# Tunable bandgap renormalization by nonlocal ultra-strong coupling in nanophotonics

## Abstract

In quantum optics, great effort is being invested in enhancing the interaction of quantum emitters with light. The different approaches include increasing the number of emitters, the laser intensity or the local photonic density of states at the location of an atom-like localized emitter. In contrast, solid-state extended emitters hold an unappreciated promise of vastly greater enhancements through their large number of vacant electronic valence states. However, the majority of valence states are considered optically inaccessible by a conduction electron. We show that, by interfacing three-dimensional (3D) solids with 2D materials, we can unlock the unoccupied valence states by nonlocal optical interactions that lead to ultra-strong coupling for each conduction electron. Consequently, nonlocal optical interactions fundamentally alter the role of the quantum vacuum in solids and create a new type of tunable mass renormalization and bandgap renormalization, which reach tens of millielectronvolts in the example we show. To present quantitative predictions, we develop a non-perturbative macroscopic quantum electrodynamic formalism that we demonstrate on a graphene–semiconductor–metal nanostructure. We find new effects, such as nonlocal Rabi oscillations and femtosecond-scale optical relaxation, overcoming all other solid relaxation mechanisms and fundamentally altering the role of optical interactions in solids.

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## Data availability

Source data are available for this paper. All other data that support the plots in this paper and other findings of this study are available from the corresponding author.

## Code availability

The code that supports the findings of this study is available from the corresponding author.

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## Acknowledgements

We thank D. Podolsky, N. Lindner and J.C. Song for their advice and for fruitful discussions regarding this paper. The research is supported by the Azrieli Faculty Fellowship, by the GIF Young Scientists’ Program and by an ERC Starter Grant.

## Author information

Authors

### Contributions

All authors made significant contributions to the manuscript.

### Corresponding authors

Correspondence to Yaniv Kurman or Ido Kaminer.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature Physics thanks Nicholas Christakis and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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## Supplementary information

### Supplementary Information

Supplementary text, methods, Figs. 1–6, Table 1 and references.

### Supplementary Video 1

The relaxation dynamics, showing the different stages by means of the map of optical excitation P(q,ω)(t) and the initial excited state probability $$P_{\bf{k}_{\mathrm{i}}}\left( t \right)$$.

## Source data

### Source Data Fig. 2

Source data for Fig. 2b. The coupling kernel for each frequency and wavenumber.

### Source Data Fig. 3

Source data for Fig. 3. a The time-dependent initial state population probability. b The optical excitations’ occupation probability.

### Source Data Fig. 4

Source data for Fig. 4. a The energy shift as a function of the initial momentum. b The tunable bandgap renormalization. c The shift in the photon absorption spectrum. d The highly expected nearfield emission spectra.

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