Superextensive electrical power from a quantum battery

Device fabrication

ITO-coated glass substrates (Asahi Glass Co. Ltd., Japan) were cleaned with isopropyl alcohol and deionised water in an ultrasonic bath. The substrates were then subjected to a UV−ozone treatment in a UV−ozone cleaner (Novascan PSD Pro) to obtain an oxygen-rich ITO surface, increasing the work function of ITO. A 75 nm thick Ag (Sigma-Aldrich) bottom mirror layer was then deposited on the ITO layer, followed by a 15 nm hole-injection HAT-CN (Luminescence Technology Co.) layer under high vacuum. From then on, all layers up to the Ag top mirror layer were deposited using a thermal evaporation method under high vacuum (<1.0 × 10⁻⁴Pa). Next, a 15 nm CuPc layer, a mixed layer of CuPc:C₆₀ (Luminescence Technology Co.), and a C₆₀ layer were sequentially deposited (the thicknesses of these layers varied for each device). A 15 nm BPhen (Luminescence Technology Co.) and a 1 nm LiF (Sigma-Aldrich) layer were then deposited as electron transport and injection layers, respectively. Finally, a 25 nm Ag (Sigma-Aldrich Solutions) top mirror layer was deposited on the LiF layer. The devices were encapsulated with cover glass and a desiccant, and then sealed with a UV-curable epoxy resin. The final device is ~0.5 × 2 cm² and has four separated 2 × 5 mm² regions of cavity, and two 2 × 5 mm² regions of the corresponding no-cavity controls. Further fabrication details can be found in the Supplementary Information.

Steady-state spectroscopy

Angle–resolved reflectometry measurements were performed with an Agilent Cary 7000 UV-Visible-NIR spectrophotometer with Universal Measurement Accessory (UMA) and xenon lamp source. Measurements across the devices were baseline-corrected and performed with standardised spot size and lamp intensity. Fixed-angle reflectometry and transmission measurements were performed with an Agilent Cary 5000 UV-Visible-NIR spectrophotometer with Diffuse Reflectance Accessory (DRA) and xenon lamp source. Measurements across the devices were baseline-corrected and performed with standardised spot size and lamp intensity. Reflectance measurements were collected at 8° angle of incidence to enable the collection of a specular and diffuse reflectance signal.

Ultrafast spectroscopy

Femtosecond transient reflection spectroscopy was used to probe the depletion of a common ground state between the LP and UP. We used two-colour pump-probe spectroscopy to circumvent pump scatter, which can obscure degenerate pump-probe measurements. The pump-probe measurement produces a differential reflectivity

where \({R}_{{ON}}\) (\({R}_{{OFF}}\)) is the probe reflectivity with (without) a pump pulse. Since an increase in probe reflectivity in response to the pump is associated with the quantum system absorbing energy (i.e., by inducing a resonant transition \(i\to f\) in the quantum battery), \(\Delta R/R\) is proportional to the population \({P}_{i}\) of the initial state \(i\)³⁸. A probe wavelength resonant with the ground to upper polariton transition was used, such that an increase in differential reflectivity provides a time-resolved measure of the ground-state population or, conversely, of the sum of all excited-state populations. We pump and probe at the lower and upper polaritons, respectively, specifically not to trigger relaxation dynamics between the polariton branches during the time-evolution of the experiment.

Both two-colour pump-probe and pump-supercontinuum-probe measurements were performed on the quantum battery devices in reflection geometry, at room temperature, in air. The two-colour femtosecond laser pulses were generated by two NOPAs (Light Conversion, Orpheus-N-2H and Orpheus-N-3H). The NOPAs were pumped by a Yb:KGW laser amplifier (Light Conversion Pharos, 1030 nm, 180 fs, 100 µJ pulses). A secondary identical amplifier is seeded by the oscillator of the first amplifier (Pharos Duo, Light Conversion). The output of the secondary amplifier was focused onto a 2 mm sapphire crystal to generate a broadband supercontinuum which spanned 500–1000 nm. Amplifier pulses were selected at a repetition rate of 33.3 kHz for all measurements. The two-colour pump-probe was used to measure the sub-100 fs dynamics, while the pump-supercontinuum-probe was used to measure spectrally broad long-time dynamics.

