Single-photon pump by Cooper-pair splitting

TL;DR

This paper proposes a novel scheme using a Cooper-pair splitter with two quantum dots coupled to local resonators, demonstrating efficient cooling and heat transfer control, with potential applications in quantum heat management.

Contribution

It introduces a new method for cooling and heat transfer control in quantum resonators using Cooper-pair splitting in a hybrid quantum dot setup.

Findings

Cross-Andreev reflection cools both resonators to ground state

Nonlocal heat transfer occurs under resonance conditions

The scheme functions as a heat-pump device for quantum resonators

Abstract

Hybrid quantum dot-oscillator systems have become attractive platforms to inspect quantum coherence effects at the nanoscale. Here, we investigate a Cooper-pair splitter setup consisting of two quantum dots, each linearly coupled to a local resonator. The latter can be realized either by a microwave cavity or a nanomechanical resonator. Focusing on the subgap regime, we demonstrate that cross-Andreev reflection, through which Cooper pairs are split into both dots, can efficiently cool down simultaneously both resonators into their ground state. Moreover, we show that a nonlocal heat transfer between the two resonators is activated when opportune resonance conditions are matched. The proposed scheme can act as a heat-pump device with potential applications in heat control and cooling of mesoscopic quantum resonators.

Equations

$$\begin{array}{rl}H=& \sum _{\alpha \sigma }ϵN_{\alpha \sigma }-\frac{{\Gamma }_S}{2}(d_{R\uparrow }^{†}{d}_{L\downarrow }^{†}-d_{R\downarrow }^{†}{d}_{L\uparrow }^{†}+H.c.)\\ +& \sum _{\alpha }{\omega }_{\alpha }{b}_{\alpha }^{†}{b}_{\alpha }+\sum _{\alpha ,\sigma }{\lambda }_{\alpha }(b_{\alpha }+b_{\alpha }^{†})N_{\alpha \sigma },\end{array}$$

$$w_{\mathrm{el},j\leftarrow i}^{\alpha ,s}$$

$$w_{\mathrm{ph},j\leftarrow i}^{\alpha ,s}$$

$${\dot{P}}_i=\sum _j{w}_{i\leftarrow j}{P}_j-\sum _j{w}_{j\leftarrow i}{P}_i,$$

$$\bar{H}=\sum _{\alpha \sigma }{\overline{ϵ}}_{\alpha }{N}_{\alpha \sigma }-\frac{{\Gamma }_S}{\sqrt{2}}(|S⟩⟨0|X+|0⟩⟨S|X^{†})+\sum _{\alpha }{\omega }_{\alpha }{b}_{\alpha }^{†}{b}_{\alpha },$$

$$\bar{\delta }\approx |p{\omega }_L\pm q{\omega }_R|,$$

$$H_{\mathrm{loc}}=\sum _{\alpha }\frac{1}{2}{\lambda }_{\alpha }\sin \bar{\theta }(b_{\alpha }{\tau }_{+}+b_{\alpha }^{†}{\tau }_{-}),$$

$${\lambda }_{\mathrm{NL}}=\frac{{\Gamma }_S{\lambda }_L{\lambda }_R}{\sqrt{2}{\omega }_L{\omega }_R}\cos \bar{\theta }.$$

$${\dot{E}}_{\alpha }^{\mathrm{ph}}$$

$${\eta }_{\mathrm{NL}}=\frac{2e|{\dot{E}}_L-{\dot{E}}_R|}{|I_S|({\omega }_L-{\omega }_R)}.$$

$$H_{\mathrm{int}}=-\frac{{\Gamma }_S}{\sqrt{2}}\left[i{\sigma }_y\Pi +{\sigma }_x\left(1+\frac{{\Pi }^2}{2}\right)\right]+\mathcal{O}({\Pi }^3),$$

$$|+⟩$$

$$|-⟩$$

$$\begin{array}{rl}\bar{H}& =\sum _{\alpha \sigma }{\overline{ϵ}}_{\alpha }{N}_{\alpha \sigma }+\frac{\bar{\delta }}{2}{\tau }_z+\sum _{\alpha }{\omega }_{\alpha }{b}_{\alpha }^{†}{b}_{\alpha }\\ & -\frac{{\Gamma }_S}{2\sqrt{2}}\left[2i{\tau }_y\Pi +(\sin \bar{\theta }{\tau }_z+\cos \bar{\theta }{\tau }_x){\Pi }^2\right]+\mathcal{O}({\Pi }^3).\end{array}$$

