these pairs of states is called a virtual qubit. The interaction (6) then describes resonant energy exchange between the virtual qubits and A.

When each subsystem is at thermal equilibrium with its respective bath, the populations of the virtual qubit states are thermally distributed at a virtual temperature

$$\begin{eqnarray}&&{T}_{v}=\displaystyle \frac{{E}_{A}}{{E}_{C}/{T}_{r}-{E}_{B}/{T}_{h}}.\end{eqnarray} \tag{ 8 }$$

That is to say, each pair of virtual qubit states is populated in the ratio ${{\rm{e}}}^{-{E}_{A}/{T}_{v}}$. As long as the parameters of the refrigerator are chosen so that ${T}_{v}\lt {T}_{r}$, subsystem A will be pushed towards a lower temperature as it equilibrates with the virtual qubits under the interaction (6). This effect is counteracted by the thermalising influence of the reservoir interacting with A, thus establishing a heat current flowing from the environment surrounding A into the refrigerator. See [38] for a more complete discussion of virtual qubits and temperatures.

To conclude this section, we briefly mention that if the assumption that ${T}_{h}\gt {T}_{r}$ is relaxed, one can arrange for T_v to take any value by adjusting the bath temperatures and energy splittings. If ${T}_{v}\gt {T}_{r}$, then the steady-state temperature of A is increased and the system operates as a heat pump. On the other hand, if ${T}_{v}\lt 0$ the machine tries to induce population inversion in the state of A, which can be thought of as the quantum analogue of a classical heat engine lifting a weight [38]. In the following, we restrict our attention to the absorption chiller mode, where ${T}_{h}\gg {T}_{r}$. However, our results could be applied equally well to the construction and study of autonomous quantum heat pumps and engines.

To analyse the performance of a refrigerator, one must choose figures of merit. The appropriate figure of merit depends on the problem at hand, as we now explain.

2.3.1. Coefficient of performance and cooling power

From one viewpoint, the refrigerator can be seen as a device which extracts heat from the environment surrounding subsystem A. Thus, the refrigerator performance is characterised by the stationary heat currents flowing to and from the reservoirs. The cooling power ${\dot{Q}}_{A}$ gives the heat current into the refrigerator from the environment of A, while ${\dot{Q}}_{B}$ gives the input power corresponding to the heat current flowing in from the hot reservoir. Therefore, the relevant figure of merit is the coefficient of performance $\epsilon ={\dot{Q}}_{A}/{\dot{Q}}_{B}$.

Note that this point of view makes sense only if the reservoirs connected to subsystems A and C are considered as separate entities. If subsystems A and C are in fact connected to the same environment, the net effect of the machine is simply to dump ${\dot{Q}}_{B}$ energy per unit time from the hot reservoir into this environment.

Assuming that the reservoirs connected to A and C are independent, we can estimate the coefficient of performance using the equations of motion for the mean local energies of subsystems A and B,

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}\langle {H}_{j}\rangle }{{\rm{d}}t}={\dot{Q}}_{j}+{\rm{i}}{{gE}}_{j}\langle {L}_{A}{L}_{B}{L}_{C}^{\dagger }-{L}_{A}^{\dagger }{L}_{B}^{\dagger }{L}_{C}\rangle \end{eqnarray} \tag{ 9 }$$

for $j=A,B$, where the second term on the right-hand side (RHS) follows from the Heisenberg equation generated by the interaction Hamiltonian (4). In the stationary state, the derivatives of the mean energies vanish, leading to the following simple expression for the coefficient of performance

$$\begin{eqnarray}&&\epsilon =\displaystyle \frac{{E}_{A}}{{E}_{B}}.\end{eqnarray} \tag{ 10 }$$

We see that the coefficient of performance grows without limit as E_A is increased while holding E_B fixed (assuming that equation (3) is always satisfied). It is important to note that our approximate analysis ignores the contribution of the interaction energy to the heat currents and is therefore only strictly correct in the weak-coupling limit of vanishingly small g [18].

2.3.2. Achievable temperature and cooling time

In the above scenario, one uses a microscopic machine to cool a macroscopic body, namely the reservoir connected to A. Perhaps a more appropriate application of a quantum refrigerator is to cool a microscopic system, namely subsystem A itself. From this viewpoint, the most important figure of merit is the achievable temperature (or more generally, the achievable energy and entropy) of subsystem A [20].

The achievable steady-state temperature can be estimated from the virtual temperature given by equation (8). This takes its minimal value when the temperature of the hot bath is large, from which we find

$$\begin{eqnarray}&&\underset{{T}_{h}\to \infty }{\mathrm{lim}}{T}_{v}=\displaystyle \frac{{E}_{A}}{{E}_{C}}{T}_{r}=\displaystyle \frac{\epsilon }{1+\epsilon }{T}_{r},\end{eqnarray} \tag{ 11 }$$

where we have used equations (3) and (10) to rewrite the virtual temperature in terms of the coefficient of performance $\epsilon$. We find that the virtual temperature is minimised when $\epsilon$ is small. This illustrates that the standard thermodynamic measures of steady-state refrigerator performance are essentially irrelevant when the task at hand is to cool a quantum system having a finite energy.

