Vacuum-Enhanced Optical Nonlinearities with Organic Molecular Photoswitches

Vacuum-enhanced optical nonlinearities with organic molecular photoswitches

Marina Litinskaya1, ∗ and Felipe Herrera2, 3, † 1Department of Physics & Astronomy and Department of Chemistry, University of British Columbia, Vancouver, Canada, V6T 1Z1 2Department of Physics, Universidad de Santiago de Chile, Av. Ecuador 3493, Santiago, Chile. 3Millenium Institute for Research in Optics MIRO, Chile. (Dated: April 6, 2018) We propose a cavity QED scheme to enable cross-phase modulation between two arbitrarily weak classical ﬁelds in the optical domain, using organic molecular photoswitches as a disordered intracavity nonlinear medium. We show that a long-lived vibrational Raman coherence between the cis and trans isomer states of the photoswitch can be exploited to establish the phenomenon of vacuum-induced transparency (VIT) in high-quality microcavities. We exploit this result to derive an expression for the cross-phase modulation signal and demonstrate that it is possible to surpass the detection limit imposed by absorption losses, even in the presence of strong natural energetic and orientational disorder in the medium. Possible applications of the scheme include the development of organic nanophotonic devices for all-optical switching with low photon numbers.

Organic chromophores and semiconducting polymers ulation between the probe and signal fields can reach are known for having very large optical nonlinearities [1], values that are orders of magnitude larger than without with applications in lasing [2], frequency conversion [3], the cavity, exceeding the detection limit imposed by ab- nonlinear microscopy [4], and all-optical switching [5]. sorption losses. Replacing the control laser by a strongly However, the typical magnitude of the nonlinear suscep- coupled vacuum represents a clear advantage for organic tibility in organic materials is still orders of magnitude materials with low laser damage thresholds [2]. Although below those needed to enable nonlinear optical effects at EIT has been measured at low temperatures in solid state the level of single photons, an important prerequisite in systems with negligible inhomogeneous broadening [34– photonic quantum technologies [6]. In recent years, the 37], condensed-phase VIT has yet to be demonstrated in demonstration of strong and ultrastrong coupling of or- systems with strong inhomogeneously broadening, such ganic molecules with the quantized electromagnetic vac- as organic molecular ensembles. uum in optical and infrared cavities [7–10] has stimulated The envisioned light-matter scheme is shown in Fig.1. the study of single-photon control of chemical reactiv- The ground electronic potential of the photoswitch (S0) ity [11–13], energy transport [14, 15], charge transport has two deep wells in the isomerization coordinate corre- [16, 17], and spectroscopy [18–21]. However, the manip- sponding to the cis and trans molecular isomers [25]. The ulation of polariton coherences to enhance the natural lowest vibrational state of the trans isomer, is the stable optical nonlinearities of organic materials still remains a ground state of the system 1 . The meta-stable ground largely unexplored topic [22–24]. state 2 corresponds to the lowest| i vibrational level of the | i We propose a scheme to observe cross-phase modula- cis isomer potential well. Thermal isomerization of state tion between two arbitrarily weak classical fields within a nanoscale cavity, using disordered organic molecular photoswitches as the intracavity medium. Molec- (a) S (a) 2 4 ular photoswitches are commonly used in photochem- | i istry due to their ability to undergo spectrally-resolved IVR (b) S1 3 photoisomerization processes [25], for applications such | i as optical information storage and controlled cataly- 31 32 ! arXiv:1804.01816v1 [quant-ph] 5 Apr 2018 s sis [26–28]. In this work, we suggest that natural ! features in the vibrational structure of organic photo- !p c switches can be exploited to enable the observation of S0 vacuum-induced transparency (VIT [29–32]), a variation 2 1 | i of electromagnetically-induced transparency (EIT [33]) | i Cis in which a cavity vacuum–rather than a strong control Trans laser– is used to drive an internal coherence in the material. We further demonstrate that under conditions of FIG. 1: Light-matter coupling scheme. (a) Energy level struc- VIT, the figure-of-merit for intracavity cross-phase mod- ture of molecular photoswitches. Straight arrows indicate near-resonant light-matter couplings at the cavity (ωc), probe (ωp) and signal (ωs) frequencies. Curly arrows indicate non- radiative molecular decay processes. (b) Representative absorption spectrum of the molecular photoswitch. The highest ∗Electronic address: [email protected] †Electronic address: [email protected] trans absorption peak corresponds to the transition S0 → S2. 2

