Abstract
The ability to generate and control strong longrange interactions via highly excited electronic states has been the foundation for recent breakthroughs in a host of areas, from atomic and molecular physics to quantum optics and technology. Rydberg excitons provide a promising solidstate realization of such highly excited states, for which recordbreaking orbital sizes of up to a micrometer have indeed been observed in cuprous oxide semiconductors. Here, we demonstrate the generation and control of strong exciton interactions in this material by optically producing two distinct quantum states of Rydberg excitons. This is made possible by twocolor pumpprobe experiments that allow for a detailed probing of the interactions. Our experiments reveal the emergence of strong spatial correlations and an interstate Rydberg blockade that extends over remarkably large distances of several micrometers. The generated manybody states of semiconductor excitons exhibit universal properties that only depend on the shape of the interaction potential and yield clear evidence for its vastly extendedrange and powerlaw character.
Introduction
Immersing an exciton into a manybody state of another species gives rise to a number of fascinating phenomena, ranging from the formation of polarons^{1} to the emergence of spinor interactions in semiconducting materials^{2,3}. However, reaching and probing the regime of strong interactions has remained challenging, whereby experimental signatures of interactions are often confined to measurements of spectral line shifts due to short ranged collisional interactions. On the other hand, the strong interactions between Rydberg excitons can act over large distances^{4}, which suggests new opportunities for probing the fundamental interactions between excitons.
In the past decades, the emergence of strong longrange interactions between highly excited Rydberg states was realized in a variety of atomic and molecular systems, ranging from trapped atoms^{5} and ions^{6} to Rydberg molecules^{7,8}. In the field of quantum optics, the strong mutual interactions between Rydberg states can mediate enhanced optical nonlinearities even between single photons^{9,10}, which leads to the realization of a wide range of applications in quantum information processing^{11,12}.
A prominent example of the underlying interaction mechanism is the case where the presence of one excited particle can perturb or even prevent the excitation of another by shifting its energy via the interaction between the two Rydberg states. This Rydberg blockade^{13} not only enables rapid saturation at very low light intensities^{14,15,16}, but can also lead to the emergence of strongly correlated manybody states of Rydberg excitations^{17,18}. While such correlations can be observed directly on a microscopic level in coldatom experiments^{19,20}, they are more difficult to access in solidstate systems.
Microscopically the Rydberg blockade can be traced back to the asymptotic interaction between neutral particles, such as atoms or excitons, which is dominated by the van der Waals potential that decreases as a simple power law, V(r) = C_{6}/r^{6}, with the interparticle distance r. For electronic ground states, however, the van der Waals coefficient C_{6} is typically very small and shortdistance exchange effects play an important role in the overall interaction that can often be described in terms of zerorange collisions^{21}. The drastic increase of C_{6} ~ n^{11} with the principal quantum number n of excited states, on the other hand, gives rise to exaggerated van der Waals interactions that can be sufficiently strong to even affect the very process of optically generating highlying Rydberg states, as observed and exploited in coldatom systems.
In this work, we demonstrate the existence of strongly correlated exciton states by realizing a binary mixture of interacting excitons in highlying quantum states with different principal quantum numbers via twocolor optical excitation in the semiconductor Cu_{2}O. Through an adequate choice of Rydberg states and laser intensities, we implement an asymmetric Rydberg blockade^{22} between excitons, where the interactions among identical excitons are of minor importance while the interactions between excitons in different quantum states result in strong spatial correlations and an extended excitation blockade of interspecies exciton pairs. Our experiments exploit this asymmetry to employ one exciton species as a probe for the presence of the other, and reveal direct spectroscopic signatures of the emerging spatial correlations between the two species as well as the distinct powerlaw character of their mutual interaction.