Because the refrigerator is out of equilibrium, the thermodynamic temperature of A may not be strictly defined. For our purposes, it is sufficient to adopt the mean energy $\langle {H}_{A}\rangle$ as a figure of merit, rather than the temperature. This also provides an adequate measure of entropy, as the von Neumann entropy of a state with mean energy $\langle {H}_{A}\rangle$ is upper-bounded by that of a Gibbs state having the same mean energy. If the cooling is subject to time constraints, the relaxation time (the time taken for $\langle {H}_{A}\rangle$ to reach its stationary value) is also a measure of performance. However, the relaxation time is a non-universal figure of merit as it may depend on the initial conditions.

Throughout the remainder of this article, we adopt the present framework, where the objective is to cool subsystem A. This viewpoint is particularly appropriate for quantum technology applications. Here, the motivation for cooling a quantum system is typically to maximise the efficiency of subsequent control operations by reducing uncertainty over the initial conditions, i.e. by minimising the entropy of the quantum system. We therefore neglect traditional efficiency measures such as the coefficient of performance $\epsilon$, choosing rather to focus on the mean energy of the subsystems constituting the refrigerator, in particular that of the target body A.

We consider a single atom or ion of mass M confined in the x direction by a harmonic potential with oscillation frequency $\nu /2\pi$. The atom is assumed to possess a pair of relevant internal electronic states $| \downarrow \rangle$ and $| \uparrow \rangle$ separated by an energy ε. The trap is placed inside an optical cavity whose axis is aligned in the x direction. The minimum of the harmonic potential is placed at a node of the electric field of a cavity mode with frequency $\omega /2\pi$, chosen such that $\omega =\varepsilon -\nu$. We assume that $\omega \sim \varepsilon$ and $\varepsilon ,\omega \gg \nu$, as appropriate for typical optical and vibrational frequencies. The geometry of the problem is depicted schematically in figure 2.

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The free Hamiltonian of the system is

$$\begin{eqnarray}&&{H}_{1}=\nu {a}^{\dagger }a+\omega {b}^{\dagger }b+\varepsilon {\sigma }^{+}{\sigma }^{-},\end{eqnarray} \tag{ 12 }$$

where the bosonic ladder operators ${a}^{\dagger }$ and ${b}^{\dagger }$, respectively, create motional quanta (phonons) and light quanta (photons), while ${\sigma }^{-}=| \downarrow \rangle \quad \langle \uparrow | ={({\sigma }^{+})}^{\dagger }$ is the atomic lowering operator. We assumed that all other electronic states and vibrational or cavity modes are far off-resonant and can be neglected.

The interaction between the atom and the cavity field in the dipole approximation reads as

$$\begin{eqnarray}&&{V}_{1}=g\mathrm{sin}[\eta (a+{a}^{\dagger })](b+{b}^{\dagger })({\sigma }^{-}+{\sigma }^{+}),\end{eqnarray} \tag{ 13 }$$

where g is the cavity coupling constant and the Lamb-Dicke parameter is defined as $\eta =\omega /\sqrt{2{{Mc}}_{0}^{2}\nu }$, with c₀ the speed of light in vacuum. The form of the interaction (13) reflects the symmetry of the problem when the harmonic potential minimum coincides exactly with an electric field node. The electric field operator then changes sign under a parity transformation of the atomic centre-of-mass coordinate. This gives rise to a selection rule allowing only transitions between motional states of opposite parity. Since the vibrational energy eigenstates have definite parity, the absorption or emission of a photon must therefore be accompanied by a change in the number of phonons.

We now show that the system approximately realises a QAR, and estimate its virtual temperature. We work in the limit $\eta \ll 1$, which requires that the cavity mode wavelength is much larger than the characteristic length scale of atomic motion. We can therefore invoke the Lamb-Dicke approximation (LDA) and expand equation (13) to first order in η. We also make the rotating wave approximation (RWA) by discarding counter-rotating terms at optical frequency, leading to

$$\begin{eqnarray}&&{V}_{1}\approx g\eta (a+{a}^{\dagger })(b{\sigma }^{+}+{b}^{\dagger }{\sigma }^{-}).\end{eqnarray} \tag{ 14 }$$

Assuming that terms counter-rotating at frequency $\pm 2\nu$ can also be neglected, we finally obtain

$$\begin{eqnarray}&&{V}_{1}\approx g\eta ({ab}{\sigma }^{+}+{a}^{\dagger }{b}^{\dagger }{\sigma }^{-}).\end{eqnarray} \tag{ 15 }$$

We therefore find under these assumptions that the system exhibits a three-body interaction of the type (4). Note that the approximation leading to equation (15) is valid only when $g\eta \ll 2\nu$. Equation (15) also assumes that the line widths of relevant transitions are much smaller than the trap frequency ν, as discussed in detail in section 3.3.

The heat reservoir is provided by coupling broadband thermal light, e.g. sunlight, at a high temperature T_h into the cavity resonator. Assuming that the thermal light source is well collimated, the electronic transition couples only to the ambient radiation field at room temperature ${T}_{r}\ll {T}_{h}$, which leads to spontaneous emission. Meanwhile, the motion of the atom undergoes intrinsic heating in the trap, for example, due to fluctuations of the trapping potential.

