The resonance energy and the transition rate of atoms, molecules, and solids were understood as their intrinsic properties in classical electromagnetism. It was later realized that these quantities are linked to the radiative coupling between the transition dipole and photon modes. Such effects can be greatly amplified in macroscopic many-body systems from virtual photon exchange between dipoles, but are often masked by inhomogeneity and pure dephasing, especially in solids. Here, we observe in both absorption and emission spectroscopy the renormalization of the exciton resonance and the radiative decay rate in transition metal dichalcogenides monolayers due to their radiative interactions. Tuning the photon mode density near the monolayer, we demonstrate control of cooperative Lamb shift, radiative decay, and valley polarization of the excitons as well as control of the charged exciton emission. Our work establishes a technologically accessible and robust experimental system for engineering cooperative matter–light interactions.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
27 November 2019: A correction was made to the disclosures section.
Excitons are collective excitations in semiconductors due to Coulomb interactions between electrons and holes. When multiple excitonic modes located within a light wavelength are subjected to an external electromagnetic field, they could react coherently and collectively to the field. Such a collective motion of electron–hole pairs forms the exciton resonance we observe experimentally. The radiative coupling to the light field can be enhanced due to the collective interaction resulting in unusually large oscillator strength and fast radiative decay rate. The strong light–matter interaction makes the excitons in semiconductors a great platform to study the physics of lasing and polariton condensate when combining with resonators and microcavities.
Therefore, it is important to understand the fundamental process of the collective excitonic excitations interacting with an external light field. When a transition dipole is coupled to external radiative channels (which we call “radiative coupling”), it is well known that its resonance energy and transition rate can be modified due to the emission and re-absorption of virtual photons . The modification of resonance properties due to coupling to radiative channels is universal and can also happen in collective systems with dipole transitions. In the 1970s, Friedberg et al.  showed that the effect becomes enhanced collectively in certain many-body systems by coherent exchange of virtual photons among the dipoles, leading to, for example, the cooperative Lamb shift [2,3] and superradiance [3–8]. Even though these effects can also be explained by optical interference between dipoles using semi-classical treatment, they have been difficult to observe experimentally in a many-body system, as dephasing and inhomogeneity in typical many-body systems tend to destroy the cooperativity. The cooperative Lamb shift in the optical domain has only been reported in a few experiments with cold atoms or ions [9–13]. In solids, it has only been observed in nuclei excited by synchrotron x rays, intersubband transitions in quantum wells, and in superconducting microwave circuits [14–16].
As exciton resonances in semiconducting thin films are collective excitations [17–19], the radiative-induced effects mentioned above should be enhanced accordingly. However, due to the dephasing and inhomogeneous broadenings in typical quantum wells, the observation of these collective effects is still obscure except special systems under strong external fields . Recently, high reflectivity was measured from a mere single layer of transition metal dichalcogenides crystal (TMDC) [21,22], owing to the large exciton–photon coupling strength, near radiative-limited linewidth and two-dimensional (2D) translational symmetry. Based on this, theoretical work suggests monolayer TMDCs may provide an easy-to-access 2D many-body system for observing and utilizing the effects of vacuum fluctuations [23,24].
In this work, using a mirror to control the radiative coupling of the dipole transition of excitons in a high-quality, monolayer TMDC, we demonstrate the cooperative Lamb shift of the excitons accompanied by modified exciton radiative decay rate. This observation suggests that the TMDC monolayer can be used as a sensitive probe of the vacuum modes [25–27]. We also demonstrate control of the charged exciton and valley polarization in the TMDCs with the tunable radiative environment. In contrast to the strong coupling and Purcell effect, where a cavity is used to modify the resonances, in this work we have a fully open system coupled to a continuous spectrum of modes.
Figure 1 shows a schematic of our system. A monolayer TMDC is placed at a distance L from a mirror made of a distributed Bragg reflector (DBR). The mirror imposes a boundary condition on the electromagnetic field in the space, creating a standing wave pattern with an electric field node at the mirror plane. Here and are the electric field amplitude and the wavenumber of the incident plane wave, and is the distance from the mirror.