2 is strongly suppressed [25], resulting in lifetimes for state| i 2 of up to a few days at room-temperature [38, 39]. We associate| i the lowest two excited electronic potentials of the molecular photoswitch, S1 and S2 [40, 41], with the states 3 and 4 , respectively. The vibrational structure of the| excitedi | potentialsi is not resolved in condensed phase, but S0 has well-defined torsional modes with frequencies in the range ωv 160 190 meV/~ [42–44]. The level structure in Fig.1 generalizes∼ − the atomic scheme introduced in Refs. [45, 46] to molecular systems with multiple ground state isomers. Our model does not require an exact knowledge of the molecular potentials. Instead, we impose a set of VIT FIG. 2: Disorder-averaged absorptive (Imhχp i) and dis- constraints on the inhomogeneously-broadened photo- VIT persive response (Rehχp i) of molecular photoswitches as a switch absorption spectrum outside the cavity, which function of the probe detuning ∆p = ωp − hω31i, for a res- we illustrate in Fig.1b. We assume that the photo- onant cavity field ωc = hω32i. (a)-(b) Uniform orientational switch spectrum has: (i) a well-resolved band associ- disorder on the vacuum Rabi frequency with mean amplitude ated with the cis transition 2 3 , resonant with Ω0 = 1.2γ. The disorder-free lineshape is shown for compari- | i → | i the cavity frequency ωc;(ii) a well-resolved band asso- son (dashed line). (c)-(d) Gaussian disorder on ω31 for σ = 6γ ciated with the trans transition 1 3 , near resonant and Ωc = 1.2γ. The VIT linewidth ΓVIT is graphically de- | i → | i fined. γ is the homogeneous absorption linewidth. with the probe frequency ωp > ωc;(iii) a well-resolved high-frequency band associated with the cis transition 2 4 , weakly driven by a blue-detuned signal field | i → | i at frequency ωs > ωp. These spectral requirements can cavity field polarization. be met using so-called orthogonal molecular photoswiches Although in general the dissipative dynamics of photoi- [47–49]. somerization involves a complex interplay between non- For a single molecular emitter, the light-matter cou- adiabatic dynamics and non-secular relaxation [51, 52], pling scheme in Fig.1 can be modelled using the Hamil- in this work we adopt a simpler Lindblad quantum mas- tonian ter equation approach to describe dissipation. We account for empty cavity decay with bare photon lifetimes Hˆ = ω aˆ†aˆ + ω 2 2 + ω 3 3 + ω 4 4 S c 21 | i h | 31 | i h | 41 | i h | assumed in the range 1/κ 1 100 ps, which can † −iωpt ∼ − +Ωc( 3 2 aˆ + 2 3 aˆ ) + Ωp( 3 1 e (1) be achieved using high-Q dielectric cavities [53]. Non- | i h | | i h | | i h | iωpt −iωst iωst + 1 3 e ) + Ωs( 4 2 e + 2 4 e ), radiative decay of the excited state S2 S1 by in- | i h | | i h | | i h | tramolecular vibrational relaxation (IVR)→ is also taken where Ω and Ω are the probe and signal semiclassi- into account, with decay lifetime 1/ΓIVR 0.1 ps [43], p s ∼ cal Rabi frequencies, respectively. Ω is the cavity vac- as well as non-radiative decay S1 S0 into the trans c → uum Rabi frequency, anda ˆ is annihilation operator of and cis ground states, at rates Γ31 and Γ32, respectively. the cavity field. In a dressed-state picture, our elec- The homogeneous probe absorption linewidth is defined tronic basis must be supplemented with the cavity states as γ = Γ31+Γ32, with typical excited state lifetimes in the to read: 1˜ 1; 0 , 2˜ 2; 1 , 3˜ 3; 0 , and range 1/γ 0.1 0.5 ps [50, 54]. Finally, we account for c c c ∼ − 4˜ 4;| 1i. ≡ The | probei | i ≡ field | thusi | drivesi ≡ | thei mate- pure dephasing of the coherence between the vibrational | i ≡ | ci rial transition 1˜ 3˜ , the signal field the transition ground states 1 and 2 at the rate Γpd, with typical | i ↔ | i | i | i 2˜ 4˜ , and the cavity field strongly admixes the dephasing times in range 1/Γpd 1 100 ps [55–58], ∼ − nearly-degenerate| i ↔ | i dressed states 2˜ and 3˜ . depending on the molecular species, solvent and temper- | i | i We model energy disorder in the Hamiltonain HˆS by ature. Having defined a master equation for the problem, defining the random transition frequencies ω31 = ω31 + we solve for the stationary reduced density matrix of the δ , ω = ω + δ and ω = ω + δ ,h wherei system to obtain an expression for disorder-free medium 31 32 h 32i 32 42 h 42i 42 ωji corresponds to the band center frequency and δji susceptibility at the probe frequency, denoted χp, to first his ai random static fluctuation that gives rise to the in- order in Ωp. The derivation can be found in the Sup- homogeneous linewidths σji. Inhomogeneous linewidths plemental Material (SM). We then obtain the disorder- averaged susceptibility χ , by integrating χ –both nu- due to energy disorder are typically in the range 150–200 h pi p meV for organic photoswitches [50], magnitudes that far merically and analytically– over random frequency con- exceed a typical homogeneous linewidth. In addition to figurations and dipole orientations. In what follows, we energy disorder, below we also model orientational disor- use the results of this procedure to describe the photo- der in the system by writing the vacuum Rabi frequency physics of inhomogeneously broadened molecular photo- as Ωc = Ω0 cos θ, where Ω0 is a constant amplitude and switches inside nanoscale optical cavities. θ is a uniformly distributed random angle between the We begin our analysis of the intracavity response by molecular transition dipole moment and the space-fixed assuming that no signal field is present (Ωs = 0). In Fig. 3

