- Open Access
Optical Signatures of Dynamical Excitonic Condensates
Phys. Rev. Lett. 135, 106902 – Published 5 September, 2025
DOI: https://doi.org/10.1103/58r8-cpzn
Abstract
We theoretically study dynamical excitonic condensates occurring in bilayers with an imposed chemical potential difference and in photodoped semiconductors. We show that optical spectroscopy can experimentally identify phase-trapped and phase-delocalized dynamical regimes of condensation. In the weak-bias regime, the trapped dynamics of the order parameter’s phase lead to an in-gap absorption line at a frequency almost independent of the bias voltage, while, for larger biases, the frequency of the spectral feature increases approximately linearly with bias. In both cases, there is a pronounced second-harmonic response. Close to the transition between the trapped and freely oscillating states, we find a strong response upon application of a weak electric probe field, compare the results to those found in a minimal model description for the dynamics of the order parameter’s phase, and analyze the limitations of the latter.
Physics Subject Headings (PhySH)
Article Text
Supplemental Material
References (73)
- D. Jérome, T. M. Rice, and W. Kohn, Excitonic insulator, Phys. Rev. 158, 462 (1967).
- L. V. Keldysh and A. N. Kozlov, Collective properties of excitons in semiconductors, Zh. Eksp. Teor. Fiz. 54, 978 (1968) [Sov. J. Exp. Theor. Phys. 27, 521 (1968)].
- A. V. Balatsky, Y. N. Joglekar, and P. B. Littlewood, Dipolar superfluidity in electron-hole bilayer systems, Phys. Rev. Lett. 93, 266801 (2004).
- J. Eisenstein and A. H. MacDonald, Bose–Einstein condensation of excitons in bilayer electron systems, Nature (London) 432, 691 (2004).
- J. Eisenstein, Exciton condensation in bilayer quantum hall systems, Annu. Rev. Condens. Matter Phys. 5, 159 (2014).
- P. Littlewood, P. Eastham, J. M. J. Keeling, F. Marchetti, B. Simons, and M. Szymanska, Models of coherent exciton condensation, J. Phys. Condens. Matter 16, S3597 (2004).
- Y. Wakisaka, T. Sudayama, K. Takubo, T. Mizokawa, M. Arita, H. Namatame, M. Taniguchi, N. Katayama, M. Nohara, and H. Takagi, Excitonic insulator state in probed by photoemission spectroscopy, Phys. Rev. Lett. 103, 026402 (2009).
- K. Seki, Y. Wakisaka, T. Kaneko, T. Toriyama, T. Konishi, T. Sudayama, N. L. Saini, M. Arita, H. Namatame, M. Taniguchi, N. Katayama, M. Nohara, H. Takagi, T. Mizokawa, and Y. Ohta, Excitonic Bose-Einstein condensation in above room temperature, Phys. Rev. B 90, 155116 (2014).
- C. Monney, H. Cercellier, F. Clerc, C. Battaglia, E. F. Schwier, C. Didiot, M. G. Garnier, H. Beck, P. Aebi, H. Berger, L. Forró, and L. Patthey, Spontaneous exciton condensation in : BCS-like approach, Phys. Rev. B 79, 045116 (2009).
- A. K. Geim and I. V. Grigorieva, Van der Waals heterostructures, Nature (London) 499, 419 (2013).
- K. S. Novoselov, A. Mishchenko, A. Carvalho, and A. H. C. Neto, 2D materials and van der Waals heterostructures, Science 353, aac9439 (2016).
- C. Jin, E. Y. Ma, O. Karni, E. C. Regan, F. Wang, and T. F. Heinz, Ultrafast dynamics in van der Waals heterostructures, Nat. Nanotechnol. 13, 994 (2018).
- V. Pareek, D. R. Bacon, X. Zhu, Y.-H. Chan, F. Bussolotti, N. S. Chan, J. P. Urquizo, K. Watanabe, T. Taniguchi, M. K. L. Man, J. Madéo, D. Y. Qiu, K. E. J. Goh, F. H. da Jornada, and K. M. Dani, Driving non-trivial quantum phases in conventional semiconductors with intense excitonic fields, arXiv:2403.08725.
- F. Boschini, M. Zonno, and A. Damascelli, Time-resolved ARPES studies of quantum materials, Rev. Mod. Phys. 96, 015003 (2024).
