Quantum-enhanced capture of photons using optical ratchet states
QQuantum-enhanced capture of photons using optical ratchet states
K. D. B. Higgins, B. W. Lovett,
2, 1, ∗ and E. M. Gauger † Department of Materials, Oxford University, Oxford OX1 3PH, United Kingdom SUPA, School of Physics and Astronomy, University of St Andrews, KY16 9SS, United Kingdom SUPA, Institute of Photonics and Quantum Sciences,Heriot-Watt University, EH14 4AS Edinburgh, United Kingdom (Dated: November 8, 2018)Natural and artificial light harvesting systems often operate in a regime where the flux of photonsis relatively low. Besides absorbing as many photons as possible it is therefore paramount to preventexcitons from annihilation via photon re-emission until they have undergone an irreversible energyconversion process. Taking inspiration from photosynthetic antenna structures, we here considerring-like systems and introduce a class of states we call ratchets: excited states capable of absorbingbut not emitting light. This allows our antennae to absorb further photons whilst retaining theexcitations from those that have already been captured. Simulations for a ring of four sites reveal apeak power enhancement by up to a factor of 35 under ambient conditions owing to a combinationof ratcheting and the prevention of emission through dark-state population. In the slow extractionlimit the achievable power enhancement due to ratcheting alone exceeds 20%.
Introduction –
The absorption of light and preventionof its reemission is essential for the efficient operationof solar energy harvesting devices [1]. From the manycauses of device inefficiency, few are as fundamental andseemingly insurmountable as energy loss via radiative re-combination: any absorption process must have a com-panion emission process. This inherent absorption ineffi-ciency is a result of the principle of ‘detailed balance’ andconstitutes a key contribution to the famous Shockley-Queisser limit [2]. However, pioneering work by Scullyshowed that it is possible to break detailed balance, givenan external source of coherence [3]; later work showedthat this can be achieved by clever internal design alone,by using an optically dark state to prevent exciton re-combination [5–7]. Such dark states are populated pas-sively if the energy separation between dark and brightstates falls into the vibrational spectrum of the absorb-ing nanostructure: dissipation then preferentially medi-ates transfer into states from which optical decay cannotoccur. This is thought to play a role in photosyntheticlight harvesters [8, 9], e.g. by means of dynamic localisa-tion reducing the effective optical dipole strength [10].However, time spent in dark states is ‘dead time’ withrespect to absorbing further photons. In organic lightharvesting systems the time needed to extract an exci-ton from a dark state and turn it into useful energy isoften orders of magnitude slower than typical light ab-sorption rates [11, 12], and this results in the loss of anysubsequently arriving photons. In this Letter, we showthat certain exciton states, which we will call ratchets ,can enhance photocell efficiency by enabling absorptionof these subsequent photons, while preventing emission.
Model –
We consider a ring of N identical two-leveloptical emitters (see Fig. 1), with nearest pairs coupledby a transition dipole-dipole interaction, governed by the Hamiltonian ( (cid:126) = 1): H s = ω N (cid:88) i =1 σ + i σ − i + S N (cid:88) i =1 ( σ + i σ − i +1 + σ + i +1 σ − i ) , (1)where ω is the bare transition energy of the sites, S isthe hopping strength, the σ ± i denote the usual raisingand lowering operators which create and destroy an exci-tation on site i , and σ N +1 = σ . For a single exciton, theeigenstates are equal superposition of excitons localizedat the different ring positions, with each being character-ized by a relative phase k i = 2 πj/N with j ∈ , ...N − H s = (cid:80) K λ K | K (cid:105) (cid:104) K | . K describes the eigenstate with energy λ K and each K now corresponds to a set of the phase factors k i ,one for each excitation [see Supplementary Information(SI) [15]]. The eigenstates fall into a series of bands,where each state within a band contains the same num-ber of excitons n = | K | , with | K | the number of elementsin set K .Optical transitions connect eigenstates differing by oneexciton with rates proportional to Γ K,K (cid:48) = |(cid:104) K (cid:48) | J + | K (cid:105)| where J ± = (cid:80) Ni =1 σ ± i . An explicit analytical expressionfor Γ K,K (cid:48) can be derived [2, 3] and is given in [15]. Onlythe fully symmetric k i = 0 single exciton eigenstate hasa dipole matrix element with the ground state, since herethe transition dipoles interfere constructively. The other N − N − a r X i v : . [ qu a n t - ph ] D ec Ratchet states Trap for exciton extractionRing antenna
FIG. 1. Left: N = 4 ring structure with attached trap.Right: Energy level diagram of this ring showing dipole-allowed optical transitions (red). Solid lines denote primaryabsorption transitions for the ratcheting cycle, others are in-dicated with thinner dashed lines. Only relevant phonon tran-sitions are shown (blue). Rarely populated levels are shownin grey. that in a photovoltaic circuit can be converted into usefulphotocurrent. Such states are examples of ratchet states | K R (cid:105) , which have the simultaneous properties:Γ − K R := (cid:88) K (cid:48) Γ K R ,K (cid:48) δ | K R | , | K (cid:48) | +1 = 0 , (2)Γ + K R := (cid:88) K (cid:48) Γ K R ,K (cid:48) δ | K R | , | K (cid:48) |− > , (3)where Γ − (+) K R is the sum of all transition rates connectingthe ratchet eigenstate | K R (cid:105) to the adjacent band below(above) and δ i,j is the Kronecker symbol.For N = 4 and S > ω + 2 S , and the threeratchet states have energies ω, ω and ω − S , see Fig. 1band [15]. In a molecular system, the ratchet states can bereached following excitation into the bright state whenthe exciton loses a small amount of energy to molecu-lar phonons. Importantly, such vibrational modes havemuch shorter wavelengths than optical modes and can of-ten be assumed to be local to each site. Unlike the photonfield, the phonon interaction then breaks the symmetryand enables intraband transitions. Dynamics –
In order to model the dynamics we usethe QuTiP package [23] to construct full open systemdynamics in Bloch-Redfield theory, for each kind of envi-ronmental interaction. We always retain the full form ofthe resulting dissipator tensor since the secular approx-imation cannot be applied when a system has bands ofclosely spaced levels which may be more closely spacedthan the corresponding dissipation rates [4, 24].First, consider the dynamics due to the light-matterinteraction Hamiltonianˆ H s − l = (cid:88) Q N (cid:88) i =1 G Q (ˆ σ + i ˆ b Q + ˆ σ − i ˆ b † Q ) , (4) where ˆ b † Q and ˆ b Q are creation and annihilation operatorsfor photons of wavevector Q . The photon spectral den-sity is constant across all transition energies, and entersinto the master equation dissipator only via the singleemitter decay rate γ o . Absorption and stimulated emis-sion terms are weighted by the appropriate Bose-Einsteinfactor N ( ω Ω , T o ) = 1 / (exp[ ω Ω / ( k B T o )] − ω Ω and at temperature T o (= 5800 K for solar radiation).Second, the exciton-phonon interaction is accountedfor through [25]ˆ H s − p = N (cid:88) i =1 (cid:88) q g q ,i ˆ σ zi (ˆ a q ,i + ˆ a † q ,i ) , (5)where for each site i ˆ a q and ˆ a † q are the creation and an-nihilation operators for phonons of wavevector q withenergy ω q and exciton-phonon coupling g q . This inter-action commutes with (cid:80) i σ zi and so only mediates tran-sitions within each excited band. For simplicity, we againassume a spectral density J ( ω ) that is approximatelyconstant across all intraband transition energies ω Π , suchthat J ( ω Π ) = γ p . We calculate appropriate phonon ma-trix elements for each transition, letting absorption andstimulated emission terms carry the appropriate Bose-Einstein factors N ( ω Π , T p ), with T p the ambient phonontemperature. Reflecting the fact that phonon relaxationtypically proceeds orders of magnitude faster than opticaltransition rates [20], we fix γ p = 1000 γ o . This separationof timescales allows population to leave the bright