The virtual qubit states in the machine are the pairs $\{| {n}_{b},\downarrow \rangle ,| {n}_{b}-1,\uparrow \rangle \}$, where $| {n}_{b}\rangle$ denotes a Fock state with n_b quanta in the cavity mode. The virtual temperature of the refrigerator is therefore

$$\begin{eqnarray}&&{T}_{v}=\displaystyle \frac{\nu }{\varepsilon /{T}_{r}-\omega /{T}_{h}}.\end{eqnarray} \tag{ 16 }$$

Because ${T}_{h}\gg {T}_{r}$ and $\nu \ll \varepsilon$, we find that

$$\begin{eqnarray}&&\displaystyle \frac{{T}_{v}}{{T}_{r}}\approx \displaystyle \frac{\nu }{\varepsilon }.\end{eqnarray} \tag{ 17 }$$

The ratio of frequencies is typically $\nu /\varepsilon \sim {10}^{-8}$ or less, implying that very low virtual temperatures are achievable.

The refrigerator operation can be understood by a straightforward analogy with laser sideband cooling. The optical cavity behaves as a filter that singles out frequencies close to the red sideband ${\omega }_{\mathrm{red}}=\varepsilon -\nu$. Pumping the cavity with thermal light increases the number of photons with the correct frequency to drive the red sideband transition. Spontaneous emission then resets the electronic state, completing the cooling cycle (see figure 3). So long as the blue sideband frequency ${\omega }_{\mathrm{blue}}=\varepsilon +\nu$ is far off-resonant, the absorption of thermal cavity photons drives the motion towards its ground state.

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To study the dynamics of the model, we employ a quantum master equation for the density operator ρ of the form

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}\rho }{{\rm{d}}t}=-{\rm{i}}[{H}_{1}+{V}_{1},\rho ]+\displaystyle \sum _{j=a,b,\sigma }{{ \mathcal L }}_{j}\rho .\end{eqnarray} \tag{ 18 }$$

The superoperators ${{ \mathcal L }}_{j}$ are dissipative contributions due to the coupling of each subsystem to its respective reservoir. The term ${{ \mathcal L }}_{a}$ describes motional heating, ${{ \mathcal L }}_{b}$ corresponds to the thermal pumping of the cavity, while ${{ \mathcal L }}_{\sigma }$ relates to spontaneous emission.

Introducing the general notation for a Lindblad dissipator

$$\begin{eqnarray}&&{ \mathcal D }[L]\rho =L\rho {L}^{\dagger }-\displaystyle \frac{1}{2}\{{L}^{\dagger }L,\rho \},\end{eqnarray} \tag{ 19 }$$

the motional heating is described by

$$\begin{eqnarray}&&{{ \mathcal L }}_{a}=\lambda (1+{\bar{n}}_{a}^{-1}){ \mathcal D }[a]+\lambda { \mathcal D }[{a}^{\dagger }],\end{eqnarray} \tag{ 20 }$$

where ${\bar{n}}_{a}={({{\rm{e}}}^{\nu /{T}_{r}}-1)}^{-1}$ is the equilibrium phonon number at room temperature. Since ${\bar{n}}_{a}^{-1}\approx 0$ to an excellent approximation under typical laboratory conditions, the dissipator (20) describes approximately linear growth of the phonon number at a constant heating rate λ. Pumping of the cavity with thermal light is described by

$$\begin{eqnarray}&&{{ \mathcal L }}_{b}=2\kappa (1+{\bar{n}}_{b}){ \mathcal D }[b]+2\kappa {\bar{n}}_{b}{ \mathcal D }[{b}^{\dagger }],\end{eqnarray} \tag{ 21 }$$

where κ is the cavity line width and ${\bar{n}}_{b}={({{\rm{e}}}^{\omega /{T}_{h}}-1)}^{-1}$ is the equilibrium photon number at temperature T_h. For simplicity, we assume here that the thermal driving is applied to both sides of the cavity. Spontaneous emission is described by the Liouvillian

$$\begin{eqnarray}&&{{ \mathcal L }}_{\sigma }={\rm{\Gamma }}(1+{\bar{n}}_{\sigma }){\displaystyle \int }_{-1}^{1}{\rm{d}}u\;{\rm{\Pi }}(u){ \mathcal D }[{{\rm{e}}}^{{\rm{i}}\eta u(a+{a}^{\dagger })}{\sigma }^{-}]+{\rm{\Gamma }}{\bar{n}}_{\sigma }{\displaystyle \int }_{-1}^{1}{\rm{d}}u\;{\rm{\Pi }}(u){ \mathcal D }[{{\rm{e}}}^{-{\rm{i}}\eta u(a+{a}^{\dagger })}{\sigma }^{+}],\end{eqnarray} \tag{ 22 }$$

with ${\bar{n}}_{\sigma }={({{\rm{e}}}^{\varepsilon /{T}_{r}}-1)}^{-1}$, while ${\rm{\Pi }}(u)$ is the angular distribution of emitted photons as a function of $u=\mathrm{cos}\theta$, where θ is the angle subtended from the x axis by the photon wave vector. Note that ${\bar{n}}_{\sigma }\approx 0$ at optical frequencies and room temperature, and therefore the absorption term on the second line of equation (22) is typically negligible.