$$\begin{array}{rl}{H}_{\text{int}}(t)& =-\sum _{\alpha }\frac{{\lambda }_{\alpha }{\Gamma }_S}{{\omega }_{\alpha }\sqrt{2}}\left(e^{i{\omega }_{\alpha }t}{b}_{\alpha }^{†}-e^{-i{\omega }_{\alpha }t}{b}_{\alpha }\right)\left(e^{i\bar{\delta }t}{\tau }_{+}-e^{-i\bar{\delta }t}{\tau }_{-}\right)\\ & -\frac{{\Gamma }_S{\lambda }_L{\lambda }_R}{\sqrt{2}{\omega }_L{\omega }_R}\left[e^{i\Omega t}{b}_L^{†}{b}_R^{†}+e^{-i\Omega t}{b}_L{b}_R-e^{i(\Delta \omega )t}{b}_L^{†}{b}_R-e^{-i(\Delta \omega )t}{b}_L{b}_R^{†}\right]\left[\sin (\bar{\theta }){\tau }_z+\cos (\bar{\theta })(e^{i\bar{\delta }t}{\tau }_{+}+e^{-i\bar{\delta }t}{\tau }_{-})\right]\\ & -\sum _{\alpha }\frac{{\Gamma }_S{\lambda }_{\alpha }^2}{2\sqrt{2}{\omega }_{\alpha }^2}\left[e^{2i{\omega }_{\alpha }t}(b_{\alpha }^{†}{)}^2+e^{-2i{\omega }_{\alpha }t}{b}_{\alpha }^2-2b_{\alpha }^{†}{b}_{\alpha }-1\right]\left[\sin (\bar{\theta }){\tau }_z+\cos (\bar{\theta })(e^{i\bar{\delta }t}{\tau }_{+}+e^{-i\bar{\delta }t}{\tau }_{-})\right]+\mathcal{O}({\lambda }_{\alpha }^3/{\omega }_{\alpha }^3).\end{array}$$

$$H_{\mathrm{RWA}}^{\bar{\delta }=\omega }=\sum _{\alpha }\frac{1}{2}{\lambda }_{\alpha }\sin (\bar{\theta })(b_{\alpha }{\tau }_{+}+b_{\alpha }^{†}{\tau }_{-}).$$

$$H_{\mathrm{RWA}}^{\bar{\delta }=\Delta \omega }=\sum _{\alpha }\frac{{\Gamma }_S{\lambda }_{\alpha }^2}{2\sqrt{2}{\omega }_{\alpha }^2}(2n_{\alpha }+1)\sin \bar{\theta }{\tau }_z+{\lambda }_{\mathrm{NL}}(b_L^{†}{b}_R{\tau }_{-}+H.c.),$$

$$H_{\mathrm{RWA}}^{\bar{\delta }=\Omega }=\sum _{\alpha }\frac{{\Gamma }_S{\lambda }_{\alpha }^2}{2\sqrt{2}{\omega }_{\alpha }^2}(2n_{\alpha }+1)\sin \bar{\theta }{\tau }_z+{\lambda }_{\mathrm{NL}}(b_L^{†}{b}_R^{†}{\tau }_{-}+H.c.).$$

Full text

I Introduction

Nonlocality Buhrman et al. (2010); Brunner et al. (2014) and quantum correlations Eckert et al. (2002) are at the heart of many quantum technologies. Monroe (2002); Ladd et al. (2010); Degen et al. (2017) In hybrid quantum-dot devices, Cooper pairs are a source of correlated electrons and their nonlocal splitting has experimentally Hofstetter et al. (2009); Herrmann et al. (2010); Hofstetter et al. (2011); Das et al. (2012); Schindele et al. (2012, 2014); Fülöp et al. (2014, 2015); Tan et al. (2015); Borzenets et al. (2016); Baba et al. (2018) and theoretically Recher et al. (2001); Chevallier et al. (2011); Rech et al. (2012); Scherübl et al. (2014); Trocha and Weymann (2015); Nigg et al. (2015); Schroer and Recher (2015); Probst et al. (2016); Domínguez and Yeyati (2016); Amitai et al. (2016); Hussein et al. (2016); Wrześniewski et al. (2017); Hussein et al. (2017); Wrześniewski and Weymann (2017); Walldorf et al. (2018); Bocian et al. (2018) drawn much attention over the last few years. In particular, the nonlocal breaking of the particle-hole symmetry in such Cooper-pair splitters (CPSs) gives rise to peculiar thermoelectric effects. Cao et al. (2015); Sánchez et al. (2018); Hussein et al. (2019); Kirsanov et al. (2019) On the other hand, mesoscopic cavity quantum electrodynamics (cQED) devices Childress et al. (2004); Cottet et al. (2017) are excellent tools for correlating few-level systems over a distance. Delbecq et al. (2013); Lambert et al. (2013); Deng et al. (2015, 2016); Ramírez-Muñoz et al. (2018); Trif and Simon (2019) Such cQED devices have applications in the readout of charge, Frey et al. (2012); Cottet et al. (2012); Viennot et al. (2014); Bruhat et al. (2016); Viennot et al. (2016); Stockklauser et al. (2017); Burkard et al. (2019) spin, Petersson et al. (2012); Viennot et al. (2015); Benito et al. (2017); Zhu et al. (2018); Mi et al. (2018a) and valley-orbit states, Kohler (2018); Mi et al. (2018b) as well as few-phonon manipulation when mechanical modes can be cooled close to the ground state. O’Connell et al. (2010); Piovano et al. (2011); Teufel et al. (2011); Stadler et al. (2016, 2017); Clark et al. (2017) A mechanism which induces nonlocal photon or phonon correlations through Cooper pair transport, implemented in a hybrid setup, bridges the gap between the study of heat flows in quantum-dot-based Cao et al. (2015); Erdman et al. (2017, 2018); Sánchez et al. (2018); Hussein et al. (2019) and circuit-QED devices. Hofer et al. (2016); Ronzani et al. (2018); Dambach et al. (2019)