Results
Experimental setting
Our experiments make it possible to probe blockade effects in a semiconductor material at an extreme range of several μm. To this end, we have implemented a twocolor excitation scheme, whereby two spectrallynarrow laser beams with different frequencies excite Rydberg excitons with different principal quantum numbers n and \(n^{\prime} \), respectively (see Fig. 1a). We use a natural Cu_{2}O crystal that is cut and polished to a thickness of 30 μm and held at a temperature of 1.3 K. The selection rules associated with the involved band symmetries allow us to excite excitonic pstates in the socalled yellow series via singlephoton absorption in the optical domain. Hereby, a pump laser generates pstate excitons with \(n^{\prime} =16\), while another weak probe field with variable frequency creates Rydberg excitons with variable principal quantum numbers n = 6, ..., 20 that sense the presence of the pump excitons via their mutual interactions (Fig. 1b). By varying the power of the pump beam, P, we can control the density of the pump excitons and monitor their effect on the probefield absorption around a given probeexciton resonance.
To achieve the required sensitivity for an accurate measurement of interaction effects, we modulate the pump laser by an optical chopper and detect the transmitted intensity of the probe laser, I, with a photodiode that is connected to a lockin amplifier and locked to the pump modulation frequency (see Fig. 1c and the Methods for details). By choosing a low modulation frequency of 3.33 kHz far below any relevant dynamical frequency scale in the system, we ensure that the resulting signal yields the spectral form of the pumpinduced change in the probe transmission ΔI ∝ I(P) − I(P = 0). Importantly, this approach makes it possible to scan the Rydberg series of the probe excitons while maintaining otherwise stable excitation conditions.
Differential probe spectra
Figure 1d summarizes the result of such measurements and shows differential probe spectra for a varying power P of the pump laser (see Supplementary Note 1 for further details). As the density, ρ, of pump excitons grows with P, the depicted power dependence carries information about the effects of interactions between pump and probe excitons. A positive signal corresponds to an increased transmission caused by the generated pump excitons, and the observed transmission peaks can be assigned to the Rydbergstate resonances of the probe excitons, as indicated in Fig. 1d. For each individual peak, we observe a linear growth of the signal with P but no measurable shift of the position, \({{{\Delta }}}_{\max }\), of the transmission peaks, see closeup in Fig. 1f. Here, Δ is the frequency detuning of the probe laser normalized to the linewidth γ of the resonance. At first glance, this comes as a surprise, as an increasing density of pump excitons would be expected to cause a larger shift of the probeexciton line, similar to the known behavior of quantumwell excitons with shortrange interactions^{23,24}. In addition, we find that the differential transmission ΔI crosses zero at the highenergy side of the resonance at a detuning Δ_{0} that is entirely independent of the pumplaser power, making it an isosbestic point. Indeed all of these features provide strong indication for the emergence of extended spatial correlations during the optical generation of excitons, as we shall see below.
Let us first assume that spatial correlations are insignificant. In this case, the interactions between pump and probe excitons would lead to a simple energy shift, Δ_{mf} = ρ∫drV(r), of the probe exciton resonance that is determined by the exciton interaction potential V(r) and increases linearly with the pumpexciton density ρ. Note that such a meanfield treatment describes many aspects of excitons with weak shortrange interactions^{25}, such as the observed nonlinear spectral properties of semiconductor microcavities^{26}, or the fluidlike behavior of excitonpolaritons in such settings^{27}. For our experiments, however, a pure meanfield picture fails to capture the essential physics of the exciton dynamics as it predicts a powerdependent position of the maxima and roots of the differential transmission. The meanfield prediction is illustrated in Fig. 1e for three different interaction strengths, and stands in stark contrast to our measurements shown in the middle panel of the same figure.