- M. Reutzel, G. S. M. Jansen, and S. Mathias, Probing excitons with time-resolved momentum microscopy, Adv. Phys. X 9, 2378722 (2024).
- Y. Murotani, C. Kim, H. Akiyama, L. N. Pfeiffer, K. W. West, and R. Shimano, Light-driven electron-hole Bardeen-Cooper-Schrieffer-like state in bulk gaas, Phys. Rev. Lett. 123, 197401 (2019).
- E. Perfetto, D. Sangalli, A. Marini, and G. Stefanucci, Pump-driven normal-to-excitonic insulator transition: Josephson oscillations and signatures of BEC-BCS crossover in time-resolved ARPES, Phys. Rev. Mater. 3, 124601 (2019).
- E. Perfetto, S. Bianchi, and G. Stefanucci, Time-resolved ARPES spectra of nonequilibrium excitonic insulators: Revealing macroscopic coherence with ultrashort pulses, Phys. Rev. B 101, 041201(R) (2020).
- D. Schmitt, J. P. Bange, W. Bennecke, A. AlMutairi, G. Meneghini, K. Watanabe, T. Taniguchi, D. Steil, D. R. Luke, R. T. Weitz et al., Formation of moiré interlayer excitons in space and time, Nature (London) 608, 499 (2022).
- J. P. Bange, D. Schmitt, W. Bennecke, G. Meneghini, A. AlMutairi, K. Watanabe, T. Taniguchi, D. Steil, S. Steil, R. T. Weitz, G. S. M. Jansen, S. Hofmann, S. Brem, E. Malic, M. Reutzel, and S. Mathias, Probing electron-hole Coulomb correlations in the exciton landscape of a twisted semiconductor heterostructure, Sci. Adv. 10, eadi1323 (2024).
- X. Zhu, P. B. Littlewood, M. S. Hybertsen, and T. M. Rice, Exciton condensate in semiconductor quantum well structures, Phys. Rev. Lett. 74, 1633 (1995).
- P. Littlewood and X. Zhu, Possibilities for exciton condensation in semiconductor quantum-well structures, Phys. Scr. 1996, 56 (1996).
- M. H. Szymanska and P. B. Littlewood, Excitonic binding in coupled quantum wells, Phys. Rev. B 67, 193305 (2003).
- M. Xie and A. H. MacDonald, Electrical reservoirs for bilayer excitons, Phys. Rev. Lett. 121, 067702 (2018).
- Z. Wang, D. A. Rhodes, K. Watanabe, T. Taniguchi, J. C. Hone, J. Shan, and K. F. Mak, Evidence of high-temperature exciton condensation in two-dimensional atomic double layers, Nature (London) 574, 76 (2019).
- L. Ma, P. X. Nguyen, Z. Wang, Y. Zeng, K. Watanabe, T. Taniguchi, A. H. MacDonald, K. F. Mak, and J. Shan, Strongly correlated excitonic insulator in atomic double layers, Nature (London) 598, 585 (2021).
- P. X. Nguyen, L. Ma, R. Chaturvedi, K. Watanabe, T. Taniguchi, J. Shan, and K. F. Mak, Perfect Coulomb drag in a dipolar excitonic insulator, Science 388, 274 (2025).
- R. Qi, A. Y. Joe, Z. Zhang, J. Xie, Q. Feng, Z. Lu, Z. Wang, T. Taniguchi, K. Watanabe, S. Tongay, and F. Wang, Perfect Coulomb drag and exciton transport in an excitonic insulator, Science 388, 278 (2025).
- X. Liu, N. Leisgang, P. E. Dolgirev, A. A. Zibrov, J. Sung, J. Wang, T. Taniguchi, K. Watanabe, V. Walther, H. Park, E. Demler, P. Kim, and M. D. Lukin, Optical signatures of interlayer electron coherence in a bilayer semiconductor, Nat. Phys. (2024), 10.1038/s41567-025-02971-0.
- D. Nandi, A. Finck, J. Eisenstein, L. Pfeiffer, and K. West, Exciton condensation and perfect coulomb drag, Nature (London) 488, 481 (2012).
- A. L. Ivanov, P. B Littlewood, and H. Haug, Bose-Einstein statistics in thermalization and photoluminescence of quantum-well excitons, Phys. Rev. B 59, 5032 (1999).