In this work, we consider a CPS in a double-quantum-dot setup with each dot linearly coupled to a local resonator, constituted by either a microwave cavity Viennot et al. (2015); Stockklauser et al. (2015); Bruhat et al. (2016); Mi et al. (2017); Stockklauser et al. (2017); Li et al. (2018); Cubaynes et al. (2019) or a mechanical oscillator, Benyamini et al. (2014); Okazaki et al. (2016); Deng et al. (2016); Wen et al. (2019); Urgell et al. (2019) see Fig. 1(a). We demonstrate that this system is a platform to obtain full control on the heat and photon exchange of two originally uncoupled cavities. This induced coupling arises from the proximity between the dots and the superconducting lead, and has a purely nonlocal origin due to cross-Andreev reflection. Subsequent, we discuss the underlying physical mechanism following the lines of Ref. Rastelli and Governale, 2019, where a single-quantum-dot system in the single-atom lasing regime has been investigated.

For large intradot Coulomb interactions, $U$, and superconducting gap, $|\Delta|\to\infty$, the proximity of the superconductor causes a nonlocal splitting (and recombination) of Cooper pairs into both dots with the pairing amplitude $\Gamma_{S}>0$. The corresponding Andreev bound states $|\pm\rangle$ are a coherent superposition of the dots’ singlet, $|S\rangle$, and empty state, —$0\rangle$. The dots are further tunnel-coupled to normal contacts, which are largely negative-voltage-biased with respect to the chemical potential $\mu_{S}=0$ of the superconductor. In this configuration, due to single-electron tunneling, the singlet state decays at rate $\Gamma$ into a singly-occupied state, $|\alpha\sigma\rangle$ ($\alpha=L,R$ and $\sigma=\uparrow,\downarrow$) and further into the empty state, see Fig. 1(b). For large dot onsite energies $\epsilon\gtrsim\Gamma_{S}$, the charge hybridization is weak ($|+\rangle\approx|S\rangle$, $|-\rangle\approx|0\rangle$), and the transitions $|+\rangle\rightarrow|\alpha\sigma\rangle$ and $|\alpha\sigma\rangle\rightarrow|-\rangle$ are faster than the opposite processes, Rastelli and Governale (2019) see Fig. 1(c). This asymmetry in the relaxation ultimately explains how to pump or absorb energy within a single mode, and how to transfer photons between the cavities. In the latter case, when the energy splitting $\delta$ between the Andreev bound states is close to the difference of the cavity frequencies, the relevant level structure of the uncoupled system is summarized in Fig. 1(d). We show below that the effective interaction couples the states $|+,n_{L}\!-\!1,n_{R}\!+\!1\rangle$ and $|-,n_{L},n_{R}\rangle$, where $n_{\alpha}$ indicates the Fock number in the resonator $\alpha$. An electron tunneling event favours transitions $|+\rangle\rightarrow|\alpha\sigma\rangle\rightarrow|-\rangle$ conserving the photon number. When the system reaches the state $|-\rangle\approx|0\rangle$, this coherent cycle restarts. When the system is in $|+\rangle$, it can again decay. During each cycle, a boson is effectively transferred from the left to the right cavity. Since the two cavities are not isolated, but naturally coupled to external baths, a steady heat flow is eventually established between the cavities.

The effect discussed above refers to a single operation point of the system. More generally, using a master equation approach, we show that the interaction between the CPS and the two resonators opens a rich set of inelastic resonant channels for the electron current through the dots, involving either absorption/emission of photons from a local cavity or nonlocal transfer processes. By tuning $\epsilon$ to match these resonances, the CPS acts as a switch allowing the manipulation of heat between the resonators. Each resonant process can be captured with good approximation by an effective Hamiltonian which is valid close to the resonance and generalizes the mechanism described above.

This work is structured as follows. After introducing our model and the employed master equation in Sec. II, we provide therein an effective Hamiltonian describing local and nonlocal transport processes. In Section III, we discuss the possibility of simultaneous cooling (and heating) of the resonators. Section IV is dedicated to the nonlocal photon transfer between them, and in Sec. V we analyze the efficiency of this transfer. Finally, we draw our conclusions in Sec. VI.

II Cooper-pair splitter coupled to resonators

We consider the effective model for two single-level quantum dots proximized by a $s$-wave superconductor, and each linearly coupled to a local harmonic oscillator. For large intradot Coulomb interaction, $U\gg|\epsilon|$, the subgap physics of the system is described by the effective Hamiltonian Hussein et al. (2016); Rozhkov and Arovas (2000); Meng et al. (2009); Eldridge et al. (2010); Braggio et al. (2011); Droste et al. (2012); Sothmann et al. (2014); Weiss and König (2017); Walldorf et al. (2018)