statesbefore reemission occurs.To complete the photovoltaic circuit, we include anadditional ‘trap site’ t , with excited state | α (cid:105) and groundstate | β (cid:105) , where excitons are irreversibly converted intowork. The trap Hamiltonian is H t = ω t ˆ σ + t ˆ σ − t , where ω t represents its transition frequency, and σ − t = | β (cid:105) (cid:104) α | .Henceforth, we shall consider the joint density matrix ofring system and trap, ρ = ρ s ⊗ ρ t . Throughout the paperwe use an incoherent hopping from a single site in thering to the trap (see SI [15] for an alternative model),and we fix the trap energy to be resonant with the statein which population is most likely to accumulate — unlessotherwise stated, this is at the bottom of the first excitonband, ω t = ω − S .The theory of quantum heat engines [5, 21, 22] providesa way for us to assess the current and voltage output ofthe ring device: The action of a load across the deviceis mimicked by the trap decay rate γ t , which varies de-pending on the load resistance. The current is simply I = eγ t (cid:104) ρ α (cid:105) ss , with (cid:104) ρ α (cid:105) ss the steady state population ofthe excited trap state. The potential difference seen bythe load is given by the deviation of the trap’s populationfrom its thermal distribution [5, 21]: eV = (cid:126) ω t + k B T p ln (cid:18) (cid:104) ρ α (cid:105) ss (cid:104) ρ β (cid:105) ss (cid:19) , (6) Parameter Symbol Default valueAtomic transition frequency ω . Hopping strength S γ o . × Hz ≡ µ eVPhonon relaxation rate γ p γ o Extraction rate γ x − γ o Photon bath temperature T o Phonon bath temperature T p
300 KTABLE I. Default model parameters; italic rows are plot pa-rameters in certain Figures, as stated in the relevant captions.FIG. 2. (a) Steady-state exciton population and (b) relativeratchet enhancement compared to the FD scenario (see text),both as functions of the optical and phononic bath temper-atures, and without any trap. The former is given in solartemperature units, K S = 5800 K, other parameters are as inTable I except that γ x = 0. where k B is Boltzmann’s constant and T p the (ambient)phonon bath temperature. To study the relationship be-tween current, voltage, and therefore power, we alter thetrapping rate γ t . Letting γ t → ∞ leads to the well-known open and short circuit limit, respectively, however,our main interest in the following is in the region nearthe optimal power output point of the device. Results –
We proceed by finding the steady state of thefollowing master equation which incorporates the opticaland phononic dissipator as well as the trap decay process:˙ ρ = − i [ H s + H t , ρ ] + D o [ ρ ] + D p [ ρ ] + D t [ ρ ] + D x [ ρ ] , (7)where D o [ ρ ], D p [ ρ ] are the Bloch Redfield dissipatorsfor the photon and phonon fields discussed above, and D t [ ρ ] = γ t ( σ − t ρσ + t − { σ + t σ − t , ρ } ) is a standard Lindbladdissipator representing trap decay. D x [ ρ ] represents ex-citon extraction via incoherent hopping at rate γ x froma single site on the ring to the trap (see Fig. 1 a).To expose the effect of the optical ratchets we nowcontrast the full model of Eq. (7) with two other artifi-cial cases that exclude the possibility of ratchets: In the first, we remove all phonon-assisted relaxation ( γ p = 0),meaning the system only undergoes optical transitionsalong the Dicke ladder of bright states [4] – we term this‘no phonons’ (NP). In the second construct, the ratchetstates are rendered fully dark by setting all upwards tran-sition matrix elements from them to zero, a scenario werefer to as ‘forced dark’ (FD). All results in this sectionare based on the steady state of the system for N = 4, ob-tained by an iterative numerical method performed withQuTiP [23], and using the default parameters from Ta-ble I unless explicitly stated otherwise. Our choice of ω = 1 . − S differs from that in previous works [5, 6]:our subtraction of 2 S ensures that the highest (brightstate) level in the first band always remains at 1.8 eVand absorbs at a fixed frequency.In Fig. 2 we display the steady-state exciton populationof the system without trapping as a function of opticaland phonon bath temperatures. We compare the ratchetmodel to FD, finding that the ratchet enhancement isgreatest in the hot photons, cold phonons regime. This isto be expected: hot photons quickly promote the systemup the excited bands via ratchets, with cold phonons al-lowing a one-way protection of gained excitation energy.For FD states at low phonon temperature the excitonpopulation plateaus near one, as most population endsup trapped in the state at the bottom of the first band(see [15] for the FD data and an extended discussion).By contrast, the ratchet states keep absorbing, allowingthe steady state population to rise toward the infinitetemperature limit of 2 (for N = 4).Using the heat engine model, we can now explore theperformance of the four site ring as an energy harvester.We will focus on the dependence of the power output ontwo parameters: the coupling S , and extraction rate γ x .For each parameter set, the trap decay rate γ t is variedfor maximum power output; example power traces as afunction of γ t can be found the SI [15].In Fig. 3 we display the absolute value of the powergenerated for the ratchet and NP scenarios, whereas inFig. 4 we display the relative power output of ratchetstates over each of the two artificial scenarios. To en-sure fair comparison, for NP we set the trap energy tobe resonant with the bright state at ω + 2 S , since theratchet states at lower energy are inaccessible here. Allthree scenarios have lower output power when the trap-ping (charge separation) rates are low, since this createsa bottleneck in the cycle and limits the size of the pho-tocurrent.However, the ratchet states hold a great advantage overthe other scenarios in this bottlenecked region, since thisis precisely the situation in which excitations need tobe held for some time by the ring before extraction ispossible. The NP scenario only performs poorly in thiscase, since excitations in all likelihood decay before beingextracted. Indeed, the relative power output of ratchet-ing over NP, shown in the left panel of Fig. 4, therefore FIG. 3. Absolute power output in the ratcheting (blue sur-face) and NP (red surface) cases as a function of (single site)extraction rate γ x and hopping strength S . Other parametersare as in Table I. For each point, the optimal trap decay rateis found by numerical search. The sets of curves projectedonto the side walls of the 3D plot are cuts through the datafor fixed values of the appropriate parameter. The two sur-faces cross in the region where the bottleneck is lifted, withratcheting generating more power for a severe and intermedi-ate bottleneck. The line at which the curves cross is projectedonto the bottom ( xy ) plane of the figure. rises to as high as a factor of 35. The right panel ofFig. 4 demonstrates the importance of ratcheting overdark state protection alone, with the ratchet model de-livering up to 20 % better performance than FD.In Fig. 4 we can see that advantage of ratcheting per-sists into a region where there is only a moderate bot-tleneck, i.e. up to γ x ∼ − eV = 10 γ o . Referring toFig. 3 we see that the ratchet power output in this regionis already close to its maximum. Even so, if the extrac-tion rate was arbitrarily tunable, then even a moderatebottleneck could be avoided completely and the main ad-vantage of ratcheting removed. However, in photosynthe-sis, creating a fast extraction rate carries with it a clearresource cost: whereas antenna systems can be compar-atively ‘cheap’, reaction centres carry a much larger spa-tial footprint, being typically embedded in membranesand requiring significant surrounding infrastructure (suchas concentration gradients produced by proton pumps).The severity of the bottleneck is likely to be inversely pro-portional to the number of reaction centres. A photocelldesign exploiting ratcheting in the moderate bottleneckregime would be likely then to generate optimal powerper unit volume of material — and similar design prin-ciples will apply to artificially designed molecular lightharvesting systems.The choice of interaction strength S is also important.Ratcheting achieves optimal results for S ∼ .