- L. V. Butov, A. C. Gossard, and D. S. Chemla, Macroscopically ordered state in an exciton system, Nature (London) 418, 751 (2002).
- Y. Zeng and A. H. MacDonald, Electrically controlled two-dimensional electron-hole fluids, Phys. Rev. B 102, 085154 (2020).
- Y. Murakami, M. Schüler, S. Takayoshi, and P. Werner, Ultrafast nonequilibrium evolution of excitonic modes in semiconductors, Phys. Rev. B 101, 035203 (2020).
- R. Hanai, P. B. Littlewood, and Y. Ohashi, Non-equilibrium properties of a pumped-decaying Bose-condensed electron–hole gas in the BCS-BEC crossover region, J. Low Temp. Phys. 183, 127 (2016).
- R. Hanai, P. B. Littlewood, and Y. Ohashi, Dynamical instability of a driven-dissipative electron-hole condensate in the BCS-BEC crossover region, Phys. Rev. B 96, 125206 (2017).
- Z. Sun, T. Kaneko, D. Golež, and A. J. Millis, Second-order Josephson effect in excitonic insulators, Phys. Rev. Lett. 127, 127702 (2021).
- Z. Sun, Y. Murakami, F. Xuan, T. Kaneko, D. Golež, and A. J. Millis, Dynamical exciton condensates in biased electron-hole bilayers, Phys. Rev. Lett. 133, 217002 (2024).
- Y. Zeng, V. Crépel, and A. J. Millis, Keldysh field theory of dynamical exciton condensation transitions in nonequilibrium electron-hole bilayers, Phys. Rev. Lett. 132, 266001 (2024).
- A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions, Rev. Mod. Phys. 68, 13 (1996).
- H. Aoki, N. Tsuji, M. Eckstein, M. Kollar, T. Oka, and P. Werner, Nonequilibrium dynamical mean-field theory and its applications, Rev. Mod. Phys. 86, 779 (2014).
- See Supplemental Material at http://link.aps.org/supplemental/10.1103/58r8-cpzn for additional info on the numerics, further supporting figures and details on the Ginzburg-Landau model, which includes Refs. [43–46].
- P. Werner, H. U. R. Strand, S. Hoshino, and M. Eckstein, Ultrafast switching of composite order in , Phys. Rev. B 95, 195405 (2017).
- S. Giuli, A. Amaricci, and M. Capone, Mott-enhanced exciton condensation in a Hubbard bilayer, Phys. Rev. B 108, 165150 (2023).
- A. M. J. Schakel, Time-dependent Ginzburg-Landau theory and duality, in Topological Defects and the Non-Equilibrium Dynamics of Symmetry Breaking Phase Transitions, edited by Y. M. Bunkov and H. Godfrin (Springer, Dordrecht, 2000), pp. 213–238.
- Z. Sun and A. J. Millis, Bardasis-Schrieffer polaritons in excitonic insulators, Phys. Rev. B 102, 041110(R) (2020).
- G. Mazza, M. Rösner, L. Windgätter, S. Latini, H. Hübener, A. J. Millis, A. Rubio, and A. Georges, Nature of symmetry breaking at the excitonic insulator transition: , Phys. Rev. Lett. 124, 197601 (2020).
- B. Chatterjee, J. Mravlje, and D. Golež, Collective modes and Raman response in , Phys. Rev. B 111, L121106 (2025).
- T. B. Boykin, R. C. Bowen, and G. Klimeck, Electromagnetic coupling and gauge invariance in the empirical tight-binding method, Phys. Rev. B 63, 245314 (2001).
- D. Golež, M. Eckstein, and P. Werner, Multiband nonequilibrium formalism for correlated insulators, Phys. Rev. B 100, 235117 (2019).
- J. Li, D. Golez, G. Mazza, A. J. Millis, A. Georges, and M. Eckstein, Electromagnetic coupling in tight-binding models for strongly correlated light and matter, Phys. Rev. B 101, 205140 (2020).
- M. Schüler, J. A. Marks, Y. Murakami, C. Jia, and T. P. Devereaux, Gauge invariance of light-matter interactions in first-principle tight-binding models, Phys. Rev. B 103, 155409 (2021).
- O. Dmytruk and M. Schiró, Gauge fixing for strongly correlated electrons coupled to quantum light, Phys. Rev. B 103, 075131 (2021).