[TABLE]

where $\hbar=1$. Here, $d_{\alpha\sigma}$ is the fermionic annihilation operator for a spin-$\sigma$ electron in dot $\alpha$, with the corresponding number operator $N_{\alpha\sigma}$ and onsite energy $\epsilon$. The interaction of the dot with the $\alpha$-oscillator of frequency $\omega_{\alpha}$ and corresponding bosonic field $b_{\alpha}$ is realized through the charge term, with coupling constant $\lambda_{\alpha}$. The relevant subspace of the electronic subsystem is spanned by six states: The empty state $|0\rangle$, the four singly-occupied states $|\alpha\sigma\rangle=d^{\dagger}_{\alpha\sigma}|0\rangle$ and the singlet state $|S\rangle=\frac{1}{\sqrt{2}}(d^{\dagger}_{R{\uparrow}}d^{\dagger}_{L{\downarrow}}-d^{\dagger}_{R{\downarrow}}d^{\dagger}_{L{\uparrow}})|0\rangle$. Triplet states and doubly-occupied states are inaccessible due to large negative voltages, see Fig. 1(a), and large intradot Coulomb repulsion. Finally, in the subgap regime, the superconductor can only pump Cooper pairs, which are in the singlet state. The states $|0\rangle$ and $|S\rangle$ are hybridized due to the $\Gamma_{S}$-term, yielding the Andreev states $|+\rangle=\cos(\theta/2)|0\rangle+\sin(\theta/2)|S\rangle$ and $|-\rangle=-\sin(\theta/2)|0\rangle+\cos(\theta/2)|S\rangle$, with the mixing angle $\theta=\arctan[\Gamma_{S}/(\sqrt{2}\epsilon)]$. We denote their energy splitting by $\delta=\sqrt{\smash[b]{4\epsilon^{2}+2\Gamma_{S}^{2}}}$.

Electron tunneling into the normal leads and dissipation for the resonators can be treated in the sequential-tunneling regime to lowest order in perturbation theory, assuming small dot-lead tunneling rates, $\Gamma\ll\Gamma_{S},k_{B}T$, and large quality factors $Q_{\alpha}=\omega_{\alpha}/\kappa_{\alpha}$ for the resonators, i.e. $\kappa_{\alpha}\ll\omega_{\alpha},k_{B}T$. Here, $\kappa_{\alpha}$ is the decay rate for the $\alpha$-resonator and $T$ is the temperature of the fermionic and bosonic reservoirs. The fermionic and bosonic transition rates between two eigenstates $|i\rangle$ and $|j\rangle$ of Hamiltonian (1) are given by Fermi’s golden rule, Benenti et al. (2017)

[TABLE]

with $f^{(s)}_{\alpha}(x)=\{\exp[s(x-\mu_{\alpha})/k_{B}T]+1\}^{-1}$ the generalized Fermi function ($s=\pm$) at chemical potential $\mu_{\alpha}$, and $n_{B}(x)=[\exp(x/k_{B}T)-1]^{-1}$ the Bose function. $E_{ji}\equiv E_{j}-E_{i}$ denotes the energy difference between two eigenstates. We use the notation $d^{(-)}_{\alpha\sigma}$ ($d^{(+)}_{\alpha\sigma}$) for fermionic annihilation (creation) operators, and correspondingly $b^{(\pm)}_{\alpha}$ for the bosonic ones. The populations $P_{i}$ of the system eigenstates obey a Pauli-type master equation of the form Sauret et al. (2004); Governale et al. (2008); Hussein et al. (2016)

[TABLE]

which admits a stationary solution given by $P^{\mathrm{st}}_{i}$. The total rates entering Eq. (4) are given by $w_{j\leftarrow i}=\sum_{\alpha,s}(w^{\alpha,s}_{\mathrm{el},j\leftarrow i}+w^{\alpha,s}_{\mathrm{ph},j\leftarrow i}).$ As mentioned before, we assume the chemical potentials of the normal leads $\mu_{\alpha}=-eV$ to be largely negative-biased, i.e., $U,|\Delta|\gg eV\gg k_{B}T,\epsilon,\Gamma_{S}$, with $V>0$ and $e>0$ denoting the applied voltage and the electron charge, respectively. In this regime, the electrons flow unidirectionally from the superconductor via the quantum dots into the leads; the temperature of the normal leads becomes irrelevant, and the rates $w_{\mathrm{el},j\leftarrow i}^{\alpha,+}$ vanish. Under these assumptions, the stationary electron current through lead $\alpha$ is simply given by $I_{\alpha}=e\Gamma\sum_{\sigma}\langle N_{\alpha\sigma}\rangle$. For a symmetric configuration, as assumed here, both stationary currents coincide, $I_{L}=I_{R}$. To evaluate the stationary current and the other relevant quantities, we diagonalize numerically Hamiltonian (1), and build the transition-rate matrices appearing in Eq. (4). The stationary populations, $P_{i}^{\mathrm{st}}$, are then found by solving the system of Eqs. (4) for $\dot{P}_{i}=0$.