05 eV inthe bottlenecked region. This dependence arises becausethe size of S determines the gap between bright and darkor ratchet states. In turn, this controls the effective rate FIG. 4. Enhancement of power output for ratchet states overNP (left, displayed as a multiplicative factor) and FD (right,displayed as a percentage), as a function of both hoppingstrength S and extraction rate γ x . Parameters are as in Fig. 3. for ‘upwards’ phonon-assisted transitions within excita-tion bands, as those rely on the absorption of a phononand are proportional to N ( ω Π , T p ). Consequently, largervalues of S entail more directed dissipation into the lowerstates of each band, boosting the occupation of ratchetstates. However at the same time, increasing S leads toa lower ratchet state energy and so a lower trap energy –and hence to a voltage drop. The trade off between thesetwo competing influences leads to a maximum in ratchetperformance.As we discuss in detail in the SI, ratcheting continues toconvey an advantage in the presence of moderate levels ofvarious real-world imperfections, such as site energy dis-order, non-radiative recombination, and exciton-excitonannihilation [15]. Conclusion –
We have investigated the light harvest-ing properties of coupled ring structures, inspired by themolecular rings that serve as antennae in photosynthesis.Considering a vibrational as well as an electromagneticenvironment allows the system to explore the full Hilbertspace rather than just the restricted subset of Dicke lad-der states. We have shown that the off-ladder states pos-sess interesting and desirable properties, which can beharnessed for enhancing both the current and power of aring-based photocell device.Several possible systems [15] could be used to observethe effect, from superconducting qubits [13] to macro-cyclic molecules [23]. Our approach generalises exist-ing concepts for dark state protection [5, 6] to arbitrarynumbers of sites and importantly includes multi-excitonstates, which introduce the ratcheting effect as a distinctadditional mechanism for enhancing the overall light-harvesting performance. The optical ratchet enhance-ment is particularly well suited to situations where ex-citon extraction and conversion represent the bottleneckof a photocell cycle.In future work it would be interesting to explore com-bining optical ratcheting with enhancements of the pri-mary absorption process, for example by exploiting thephenomena of stimulated absorption [28] or superabsorp-tion [4].The authors thank William B. Brown for insightfuldiscussions. This work was supported by the EPSRC andthe Leverhulme Trust. BWL thanks the Royal Society fora University Research Fellowship. EMG acknowledgessupport from the Royal Society of Edinburgh and theScottish Government. ∗ [email protected] † [email protected][1] B. Kippelen and J.-L. Bredas, Energy and EnvironmentalScience , 251 (2009).[2] W. Shockley and H. J. Queisser, Journal of AppliedPhysics , 510 (1961).[3] M. O. Scully, Physical Review Letters , 207701(2010), URL http://link.aps.org/doi/10.1103/PhysRevLett.104.207701 .[5] C. Creatore, M. A. Parker, S. Emmott, and A. W.Chin, Physical Review Letters , 253601 (2013), URL http://link.aps.org/doi/10.1103/PhysRevLett.111.253601 .[6] Y. Zhang, S. Oh, F. H. Alharbi, G. S. Engel, and S. Kais,Phys. Chem. Chem. Phys. , 5743 (2015), URL http://dx.doi.org/10.1039/C4CP05310A .[6] M. I. Yasuhiro Yamada, Youhei Yamaji, ArXiv e-printsp. arXiv:1502.07341 (2015).[7] A. Fruchtman, R. G´omez-Bombarelli, B. W. Lovett, andE. M. Gauger, Phys. Rev. Lett. , 203603 (2016), URL http://link.aps.org/doi/10.1103/PhysRevLett.117.203603 .[8] K. Mukai, S. Abe, and H. Sumi, The Jour-nal of Physical Chemistry B , 6096 (1999),http://dx.doi.org/10.1021/jp984469g, URL http://dx.doi.org/10.1021/jp984469g .[9] R. Pearlstein and H. Zuber, in Antennas and ReactionCenters of Photosynthetic Bacteria , edited by M. Michel-Beyerle (Springer Berlin Heidelberg, 1985), vol. 42 of
Springer Series in Chemical Physics , pp. 53–61, ISBN978-3-642-82690-0, URL http://dx.doi.org/10.1007/978-3-642-82688-7_7 .[10] A. F. Fidler, V. P. Singh, P. D. Long, P. D. Dahlberg,and G. S. Engel, Nat Commun (2014), URL http://dx.doi.org/10.1038/ncomms4286 .[11] D. Caruso and A. Troisi, Proceedings of the Na-tional Academy of Sciences .[12] R. E. Blankenship, Molecular Mechanisms of Photosyn-thesis (Blackwell Science, 2002), 1st ed.[1] P. Jordan and E. P. Wigner, Z.Phys. , 631 (1928).[2] T. Tokihiro, Y. Manabe, and E. Hanamura, Physical Re-view B , 2019 (1993), URL http://link.aps.org/ doi/10.1103/PhysRevB.47.2019 .[15] Please see supplementary information for details. [3] F. C. Spano, Physical Review Letters , 3424(1991), URL http://link.aps.org/doi/10.1103/PhysRevLett.67.3424 .[7] H. P. Breuer and F. Petruccione, The Theory of OpenQuantum Systems (Oxford University, 2002).[4] K. D. B. Higgins, S. C. Benjamin, T. M. Stace, G. J. Mil-burn, B. W. Lovett, and E. M. Gauger, Nat Commun (2014), URL http://dx.doi.org/10.1038/ncomms5705 .[19] E. Wientjes, J. Renger, A. G. Curto, R. Cogdell, andN. F. van Hulst, Nat Commun (2014), URL http://dx.doi.org/10.1038/ncomms5236 .[20] D. Brinks, R. Hildner, E. M. H. P. van Dijk, F. D. Stefani,J. B. Nieder, J. Hernando, and N. F. van Hulst, Chem.Soc. Rev. , 2476 (2014), URL http://dx.doi.org/10.1039/C3CS60269A .[21] K. E. Dorfman, D. V. Voronine, S. Mukamel,and M. O. Scully, Proceedings of the Na-tional Academy of Sciences .[22] M. O. Scully, K. R. Chapin, K. E. Dorfman, M. B.Kim, and A. Svidzinsky, Proceedings of the Na-tional Academy of Sciences .[23] J. Johansson, P. Nation, and F. Nori, ComputerPhysics Communications , 1234 (2013), ISSN 0010-4655, URL .[24] P. R. Eastham, P. Kirton, H. M. Cammack, B. W. Lovett,and J. Keeling, Phys. Rev. A , 012110 (2016)[25] G. D. Mahan, Many Particle Physics (Physics of Solidsand Liquids) , 3rd ed. (Springer, 2000)[13] J. A. Mlynek, A. A. Abdumalikov, C. Eichler, andA. Wallraff, Nature Communications , 5186 EP (2014),URL http://dx.doi.org/10.1038/ncomms6186 .[23] M. C. O’Sullivan, J. K. Sprafke, D. V. Kondratuk,C. Rinfray, T. D. W. Claridge, A. Saywell, M. O. Blunt,J. N. O’Shea, P. H. Beton, M. Malfois, et al., Na-ture , 72 (2011), URL http://dx.doi.org/10.1038/nature09683 .[28] G. E. Alexey Kavokin, Natural Science , 63 (2010). Supplemental Information:Quantum-enhanced capture of photonsusing optical ratchet states