Unfortunately, the single-cavity refrigerator suffers from several severe practical limitations. The most important of these is due to line broadening. The picture illustrated in figure 3, described by the Hamiltonian (15), is valid when the sideband transitions are 'sharp' in the sense of having a well-defined frequency. However, thermal dissipation implies some unavoidable energy uncertainty due to the finite lifetime of the states involved. Significant cooling is only possible in the sideband-resolved regime, where the frequency uncertainty of the relevant transitions is much less than ν. Otherwise, line broadening brings the blue sideband transition partially onto resonance, leading to heating rather than cooling. In particular, this means that we must have $\lambda ,\kappa ,{\rm{\Gamma }}\lt \nu$ for effective refrigeration.

To illustrate the effect of line broadening, we compute the steady-state phonon occupation ${n}_{a}(\infty )=\mathrm{Tr}[{a}^{\dagger }a{\rho }_{\infty }]$, where ${\rho }_{\infty }$ is the stationary quantum state satisfying ${\rm{d}}{\rho }_{\infty }/{\rm{d}}t={ \mathcal L }{\rho }_{\infty }=0$, with ${ \mathcal L }$ given by the RHS of equation (18). The problem is simplified by taking the interaction Hamiltonian (14) under the LDA and RWA, and making the approximation ${\bar{n}}_{a}^{-1}\approx 0\approx {\bar{n}}_{\sigma }$, leaving just five free parameters governing the phonon population dynamics: λ, η, κ, ${\bar{n}}_{b}$ and Γ. We compute the stationary state by representing ${ \mathcal L }$ as a matrix and solving the eigenvalue equation ${ \mathcal L }{\rho }_{\infty }=0$. The integral in equation (22) is numerically approximated by a trapezoidal rule. We take ${\rm{\Pi }}(u)=3(1+{u}^{2})/8$, as appropriate for a point dipole aligned perpendicularly to the cavity axis. Sampling a grid of 100 evenly spaced points in the interval $u\in [-1,1]$ is sufficient to obtain convergence. The resulting Liouvillian matrix has low sparsity and is therefore challenging to diagonalise, which limits the achievable Hilbert space dimension considerably. We use 21 phonon states and four cavity photon states in total. The results are therefore quantitatively inaccurate, but suffice to obtain qualitative trends.

The qualitative dependence of ${n}_{a}(\infty )$ on the dissipation rates κ and Γ is plotted for some example parameters in figure 4. We see that the optimum operating regime is $\kappa ,{\rm{\Gamma }}\ll \nu$, as expected. The performance deteriorates rapidly as Γ or κ is increased above the trap frequency ν. Increasing the spontaneous emission rate has a particularly adverse effect, because the recoil momentum of emitted photons leads to further motional heating (see equation (22)). This is highly problematic, because the spontaneous emission rate in atomic two-level systems is fixed by nature and may be much larger than a typical vibrational frequency, on the order of tens or hundreds of megaherz.

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The deleterious effect of line broadening is worsened when the trapping potential minimum is not placed exactly on the electric field node of the cavity. Outside of the sideband-resolved regime, we found that the system is remarkably sensitive to small misalignments of the trapping potential: displacements of a few nanometres away from the cavity field node lead to an almost complete disappearance of the cooling effect. This can be understood as follows. Away from the node, cavity photons may be emitted and absorbed without affecting the vibrational state of the atom. Photon absorption in particular depletes the cavity field and reduces the effective temperature of the hot reservoir. Although these transitions, which occur at the so-called carrier frequency ε, are off-resonant in principle, they become important when levels are broadened.

Finally, to enforce the resonance condition $\omega =\varepsilon -\nu$, it is necessary to stabilise the length of the cavity to prevent frequency drift. This may be achieved, for example, by continuously driving the cavity with a laser field and using the Pound-Drever-Hall technique [39]. Importantly, this stabilisation can be performed using other polarisation modes or different cavity harmonics from those directly relevant for the refrigerator's operation. Of course, this use of an external laser field means that the machine is not truly autonomous. However, the laser does not supply any work used directly for cooling; rather, its role is to provide a stable frequency reference.

As in section 3.1, we consider a harmonically trapped atom or ion of mass M, possessing a pair of electronic states $| \downarrow \rangle$ and $| \uparrow \rangle$ separated by energy ε. In this section, we explicitly model the atomic motion in both the x and y directions, although it will shortly be shown that the y coordinate decouples from the dynamics for our chosen configuration. For simplicity of presentation, we make the inessential assumption of equal oscillation frequencies in both the x and y directions, given by $\nu /2\pi$.

The atom is placed inside a pair of optical cavities, b and c, with axes aligned in the x and y direction, respectively. These cavities have relevant modes at frequencies ${\omega }_{b}/2\pi$ and ${\omega }_{c}/2\pi$. The minimum of the trap potential is placed a distance d_b from a node of the electric field in cavity b, and a distance d_c from an anti-node of cavity c. The geometry of the problem is indicated in figure 5.