In order to explain our numerical results, we perform the Lang-Firsov polaron transformation to Hamiltonian (1). Lang and Firsov (1962); Brandes and Kramer (1999); Brandes (2005) For an operator $O$, we define the unitary transformation $\bar{O}=e^{\xi}Oe^{-\xi}$, with $\xi=\sum_{\alpha\sigma}\Pi_{\alpha}N_{\alpha\sigma}$ and $\Pi_{\alpha}=(\lambda_{\alpha}/\omega_{\alpha})(b^{\dagger}_{\alpha}-b_{\alpha})$. The polaron-transformed Hamiltonian reads then

[TABLE]

with $\bar{\epsilon}_{\alpha}=\epsilon-\lambda_{\alpha}^{2}/\omega_{\alpha}$ and $X=\exp(\sum_{\alpha}\Pi_{\alpha})$.111Notice that the polaron transformation also gives rise to a modification of the tunneling Hamiltonian, which describes inelastic dot-lead tunneling events scaling with $\Gamma(\lambda_{\alpha}/\omega_{\alpha})^{2}$. However, in our operative regime, such processes can be neglected since $\Gamma\ll\Gamma_{S}$ and $\lambda_{\alpha}\ll\omega_{\alpha}$ ensure $\Gamma(\lambda_{\alpha}/\omega_{\alpha})^{2}\ll\Gamma\ll\Gamma_{S}$. Equation (5) contains a transverse charge-resonator interaction term to all orders in the couplings $\lambda_{\alpha}$. Intriguingly, this coupling has a purely nonlocal origin stemming from the cross-Andreev reflection. By expanding $X$ in powers of $\Pi\equiv\sum_{\alpha}\Pi_{\alpha}$ assuming small couplings $\lambda_{\alpha}\ll\omega_{\alpha}$, and moving to the interaction picture with respect to the noninteracting Hamiltonian, we can identify a family of resonant conditions given by

[TABLE]

with $p,q$ nonnegative integers, as discussed in Appendix A. Here, $\bar{\delta}=\sqrt{\smash[b]{4\bar{\epsilon}^{2}+2\Gamma_{S}^{2}}}$ is the renormalized energy splitting of the Andreev states due to the polaron shift, with $\bar{\epsilon}=\epsilon-\sum_{\alpha}\frac{\lambda_{\alpha}^{2}}{2\omega_{\alpha}}$. The renormalized mixing angle reads $\bar{\theta}=\arctan[\Gamma_{S}/(\sqrt{2}\bar{\epsilon})]$. Around the conditions stated in Eq. (6), a rotating-wave approximation yields an effective interaction of order $p+q$ in the couplings $\lambda_{\alpha}$. Hereafter, we discuss in detail the resonances at $\bar{\delta}=\omega_{L}=\omega_{R}$ and $\bar{\delta}=\omega_{L}-\omega_{R}$ corresponding to one- and two-photon processes, respectively. They can be fully addressed by expanding $X$ up to second order in $\lambda_{\alpha}/\omega_{\alpha}$ and subsequently performing a rotating-wave approximation, see Appendix A.

III Simultaneous cooling and heating

For $\bar{\delta}=\omega_{L}=\omega_{R}$, one can achieve simultaneous cooling as well as heating of both resonators, which is already described by the first order terms in $\lambda_{\alpha}$ of Eq. (5). Here, we consider two identical resonators and tune the dot levels $\epsilon$ around the resonance condition $\bar{\delta}\approx\omega_{\alpha}$, i.e., $\bar{\epsilon}=\pm\sqrt{\smash[b]{\omega_{\alpha}^{2}-2\Gamma_{S}^{2}}}/2$. The effective first-order interaction Hamiltonian reads after a rotating-wave approximation

[TABLE]

as we show in Appendix A. The operators $\tau_{+}=|+\rangle\langle-|$ and $\tau_{-}=|-\rangle\langle+|$ describe the hopping between the two-level system formed by the states $|+\rangle$ and $|-\rangle$, coupled to the modes through a transverse Jaynes-Cummings-like interaction. The effective coupling is proportional to $\sin\bar{\theta}=\sqrt{2}\Gamma_{S}/\bar{\delta}$, and, thus, a direct consequence of the nonlocal Andreev reflection. The effective interaction in Eq. (7) coherently mixes the three states $|+,n_{L},n_{R}\rangle,|-,n_{L}+1,n_{R}\rangle$, and $|-,n_{L},n_{R}+1\rangle$ which are degenerate for $H_{\mathrm{loc}}=0$. When $|\epsilon|\gtrsim\Gamma_{S}$, the hybridization between the charge states is weak. The sign of $\epsilon$ changes the bare dots’ level structure: For $\epsilon<0$, $|+\rangle\approx|0\rangle$ and $|-\rangle\approx|S\rangle$, whereas for $\epsilon>0$, $|+\rangle\approx|S\rangle$ and $|-\rangle\approx|0\rangle$. In the latter case, the chain of transitions $|+\rangle\rightarrow|\alpha\sigma\rangle\rightarrow|-\rangle$ is faster than the opposite process, see Fig. 1(c). For $\epsilon<0$, energy is pumped into the modes. Conversely, for $\epsilon>0$, we can achieve simultaneous cooling of the resonators. In Fig. 2, we show the stationary electron current $I_{\alpha}$ [calculated using the full Hamiltonian (1)], together with the average photon number $\bar{n}_{\alpha}=\langle b^{\dagger}_{\alpha}b_{\alpha}\rangle$ of the corresponding resonator, as a function of $\epsilon$. The broad central resonance of width $\Gamma_{S}$ corresponds to the elastic current contribution mediated by the cross-Andreev reflection. The additional inelastic peak at negative $\epsilon$ is related to the emission of photons in both resonators at $\bar{\delta}\approx\omega_{\alpha}$. At finite temperature, a second sideband peak emerges at positive $\epsilon$, where the resonators are simultaneously cooled down. The cavities are efficiently cooled into their ground state for a wide range of values of $\Gamma_{S}$, as can be appreciated in the inset of Fig. 2(b). The optimal cooling region is due to the interplay between the effective interaction with the resonator—which vanishes for small $\Gamma_{S}$—and the hybridization of the empty and singlet state, which increases as $\epsilon$ approaches the Fermi level of the superconductor and reduces the asymmetry of the transitions $|\pm\rangle\leftrightarrow|\alpha\sigma\rangle$.