RING STRUCTURE DIAGONALISATION
The Jordan-Wigner transformation maps a Pauli spin1/2 system onto a ‘hard-core’ boson model [S1]. Thisleads to a bosonic description of the collective excitonswhilst maintaining Pauli’s exclusion principle, which for-bids double excitation of a single site. In our case each‘spin’ represents one of N identical optical emitters /absorbers and the spin’s ‘up’ / ‘down’ projection alongthe z -axis denotes the presence / absence of an excitonon the respective site. We assume a ring-like geometri-cal arrangement as shown in Fig. 1a of the main paper.Considering a ring rather than a spin chain with free endsintroduces the additional complexity of an alternating pe-riodic or anti-periodic boundary condition depending onthe number of excitations [S2].The Jordan-Wigner transformation is achieved by in-troducing bosonic creation and annihilation operators,which are defined in terms of the Pauli spin raising andlowering operators as follows: c † l = σ + l e iπ (cid:80) l − j =1 σ + j σ − j , (S1) c l = e − iπ (cid:80) l − j =1 σ + j σ − j σ − l . (S2)Here, the e iπ (cid:80) n − σ + j σ − j factors are traditionally referredto as ‘strings’, and their inclusion is necessary for produc-ing the correct anti-commutation relation for fermions,which also apply to hard-core bosons: { c α , c † β } = δ α,β , (S3) { c † α , c † β } = { c α , c β } = 0 . (S4)Performing the transform (S1, S2) to the Hamiltonian ofthe main paper given by Eq. (1) yields: H s = ω ˆ n − S N (cid:88) j =1 ( c † j c j +1 + c † j +1 c j )+ S ( e iπ ˆ n + 1)( c † N c + c † c N ) , (S5)where ˆ n = (cid:80) Nj =1 c † j c j = (cid:80) Nj =1 σ + j σ − j counts the num-ber of excitons on the ring. The final term in Eq. (S5)has been introduced to implement the required alternat-ing boundary condition [S2]. Note that this elegant andcompact form of the Hamiltonian double-counts the in-teraction term for N = 2, but this is of no concern heresince we are only interested in N ≥ e iπ ˆ n commutes with the Hamilto-nian (S5), so that the eigenvalues of e iπ ˆ n are good quan-tum numbers. The system can thus be partitioned intotwo parity subspaces of even and odd exciton number n , each of which can be solved separately. We complete thediagonalisation with the help of the Fourier transform c † k = √ N (cid:80) Nj =1 e ik j c † j , (S6) c k = √ N (cid:80) Nj =1 e − ik j c j , (S7)where the k j are phase factors defined by k j = πN j ( n odd) , (S8) k j = π (2 j +1) N ( n even) . (S9)Each of the above N distinct k -values corresponds toa single exciton state of the system. The eigenstates ofthe full multi-exciton system are given by combinationsof these states. The eigenstates are thus defined by sets K whose elements are the k ’s, and where the number ofelements in K represents the number of excitons in thestate ( | K | = n ). The Pauli exclusion principle manifestsitself by allowing only different single exciton states toform a multiple exciton state. The eigenvalues are thengiven by [S2] λ K = (cid:88) k ∈ K ( ω − S cos k ) (S10)with corresponding eigenstates | K (cid:105) = c † k ...c † k n | (cid:105) , (S11)where | (cid:105) is the ground state. Obviously, Eq. (S11) yieldsan eigenstate with n excitations and each band comprisesa total of N ! n !( N − n )! possible states. In its diagonalisedform the Hamiltonian Eq. (S5) is then simply given by H s = (cid:88) K λ K | K (cid:105) (cid:104) K | . (S12) OPTICAL TRANSITON MATRIX ELEMENTS
The optical transition operator ˆ J ± = (cid:80) Ni =1 ˆ σ ± i con-nects eigenstates differing by one exciton. The transitionrates are proportional to Γ K,K (cid:48) = |(cid:104) K (cid:48) | J + | K (cid:105)| and canbe explicitly evaluated with the help of Slater determi-nants, albeit involving some tedious calculations [S2, S3],yielding:Γ K,K (cid:48) = (cid:12)(cid:12)(cid:12)(cid:12) n N ( − n + ) δ (cid:88) j k (cid:48) j , (cid:88) i k i (S13)Π i>i (cid:48) ( e ik i − e ik i (cid:48) )Π j>j (cid:48) ( e − ik (cid:48) j − e − ik (cid:48) j (cid:48) )Π ni =1 Π n +1 j =1 (1 − e i ( k (cid:48) j − k j ) ) (cid:12)(cid:12)(cid:12)(cid:12) , where δ ( x, y ) is 1 if x = y + 2 πm for integer m andzero otherwise. The indices i, i (cid:48) , j, j (cid:48) inside the sums andproducts of the above expression refer to the elements ofunique pairs of k -values composing K or K (cid:48) (see [S2] forfull details). (a) (b) (c) FIG. S1. Number of excitons in the ring molecule in the steady state, with no attached trap. (a) Ratchets model; (b) forceddark (FD) model; (c) no phonon (NP) model. All three panels use the same colour mapping for ease of direct visual comparison.Panel (a) plots the same data also displayed in Fig. 2a in the main text. Parameters are from Table I in the main text.