Such qualitative discrepancies indicate that emerging correlations between the interacting excitons play a significant role during their optical generation. We can explore this further by theoretically considering the correlated excitation dynamics of probe excitons in a background of pump excitons with a density ρ. As the interaction, V(r), between the pump and probeexcitons shifts the energy of excitonpair states, it leads to strong spatial correlations following a pumpprobe correlation function, that assumes a particularly simple form
to lowest order in the probe intensity (see Supplementary Notes 3 and 4). The function g^{(2)}(r) yields the probability to generate a probe exciton in the vicinity of a pump exciton at a distance r. As illustrated in Fig. 1h, this probability vanishes rapidly as the distance between excitons falls below a critical radius r_{bl} that is determined by the linewidth γ and frequency detuning Δ (see Fig. 1a) of a given Rydbergstate resonance. In our experiments, the range of this exciton blockade can take on remarkably large values of several μm that would significantly affect the probebeam transmission as expressed by the change of the absorption coefficient
relative to the absorption α_{0} in the absence of the pumpbeam excitons. For low absorption, the proportionality ΔI ∝ Δα = α_{0} − α affords direct comparisons of our differential transmission measurements with the prediction of Eqs. (1) and (2). As demonstrated in Fig. 1g, our theory for the correlated exciton dynamics indeed reproduces the essential features of our observations. In particular, it yields a maximum and an isosbestic point on the blue side of the differential transmission spectrum that is independent of the pumpbeam intensity. This characteristic behavior is also evident from Eq. (2), since the spectral shape of α_{0}(Δ) − α(Δ) is solely determined by the exciton interaction and pair correlation function, while the pumpexciton density merely enters as a linear prefactor, which reproduces the observed overall linear scaling of the signal with the pumpbeam power.
Rydberg scaling of the interaction
While these characteristic features already provide clear evidence for the Rydberg blockade and emergence of strong exciton correlations, we can also obtain more detailed information about the underlying van der Waals interaction by analyzing the maxima of the differential transmission around each Rydbergstate resonance. To this end, we have recorded the pumppower dependence of the maximum signal strength, which exhibits a linear scaling ΔI = βP for low pumpbeam intensities with a slope \(\beta (n,{n}^{\prime})\) that depends on the principal quantum number of both involved Rydberg states (see Supplementary Notes 2 and 5). By evaluating the integral in Eq. (2) for nearresonant excitation, Δ/γ ≪ 1, one finds that the slope should scale as \(\beta \propto {r}_{{\rm{bl}}}^{3}\). Here, the blockade radius \({r}_{{\rm{bl}}}={\left(\frac{{C}_{6}}{\gamma /2}\right)}^{1/6}\) corresponds to the distance below which the van der Waals interaction \(V({r}_{{\rm{bl}}})={C}_{6}/{r}_{{\rm{bl}}}^{6}\) starts to exceed the width of the probeexciton resonance. By scaling the observed slope with the measured exciton linewidths γ and absorption strengths α_{0}, we can thus probe the statedependence of the van der Waals coefficient, C_{6} ∝ β^{2}γ (see Supplementary Note 5).
Our experimentally determined values are shown in Fig. 2 and indicate a rapid increase of the interaction strength by about three orders of magnitude over the probed range of principal quantum numbers. From the characteristic scaling laws for the level spacings and transition matrix elements of hydrogenic Rydberg states one would expect a characteristic scaling of the interstate van der Waals interaction as (see Supplementary Note 5)
which is shown by the black line in Fig. 2. Indeed, our measurements show such a rapid n^{7}increase of the interaction for lowlying states (\(n\ll {n}^{\prime}\)) and also indicate a crossover to a slower increase, ~n^{4}, once the excitation level of the probeexciton exceeds that of the pump excitons at \({n}^{\prime}=16\) (\(n\gg {n}^{\prime}\)) in accordance with the model prediction.
Universal spectral shape
The peak height of the differential transmission, therefore, demonstrates the powerlaw scaling of the interaction strength with the principal quantum number of the excitons. In addition, however, its spectral shape also carries spatial information about the powerlaw decay of the interaction potential as a function of the distance r between the excitons. In fact, Eqs. (1) and (2) predict that the scaled energy difference \({{\Delta }}E/\hslash \gamma =({{{\Delta }}}_{0}{{{\Delta }}}_{\max })/\gamma \) between the positions of the maximum and the isosbestic point of the transmission signal (see Fig. 1c) should depend only on the form and strength of the interaction potential. For pure powerlaw potentials, ΔE/ℏγ even becomes independent of the interaction strength and assumes a characteristic universal value for each powerlaw exponent, which is given by ΔE/ℏγ = 0.45 for the van der Waals interaction, V(r) ~ 1/r^{6}. As shown in Fig. 3, our measurements indeed approach this universal value for high principal quantum numbers n of the Rydberg excitons. The spectral line shape of lower lying states is stronger affected by excitonphonon coupling, which leads to asymmetric broadening that causes small deviations from the universal behavior. These deviations are remarkably well captured by corrections based on Fanoresonance theory^{28} as shown by the red line in Fig. 3 (see Supplementary Note 6).