2, we show the corresponding disorder-averaged suscep- merit η Φ/αL = Re χp /2Im χp to quantify the de- VIT ≡ h i h i tibility χp as a function of the mean probe detun- gree by which it is possible to interferometrically resolve ing ∆ h ω i ω , for the representative decay ratios the nonlinear phase shift Φ at frequency ω . Detectable p ≡ p − h 31i p Γpd/γ = 0.01 and κ/γ = 0.0001. We numerically average nonlinear signals require η > 1. the homogeneous response over a large number of disor- We developed an analytical approximation for the der configurations, separately studying orientational dis- disorder-averaged susceptibility χp in the presence of order (panels a and b) and energy disorder (panels c and both probe and signal fields. Weh integratei the disorder- d). For energy disorder, we take the random frequency free susceptibility over the random fluctuations (δ31, δ42) fluctuations δji from independent Gaussian distributions assuming that they belong to independent Lorentzian with the same standard deviation, i.e. σji = σ. We distributions with different widths. We use Lorentzians also assume that the cavity detuning ∆c ωc ω32 instead of more realistic Gaussian distributions [44, 50] vanishes. Figure2 clearly shows the characteristic≡ − h sig-i because the latter becomes analytically intractable for natures of VIT despite disorder: strong suppression of the cross-phase modulation signal that we seek to de- the probe absorption (transparency) and steep disper- scribe. Our approach is motivated by previous work on sion (slow light), over a narrow frequency window around Doppler-broadened EIT [60, 61], and is justified below two-photon resonance condition ∆p ∆c = 0 [33]. More- by comparison with a Gaussian numerical averaging. We over, panels2a-b show that uniform− orientational disor- show in the SM, that the mean susceptibility as a func- der does not significantly alter the photoswitch response tion of the probe detuning x ∆p can be written in the from the disorder-free case, reinforcing previous studies form ≡ on the minor importance of orientational disorder in or- xA (x) Ω2(x x ) ganic cavities [59], a result that can also be understood s c s 2 ηp(x) = − 2 − , (2) analytically (see SM for details). h i −Σ31As(x) + Ωc (γ21 + γs)

Panels2c,d show that even for strong inhomogeneous where γ21 (κ/2+Γpd) is the decay rate of the coherence broadening due to energy disorder (σ = 6γ), a narrow ≡ between dressed states 1˜ and 2˜ .Σ31 (γ/2 + σ31) VIT absorption dip with narrow linewidth Γ [60, 61] | i | i ≡ VIT is the total probe trans absorption linewidth, with σ31 can still persist on top of a broader Gaussian absorption being the inhomogeneous contribution. We have defined background, under two-photon resonance. For systems 2 2 the function As(x) (x xs) +(γ21 +γs) , characterized with arbitrary σ, which is the dominant energy scale in ≡ − by the shift xs λs∆s and the width γs λsΣ41, written molecular photoswitches [50], we can achieve VIT as long in terms of the≡ dimensionless signal parameter≡ as the frequency hierarchy κ Γpd Ωc . γ holds. We ∼ 2 2 2 find survival of VIT to inhomogeneous broadening un- λs Ωs/(∆s + Σ41), (3) der the assumption that the static energy fluctuations ≡ with ∆ = ω ω being the mean signal detuning of states 1˜ and 2˜ are equal, i.e., δ31 = δ32 to a very s s 42 | i | i and Σ κ/2 +− Γ h i/2 + σ , where σ is the inhomo- good approximation. The quality of VIT degrades as we 41 ≡ IVR 42 42 allow the ground isomer potential minima to fluctuate in- geneous linewidth of the cis absorption band. Equation dependently. This sets a constraint on the contribution (2) shows that within our Lorentzian disorder model, the of inhomogenous broadening on the vibrational Raman inhomogeneous linewidths σ31 and σ42 simply add up to the corresponding homogeneous linewidths, allowing for frequency ω21 to be much smaller than the excited state a transparent interpretation of the nonlinear system pho- linewidth γ. If ω21 were to fluctuate significantly, the intracavity Raman dark state D = √a 1˜ + √1 a 2˜ tophysics. (0 < a < 1), generated under VIT| i conditions| i [33],− would| i Outside the cavity (Ωc = 0), Eq. (2) simply reduces rapidly dephase, breaking the destructive interference ef- to the linear response result ηp = ∆p/2Σ31, which near resonance gives η h1i due− to the absorption fect that causes the VIT absorption dip. However, given h pi that both cis and trans isomers belong to the same elec- of the probe. In the presence of the cavity vacuum, but without signal driving (Ω = 0), the figure-of-merit tronic potential, it can be expected that δ31 δ32 /γ 1, s independent of the molecular solvent. | − | ηp(Ωs = 0) ηVIT can be considered as the degree ofh coherencei in ≡ the h responsei of the medium to the probe Having established the physical conditions for achiev- field. We show in the SM that in this VIT-regime, Eq. ing VIT in disordered molecular photoswitches, we now (2) reduces to discuss a cross-Kerr scheme involving the simultaneous driving of the intracavity medium by both probe and Ω2x η (x) c 1, (4) signal fields. We focus on the phase shift Φ experienced h VIT i ≈ 2(Σ x2 + γ Ω2) by the probe field due to co-propagation with a signal 31 21 c field over a path length L. Attenuation at the probe under the conditions γ21 x Ωc. The large figure-of- frequency is quantified by the absorption coefficient α. merit predicted by Eq. (4) is a direct consequence of the Under conditions of VIT, both Φ and α not only de- transparency established for the probe field by the cavity pend on the probe and signal field frequencies and Rabi vacuum. frequencies, but also on the cavity vacuum parameters For a system dominated by inhomogeneous broaden- [29, 30]. Following Ref. [62], we define the figure-of- ing (γ σ ) inside a high-Q microcavity (κ Γ ), 31 pd 4

Eq. (4) has a maximum at the probe detuning x∗ = p Ωc Γpd/σ31, blue-shifted from the VIT absorption min- imum, giving the optimal ﬁgure-of-merit p ηmax = Ω / 16 Γ σ . (5) h VIT i c pd 31