We determined and optimised the pulse durations using auto- and cross-correlation measurements (Figs. S6 and S7). The pump pulses from the 2H-NOPA had an average pulse width (Gaussian FWHM) of 34 ± 1 fs. The probe pulses from the 3H-NOPA had an average pulse width of 32 ± 3 fs across all measurements. We used the measured cross-correlation widths (~50 fs) for the instrument response function in the theoretical model. The pump and probe beams were focused onto the QB devices with an almost collinear geometry (~4° between beam paths) with a FWHM spot diameter of 32 µm for the pump, and 42 µm for the probe. We have checked that the relative sizing between the pump and probe does not change the dynamics of the \(\Delta R/R\) signal. The QB devices were rotated 25° to the incoming pump and probe pulses so the reflected probe beam could be spatially isolated and re-collimated. The probe was then directed into a spectrometer with a high-speed CCD (Entwicklungsbuero Stresing, FLC3030) triggered by the laser amplifier. To measure the \(\Delta R/R\) signal, the pump was modulated by a mechanical chopper operating at 16.67 kHz (SciTec Instruments, 310CD). To scan the \(\Delta R/R\) signal as a function of time, the probe beam was sent through optical retroreflector delay lines, which extended to 8 ns for the supercontinuum probe pulses (Newport, DL325) or to 0.8 ns for the NOPA probe pulses (Newport, ESP300). Longer supercontinuum probe delay times out to ~16 μs were achieved by electronically triggering the secondary amplifier from later oscillator pulses (period of 13.3 ns) relative to the first amplifier. After each pump pulse is delivered to the devices, they show a \(\Delta R/R\) signal out to ~10 ns (Fig. S10), and the next pump pulse arrives ~66 μs later.

Since the chopper modulates at half the repetition rate of the pump laser, sequential probe-only and pump-probe pulses are incident on the devices, thus enabling the acquisition of shot-to-shot \(\Delta R/R\). The two-colour pump-probe measurements were scanned over the probe delay of −5 to 10 ps, with −200 to 400 fs in steps of 10 fs. At each probe delay time, 5000 shot-to-shot \(\Delta R/R\) spectra are averaged. Measurements across the entire probe delay were repeated 5 times under identical experimental conditions, and the average of these measurements was taken, with the 1σ standard deviation as the error bars.

To compare the ultrafast dynamics across different devices, it was important to keep a constant ratio of incident photons to absorber molecules in the laser volume. To achieve this, the fluence and wavelength of the pump-probe pulses were varied to account for the steady-state reflection spectrum of each device (see Supplementary Information). The fluence of the pump as a function of \(\Delta R/R\) intensity followed a linear dependence for all fluences used in the experiments (see Supplementary Information), ensuring higher-order effects induced by the pump are negligible in the measurements.

External quantum efficiency measurements

External quantum efficiency of the devices was calculated as the ratio of the number of photons absorbed by the device to the number of photons that participate in generating a photocurrent. This is given by \(\hslash \omega I/{eP}\), where \(I\) is the photocurrent, \(\hslash \omega\) is the photon energy, \(P\) is the power of incident photons, and \(e\) is the electron charge.

External quantum efficiency spectroscopy was performed at the RMIT Nanostructures Laboratory using a PV Measurements QEXL quantum efficiency instrument with Xenon arc lamp and monochromator, calibrated with a Si photodiode at 300–1100 nm. Internal quantum efficiency was calculated by dividing the experimentally measured external quantum efficiency by the absorption at each wavelength, obtained by fixed-angle UV-Visible-NIR reflectance and transmission spectroscopy on an Agilent Cary 5000 UV-Vis-NIR spectrophotometer with DRA and xenon arc lamp.

Discharging power measurements

I-V measurements were collected at the RMIT Nanostructures Laboratory using a Keithley 2636B SYSTEM SourceMeter® and a collimated 625 nm LED source (ThorLabs M625L4) at 10 mW/cm², with a resolution of 0.01 V. Measurements proceeded by first illuminating the device with the LED source, then varying the bias voltage from 1.5 V to −1.5 V. The power of each device was calculated at the point where the product of current and voltage is at a maximum, i.e., the maximum power point, according to P = IV.