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The free Hamiltonian of the system is

$$\begin{eqnarray}&&{H}_{2}=\nu {a}_{x}^{\dagger }{a}_{x}+\nu {a}_{y}^{\dagger }{a}_{y}+{\omega }_{b}{b}^{\dagger }b+{\omega }_{c}{c}^{\dagger }c+\varepsilon {\sigma }^{+}{\sigma }^{-},\end{eqnarray} \tag{ 23 }$$

where ${a}_{x}^{\dagger }$ (${a}_{y}^{\dagger }$) creates motional excitations in the x direction (y direction), ${b}^{\dagger }$ (${c}^{\dagger }$) creates photons in the cavity parallel to the x axis (y axis), and ${\sigma }^{-}=| \downarrow \rangle \quad \langle \uparrow | ={({\sigma }^{+})}^{\dagger }$. The light–matter interaction Hamiltonian reads as

$$\begin{eqnarray}&&{V}_{2}={g}_{b}\mathrm{sin}[{\delta }_{b}+{\eta }_{b}({a}_{x}+{a}_{x}^{\dagger })](b+{b}^{\dagger })({\sigma }^{-}+{\sigma }^{+})+{g}_{c}\mathrm{cos}[{\delta }_{c}+{\eta }_{c}({a}_{y}+{a}_{y}^{\dagger })](c+{c}^{\dagger })({\sigma }^{-}+{\sigma }^{+}),\end{eqnarray} \tag{ 24 }$$

where g_j is the coupling constant, ${\eta }_{j}={\omega }_{j}/\sqrt{2{{Mc}}_{0}^{2}\nu }$ is the Lamb-Dicke parameter, and ${\delta }_{j}={d}_{j}{\omega }_{j}/{c}_{0}$ is the dimensionless misalignment for cavity $j=b,c$. As before, all other cavity and vibrational modes and electronic states are assumed be far off-resonant.

Assuming that ${\eta }_{j},{\delta }_{j}\ll 1$, we expand equation (24) to first order in small quantities and make the RWA, which yields

$$\begin{eqnarray}&&{V}_{2}\approx {\tilde{g}}_{b}{\eta }_{b}({a}_{x}+{a}_{x}^{\dagger })(b{\sigma }^{+}+{b}^{\dagger }{\sigma }^{-})+{\tilde{g}}_{c}(c{\sigma }^{+}+{c}^{\dagger }{\sigma }^{-})+{h}_{b}(b{\sigma }^{+}+{b}^{\dagger }{\sigma }^{-}),\end{eqnarray} \tag{ 25 }$$

where ${\tilde{g}}_{b/c}={g}_{b/c}\mathrm{cos}{\delta }_{b/c}$ and ${h}_{b}={g}_{b}\mathrm{sin}{\delta }_{b}$. We see that to lowest order, the excitation of phonons in the y direction is suppressed close to the anti-node of cavity c. The motion in the y direction is therefore neglected from here on. To simplify the notation, we also set ${a}_{x}=a$ and ${\eta }_{b}=\eta$.

We demand that the cavities be tuned to two-photon resonance with the red sideband, ${\omega }_{c}-{\omega }_{b}=\nu$, yet detuned from the carrier by an amount ${\rm{\Delta }}={\omega }_{c}-\varepsilon$, where $\varepsilon \gg | {\rm{\Delta }}| \gg {g}_{b/c}$. Direct excitation of the internal state of the atom and the associated spontaneous emission is thus strongly suppressed. However, due to the resonance condition ${\omega }_{c}={\omega }_{b}+\nu$, the cavities can coherently exchange photons, assisted by the creation or destruction of phonons. In the following subsection, we show that this process is described by an effective interaction of the form

$$\begin{eqnarray}&&{V}_{\mathrm{eff}}=k({{abc}}^{\dagger }+{a}^{\dagger }{b}^{\dagger }c),\end{eqnarray} \tag{ 26 }$$

which obviously corresponds to the general expression (4).

The refrigerator is powered by pumping cavity mode b with hot thermal light at temperature T_h, while cavity mode c couples to the radiation field at room temperature T_r. The virtual qubit states for this system are the pairs $\{| {n}_{b},{n}_{c}\rangle ,| {n}_{b}-1,{n}_{c}+1\rangle \}$. The virtual temperature is given by

$$\begin{eqnarray}&&{T}_{v}=\displaystyle \frac{\nu }{{\omega }_{c}/{T}_{r}-{\omega }_{b}/{T}_{h}}\approx \displaystyle \frac{\nu }{{\omega }_{c}}{T}_{r},\end{eqnarray} \tag{ 27 }$$

because ${T}_{h}\gg {T}_{r}$ and ${\omega }_{c}\gg \nu$, and we see again that very low virtual temperatures can be obtained.

The operation of the refrigerator can be understood by analogy with Raman laser sideband cooling (see figure 6). Addressing the red sideband with a two-photon transition avoids populating the fast-decaying excited electronic state. The line width of the transition is therefore determined by the cavity decay rates, which in principle may be made much smaller than the spontaneous emission rate. This makes achieving the sideband-resolved regime a feasible prospect in this system.