With typical photoswitch parameters Γpd 1 THz and σ 50 THz, reaching ηmax > 1 would∼ require 31 ∼ h VIT i Ωc 120 meV. Vacuum Rabi frequencies of this order of magnitude≥ and above can be obtained either by strongly confining the vacuum field to picoscale dimensions [63], or by exploiting collective Rabi coupling in larger cavities FIG. 3: Nonlinear figure-of-merit hηpi for different signal pa- [53]. rameters λs. (a) hηpi as a function of probe detuning ∆p for We next discuss the behaviour of Eq. (2) under simul- λs = 0 (blue line), λs = 0.5 (red) and λs = 1 (green). Solid taneous probe and signal driving (Ωs > 0). We refer to curves corresponds to a Gaussian disorder model and dashed this as the VIT-Kerr regime, where the figure-of-merit is lines to Lorentzian disorder. Arrows indicate the maximum directly related with the magnitude of a cross-phase mod- figure-of-merit hηmaxi for the Gaussian case. (b) hηmaxi as ulation signal [62]. Coupling to the signal field tends to a function of λs for Gaussian (σ = 5γ) and Lorentzian dis- destroy VIT, by weakly admixing the dark state D (see order distributions with equal width (FWHM). The shaded region corresponds to signal-to-noise ratios below unity for above), with the fast-decaying excited dressed state| i 4˜ . | i cross-phase modulation. In both panels we set γ21 = 0.002γ The maximum achievable figure-of-merit ηmax subject and Ωc = 0.8γ. to signal driving therefore cannot exceed theh VITi bound max in Eq. (5), i.e., ηmax ηVIT for any λs, with the equality sign holdingh onlyi ≤ when h λi = 0. s case. In the regime where κ < √NΩc < γ, we can replace We can use Eq. (2) to find the constraints on x and s one-body terms of the form (Ωc 3i 2i aˆ + H.c.) with γs that allow the nonlinear figure-of-merit to approach √ | i h | max the collective coupling term ( NΩc αcis Gcis aˆ+H.c.), the VIT limit ηVIT (see SM for details). For instance, PN | i h | h i where αcis = 21, 22, , 3i, , 2N /√N is a in realistic organic systems dominated by inhomogeneous | i i=1 | ··· ··· i totally-symmetric collective excitation [21], and Gcis broadening with Σ41 Σ31 σ and assuming Ωc/ ∆s | i ≡ p ≈ ≈ | | ≤ 21, 22, , 2N is the cis ground state of the ensemble. γ21/σ, our Lorentzian disorder model predicts that it |The signal··· fieldi thus weakly drives the coherence between is possible to reach ηmax & 1 under a single constraint h i the cis dressed ground state Gcis 1c and state 4i 1c on the dimensionless signal parameter, given by on the i-th molecule. Our single-emitter| i | i results for| i |χ i h pi thus remain valid after replacing Ω by √NΩ . λs . γ21/σ 1. (6) c c In summary, we propose an organic cavity scheme to Small values of λs are obtained by using weak signal achieve large cross-phase modulation signals using ar- fields, large (blue) signal detunings, or both. bitrarily weak probe and signal fields. The scheme ex- We can now compare the predictions of our Lorentzian ploits a long-lived vibrational coherence between the cis disorder model in Eq. (2) with a more realistic Gaussian and trans ground vibrational states in a class of chro- model. Figure3a shows the η(x)-profiles predicted by the mophores known as molecular photoswitches, to establish Lorentzian model and the result of numerically integrat- conditions for vacuum-induced transparency inside high- ing over a large number of Gaussian configurations for δ31 Q optical microcavities. The predicted Kerr nonlinearity and δ42, again assuming that δ31 = δ32 as in Fig.2. We is found to exceed the corresponding cavity-free values by find that even when the Gaussian and Lorentzian noise orders of magnitude, even in the presence of strong en- distributions have equal widths (FWHM), Gaussian dis- ergy and orientational disorder in the organic medium. order consistently allows for higher values of ηmax . This Our results may thus pave the way for the development behaviour is captured in panel3b, which clearlyh i shows of novel integrated nanophotonic devices for all-optical that for our choice of material linewidths, a system with switching with narrow laser pulses at low power levels, Gaussian disorder can surpass the limit η = 1 for using organic instead of inorganic materials [64] as the h maxi values of λs that are nearly an order of magnitude higher coherently driven nonlinear medium. than those predicted by Eq. (6), which reduces the ex- perimental constraints on the signal field. Figure3b also Acknowledgments shows that the Gaussian asymptote on ηmax as λs 0, is about two times higher than the Lorentzianh i bound→ from Eq. (5). We also find that for values of λs of or- We thank Stéphane Kéna-Cohen and Yaroslav Is- der 10−2 and above, the predictions of the Gaussian and polatov for comments. M.L. also thanks the De- Lorentzian disorder models become indistinguishable. partment of Physics at USACH for hospitality dur- Finally, we note that the single-emitter Hamiltonian ing early stages of this work. F.H. is supported HˆS in Eq. (1) can be generalized to the many-particle by PAI 79140030, FONDECYT Iniciación 11140158, 5

Proyectos Basal USA 1555-VRIDEI 041731, and Millenium Institute for Research in Optics.