STEADY STATE EXCITON POPULATIONS
In Fig. 2 of the main text we display the steady-stateexciton population in the ratchets model as a functionof phonon and photon temperatures, and also the rel-ative population of the ratchets model over the forceddark (FD) scenario. To complete the picture, in Fig. S1we show the absolute values of the steady-state excitoncount for all three scenarios. In the no phonons (NP)case, as expected there is no dependence on the phonontemperature. Indeed, since there is no dark-state protec-tion in the NP case, we find a lower exciton populationthan in the other two cases; the difference is more pro-nounced at lower phonon bath temperature, when darkstates are most likely to be populated. Naively, one mightexpect the surface for the FD scenario to plateau at one,however, the ladder of bright states remains availablein this case and via it population can access and gettrapped in higher lying dark states above the single ex-citon band. We expect this to happen for high opticaland low but finite phonon temperature, and indeed inthis region the surface rises above one. Note that unlikepanels (a) and (c) in Fig. S1, (b) starts at a phonon bathtemperature of T = 75 K due to numerical instabilities inour steady-state solver. At much lower phonon temper-ature, where phonon-assisted relaxation becomes effec-tively uni-directional, we expect the FD surface to droptowards one, as it gets increasingly unlikely for popula-tion to bypass or escape the single excitation subspace.Using alternative methods, we have verified this is indeedthe case (not shown). FIG. S2. Absolute power output in the ratcheting (blue sur-face) and FD (red surface) cases as a function of (single site)extraction rate and F¨orster coupling. Here the collective ex-traction model is used. Other parameters are from Table Iand thus identical to those of Fig. 4 of the main text. Thesets of curves projected onto the side walls of the 3D plotare cuts through the data for fixed values of the appropri-ate parameter. For each point, the optimal trap decay rateis found by numerical search. The two surfaces cross in theregion where the bottleneck is lifted, with ratcheting generat-ing more power for a severe and intermediate bottleneck. Theline at which the curves cross is projected onto the bottom( xy ) plane of the figure. FIG. S3. For the collective extraction model, enhancementof power output for ratchet states over NP (left, displayed asa multiplicative factor) and FD (right, displayed as a percent-age), as a function of both hopping strength S and extractionrate γ x . The simulations use the same model and parametersas Fig. S2. EXTRACTION MECHANISMS
Our photocell cycle relies on the extraction of the ex-citons from the ring to the trap site. Below we give anexplicit description of two different mechanisms and mod-els, one of which features in the main text, the otherwhose results are displayed for comparison in this sup-plement.For the incoherent hopping process that we use in themain text we assume that excitons are extracted incoher-ently from a single site of the ring to the trap. We thensimply include a dissipator: D x [ ρ ] = γ x D [ σ − σ + t , ρ ] . (S14)For optimal extraction the trap’s energy is resonant withthe desired extraction level, i.e. for the ratcheting andFD cases, the lowest state in the first exciton band, ω t = ω − S ; for the NP case, the trap is instead made resonantwith the bright state, ω t = ω + 2 S . For consistency anda fair comparison across the different mechanisms, theenergy of the trap site is also adjusted in the same wayfor the extraction mechanism described next.We use the single site extraction process in the pa-per since it is relatively straightforward to implement.However, it is not necessarily the most efficient — soin this supplement we will also consider the possibilityof extraction directly from particular eigenstates of thering. This can sometimes be achieved by suitable posi-tioning of the trap: e.g. placing the trap at the ring’scentre with equal coupling to all other sites allows ex-traction from symmetric states [S4]. Unfortunately, thisdoes not straightforwardly generalise to the more com- V P ( p W ) ratchets R max forced-darkFD max no phononsNP max γ T (Hz) FIG. S4. Power (solid) and its limit (dashed) as a functionof voltage based on single site incoherent Lindblad extrac-tion. Three scenarios are shown, ratcheting (blue), forced-dark (FD, green) and NP (red). The power limit is givenby the maximum possible voltage multiplied by the current P max = I ( ω + 2 S ) [S5]. Model parameters are from Table Iof the main text. Each plot is found by varying γ t indepen-dently for the different models. The upper x -axis shows how γ t varies in the ratchets model over many orders of magnitudefor relatively small changes on the voltage scale. plex anti-symmetric dark states. For the special casesof N = 2 , C i . In the case of the ratchets andthe FD model, the ˆ C i move population from the loweststate of a particular band of ˆ H s into the lowest state inthe exciton band below, exciting the trap in the process.In the NP case, the ˆ C i go from the highest state of oneband to the highest of the band below (since these arethe only states accessed in this model). We incorporatethis process into our model through the dissipator D x, col [ ρ ] = γ x (cid:88) i D [ ˆ C i , ρ ] , (S15)where, once again, γ x is the effective extraction rate.Once on the trap, excitons are assumed to undergo chargeseparation at rate γ t , which we model as an irreversibledecay process.Fig. S2 shows the absolute power for the ratchets andNP model for the collective trapping model. This plot isequivalent to Fig. 3 of the main paper which displayedthe same quantities for the single site incoherent trap-ping model. Similarly, Fig. S3 shows the relative powerenhancement of ratcheting over both artificial models forthe collective trapping model — again this is the equiv-alent plot to Fig. 4 in the main paper but here for the V P ( p W ) γ x = γ o / ratchetsforced-darkno phonons 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 V P ( p W ) γ x = γ o ratchetsforced-darkno phonons1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 V P ( p W ) γ x =100 γ o ratchetsforced-darkno phonons 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 V P ( p W ) γ x =10 γ o ratchetsforced-darkno phonons FIG. S5. Power-Voltage characteristics for the single site incoherent extraction model. Parameters are as in Table I of themain paper except for γ x . Each panel compares the three models (ratchets, NP, FD) for a different extraction rate γ x . Theadvantage of ratcheting is most apparent for slower extraction rates, i.e. in the bottleneck region. collective extraction model. Unsurprisingly, the absolutepower extracted in the collective model is larger than theincoherent model. This is because we are here extractingdirectly from the ratchet or dark states and thus have ac-cess to the ‘entire’ exciton rather than just its projectiononto a single site of the ring. The relative advantage ofratcheting over FD has a similar form for both extractionmechanisms, though it is smaller for the collective model. POWER-VOLTAGE CURVES
In Figs. 3 and 4 of the main paper we are displayingmaximum output power as a function of S and γ x . Eachpoint in those plots is a result of generating a power-voltage curve for each choice of parameter, and findingthe maximum power. Each such curve is generated byvarying the trap decay rate γ t . We show examples ofthese power-voltage curves in Fig. S4, considering theusual three scenarios: ratchets, FD and NP. On the upper x -axis we also show the values of γ t corresponding to eachvoltage taking the example of the ratchets case; as shown γ t varies over orders of magnitude for only modest voltageshifts. These curves make use of the parameters specifiedin Table I of the main text, and a comparison with Figs. 3and 4 of the main paper shows that S = 0 .