The effects of radiative coupling on a homogeneous exciton resonance in a 2D material can be modeled by the following effective Hamiltonian in the rotating wave approximation:Supplement 1, the solution for ( is the radiative decay rate for a free-standing monolayer) can be written as 28,29]. This is expected since the collective Lamb shift and superradiance physics have a semi-classical description as pointed out by Friedberg et al. .
The renormalized exciton radiative rate equals at the anti-node () and zero at the node () of the electric field. This is due to modification of exciton–photon coupling through the local electric field. At the anti-node, the local field enhances both the absorption and the emission rate, similar to the Purcell effect in a cavity configuration. At the node, the local electric field is suppressed; therefore, radiative decay of the exciton excitation is suppressed. The renormalized exciton energy is shifted from due to coupling with the photon field. This energy shift in a collective excitation system has been named as the “cooperative Lamb shift.” It shares the same origin as the Lamb shift but can be as large as the radiative linewidth , due to the cooperative enhancement. In a radiative-limited sample, the renormalization on the resonance energy and the radiative decay rate can be directly observed through spectra of the exciton in the frequency domain.
An alternative approach is to treat the renormalization as the result of interaction between the monolayer dipole moment and its image dipole [Fig. 1(b)]. In our case, the image dipole has a phase difference compared with the original dipole due to the boundary condition set by the mirror. The renormalization of the radiative decay rate can be understood as follows: when the two dipole sheets are separate by (the node condition), the radiated fields generated by them destructively interfere, leading to suppression in emission and a polariton of infinite lifetime . When their distance is (the anti-node condition), the radiated fields constructively interfere leading to enhanced emission rate and short exciton lifetime. The cooperative Lamb shift can be understood through the dipole–dipole interaction: the emitted field from the image dipole at the original dipole has a phase shift of , where the comes from the phase relation between the dipole and its radiation and is the phase accumulation during the propagation. Due to the dipole interaction , the energy is lower if and the energy is higher if , leading the shift of exciton energy.
Note that the photon mode structure applies to not only classical fields but also the vacuum fluctuations. The structure of the vacuum fluctuations can be measured through its effect on a dipole emitter, such as an exciton in solids. Given a transition dipole, the radiative decay of the excited state to the ground state is proportional to the strength (spectral density) of electromagnetic fluctuations near the resonance frequency that are present in the environment. Therefore, measuring the lifetime of the excitation probes the local strength of vacuum fluctuations at the transition frequency. In effect, the monolayer can act as a local vacuum field analyzer [25,27].
Observation of the cooperative Lamb shift and corresponding modulation of the radiative decay rate requires a homogeneous ensemble of emitters with nearly radiative-limited line broadening, which is challenging to realize in conventional semiconductors. In this work, using a monolayer TMDC, we observe the renormalization of both exciton energy and linewidth broadening due to coupling with an external radiation channel. To measure the renormalization of the exciton mode, we change the distance L between the monolayer and the mirror to modulate the local electric field at the 2D exciton position. We place an hBN-encapsulated monolayer on a sapphire substrate in front of the DBR mirror, whose position is controlled by a piezo-electric stage, as illustrated in Fig. 1. This setup allows in situ spectroscopy measurement of the same piece of while we tune the local electric mode profile. The monolayer-to-mirror distance is calibrated by measuring the interference pattern between the sapphire and the DBR (see Supplement 1 for detail).
First, we measure the reflection contrast spectra of the monolayer as it is moved through the standing wave of the electric field profile, using a weak femtosecond laser with bandwidth covering the exciton absorption peak. The excited exciton density is kept below to ensure the linewidth broadening due to exciton–exciton interactions is negligible [22,30]. As shown in Fig. 2(a), the absorption dip of the A-exciton at 1660 meV is strongly modulated, following the period of the standing wave profile. Figure 2(b) shows several reflection contrast spectra [horizontal line cuts of Fig. 2(a)]. The absorption dips are fit very well by Lorentzian functions (gray dashed curves), indicating minimal inhomogeneous broadening in the sample. The absorption depth is tuned from as low as 4% to as high as 99% at node and anti-node positions of the field, respectively. In the node region, we can hide the monolayer from the classical probe field even though it sits fully exposed in an open space. In the anti-node region, we can enhance the absorption to achieve the critical coupling condition where all photon energy is dumped into exciton energy, giving nearly 100% absorption .