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Appendix A: Derivation of the disorder-free susceptibility for organic photoswitches

1. Optical Bloch Equations for Cavity-Dressed States

We model the dynamics of the molecule-cavity reduced density matrixρ ˆS by a quantum master equation of the form (~ = 1 throughout) d ρˆ = i[Hˆ , ρˆ ] + [ˆρ] + [ˆρ ] + [ˆρ ] + [ˆρ ] (A1) dt − S S Lκ LS2 S LS1 S Dpd S where the Hamiltonian HˆS is given by Eq. (1) in the main text. The ﬁrst dissipator in Eq. (A1) corresponds to cavity decay due to photon leakage into the far ﬁeld at the rate κ, given by the Lindblad form

[ˆρ ] = (κ/2) 2ˆaρˆ aˆ† aˆ†aˆρˆ ρ aˆ†aˆ , (A2) Lκ S S − S − s

The second dissipator represents the decay of the excited potential S2 to S1 via intramolecular vibrational relaxation (IVR) at the rate ΓIVR, given by

[ˆρ ] = (Γ /2) (2 3 4 ρˆ 4 3 4 4 ρˆ ρˆ 4 4 ) (A3) LS2 S IVR | i h | S | i h | − | i h | S − S | i h | The third dissipator represents non-radiative decay through a conical intersection followed by IVR, from the excited- potential S1 to the cis and trans manifolds in the ground potential S0, given by

[ˆρ ] = (Γ /2) (2 1 3 ρˆ 3 1 3 3 ρˆ ρˆ 3 3 ) LS1 S 31 | i h | S | i h | − | i h | S − S | i h | +(Γ /2)(2 2 3 ρˆ 3 2 3 3 ρˆ ρˆ 3 3 ) (A4) 32 | i h | S | i h | − | i h | S − S | i h | We ﬁnally introduce an ad-hoc non-Lindblad term in Eq. (A1) to describe pure dephasing of the vibrational coherence between ground vibrational states in the trans and cis manifolds of S0 at the rate Γpd, given by

[ˆρ ] = Γ ( 1 1 ρˆ 2 2 + 2 2 ρˆ 1 1 ) (A5) Dpd S − pd | i h | S | i h | | i h | S | i h | We neglect any contribution from the cis-to-trans thermal isomerization reaction in the master equation, given that the isomerization rate Γ21 is negligibly low in comparison with all other dynamical processes in the problem. Before proceeding with the derivation of the probe susceptibility χp from the quantum master equation [Eq. (A1)], we introduce a convenient notation for the matrix elements ofρ ˆS that reads

ρmn(t) = i; m ρˆ(t) j; n , (A6) ij h c| | ci where i and j represent molecular states (i, j = 1, 2, 3, 4). m and n represent cavity Fock states with photon | i | i | ci | ci numbers mc and nc, respectively. We also define slowly-varying amplitudes for selected elements of reduced density 00 matrix, to remove fast oscillations from the equations of motion. For the one-photon coherence ρ13(t), we define 00 00 00 −iωpt the slowly-varying amplitude σ13(t) by the relation σ13 = ρ13 e . We also define the slowly-varying variables 01 −iωpt 01 01 01 01 −i(ωp+ωs)t 01 01 −iωst 01 11 −iωst 11 σ12 = e ρ12, σ32 = ρ32, σ14 = e ρ14, σ34 = e ρ34, and σ24 = e ρ24. In terms of these slowly- varying amplitudes, we obtain from Eq. (A1) the following coherence equations of motion

σ˙ 00 = i(ω ω ) σ00 γ σ00 iΩ (σ00 σ00) + iΩ σ01 (A7a) 13 31 − p 13 − 31 13 − p 33 − 11 c 12 σ˙ 01 = i(ω + ω ω )σ01 γ σ01 iΩ σ01 + iΩ σ00 + iΩ σ01 (A7b) 12 21 c − p 12 − 21 12 − p 32 c 13 s 14 σ˙ 01 = i(ω ω )σ01 γ σ01 iΩ (σ11 σ00) iΩ σ01 + iΩ σ01 (A7c) 32 − 32 − c 32 − 32 32 − c 22 − 33 − p 12 s 34 σ˙ 01 = i(ω + ω ω ω )σ01 γ σ01 iΩ σ01 + iΩ σ01 (A7d) 14 41 c − p − s 14 − 41 14 − p 34 s 12 σ˙ 01 = i(ω + ω ω )σ01 γ σ01 iΩ σ01 iΩ σ11 + iΩ σ01 (A7e) 34 43 c − s 34 − 43 34 − p 14 − c 24 s 32 σ˙ 11 = i(ω ω )σ11 γ σ11 iΩ σ01 + iΩ (σ11 σ11) (A7f) 24 42 − s 24 − 42 24 − c 34 s 22 − 44 8 where we introduce the decay rates

γ = Γ /2 + Γ /2 γ/2 (A8a) 31 31 32 ≡ γ21 = κ/2 + Γpd (A8b)

γ32 = κ/2 + Γ31/2 + Γ32/2 (A8c)

γ43 = κ/2 + ΓIVR/2 (A8d)

γ42 = κ + γIVR/2 (A8e)

γ41 = γ43. (A8f)

Equations (A8a) and (A8b) deﬁne the homogeneous probe and Raman linewidths γ and γ21, in the notation from the main text. The vibrational pure dephasing rate Γpd determines the cavity-dressed Raman lifetime 1/γ21 for high-Q cavities with κ γ. In deriving Eqs. (A7a)-(A7f), we neglect the contribution of states such as 2, 0c , 3, 1c , or 4, 0c , which are neither populated nor driven under our imposed assumptions of stationarity and weak| signali | andi probe| driving.i Accounting 11 for such states would result, for example, in the addition of an extra term proportional to σ13 in the right-hand side 00 of equation (A7a) forσ ˙ 13, term that can be shown to vanish in the stationary limit. In other words, the set of Eqs. (A7a-f) do not correspond to a complete description of the system coherences, but can be considered as a minimal set of equations of motion that can account for the non-linear optical response of our system of interest.