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Note that the cavity lengths must be actively stabilised in order to avoid frequency drift away from the resonance condition ${\omega }_{c}-{\omega }_{b}=\nu$, which can be performed non-invasively using a laser, as described in section 3.3.

We write the density operator of the full system including the electronic degrees of freedom as χ. This satisfies the master equation

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}\chi }{{\rm{d}}t}=-{\rm{i}}[{H}_{2}+{V}_{2},\chi ]+\displaystyle \sum _{j=a,b,c,\sigma }{{ \mathcal L }}_{j}\chi ,\end{eqnarray} \tag{ 28 }$$

where ${{ \mathcal L }}_{a}$ and ${{ \mathcal L }}_{\sigma }$ are respectively defined by equations (20) and (22). The coupling of cavity b to the external electromagnetic field is described by

$$\begin{eqnarray}&&{{ \mathcal L }}_{b}={\kappa }_{b}(2+{\bar{n}}_{b}){ \mathcal D }[b]+{\kappa }_{b}{\bar{n}}_{b}{ \mathcal D }[{b}^{\dagger }],\end{eqnarray} \tag{ 29 }$$

where ${\kappa }_{b}$ is the cavity line width and ${\bar{n}}_{b}={({{\rm{e}}}^{{\omega }_{b}/{T}_{h}}-1)}^{-1}$ is the equilibrium photon number at temperature T_h. Equation (29) represents thermal driving applied to only one side of cavity b, while the other side couples to the vacuum (which approximates the electric field at room temperature). Cavity c couples to the environment via the Liouvillian

$$\begin{eqnarray}&&{{ \mathcal L }}_{c}=2{\kappa }_{c}(1+{\bar{n}}_{c}){ \mathcal D }[c]+2{\kappa }_{c}{\bar{n}}_{c}{ \mathcal D }[{c}^{\dagger }],\end{eqnarray} \tag{ 30 }$$

where ${\kappa }_{c}$ is the corresponding line width, and ${\bar{n}}_{c}={({{\rm{e}}}^{{\omega }_{c}/{T}_{r}}-1)}^{-1}$, with ${\bar{n}}_{c}\approx 0$ for optical frequencies at room temperature.

We note that direct excitation of the electronic degrees of freedom is suppressed by the large detuning $| {\rm{\Delta }}| \gg {\tilde{g}}_{b/c},{h}_{b}$. Furthermore, Γ will typically be the largest dissipative frequency scale in the system, so that correlations between the electronic degrees of freedom and the rest of the system decay rapidly on the time scales relevant for the dynamics of the atomic motion. These assumptions enable us to simplify the model by adiabatically eliminating the excited electronic state within a Born-Markov approximation.

Using standard projection operator techniques [41–44], a master equation describing the reduced density matrix $\rho (t)={\mathrm{Tr}}_{\sigma }[\chi (t)]$ of the motional and cavity modes in the electronic ground state manifold is derived in appendix A. The result is

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}\rho }{{\rm{d}}t}=-{\rm{i}}[{H}_{{abc}}+\delta {H}_{{abc}}+{V}_{\mathrm{eff}},\rho ]+{{ \mathcal L }}_{\mathrm{se}}\rho +\displaystyle \sum _{j=a,b,c}{{ \mathcal L }}_{j}\rho ,\end{eqnarray} \tag{ 31 }$$

where

$$\begin{eqnarray}&&{H}_{{abc}}=\nu {a}^{\dagger }a+{\omega }_{b}{b}^{\dagger }b+{\omega }_{c}{c}^{\dagger }c.\end{eqnarray} \tag{ 32 }$$

The Hamiltonian $\delta {H}_{{abc}}$ is a small Lamb-shift contribution which renormalises the energy levels of H_abc. The interaction term is of the form

$$\begin{eqnarray}&&{V}_{\mathrm{eff}}=k({{abc}}^{\dagger }+{a}^{\dagger }{b}^{\dagger }c).\end{eqnarray} \tag{ 33 }$$

In the limit $| {\rm{\Delta }}| \gg {\rm{\Gamma }}$, the effective coupling constant is found to be $k={\tilde{g}}_{b}{\tilde{g}}_{c}\eta /{\rm{\Delta }}$. The generator ${{ \mathcal L }}_{\mathrm{se}}$ describes additional dissipative processes due to spontaneous emission from the excited state, which are suppressed by a factor of order ${\rm{\Gamma }}/{\rm{\Delta }}$ relative to the coherent coupling k. Full expressions for all parameters entering equation (31) can be found in the appendix.