IV Nonlocal photon transfer

By keeping terms up to second order in $\lambda_{\alpha}$ in Eq. (5), we can describe the resonances around $\bar{\delta}=\omega_{L}-\omega_{R}$ and $\bar{\delta}=\omega_{L}+\omega_{R}$. Assuming without loss of generality $\omega_{L}>\omega_{R}$, a rotating-wave approximation yields the effective interaction terms $H^{(-)}_{\mathrm{NL}}=\lambda_{\mathrm{NL}}(b^{\dagger}_{L}b_{R}\tau_{-}+\mathrm{H.c.})$ for $\bar{\delta}\approx\omega_{L}-\omega_{R}$, and $H^{(+)}_{\mathrm{NL}}=\lambda_{\mathrm{NL}}(b_{L}b_{R}\tau_{+}+\mathrm{H.c.})$ for $\bar{\delta}\approx\omega_{L}+\omega_{R}$, see Appendix A. These terms show that the two resonators become indirectly coupled through the charge states, with the strength

[TABLE]

We remark that this interaction is, as well, purely nonlocal. $H^{(+)}_{\mathrm{NL}}$ describes the hybridization of the states in the subspace $|+,n_{L}-1,n_{R}-1\rangle$ with $|-,n_{L},n_{R}\rangle$, through which photons at different frequencies are simultaneously absorbed (emitted) from (into) both cavities. Conversely, the term $H^{(-)}_{\mathrm{NL}}$ describes processes by which the superconductor mediates a coherent transfer of photons between the resonators, by coupling the subspaces $|+,n_{L}-1,n_{R}+1\rangle$ and $|-,n_{L},n_{R}\rangle$, see Fig. 1(d). Notice that this effect vanishes if the two resonators are of the same frequency, as it would require $\bar{\delta}=0$ and, thus, $\Gamma_{S}=0$. In Fig. 3(a), we report the electronic current, again calculated with the full interaction, assuming two different resonator frequencies. In addition to the sideband peaks close to $\bar{\delta}=\omega_{L}$ and $\bar{\delta}=\omega_{R}$, we can identify higher-order multiphoton resonances (e.g., $\bar{\delta}=2\omega_{R}$, where the cooling cycle involves the absorption of two photons from the same cavity) which can be described in a similar way with a rotating-wave approximation, see Appendix A. Moreover, we observe the second-order peaks described by $H_{\mathrm{NL}}^{(\pm)}$ which are responsible for processes involving both resonators. The inset of Fig. 3(c) reports the average occupation of the resonators in the vicinity of the resonance $\bar{\delta}=\omega_{L}-\omega_{R}$, where the right mode is heated and the left one is cooled. The shape of these resonances differs from the first-order peaks (which are well approximated by Lorentzians): We show in Appendix A how the second-order Hamiltonian contains indeed an additional term proportional to $\sin(\bar{\theta})(2n_{\alpha}+1)\tau_{z}$, which causes both a small frequency shift for each resonator (yielding a double-peak structure) and a small renormalization of the splitting $\bar{\delta}$ between the Andreev bound states. Nevertheless, this corrections do not alter the main physics captured by $H_{\mathrm{NL}}^{(-)}$.

V Heat transfer and efficiency

To quantify the performance of both cooling and nonlocal photon transfer, we calculate the stationary heat current Erdman et al. (2017, 2018); Benenti et al. (2017)

[TABLE]

flowing from the bosonic reservoir $\alpha$ to the corresponding resonator. It is negative (positive) when the resonator is cooled (heated), and vanishes for an oscillator in thermal equilibrium. As a figure of merit for local cooling, we can estimate the number of bosonic quanta subtracted from the resonator on average per unit time, and compare it to the rate at which Cooper pairs are injected into the system. The latter rate is given by $|I_{S}|/2e$ with $I_{S}=-(I_{L}+I_{R})$ being the Andreev current through the superconductor found from current conservation. Consequently, the local cooling efficiency around $\bar{\delta}=\omega_{\alpha}$ can be defined as $\eta^{(\alpha)}_{\mathrm{loc}}=\frac{2e|\dot{E}_{\alpha}|}{|I_{S}|\omega_{\alpha}}$. Similarly, around $\bar{\delta}=\omega_{L}-\omega_{R}$, we define the heat transfer efficiency

[TABLE]

Figures 3(b) and (c) show $\eta^{(L)}_{\mathrm{loc}}$ and $\eta_{\mathrm{NL}}$, respectively, as a function of $\epsilon$ close to the corresponding resonances. In both cases, we obtain high efficiencies close to 90%: Approximately one photon is absorbed from each cavity (local cooling) or transferred from the left to the right cavity (nonlocal transfer) per Cooper pair. The efficiency is essentially limited by two factors: (i) an elastic contribution to the current [the broad resonance of linewidth $\propto\!\Gamma_{S}$ in Figs. 2(a) and 3(a)] where electrons flow without exchanging energy with the cavities; (ii) a finite fraction of the injected electrons acting against the dominant process (cooling or photon transfer), as illustrated by the dashed blue arrows in Fig. 1(d). Both processes augment with increasing $\Gamma_{S}$ and are a byproduct of the finite hybridization between the empty and the singlet state which, however, is crucial for achieving a nonzero efficiency.