2. Homogeneously-broadened intracavity susceptibility

00 We look for the stationary one-photon probe coherence σ13 from Eqs. (A7). The medium polarization at the probe 00 frequency is then given by P (ωp) = d13σ , from semiclassical closure. Using Ωp = d31Ep/2~ and P (ωp) = χpEp, 31 − the probe susceptibility χp for homogeneously-broadened organic molecular photoswitches can be written as K χp = 0 00 , (A9) (∆31 + I0) + i(γ31 + I0 )

2 where K = d /2~ is proportional to the oscillator strength of the cavity-free probe absorption peak, ∆31 = ωp ω31 −| 13| − is the probe detuning from the trans transition. In the main text we use the notation ∆31 ∆p. 0 00 ≡ In Eq. (A9), we have introduced the complex nonlinear quantity I I + iI , given by 0 ≡ 0 0 Ω2(∆ + iγ )(iγ ∆ ) I = c 41 41 32 − 32 , (A10) 0 (iγ ∆ )Ω2 + (∆ + iγ )Ω2 (∆ + iγ )(∆ + iγ )(iγ ∆ ) 32 − 32 s 41 41 p − 21 21 41 41 32 − 32 where ∆ = ω ω is the cavity detuning from the cis absorption resonance, ∆ = ω ω ω is the two-photon 32 c − 32 21 p − c − 21 (Raman) detuning, and ∆41 = ωp ωc + ωs ω41 is the three-photon detuning. The term in the denominator 2 − − proportional to Ωp describes self-induced transparency, and is negligibly small in low-Ωp limit. In what follows we use a simpliﬁed formula

Ω2(∆ + iγ ) I = c 41 41 . (A11) 0 Ω2 (∆ + iγ )(∆ + iγ ) s − 21 21 41 41

By setting Ωs = 0 in Eq. (A11), the susceptibility χp in Eq. (A9) reduces to the standard VIT form [29, 30]

K(∆ + iγ ) χVIT = 21 21 . (A12) p (∆ + iγ )(∆ + iγ ) Ω2 31 31 21 21 − c For later convenience, we introduce the dimensionless parameter

2 Ωc λc = 2 2 , (A13) (∆21 + γ21) to rewrite Eq. (A12) as

K χVIT = . (A14) p (∆ λ ∆ ) + i(γ + λ γ ) 31 − c 21 31 c 21 9

VIT Appendix B: Derivation of hχp i for disordered photoswitches without signal driving

VIT In this Appendix, we derive expressions for the intracavity mean probe susceptibility χp , separately averaged over orientational and energy disorder in the organic photoswitch medium. h i

1. VIT with orientational disorder

For an organic system with orientational disorder, disorder average can be carried out by an integration of the form χVIT = R π/2 dθ χVIT(cos θ)P (θ), where χVIT is the uniform susceptibility given by Eq. (A12), and P (θ) = 1/2π, h p iθ −π/2 p p with π/2 θ π/2 is a uniform distribution for the molecular cis transition dipole moment with respect to the − ≤ ≤ cavity ﬁeld polarization. By writting the vacuum Rabi frequency as Ωc(θ) = Ω0 cos θ, we can express the mean susceptibility χVIT as h p iθ 1 Z 2π dθ χVIT = K(∆ + iγ ) , h p iθ 21 21 2π cos2 θ (Z + iZ ) 0 − 1 2 i sgn(Z2) = K(∆21 + iγ21)p (B1) (Z + iZ )(1 (Z + iZ )) 1 2 − 1 2 2 2 where Z1 = (∆31∆21 γ31γ21)/Ω0 and Z2 = (∆31γ21 + ∆21γ31)/Ω0. Assuming for simplicity that γ21 = 0 and rearranging terms, we arrive− at the expression VIT ∆21 B + i(A + Ω) χ θ K | | , (B2) h p i ≡ − √2 Ω√A + Ω

2 2 2 2 2 where we have defined A = ∆21 ∆31(Ω0 ∆21∆31) + ∆21γ31 , B = ∆21γ31(Ω0 2∆21∆31), and Ω = √A + B . Equation (B2) is exact within the assumptions− involved in Eqs. (A7a)-(A7f), and− has excellent agreement with the numerical integration of Eq. (A9) over a large number of angular disorder configurations, as shown in Figure B.1a. The comparison of the VIT linewidth ΓVIT for a system with orientational disorder and the disorder-free case in panel B.1b demonstrates the fluctuations of the dipole orientation preserves the linear scaling ΓVIT Ωc characteristic of homogeneously-broadened EIT [1,2], but with a higher slope. ∼

VIT FIG. B.1: VIT lineshape for uniform orientational disorder. (a) Average Im[hχp i] as a function of probe detuning ∆p for Ω0 = 1.2 γ. Solid line corresponds to the analytically expression in Eq. (B2), and circles correspond to numerical integration over a large number of disorder conﬁgurations. The homogeneous susceptibility is also shown for comparison (dashed line). (b) Scaling of the VIT linewidth ΓVIT with the Rabi amplitude Ω0, obtained both analytically (solid line) and numerically (crosses). The homogeneous scaling is also shown for comparison (dashed line).