We now summarise the approximations underlying equation (31). The assumption of negligible population of the excited electronic state is valid so long as the detuning is sufficiently large, i.e.

$$\begin{eqnarray}&&{\tilde{g}}_{b}\eta ,{h}_{b},{\tilde{g}}_{c}\ll | {\rm{\Delta }}| .\end{eqnarray} \tag{ 34 }$$

We also neglected the motional recoil due to spontaneous emission. This is justified when the spontaneous emission is isotropic and the system is deep in the Lamb-Dicke regime, so that

$$\begin{eqnarray}&&{\eta }^{2}{\rm{\Gamma }}\lesssim {\tilde{g}}_{b}\eta ,{h}_{b},{\tilde{g}}_{c}.\end{eqnarray} \tag{ 35 }$$

The Born-Markov assumption requires the memory time of the electronic degrees of freedom to be much shorter than the characteristic time scales of the effective evolution, which implies

$$\begin{eqnarray}&&k,{\kappa }_{b/c},\delta E\ll {\rm{\Gamma }},\end{eqnarray} \tag{ 36 }$$

where $\delta E$ represents any energy shift appearing in $\delta {H}_{{abc}}$. Finally, to put the master equation (31) into Lindblad form, we must perform a rotating wave approximation, valid when

$$\begin{eqnarray}&&k,{\kappa }_{b/c},\delta E\ll \nu ,\end{eqnarray} \tag{ 37 }$$

which corresponds to the definition of the sideband-resolved regime for this system.

In this subsection, we characterise the performance of the refrigerator in terms of the steady-state phonon occupation ${n}_{a}(\infty )$, focusing specifically on cooling using sunlight as an energy source. We take representative parameters pertaining to ¹⁷¹Yb⁺(listed in table 1). This species is a good choice due to its high mass and correspondingly small photon recoil, in addition to the existence of a closed dipole-allowed cooling transition. However, one could equally well consider other species of ion or neutral atom.

Table 1.  Parameters used in numerical calculations.

Parameter	Symbol	Value
Hot temperature	Th	5800 K
Room temperature	Tr	300 K
Trap frequency	$\nu /2\pi$	5 MHz
Lamb-Dicke parameter	η	0.041
Carrier frequency	$\varepsilon /2\pi$	810 THz
Spontaneous emission rate	${\rm{\Gamma }}/2\pi$	20 MHz
Trap heating rate	λ	10 quanta/s
Cavity misalignment	${d}_{b}={d}_{c}$	10 nm

From here on, we set ${g}_{b/c}=g$ and ${\kappa }_{b/c}=\kappa$ for simplicity. We compute ${n}_{a}(\infty )=\mathrm{Tr}[{a}^{\dagger }a{\rho }_{\infty }]$ by solving ${\rm{d}}{\rho }_{\infty }/{\rm{d}}t={ \mathcal L }{\rho }_{\infty }=0$, with ${ \mathcal L }$ defined by the RHS of equation (31). We use a truncated Hilbert space with 71 phonon states and four states per cavity mode. Such a small Hilbert space dimension for the cavity modes is justified because the mean number of cavity photons in the steady state is ${n}_{b}(\infty )\approx {10}^{-3}$ and ${n}_{c}(\infty )\lt {10}^{-4}$ in cavities b and c, respectively, for all parameters considered. We have checked that decreasing the Hilbert space dimension leads to negligible changes in the results.

Our predictions for ${n}_{a}(\infty )$ are shown in figure 7. We observe that sunlight at ${T}_{h}=5800\quad {\rm{K}}$ is sufficient to drive the phonon almost to its ground state, so long as the effective coupling constant k is sufficiently large. In figure 7(a) we show that, in the regime of effective cooling, the steady-state phonon occupation is reduced by increasing κ for fixed g. Nevertheless, κ must remain smaller than ν for the system to remain in the sideband-resolved regime (equation (37)), which represents a key factor limiting the achievable steady-state phonon occupation. In figure 7(b) we demonstrate that increasing Δ for fixed k can improve performance by suppressing incoherent effects associated with spontaneous emission.

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In the limit of $| {\rm{\Delta }}| \gg {\rm{\Gamma }}$ and $k\ll \kappa$, we can give a rough analytical estimate of the relaxation time. In this regime, we derive an effective evolution equation for the motional degrees of freedom by tracing over the cavity modes, as shown in appendix B. This approximate equation of motion can be solved to give the phonon population as a function of time,

$$\begin{eqnarray}&&{n}_{a}(t)={n}_{\infty }+{{\rm{e}}}^{-\gamma t}({n}_{0}-{n}_{\infty }),\end{eqnarray} \tag{ 38 }$$

where ${n}_{\infty }=\lambda /\gamma$ is the steady-state phonon number, ${n}_{0}={n}_{a}(0)$ is the initial population, and the relaxation rate is $\gamma ={k}^{2}{\bar{n}}_{b}/\kappa$.

In this subsection, we generalise to the scenario where multiple atoms are trapped inside the cavities. In particular, we focus on ion-trap systems, where the Coulomb interaction couples the motion of the different ions. The normal vibrational modes of the system are thus small collective oscillations about the mechanical equilibrium. We now show that if N ions of the same species are placed inside the crossed-cavity refrigerator, an N-fold enhancement of the coupling between photons and phonons can be obtained.