VI Conclusions

We have analyzed a CPS in a double-quantum-dot setup, with local charge couplings to two resonators. We have demonstrated that Cooper-pair splitting can generate a nonlocal transfer of photons and heat from one oscillator to the other, resulting in a stationary energy flow. Such energy flows can also be channeled to cool or heat locally a single cavity. Hence, our system constitutes a versatile tool to fully inspect heat exchange mechanisms in hybrid systems, and is a testbed for quantum thermodynamics investigations involving both electronic and bosonic degrees of freedom. Due to the single-photon nature of the coherent interactions, this can also be extended to achieve few-phonon control and manipulation, Okamoto et al. (2013); Zhu et al. (2017) e.g., by implementing time-dependent protocols for the dots’ gate voltages to tune dynamically the strength of the nonlocal features. Further practical applications include high-efficiency nanoscale heat pumps and cooling devices for nanoresonators.

A discussion on the experimental feasibility of our setup is in order. For single quantum dots coupled to microwave resonators, $\lambda_{\alpha}/(2\pi)$ can reach 100 MHz, with resonators of quality factors $Q\!\!\sim\!\!10^{4}$ and frequencies $\omega_{\alpha}/(2\pi)\!\!\sim\!\!7$ GHz. Bruhat et al. (2016); Stockklauser et al. (2017) For mechanical resonators, coupling strengths of $\lambda_{\alpha}/(2\pi)\!\!\sim\!\!100$ kHz for frequencies of order $\omega_{\alpha}/(2\pi)\!\!\sim\!\!1$ MHz and larger quality factors up to $10^{5}\!\!-\!\!10^{6}$ have been reported. Okazaki et al. (2016) In a double-quantum-dot Cooper-pair splitter setup, the cross-Andreev reflection rate is approximately $\Gamma_{S}\!\!\lesssim\!\!\sqrt{\Gamma_{SL}\Gamma_{SR}}$, when the distance between the dots is much shorter than the coherence length in the superconducting contact.Hussein et al. (2019) Here, $\Gamma_{S\alpha}$ is the local Andreev reflection rate which can reach several tens of $\mu$eV, becoming comparable to the typical microwave resonator frequencies (thus allowing $\Gamma_{S}\lesssim\omega_{\alpha}$) while being order of magnitudes lower than the superconducting gap $\Delta$.Gramich et al. (2015) Therefore, the regime of parameters we considered lies within the range of state-of-the-art technological capabilities. Moreover, experiments involving Cooper-pair splitters Hofstetter et al. (2009); Herrmann et al. (2010); Hofstetter et al. (2011); Das et al. (2012); Schindele et al. (2012, 2014); Fülöp et al. (2014, 2015); Tan et al. (2015); Borzenets et al. (2016); Baba et al. (2018) or mesoscopic cQED devices with microwave cavities Frey et al. (2012); Delbecq et al. (2013); Deng et al. (2015); Stockklauser et al. (2017); Viennot et al. (2015); Mi et al. (2017); Li et al. (2018); Cubaynes et al. (2019) and mechanical resonators Benyamini et al. (2014); Okazaki et al. (2016); Deng et al. (2016); Urgell et al. (2019) are of appealing and growing interest, and therefore promising candidates for the implementation of the system described here.

Acknowledgements.

This research was supported by the German Excellence Initiative through the Zukunftskolleg and by the Deutsche Forschungsgemeinschaft through the SFB 767. R.H. acknowledges financial support from the Carl-Zeiss-Stiftung.

Appendix A Polaron-transformed Hamiltonian and effective nonlocal interaction

We report here the derivation of the effective interactions that explain the local cooling or heating, and the nonlocal photon transfer mechanisms. The starting point is the polaron-transformed Hamiltonian given in Eq. (5) of the main text. For small coupling strengths $\lambda_{\alpha}\ll\omega_{\alpha}$, we expand the operators $X$ and $X^{\dagger}$ up to second order in $\lambda_{\alpha}$. The dots-cavities interaction term is

[TABLE]

with $i\Pi=\sum_{\alpha}i\Pi_{\alpha}$ the generalized total momentum, $\sigma_{x}=|0\rangle\langle S|+\text{H.c.}$ and $\sigma_{y}=-i|0\rangle\langle S|+\text{H.c.}$ The $\sigma_{x}$-term describes tunneling between the empty and the singlet state due to the superconductor, and is already present in Hamiltonian (1) of the main text. Diagonalizing the bare electronic part leads to the hybridized charge states