2. VIT with energy disorder

We average (A14) over energy disorder, by writing ∆31 = ∆31 +δ31, and integrating over a Lorentzian distribution 2 h 2 i of the energy fluctuation given by P (δ31) = σ3/[π(δ31 + σ3)], with σ3 being the width of the energy (detuning) 10 distribution. Note that our assumption of correlated energy fluctuations of the states 1 and 2 implies that ∆ = | i | i 21 ωp ωc ω21 is a well-defined quantity that does not fluctuate over the molecular ensemble. The resulting integral − − VIT has one pole (z = iσ3) in the upper half of the complex plane. The pole arising from the denominator of χp in Eq. (A14) always lays in the lower half-plane. We calculate the integral by closing the integration path in the upper half-plane and applying Cauchy’s theorem. The result gives

K χVIT = , (B3) h p iE (∆ λ ∆ ) + i(Σ + λ γ ) 31 − c 21 31 c 21 where we have defined the combined linewidth Σ31 = γ31 + σ3. Comparing this result with the homogeneous susceptibility in Eq. (A14) we see that the energy disorder average with static ω21 results in an additive contribution to the probe transition linewidth given by σ3. This explains the survival of VIT conditional on the smallness of γ21 and the fact that it is not sensitive to the decay rate γ31, which is eliminated by the opening of the transparency window. The analytical expression in Eq. (B3) is in qualitative agreement with the results in Figure2c,d, which were obtained numerically by averaging Eq. (A12) over a large number of detuning configurations δ31, distributed according to a Gaussian probability density. Small quantitative discrepancies arise due to the difference between Gaussian and Lorentzian distributions. Gaussian distributions are more relevant for the energy disorder than Lorentzians. However, analytically averaging Eq. (A12) over a Gaussian distribution results in a much less intuitive answer than our result in Eq. (B3), in terms of integral special functions. Our Lorentzian ansatz can thus also provide a reasonably accurate description of the system response.

Appendix C: Derivation of hχpi for photoswitches with energy disorder

Starting from the expression for the disorder-free susceptibility in Eq. (A9), with I0 as given in Eq. (A11), we introduce the random transition frequencies ω31 = ω31 δ31, ω32 = ω32 δ32, and ω42 = ω42 δ42, where ω represents a mean value and δ a random statich ﬂuctuationi − describedh byi − the normalized distributionh i − function h i P (δ). These random frequencies give rise to the random detunings ∆31 ωp ω31 + δ31 = ∆31 + δ31 and ∆ ω ω + ω ( ω δ ω + δ + ω δ ) = ∆ + (δ≡ δ−+ h δ i), where ∆h iis the mean 41 ≡ p − c s − h 42i − 42 − h 32i 32 h 31i − 31 h 41i 42 − 32 31 h 41i three-photon detuning. At this point we make an assumption that the transition frequencies ω31 and ω32 undergo identical ﬂuctuations, i.e., δ31 = δ32, and δ42 = δ41. We then use ∆41 ∆41 + δ41, with ∆21 = ωp ωc ω21 being a deterministic quantity. ≡ h i − − We further assume that the random energy shifts δ31 and δ41 are distributed according to the Lorentzian function: 2 2 2 2 P (δ31) = (1/π)σ3/(δ31 + σ3) and P (δ41) = (1/π)σ4/(δ41 + σ4). Under these assumptions, we can obtain the disorder- averaged susceptibility from the integral

Z ∞ Z ∞ χp = dδ31 dδ41 [χp(δ31, δ41)P (δ31)P (δ41)] (C1) h i −∞ −∞ σ σ Z ∞ Z ∞ dδ dδ D (D + δ ) Ω2 = K 3 4 31 41 21 41 41 − s , π2 [δ2 + σ2][δ2 + σ2] × D (D + δ )(D + δ ) Ω2(D + δ ) Ω2(D + δ ) −∞ −∞ 31 3 41 4 21 31 31 41 41 − s 31 31 − c 41 41 where we have deﬁned the complex detunings D21 ∆21 + iγ21, D31 = ∆31 + iγ31 and D41 = ∆41 + iγ41. Carrying out the double contour integration in Eq. (≡C1 h), wei obtain the followingh result:i h i

K χp E = 0 00 , (C2) h i (∆31 + IE) + i(Σ31 + IE)

0 00 where the complex quantity IE = IE + iIE, can be written as

Ω2(∆ + iΣ ) I = c 41 41 . (C3) E Ω2 (∆ + iγ )(∆ + iΣ ) s − 21 21 41 41 with Σ31 = γ31 + σ3, and Σ41 = γ41 + σ4, as before. The disorder-averaged equations (C2) and (C3) diﬀer from the homogeneous equations (A9) and (A11) only by the magnitude of the broadenings, which now account for the disorder. 11

Appendix D: Optimal nonlinear coherent signals for photoswitches with energy disorder

As discussed in the main text, we characterize the nonlinear signal coherence by the ﬁgure-of-merit η = Re[χp]/(2Im[χp]). Detectible coherent (dispersive) signals require η > 1. Our goal is to understand whether η can still exceed unity in the presence of energy disorder for an intracavity photoswitch medium.

1. Figure-of-merit in the absence of the signal ﬁeld: Pure VIT regime

VIT VIT VIT In the absence of the signal ﬁeld (Ωs = 0), we focus on the properties of ηVIT = Re[χp ]/(2Im[χp ]), with χp given by the disorder-free expression in Eq. (A14). ηVIT clearly does not characterize the nonlinear phase shift of the probe in the presence of the signal ﬁeld, but can still be considered to quantify the probe coherence under conditions VIT VIT of VIT. Extracting Re[χp ] and Im[χp ] from Eq.(A14) we obtain

∆31 λc∆21 ηVIT = − . (D1) −2(γ31 + λcγ21)

We denote the probe detuning as as ∆31 x, assume that the cavity is on resonance with the 2 3 transition, so ≡ 2 2 2 | i → | i that ∆21 = x, and use the deﬁnition of the cavity parameter λc(x) = Ωc /(x + γ21)(A13) to rewrite the equation for ηVIT in the form