Theoretical model

The reflectance spectra were modelled using a coupled oscillator Hamiltonian

where a^(†) annihilates (creates) excitations of the confined photon field with frequency Δ_c and X_αβ = |α〉〈β| are Hilbert operators that work on the low-lying electronic levels of the CuPc absorber. The indices \({\rm{\alpha }},{\rm{\beta }}\in \left\{\mathrm{0,1,2}\right\}\) enumerate the ground \({S}_{0}\), first-excited \({S}_{1}^{0}\) and second-excited \({S}_{1}^{1}\) levels, respectively, with energies \(0\), \({\Delta }_{1}\) and \({\Delta }_{2}\) as in Fig. 3b. Since we do not probe the steady-state reflectance of our devices in the near infra-red, the triplet state dynamics do not contribute to the simulated reflectance, thus we do not consider their population in our coupled oscillator model. The collective light–matter coupling constant \({g}_{{co}}\) quantifies the interaction of the CuPc ensemble electric dipole with the quantised cavity radiation field and is related to the bare coupling \(g\) via \({g}_{{co}}=\sqrt{N}g\). The steady-state reflectance of the cavity was simulated within a Fermi golden rule approach

involving the coupled oscillator eigenstates |ϕ_μ〉 with corresponding energies \({{\rm{\epsilon }}}_{{\rm{\mu }}}\) that depend upon the microscopic parameters of the Hamiltonian (1). The intensity \({I}_{0}\) and combined homogeneous and inhomogeneous broadening \({\rm{\sigma }}\) are fitting parameters, and \({\rm{\nu }}\) is the frequency of incident photons external to the quantum battery.

The pertinent dynamics of the ultrafast charging experiments are captured by the Tavis–Cummings Hamiltonian

written in the rotating frame of the laser. We assume a Gaussian temporal profile \({\rm{\eta }}\left(t\right)=\exp \left[-{\left(\left(t-{t}_{0}\right)/2{\rm{\sigma }}\right)}^{2}\right]/{\rm{\sigma }}\sqrt{2{\rm{\pi }}}\) for the pulse envelope and FWHM of \(34\) fs. The dissipative interactions between the cavity quantum battery and its environment are accounted for with the Lindblad master equation

where \({\mathcal{L}}\left[O\right]=O{\rm{\rho }}{O}^{\dagger }-\frac{1}{2}\left\{{O}^{\dagger }O,{\rm{\rho }}\right\}\) are Lindblad jump operators. In Eq. (4), we account for the rate of photon loss from the cavity \({\rm{\kappa }}\), the rate of non-radiative relaxation from the singlet \({\gamma }^{-}\) and triplet \({\gamma }_{T}^{-}\) states, intersystem crossing \({{\rm{\gamma }}}_{\text{ISC}}\) and singlet dephasing \({\gamma }^{z}\).

The ultrafast transient reflectance experiments provide a measure of the summed excited-state CuPc populations \(\Delta R/R\propto \left\langle {X}_{{TT}}\right\rangle +\left\langle {X}_{11}\right\rangle +\left\langle {X}_{22}\right\rangle\). The populations \(\left\langle {X}_{{\rm{\alpha }}{\rm{\alpha }}}\right\rangle\) for a molecule in the ensemble are obtained by numerically solving a hierarchical set of coupled equations of motion for the molecular and cavity degrees of freedom, which, in general, do not close. We enforce the closure of these equations by approximating the time-evolution of three-body correlators as \(\left\langle {ABC}\right\rangle =\left\langle {AB}\right\rangle \left\langle C\right\rangle +\left\langle {AC}\right\rangle \left\langle B\right\rangle +\left\langle A\right\rangle \left\langle {BC}\right\rangle -2\left\langle A\right\rangle \left\langle B\right\rangle \left\langle C\right\rangle\), thus capturing the pertinent dynamics with corrections scaling as \(O(1/N)\). By solving for the populations 〈X_αα〉, we obtain the time-evolution of the quantum battery energy density \(E\left(t\right)={\rm{\hslash }}\left({\Delta }_{1}\left\langle {X}_{11}\right\rangle +{\Delta }_{2}\left\langle {X}_{22}\right\rangle +{\Delta }_{T}\left\langle {X}_{{TT}}\right\rangle \right)\), the charging time \(\tau\) (defined by \(E\left(\tau \right)={E}_{\max }/2\)) and the maximum charging power \({P}_{\max }={E}_{\max }/\tau\), where \({E}_{\max }\) is the maximum energy density stored in the device.