[TABLE]

with the mixing angle $\bar{\theta}$ and the energy splitting $\bar{\delta}$ defined in the main text. By introducing the Pauli matrices $\tau_{x}=\tau_{+}+\tau_{-}$, $\tau_{y}=-i(\tau_{+}-\tau_{-})$, $\tau_{z}=[\tau_{+},\tau_{-}]$ with $\tau_{+}=|+\rangle\langle-|$ and $\tau_{-}=|-\rangle\langle+|$, we can express the Hamiltonian (5) of main text to second order by

[TABLE]

We now move to the interaction picture with respect to the noninteracting Hamiltonian $H_{0}=\sum_{\alpha\sigma}\bar{\epsilon}_{\alpha}N_{\alpha\sigma}+\frac{\bar{\delta}}{2}\tau_{z}+\sum_{\alpha}\omega_{\alpha}b^{\dagger}_{\alpha}b_{\alpha}$. By recalling the definition of $\Pi$, we obtain in the interaction picture the Hamiltonian

[TABLE]

Here, we have introduced $\Omega=\omega_{L}+\omega_{R}$ and $\Delta\omega=\omega_{L}-\omega_{R}$. Hamiltonian (15) contains all the terms that lead to cooling, heating, and nonlocal photon transfer. To isolate these features, we will focus on the relevant resonances $\bar{\delta}\approx\omega_{\alpha}$, $\bar{\delta}\approx\Omega$, and $\bar{\delta}\approx\Delta\omega$. First, let us consider two identical resonators of frequency $\omega_{\alpha}=\omega$ and tune $\epsilon$ such that $\bar{\delta}=\omega$. Notice that this can be fulfilled by two values of $\epsilon$, of opposite sign. In the following, we restrict Eq. (15) to first order in $\lambda_{\alpha}$, and then discard the fast-oscillating terms by performing a standard rotating-wave approximation (RWA). Thus, we obtain the time-independent interaction Hamiltonian given by Eq. (7) in the main text,

[TABLE]

We have used here the resonance condition $\omega=\bar{\delta}$ and the relation $\sin\bar{\theta}=\sqrt{2}\Gamma_{S}/\bar{\delta}$.

Let us now consider the nonlocal resonance, $\bar{\delta}=\Delta\omega$. A peculiarity is here, that we have to go to second order in $\lambda_{\alpha}$, since the first-order terms become in the RWA fast rotating and, thus, average to zero. The corresponding effective Hamiltonian reads

[TABLE]

with $n_{\alpha}=b_{\alpha}^{\dagger}b_{\alpha}$ the photon number operator, and $\lambda_{\mathrm{NL}}$ stated in Eq. (8) of the main text. The second term corresponds to the interaction $H_{\mathrm{NL}}^{(-)}$ (main text), and is responsible for the coherent transfer of photons between the cavities, leading to a stationary energy flow. The first term in Eq. (17) proportional to $n_{\alpha}\tau_{z}$ can be seen as a dispersive shift of the cavity frequencies, which depends on the Andreev bound state. As the quantities reported in Fig. 3 of the main text are averages calculated from the density matrix, this translates into a fine double-peak structure of the nonlocal resonance, see Fig. 3(c) of the main text. Further, the additional term proportional to $\tau_{z}$ renormalizes the level splitting $\bar{\delta}$ and, therewith, the resonance condition, $\bar{\delta}=\Delta\omega$.

Considering the condition $\bar{\delta}=\Omega$, we obtain the effective RWA Hamiltonian

[TABLE]

Here, the relevant interaction ($H^{(+)}_{\mathrm{NL}}$ of the main text) describes absorption (and emission) from both cavities simultaneously while flipping the Andreev state. So, this second-order effect may entail simultaneous cooling, $\epsilon>0$, and heating, $\epsilon<0$, of both cavities.

From the last line of Eq. (15), one can infer an effective RWA Hamiltonian governing the resonance condition $\bar{\delta}\approx 2\omega_{\alpha}$. It is similar to Eq. (16), but involves absorption and emission of two photons from the same cavity. Indeed, this two-photon resonance is also observable in Fig. 3(a) of the main text and yields cavity cooling for $\epsilon>0$ and heating for $\epsilon<0$, respectively.

By including terms up to $n$-th order in $\Pi$ in Eq. (14), one obtains terms $(b_{\alpha})^{n}$ and $(b^{\dagger}_{\alpha})^{n}$, which, after moving to the interaction picture and performing a suitable RWA, will yield $n$-photon local absorption/emission processes. The expansion contains also terms of the form $(b_{\alpha}^{\dagger})^{p}(b_{\bar{\alpha}})^{q}$ and $(b_{\alpha}^{\dagger})^{p}(b^{\dagger}_{\bar{\alpha}})^{q}$ together with their Hermitian conjugates, with $p+q=n$ ($\bar{\alpha}=R$ if $\alpha=L$ and vice versa). The former terms describe the coherent transfer of $|p-q|$ photons between the cavities, while the latter describes coherent emission and re-absorption of $p$ and $q$ photons from the $\alpha$ and $\bar{\alpha}$ cavity, respectively. The general (approximate) resonance condition thus reads $\bar{\delta}\approx|p\omega_{L}\pm q\omega_{R}|$, stated in Eq. (6) in main text. If either $p$ or $q$ is zero, the resonance corresponds to local cooling/heating of the cavities.