2 2 2 x(x + γ21) Ωc x ηVIT(x) = 2 2 − 2 . (D2) −2[γ31(x + γ21) + γ21Ωc ]

In the absence of the cavity ﬁeld (Ω = 0), we have η (x) = x/2γ 1 for x γ . Inspection of c free space − 31 31 Eq. (D2) reveals that the VIT enhancement, i.e. ηVIT > 1, can be reached only when γ21 is very small compared to p γ31, and x is at most of the order of Ωc γ21/γ31. Assuming the hierarchy of the energy scales γ21 x Ωc, γ31, we can write

2 Ωc x x ηVIT(x) 2 2 O 1. (D3) ≈ 2[γ31x + γ21Ωc ] ∼ γ21 p This function has a maximum at the optimal probe detuning x∗ = Ωc γ21/γ31. Substituting x∗ into Eq. (D3) gives an optimal ﬁgure-of-merit for the disorder-free case of the form

max Ωc ηVIT = . (D4) 4√γ21γ31

The disorder averaged optimal ﬁgure-of-merit can be obtained within our Lorentzian disorder ansatz by simply replacing γ with Σ γ + σ from Eq. (D4), to read 31 31 ≡ 31 3

max Ωc ηVIT = . (D5) h i 4√γ21Σ31

2. Figure-of-merit in the presence of the signal ﬁeld: VIT-Kerr regime

Starting with the disorder-free susceptibility χp in Eq. (A9) and I0 in Eq. (A11), we can write the ﬁgure-of-merit η as

Ω2(∆ λ ∆ ) ∆ c 21 − s 41 31 − (∆ λ ∆ )2 + (γ + λ γ )2 η = 21 − s 41 21 s 41 , (D6) − Ω2(γ + λ γ ) 2 γ + c 21 s 41 31 (∆ λ ∆ )2 + (γ + λ γ )2 21 − s 41 21 s 41 2 2 2 where we have deﬁned the dimensionless signal parameter λs = Ωs/(∆41 +γ41). As required for consistency, Eq. (D6) reduces to η in Eq. (D1), in the limit λ 0. VIT s → 12

FIG. D.1: Signal parameter λs as a function of signal detuning ∆s/γ and Rabi frequency Ωs/γ, for Σ41 = 6γ.

In order to understand the scaling of the disorder-averaged figure-of-merit η with the characteristic variables of h i the signal field (Ωs, ∆s), we first rewrite Eq. (D6) using x ∆31, ∆21 = x, and ∆41 = x+∆s, where ∆s = ωs ω42 . Assuming that x ∆ . We define the dimensionless x-independent≡ signal parameter −h i | | | s| 2 Ωs λs = 2 2 . (D7) ∆s + γ41 Finally, within our Lorentzian disorder model, we make the substitutions γ Σ γ + σ and γ Σ 31 → 31 ≡ 31 3 41 → 41 ≡ γ41 + σ4 in Eq. (D6) to obtain a mean figure-of-merit in the form

x[(x x )2 + (γ + γ )2] Ω2(x x ) η(x) = − s 21 s − c − s , (D8) h i −2 Σ [(x x )2 + (γ + γ )2] + (γ + γ )Ω2 31 − s 21 s 21 s c where we have introduced the shift parameter xs and the linewidth parameter γs, deﬁned as

2 2 Ωs∆s ΩsΣ41 xs λs∆s = 2 2 and γs λsΣ41 = 2 2 . (D9) ≡ ∆s + Σ41 ≡ ∆s + Σ41 The comparison of Eq. (D8) with Eq. (D2) reveals that in the presence of the signal field, the figure-of-merit η has the same functional form as in the case where no signal field is present (pure VIT). The net effect of the signalh i field is to blue-shift the optimal probe frequency by xs (for ∆s 0) and broaden the η(x)-profile by γs. We can thus follow the same analysis as before, to show that the mean figure-of-merit≥ η(x) in the presence of the signal field, is max h i upper bounded by ηVIT = Ωc/(4√γ21Σ31). Finally, the similarityh betweeni the VIT and Kerr schemes shows that large values of the figure-of-merit η can be p h i obtained under the conditions γs γ21 and xs x∗ Ωc γ21/Σ31, which can be summarized into the signal field parameter bound ≤ | | ≤ ≡

2 2 r Ωs γ21 Ωs ∆s γ21 2 2 , and 2 | 2| Ωc . (D10) ∆s + Σ41 ≤ Σ41 ∆s + Σ41 ≤ Σ31 which reduce into the single constraint in Eq. (6) of the main text, when assuming that with Σ41 Σ31 σ and p ≈ ≈ Ωc/ ∆s γ21/σ. As Fig. D.1 illustrates (σ4 ﬁxed), high VIT coherence for the Kerr nonlinearity (λs 1) can be reached| | by ≤ either:

Decreasing Ωs, effectively switching the signal off. The signal field party destroys VIT, as it is not protected by • the transparency window. Increasing ∆ , making the incoherent signal field less resonant and therefore less absorptive. • s 13

Increasing σ4, eﬀectively distributing the signal beam among many frequencies, many of which are far-detuned • from ω and therefore less absorptive. h 24i

(1) J. Gea-Banacloche, Y.-Q. Li, S.-Z. Jin, and M. Xiao, Phys. Rev. A 51, 576, 1995. (2) A. Javan, O. Kocharovskaya, H. Lee, and M. Scully, Physical Review A 66, 013805, 2002.