The free Hamiltonian of the system is

$$\begin{eqnarray}&&{H}_{2}=\nu {a}^{\dagger }a+{\omega }_{b}{b}^{\dagger }b+{\omega }_{c}{c}^{\dagger }c+\displaystyle \sum _{j=1}^{N}{\varepsilon }_{j}{\sigma }_{j}^{+}{\sigma }_{j}^{-}.\end{eqnarray} \tag{ 39 }$$

Here, ${a}^{\dagger }$ creates a phonon of a normal mode with frequency $\nu /2\pi$, ${\sigma }_{j}^{-}$ is the atomic lowering operator for atom j and we have allowed for variations of the electronic transition frequencies ${\varepsilon }_{j}$, due to inhomogeneous magnetic fields, for example. All other modes and electronic states are assumed to be off-resonant. The interaction Hamiltonian in the LDA and RWA reads as

$$\begin{eqnarray}&&{V}_{2}=\displaystyle \sum _{j=1}^{N}\{{\tilde{g}}_{b}({{\bf{r}}}_{j}){\eta }_{j}(a+{a}^{\dagger })(b{\sigma }_{j}^{+}+{b}^{\dagger }{\sigma }_{j}^{-})+{\tilde{g}}_{c}({{\bf{r}}}_{j})(c{\sigma }_{j}^{+}+{c}^{\dagger }{\sigma }_{j}^{-})+{h}_{b}({{\bf{r}}}_{j})(b{\sigma }_{j}^{+}+{b}^{\dagger }{\sigma }_{j}^{-})\},\end{eqnarray} \tag{ 40 }$$

where ${\tilde{g}}_{b/c}({{\bf{r}}}_{j})={g}_{b/c}({{\bf{r}}}_{j})\mathrm{cos}{\delta }_{b/c}({{\bf{r}}}_{j})$, ${h}_{b}({{\bf{r}}}_{j})={g}_{b}({{\bf{r}}}_{j})\mathrm{sin}{\delta }_{b}({{\bf{r}}}_{j})$, with ${g}_{b/c}({{\bf{r}}}_{j})$ the cavity coupling constants for the ion with equilibrium position ${{\bf{r}}}_{j}$, while ${\eta }_{j}$ are the Lamb-Dicke parameters and ${\delta }_{b/c}({{\bf{r}}}_{j})={d}_{b/c}({{\bf{r}}}_{j}){\omega }_{b/c}/{c}_{0}$ are the dimensionless misalignments, where ${d}_{b}({{\bf{r}}}_{j})$ (${d}_{c}({{\bf{r}}}_{j})$) is the distance in the x direction (y direction) between ${{\bf{r}}}_{j}$ and the field node of cavity b (anti-node of cavity c).

We assume again that the cavities are detuned from the electronic transition frequencies, ${\omega }_{c}={\varepsilon }_{j}+{{\rm{\Delta }}}_{j}={\omega }_{b}+\nu$, with ${\varepsilon }_{j}\gg | {\rm{\Delta }}{| }_{j}\gg {g}_{b/c}({{\bf{r}}}_{j})$. After adiabatically eliminating the electronic excited states according to the procedure in appendix A, we find an effective interaction of the form

$$\begin{eqnarray}&&{V}_{\mathrm{eff}}={k}_{\mathrm{col}}({{abc}}^{\dagger }+{a}^{\dagger }{b}^{\dagger }c).\end{eqnarray} \tag{ 41 }$$

In the limit $| {{\rm{\Delta }}}_{j}| \gg {\rm{\Gamma }}$, the collective coupling constant is found to be

$$\begin{eqnarray}&&{k}_{\mathrm{col}}=\displaystyle \sum _{j=1}^{N}\displaystyle \frac{{\tilde{g}}_{b}({{\bf{r}}}_{j}){\tilde{g}}_{c}({{\bf{r}}}_{j}){\eta }_{j}}{{{\rm{\Delta }}}_{j}}.\end{eqnarray} \tag{ 42 }$$

The effective collective coupling can be either enhanced or suppressed compared with the single-particle case, depending on the symmetry of the normal mode in question. For example, let us take N = 2 and assume that ${\tilde{g}}_{b/c}({{\bf{r}}}_{1})={\tilde{g}}_{b/c}({{\bf{r}}}_{2})={\tilde{g}}_{b/c}$ and ${{\rm{\Delta }}}_{1}={{\rm{\Delta }}}_{2}={\rm{\Delta }}$. For the stretch mode, with ${\eta }_{1}=-{\eta }_{2}$, we find that ${k}_{\mathrm{col}}=0$. On the other hand, for the centre-of-mass mode, with ${\eta }_{1}={\eta }_{2}=\eta$, we find a two-fold enhancement of the coupling, i.e. ${k}_{\mathrm{col}}=2k$, where $k={\tilde{g}}_{b}{\tilde{g}}_{c}\eta /{\rm{\Delta }}$ is the single-ion effective coupling. In general, the centre-of-mass oscillations experience enhanced collective coupling, as for this vibrational mode ${\eta }_{j}$ has the same sign for all j.

Note that the rate of heating may also increase with the number of ions. For example, the heating rate for the centre-of-mass mode increases proportionally to N, assuming spatially correlated trap potential fluctuations [45]. However, according to the estimate given by equation (38), the steady-state phonon occupation and relaxation time scale as ${k}_{\mathrm{col}}^{-2}$. We therefore expect an N-fold enhancement of the refrigerator's performance when cooling the centre-of-mass mode of N ions.