02 eV and γ x = 10 − eV are in fact not optimal for showing the fullratcheting advantage.Further examples of P − V curves are shown in Fig. S5,each for different single site incoherent extraction rates,fixing S = 0 .
02 eV. We can see here the advantage ofratcheting in the intermediate and severe bottleneck re-gions. For very fast extraction, any absorbed photon getsimmediately transferred to the trap, so that neither dark-state protection nor ratcheting offer any advantage (thesystem is already operating as efficiently as possible). Inthis case NP achieves the same current as ratchets andFD, but ends up producing a higher resultant power asits trap transition frequency – and thus effective voltage– is higher, as explained in Sec. .
FIG. S6. Left: absolute output power as a function of S (for fixed γ x = 10 − eV). Right: absolute output power as afunction of γ x (for fixed S = 0 .
02 eV). Curves for all threescenarios – ratchets (R), NP, FD – are displayed. The twoextraction mechanisms of incoherent single site (inc) and in-coherent collective (col) are also shown. All other parametersare as specified in Table I of the main text.
EXTRACTION BOTTLENECK 2D PLOTS
In Fig. S6 we show several 2D sections of the surfacesin Fig. 3 of the main paper and Fig. S2 – i.e. we showabsolute output power as a function of S (for fixed γ x =10 − eV) in the left panel and as a function of γ x (forfixed S = 0 .
02 eV) in the right panel. These are displayedfor all three scenarios (ratchets, FD, NP) and for bothcollective and incoherent extraction mechanisms.
IMPERFECTIONSDisorder
To model disorder, we generate random site energiesthat are normally distributed around the transition fre-quency ω + 2 S = 1 . σ . The I, V value pairs for each trap decay frequency γ t are av-eraged over 100 different implementations, and are basedon the Bloch-Redfield dissipator [S7] for the optical andphonon interactions.Results for different degrees of disorder are shown inthe left of Fig. S7. For small levels of disorder σ < S ,there is little deviation from the ideal case, and 100 im-plementations are indeed enough to give fairly smoothcurves. Once σ approaches S (0.02 eV corresponds to a σ of around 1.14%), however, not only is the achievablepower with ratchets visibly reduced but our statisticalensemble is also not large enough to smooth out thoselarger performance variations, resulting in jagged-lookingcurves.Interestingly, the performance of the NP case improveswith increasing disorder. This may be due to the fact thateven in asymmetric circumstances eigenstates with a var- ied range of interesting optical properties may arise [S8]in the fully accessible Hilbert space, and the absorptiondynamics is no longer restricted to only the single lowestladder transition.We do not show the FD scenario here, as the pres-ence of disorder makes it impossible to properly distin-guish between allowed ‘ladder’ and FD ‘forbidden’ opti-cal transitions. However, we note that random samplingfor small levels of disorder (well below relative site en-ergy fluctuations on the order of a percent) – where thesymmetries and properties of system eigenstates remainlargely unperturbed – indicates that FD always performsworse than the full ratchets model, as expected.The right panel of Fig. S7 shows the averaging froma single up to 1000 implementations of randomness for σ = S/
2. Interestingly, the single run has some data-points which can even exceed the ideal, fully ordered,ratcheting performance — though the ensemble averagepower is clearly below that in the fully ordered case. Wesuspect these outliers correspond to highly efficient asym-metric configurations: For the case of a dimer and dark-state protection it has recently been established that cer-tain asymmetric arrangements can potentially outper-form symmetric ones [S8]. However, the unravelling ofthe precise interplay of mechanisms at play will be morecomplicated in this larger system, but would be an inter-esting line of investigation for a future study.Note that we keep our extraction and trap model unal-tered for our disorder calculations, i.e. we extract with atrap energy of ω t = 1 . ± S for the NP and ratchetscase, respectively. This simplifying choice may in generalentail slightly over- or underestimating the effective volt-age of our trap. However, the relative voltage discrepancyarising from this assumption is expected to be limited byenergy shifts due to disorder given by σ . As seen in bothpanels of Fig. S7, disorder in the model affects the poweroutput for both ratchets and NP much more significantlythan that, meaning our results are clearly dominated byother processes as opposed to just being an artefact ofour simplified trapping model. Non-radiative decay
We model non-radiative decay by including a full setof uni-directional Lindblad lowering processes acting inthe site basis (as opposed to the collective optical decayevents which are present due to the interaction with theshared electromagnetic environment).Increasing non-radiative decay rates significantly de-creases the achievable current and power output ofthe photocell as shown in Fig. S8. Interestingly, non-radiative decay does not affect all three scenarios equally,and the ratcheting advantage diminishes with larger non-radiative recombination rate. The ratchets states enjoyno special protection against non-radiative decay act- increasing disorder : V P ( p W ) ratchetsno phonons FIG. S7. Effect of site energy disorder on the achievable power.
Left:
The darkest blue and red curves represent the ideal caseof no disorder, the lighter curves with dots correspond to random site energy fluctuations with a standard deviation from theidealised case of 0.1%, 0.33%, 1%, and 2%. All these curves are averaged over 100 different randomly selected configurations.
Right:
Convergence of disorder curves for σ = S/ V P ( p W ) γ d = γ o / ratchetsforced-darkno phonons 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 V P ( p W ) γ d = γ o ratchetsforced-darkno phonons 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 V P ( p W ) γ d =100 γ o ratchetsforced-darkno phonons FIG. S8. Non-radiative decay processes acting independently on each site, with effective rates as shown above each panel. Allother parameters are from Table I of the main text. Aggressive non-radiative recombination severely reduces the achievablepower output of all three models, ratchets, NP, and FD, lending an edge to NP where excitations are never stored for long inexcited levels. ing on individual sites, and when they carry more thanone excitation the impact of non-radiative decay is feltmore acutely. However, the combined ratcheting anddark state protection advantage persists as long as non-radiative decay does not become the dominant loss chan-nel.