Discussion
The combined analysis of our pumpprobe measurements thus offer broad insight into the microscopic mechanisms of exciton interactions, and provide a first experimental case for the action of longrange electrostatic van der Waals interactions between excitons in a semiconductor. Being sensitive to the shape and strength of the underlying interaction potential, our scheme makes it possible to discriminate between different types of interactions and clearly excludes direct dipoledipole interactions with V(r) ~ 1/r^{3} (see Fig. 3). One can explore this capability further by pumping the system above the band gap to create free charges instead of initial excitons (see Supplementary Note 7). The resulting pumpprobe spectra give clear evidence for a rapid dynamical recombination of the produced electronhole plasma into Rydbergexciton states similar to the behavior of ultracold atomic plasmas^{29,30}. Such elementary relaxation processes could become directly accessible to realtime measurements by operating the presented pumpprobe approach with short laser pulses. Indeed, this offers an exciting outlook on the developed method, which, when operated below the band gap, would open a new experimental window into the nonequilibrium dynamics of strongly interacting excitons in a semiconductor. Such temporal control could also make advanced studies of quantum mixtures possible, and, for example, enable the investigation of impurity physics and formation of exotic polarons^{31} in mixtures of ground and Rydbergstate excitons. Generally, the demonstrated ability to enhance interactions and realize spatially extended blockade effects between different exciton states, while maintaining weak intrastate nonlinearities, suggests interesting applications, such as highly efficient fewphoton switches^{9} that could be implemented with optical resonators in nearterm experiments.
Methods
Experimental Setup
We use a twocolor pumpprobe setup employing two stabilized dyelasers with a narrow linewidth of 5 neV that serve as pump and probe beams. The studied sample is a Cu_{2}O slab with a thickness of L = 34 μm cooled down to 1.35 K inside a liquid helium bath and mounted free of strain. Both lasers are in perfect spatial overlap. To assure a homogeneously distributed pump exciton density, the pump laser’s beam waist is set to 300 μm while the probe’s waist is set to 100 μm, both measured at full width at half maximum. The probe power is kept as low as 1 μW to ensure negligible interactions among probe excitons. The pump beam is periodically switched on and off by an optical chopper blade with a frequency of 3.33 kHz. The transmitted probe intensity is detected with a photodiode connected to the signal input (sig.) of a lockin amplifier (see Fig. 1c). The reference signal (ref.) is provided by the optical chopper and contains the modulation frequency of the pump beam. The lockin amplifier mixes both signals which yields a frequencyindependent part that is proportional to the pumpinduced change in the probe beam transmission ΔI. Due to the used low modulation frequency this serves as a quasiCW signal.
Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
We would like to thank Sjard Ole Krüger for fruitful discussions. This work was supported by the Carlsberg Foundation through the ‘Semper Ardens’ Research Project QCooL, by the DFG through the SPP 1929 GiRyd (project numbers: 316133134, 316159498 and 316214921) and the TRR 160 (project number: 249492093, project: A8), by the DFG and TU Dortmund University by the funding programme Open Access Publishing, by the European Commission through the H2020FETOPEN project ErBeStA (No. 800942), by the DNRF through a Niels Bohr Professorship to TP and the DNRF Center of Excellence "CCQ” (Grant agreement no.: DNRF156) and by the NSF through a grant for the Institute for Theoretical Atomic, Molecular, and Optical Physics at Harvard University and the Smithsonian Astrophysical Observatory.
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M.B. and M.A. designed the experiment. J.H. performed the experiments. T.P., S.S. and V.W. developed the theory. V.W. and J.H. analyzed the data. All authors wrote and commented on the manuscript.
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Heckötter, J., Walther, V., Scheel, S. et al. Asymmetric Rydberg blockade of giant excitons in Cuprous Oxide. Nat Commun 12, 3556 (2021). https://doi.org/10.1038/s4146702123852z
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DOI: https://doi.org/10.1038/s4146702123852z
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