We summarize in Fig. 2(c) the modulation of the absorption depth, linewidth, and the resonance energy of the exciton as a function of mirror distance. We use the absorption depth to determine whether the monolayer is located at a node or anti-node [vertical gray dashed lines in Fig. 2(c) indicate the anti-nodal positions]. In excellent agreement with predictions by Eq. (3), both the linewidth and energy of the exciton resonance show periodic modulations with a phase shift relative to each other and a modulation amplitude different by about a factor of 2.
The linewidth of the exciton changes by 2 meV, from 5.5 meV at an anti-node to 3.5 meV at a node (middle panel), corresponding to twice the non-renormalized radiative linewidth . Fitting the linewidth modulation with Eq. (3a) after including a constant offset (blue dashed curve), we obtain and . The offset of accounts for contributions from other broadening mechanisms, including inhomogeneous, non-radiative, and pure dephasing broadening. The main contribution of this background broadening is believed to be the exciton–phonon scattering in this study due to elevated temperature (). As shown by a previous study , the increased homogeneous broadening is about 3 meV at 50 K, which is consistent with our background linewidth. The agrees with the radiative linewidth measured from linear reflection [21,22] and 2D spectroscopy . Similar linewidth modulations due to modified dipole–vacuum coupling have been observed in atomic systems [10,11] and superconducting qubits [15,25,31] in absorption spectra, but have not been demonstrated in a solid-state system due to the small radiative linewidth compared with the total linewidth in typical semiconductor materials.
The cooperative Lamb shift of the exciton resonance due to the coupling between the exciton and its radiative channel [lower panel of Fig. 2(c)] follows the same period of the linewidth modulation but displaced in phase. The shift is zero at both the nodes and anti-nodes of the field, as predicted by Eq. (3b). Fitting the modulation of the exciton energy with Eq. (3b), we obtain the magnitude of the modulation , consistent with the result from the linewidth modulation. Such cooperative Lamb shift has been predicted theoretically  but demonstrated only in atomic and superconducting qubit systems recently [9,13,18,19]. Note that the observed linewidth and cooperative Lamb shift modification is of the order of 1 meV (), which is much larger compared with atomic and superconducting qubits systems (tens of megahertz or smaller), because of the extraordinary oscillator strength of the collectively coupled excitons.
Compared to absorption, incoherent photoluminescence (PL) stems from the coupling of incoherent exciton polarization to vacuum fluctuations, which cannot be described semi-classically. It is, therefore, interesting to test if the same renormalization effect appears in the emission spectrum of the exciton. Figure 3(a) shows a few examples of the PL spectra with varying L under the excitation of a continuous-wave laser at 532 nm. The features at 1662 meV and 1632 meV are due to exciton and trion emission, respectively. The linewidth of exciton emission measured at anti-nodes is clearly narrower than that measured at nodes [Fig. 3(b)], indicating that the same radiative decay rate renormalization is present in PL spectra. We summarize in Fig. 3(c) modulation of the PL intensity, linewidth, and resonance energy of the exciton peak. Reflection spectra are taken together to identify the node positions for the exciton wavelength, which are labeled as vertical gray dashed lines in Fig. 3(c). The PL intensity is complicated by the factor that the absorption of the 532 nm pump laser is also modulated by the monolayer–mirror distance L. As a result, we observe PL intensity minima when the monolayer is located at the nodes of the field of either 532 nm or 750 nm, indicated by green arrows and gray dashed lines, respectively. The suppression of PL at the nodes of 750 nm light arises from the optical interaction between the monolayer and its image dipole. To estimate the suppression/enhancement effect, we take the ratio of PL intensity at a node (step 16) versus an anti-node (step 27) and compare the ratio to when no mirror is present. (Both points are around anti-node of the 532 nm light.) In our measurements, the ratio can be as low as 5% and as high as 175%, showing control of PL quantum efficiency through radiative decay rate modulation.