Exciton-exciton annihilation
We now discuss the impact of exciton-exciton annihi-lation (EEA) on the ratcheting efficiency enhancement.EEA is the result of a two step dynamical process [S9].First, excitons in adjacent sites on the ring can fuse to-gether, creating an excitation on a single chromophorethat has twice the energy of the first excited exciton stateso far considered. The mechanism for this is, like excitontransfer, F¨orster coupling – i.e. a dynamic dipole-dipole V P ( p W ) γ a = γ o / ratchetsforced-darkno phonons 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 V P ( p W ) γ a = γ o ratchetsforced-darkno phonons 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 V P ( p W ) γ a =100 γ o ratchetsforced-darkno phonons FIG. S9. The effect of exciton-exciton annihilation at an effective rate as shown above each panel. All other parameters arefrom Table I of the main text. As expected neither NP nor FD are much affected, and notably the ratcheting advantage onlyfully vanishes once the exciton-exciton annihilation rate exceeds the spontaneous emission rate by an order of magnitude ormore. interaction between the adjacent excitations. The sec-ond step is fast non-radiative decay of the double energystate on the excited chromophore back down to the reg-ular single exciton state.The Hamiltonian underlying the first step might beexpressed by using an extension of the notation we haveemployed in the main paper. We now allow three levelsper chromophore: the already-considered ground state | G (cid:105) and low energy exciton state | E (cid:105) , as well as thenewly introduced double energy excitation | D (cid:105) . Withthese state labels the Pauli operator we use in the maintext is σ + = | E (cid:105) (cid:104) G | , and now we define a new spin raisingoperator η + = | D (cid:105) (cid:104) E | . The Hamiltonian then becomes: H s (cid:48) = ω N (cid:88) i =1 ( σ + i σ − i + 2 η + i η − i )+ (S16) S N (cid:88) i =1 ( σ + i σ − i +1 + σ + i η − i +1 + H.c. ) , (S17)where we have made the simplifying assumption that thetwo excitation energy transfer has the same strength ( S )as the single excitation one. This is reasonable, implyinga similar transition dipole from | G (cid:105) to | E (cid:105) as from | E (cid:105) to | D (cid:105) . By diagonalizing Eq. (S16) for an N = 4 system,in a basis of states {| G (cid:105) , | E (cid:105) , | D (cid:105)} on each site, we areable to straightforwardly obtain the new eigenstates ofthe system. Again, since the Hamiltonian preserves to-tal excitation number (assuming that | D (cid:105) states count astwo excitations), then the eigenstates fall into a series ofbands, each corresponding to a different excitation num-ber. The zero excitation ground state and the four singleexcitation eigenstates are obviously unaffected by the | D (cid:105) states, but the second band now consists of ten, not six,possible states, which are now combinations of both | D (cid:105) states on single sites and | E (cid:105) states on two sites. The lowest energy state in this second band (let us label it |G(cid:105) ) would be reached following rapid relaxation follow-ing ratchet action from the first band. We have calcu-lated that |G(cid:105) contains only a 16% | D (cid:105) state character(by probability). Considering that population of the | D (cid:105) state is crucial for EEA to occur, we can conclude thatEEA will be significantly suppressed from |G(cid:105) .Moreover, we have calculated the electric transitiondipole moment between |G(cid:105) and all states in the firstband. The dipole is zero for transitions into all but oneof the first band states, and only 18% of the dipole ofa single uncoupled chromophore for the remaining state.By contrast, if the | D (cid:105) states are not included, the equiv-alent transition dipole is 41% of the single chromophorevalue. Thus, we can conclude that the lowest state of thesecond band is more protected from optical re-emissionif the | D (cid:105) states are included in the model. This meansthat including the possibility of doubly-excited sites sur-prisingly lends a certain degree of extra ‘dark-state pro-tection’ character to the second excitation band, allowingmore time for exciton harvesting, and thus improving theefficiency of the device.There are materials that do not quickly decay throughnon-radiative transition to the lowest energy state of agiven band - these materials then do not obey Kasha’srule [S10], which states that optical emission is alwaysfrom the lowest energy state of a band. For these ma-terials, the reasoning above regarding the compositionof the lowest second band state is less relevant. How-ever, in that case, the second step involved in EEA - thenon-radiative decay of the | D (cid:105) state - is also inhibited,and this too leads to an inhibition of EEA and so tomore efficient energy harvesting devices. Indeed, findingmaterials that violate Kasha’s rule for novel optoelec-tronic applications such as this is a very active area ofresearch [S11, S12].As discussed above, we expect the |G(cid:105) to have sup-pressed EEA, but we do not expect it to disappear com-pletely. We therefore look at its impact by introducingEEA it into our model by means of one-way Lindbladoperators taking population from all doubly or higherexcited states to the highest energy (ladder) state in thesingle excitation subspace. In Fig. S9 we show power-voltage characteristics with such EEA taken into account.Each plot compares the ratchets, FD and NP models, forthe value of the annihilation rate indicated above it.As expected, neither the NP case nor the FD scenarioare visibly affected. As annihilation increases, the maxi-mal current achieved by the ratchets case eventually re-duces to that of FD — but, importantly, the ratchetingadvantage only fully vanishes once the exciton-excitonannihilation rate exceeds the spontaneous emission rateby an order of magnitude or more. On close numericalinspection, FD and NP also suffer very slightly underEEA, because in both those cases the ladder of brightstates remains available and there is thus slight chanceof multiple excitation. However, unlike in the case ofnon-radiative recombination, annihilation never kills thedark-state protection advantage. DEMONSTRATION SYSTEMS