The renormalization effect on the exciton decay rate can be observed in the frequency domain more directly. The third panel in Fig. 3(b) shows the PL linewidth as a function of monolayer–mirror distance. The PL linewidth follows only the mode profile of the 750 nm standing wave and not that of the 532 nm excitation laser. It changes from about 4.5 meV at the anti-nodes to about 2.2 meV at the nodes. Correspondingly, a cooperative Lamb shift of 1.1 meV is measured in PL [lower panel of Fig. 3(b)]. Both results are in agreement with absorption measurements. Therefore, we demonstrate the same exciton renormalization due to radiative coupling in the photoluminescence, which is the spontaneous emission caused by the coupling between the exciton mode and the vacuum fluctuation.
The dramatic effects of the radiative coupling on the TMDC excitons observed above demonstrate the possibility to control optical properties of TMDCs by vacuum engineering. We give two other examples, where we use the same tunable mirror approach to tune the charged exciton and valley polarization via the vacuum–matter coupling.
Charged exciton, or trions, are pronounced in TMDCs due to the strong Coulomb interaction in a two-dimensional film. Since the exciton and trion wavelengths are well separated, we can selectively enhance either the exciton or the trion emission by tuning the vacuum field strength at the respective wavelength. Figure 4(a) shows two PL spectra from the same monolayer, but at different mirror locations. The emission spectra show very different exciton/trion intensity ratios despite being measured from the same position on the monolayer and at a fixed charge doping. This result shows that the ability to shape the vacuum–matter coupling allows us to control the interference effect for both exciton and trion emission. We show in Fig. 4(b) the mirror–monolayer distance dependence of the PL intensities of the exciton, the trion, and their ratio. The suppression of exciton emission is observed at the node for 750 nm light indicated by the dashed gray lines while the suppression of trion emission is observed at the node for 780 nm light indicated by the orange arrows. The ratio of exciton and trion emission intensity changes over 2 orders of magnitude, from 0.02 to 2.48. This simple technique can be utilized to various 2D materials applications to enhance or suppress the transition of interest.
Another important property of TMDC materials is the valley degree of freedom. However, the valley polarization of the excitons is often complicated by several competing decay and exchange mechanisms. Considering the two dominant mechanisms—the radiative decay and intervalley exchange interaction—we can formulate the valley polarization, quantified by the degree of circular polarization (DOCP) as32]. For this study, we use an encapsulated sample and a 633 nm laser with polarization for excitation. The exciton PL spectra around 1.74 eV with anti-node, node, and no mirror conditions are shown in Fig. 5(a). The measured DOCP of exciton as a function of mirror distance [Fig. 5(b)] shows oscillation behavior in accord with the reflection contrast modulation. The tuning of vacuum fluctuation through mirror distance allows us to change the radiative decay rate of the exciton, resulting in enhancement (suppression) of the DOCP at anti-nodes (nodes). Given that the measured DOCP without a mirror present is 37% [blue dashed line in Fig. 5(b)], the modulation should be from 0% to 54% based on Eq. (4) when the radiative decay rate is tuned from 0 to . However, we observe a modulation from 25% to 43% experimentally. The discrepancy might come from other depolarization mechanisms or effective radiative lifetime from dark excitons in the monolayer and requires further studies. Nonetheless, the modulation via simply a mirror is already comparable with other reports using microcavities [32–34].
In conclusion, we observe the renormalization of exciton resonance energy and radiative linewidth in TMDC monolayers due to coupling to radiation fields. This effect has only been observed in atomic and superconducting qubit systems before. The tightly bound exciton in TMDC monolayers leads to an exceptionally large radiative linewidth, enabling pronounced effects of radiative coupling engineering, manifested as a large cooperative Lamb shift, strong modulation of the radiative linewidth, and control over trion and valley degrees of freedom. We note that the observed phenomena could also be explained by the interference effect of multiple reflections from our sample structure . Our study shows intriguing collective physics of radiative effects on excitonic many-body systems and could pave the way for future quantum optics research with 2D materials.
Army Research Office (W911NF-17-1-0312); Ministry of Science and Technology, Taiwan (106-2917-I-564-021).
Y.-H. C. acknowledges the support from the Minister of Science and Technology (Taiwan).
The authors declare no conflicts of interest.
See Supplement 1 for supporting content.
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