In this section we mention some possible systems inwhich the action of ratchet states might be initially de-tected.Ideally, we require a system composed of at least fourcoupled, radiation-absorbing elements, and where thespacing between each element is either much less thanthe wavelength of light, or else where the elements canbe placed such that the radiation with which they inter-act is in phase with all of them. This latter conditionallows for the possibility of using systems placed an in-teger number of wavelengths apart, and would include,for example, superconducting qubits interacting with mi-crowave cavity fields. For example, two such qubits havebeen used to demonstrate super-radiance [S13]. Includ-ing two more would enable ratchets states to be reached,initially by driving the bright excitation and then by al-tering single qubit phases through local manipulation.A system closer in spirit to what we have describedhere would involve coupled optical absorbers. For exam-ple, NV − centers in diamond are established as systemson which to test quantum information protocols [S14]and fundamental features of quantum theory [S15]. NVscan be individually addressed optically, and have spin se-lective coupling to light that allows optical measurementof their spin state. Moreover, they have been producedin patterns with a precision of around λ/
10, where λ isthe wavelength at which they emit [S16]. The prospectof collective photon emission (superrradiance) with mul- tiple NV centres has recently been studied [S17].NVs have strain dependent splitting so would havesome disorder, though as we have seen in Section above,this does not mean that ratchets states lose all function.However, it may be that silicon vacancies (SiVs) in dia-mond [S18], or alternatively (di)vacancies in SiC, makebetter test systems. For example, SiVs in diamond canbe placed with λ/
20 precision and are very stable, withinhomogeneous broadening heavily suppressed by theirinversion symmetry. For all these kinds of emitter, theF¨orster coupling could be enhanced using a waveguide orcavity to promote the interaction of several centers with acommon optical mode [S19]. Transient luminescence andabsorption techniques could be used to track the lightexcitation in time as it moves through the system, anddemonstrate behavior consistent with ratcheting.Quantum dots are another system that can ex-hibit strong optical coupling, and could for examplebe arranged in sub-wavelength patterns using DNAorigami [S20]. Evidence of collective photon emission oflateral quantum dot ensembles has also already been re-ported [S21].Of course, the final device likely to be used in a realphotocell would be a molecular system. Such systemscan be made with high reproducibility, and such thatthey are very closely spaced with respect to their emis-sion wavelength. H-aggregates of chromophores have apositive F¨orster coupling that shifts the bright (absorb-ing) state up in energy, as is the case in our model [S22].Alternatively, light-harvesting π -conjugated supramolec-ular rings can be made with twelve or more repeatingunits [S23], specifically to mimic the ring structures foundin biology. Indeed, the symmetry of this structure gener-ates a dark S − S transition in the absence of vibrationaldistortion [S24]. Further design optimization of both theindividual systems and rigidity of the ring could result ina system ready to meet the requirements for ratcheting. ∗ [email protected] † [email protected][S1] P. Jordan and E. P. Wigner, Z.Phys. , 631 (1928).[S2] T. Tokihiro, Y. Manabe, and E. Hanamura, PhysicalReview B , 2019 (1993), URL http://link.aps.org/doi/10.1103/PhysRevB.47.2019 .[S3] F. C. Spano, Physical Review Letters , 3424(1991), URL http://link.aps.org/doi/10.1103/PhysRevLett.67.3424 .[S4] K. D. B. Higgins, S. C. Benjamin, T. M. Stace,G. J. Milburn, B. W. Lovett, and E. M. Gauger, NatCommun http://dx.doi.org/10.1038/ncomms5705 .[S5] C. Creatore, M. A. Parker, S. Emmott, and A. W. Chin,Physical Review Letters , 253601 (2013), URL http://link.aps.org/doi/10.1103/PhysRevLett.111.253601 . [S6] Y. Zhang, S. Oh, F. H. Alharbi, G. S. Engel, andS. Kais, Phys. Chem. Chem. Phys. , 5743 (2015),URL http://dx.doi.org/10.1039/C4CP05310A .[S7] H.-P. Breuer and F. Petruccione, The Theory of OpenQuantum Systems (OUP, 2002).[S8] A. Fruchtman, R. G´omez-Bombarelli, B. W. Lovett,and E. M. Gauger, Phys. Rev. Lett. , 203603(2016), URL http://link.aps.org/doi/10.1103/PhysRevLett.117.203603 .[S9] V. May, The Journal of Chem-ical Physics , 054103 (2014),http://aip.scitation.org/doi/pdf/10.1063/1.4863259,URL http://aip.scitation.org/doi/abs/10.1063/1.4863259 .[S10] M. Kasha, Discuss. Faraday Soc. , 14 (1950), URL http://dx.doi.org/10.1039/DF9500900014 .[S11] K. Yanagi and H. Kataura, Nat Photon , 200 (2010),URL http://dx.doi.org/10.1038/nphoton.2010.77 .[S12] H. Ghosh, Chemical Physics Letters , 431 (2006),ISSN 0009-2614, URL .[S13] J. A. Mlynek, A. A. Abdumalikov, C. Eich-ler, and A. Wallraff, Nature Communications ,5186 EP (2014), URL http://dx.doi.org/10.1038/ncomms6186 .[S14] G. Waldherr, Y. Wang, S. Zaiser, M. Jamali, T. Schulte-Herbruggen, H. Abe, T. Ohshima, J. Isoya, J. F. Du,P. Neumann, et al., Nature , 204 (2014), URL http://dx.doi.org/10.1038/nature12919 .[S15] B. Hensen, H. Bernien, A. E. Dreau, A. Reiserer,N. Kalb, M. S. Blok, J. Ruitenberg, R. F. L. Ver-meulen, R. N. Schouten, C. Abellan, et al., Nature , 682 (2015), URL http://dx.doi.org/10.1038/nature15759 .[S16] W. F. Koehl, B. B. Buckley, F. J. Heremans, G. Calu-sine, and D. D. Awschalom, Nature , 84 (2011),URL http://dx.doi.org/10.1038/nature10562 .[S17] T. Hill, B. C. Sanders, H. Deng,http://arxiv.org/abs/1610.00679 (2016).[S18] A. Sipahigil et al., http://arxiv.org/abs/1608.05147(2016).[S19] G. Kurizki, A. Kofman, and V. Yudson, Phys. Rev. A , R35 (1996).[S20] H. Bui, C. Onodera, C. Kidwell, Y. Tan, E. Graugnard,W. Kuang, J. Lee, W. B. Knowlton, B. Yurke, and W. L.Hughes, Nano Letters , 3367 (2010), URL http://dx.doi.org/10.1021/nl101079u .[S21] M. Scheibner, T. Schmidt, L. Worschech, A. Forchel,G. Bacher, T. Passow, and D. Hommel, Nat Phys ,106–110 (2007), URL .[S22] J. L. McHale, The Journal of Physical Chemistry Let-ters , 587 (2012), URL http://dx.doi.org/10.1021/jz3000678 .[S23] M. C. O’Sullivan, J. K. Sprafke, D. V. Kondratuk,C. Rinfray, T. D. W. Claridge, A. Saywell, M. O.Blunt, J. N. O’Shea, P. H. Beton, M. Malfois, et al.,Nature , 72 (2011), URL http://dx.doi.org/10.1038/nature09683 .[S24] J. K. Sprafke, D. V. Kondratuk, M. Wykes, A. L.Thompson, M. Hoffmann, R. Drevinskas, W.-H. Chen,C. K. Yong, J. K¨arnbratt, J. E. Bullock, et al., Journalof the American Chemical Society , 17262 (2011),URL http://dx.doi.org/10.1021/ja2045919http://dx.doi.org/10.1021/ja2045919