- P. Werner, J. Li, D. Golež, and M. Eckstein, Entropy-cooled nonequilibrium states of the Hubbard model, Phys. Rev. B 100, 155130 (2019).
- H. Yu, Y. Wang, Q. Tong, X. Xu, and W. Yao, Anomalous light cones and valley optical selection rules of interlayer excitons in twisted heterobilayers, Phys. Rev. Lett. 115, 187002 (2015).
- D. A. Ruiz-Tijerina and V. I. Fal’ko, Interlayer hybridization and moiré superlattice minibands for electrons and excitons in heterobilayers of transition-metal dichalcogenides, Phys. Rev. B 99, 125424 (2019).
- C. Shao, T. Tohyama, H.-G. Luo, and H. Lu, Numerical method to compute optical conductivity based on pump-probe simulations, Phys. Rev. B 93, 195144 (2016).
- M. Eckstein and M. Kollar, Theory of time-resolved optical spectroscopy on correlated electron systems, Phys. Rev. B 78, 205119 (2008).
- Z. Lenarčič, D. Golež, J. Bonča, and P. Prelovšek, Optical response of highly excited particles in a strongly correlated system, Phys. Rev. B 89, 125123 (2014).
- R. Guseĭnov and L. Keldysh, Nature of the phase transition under the conditions of an “excitonic” instability in the electronic spectrum of a crystal, Zh. Eksp. Teor. Fiz. 63, 2255 (1972) [J. Exp. Theor. Phys. 36, 1193 (1973)].
- D. Golež, Z. Sun, Y. Murakami, A. Georges, and A. J. Millis, Nonlinear spectroscopy of collective modes in an excitonic insulator, Phys. Rev. Lett. 125, 257601 (2020).
- Y. Murakami, D. Golež, T. Kaneko, A. Koga, A. J. Millis, and P. Werner, Collective modes in excitonic insulators: Effects of electron-phonon coupling and signatures in the optical response, Phys. Rev. B 101, 195118 (2020).
- L. Keldysh, The electron-hole liquid in semiconductors, Contemp. Phys. 27, 395 (1986).
- H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 2009).
- Y. Murakami, D. Golež, M. Eckstein, and P. Werner, Photoinduced enhancement of excitonic order, Phys. Rev. Lett. 119, 247601 (2017).
- G. Chiriacò, A. J. Millis, and I. L. Aleiner, Negative absolute conductivity in photoexcited metals, Phys. Rev. B 101, 041105(R) (2020).
- D. Golež, L. Boehnke, H. U. R. Strand, M. Eckstein, and P. Werner, Nonequilibrium : Antiscreening and inverted populations from nonlocal correlations, Phys. Rev. Lett. 118, 246402 (2017).
- T. Kaneko, Z. Sun, Y. Murakami, D. Golež, and A. J. Millis, Bulk photovoltaic effect driven by collective excitations in a correlated insulator, Phys. Rev. Lett. 127, 127402 (2021).
- S. Mathias, S. Eich, J. Urbancic, S. Michael, A. Carr, S. Emmerich, A. Stange, T. Popmintchev, T. Rohwer, M. Wiesenmayer et al., Self-amplified photo-induced gap quenching in a correlated electron material, Nat. Commun. 7, 12902 (2016).
- C. Monney, M. Puppin, C. W. Nicholson, M. Hoesch, R. T. Chapman, E. Springate, H. Berger, A. Magrez, C. Cacho, R. Ernstorfer, and M. Wolf, Revealing the role of electrons and phonons in the ultrafast recovery of charge density wave correlations in , Phys. Rev. B 94, 165165 (2016).
- K. Katsumi, A. Alekhin, S.-M. Souliou, M. Merz, A.-A. Haghighirad, M. Le Tacon, S. Houver, M. Cazayous, A. Sacuto, and Y. Gallais, Disentangling lattice and electronic instabilities in the excitonic insulator candidate by nonequilibrium spectroscopy, Phys. Rev. Lett. 130, 106904 (2023).
- S. Mor, M. Herzog, D. Golež, P. Werner, M. Eckstein, N. Katayama, M. Nohara, H. Takagi, T. Mizokawa, C. Monney, and J. Stähler, Ultrafast electronic band gap control in an excitonic insulator, Phys. Rev. Lett. 119, 086401 (2017).
- M. Schüler, D. Golež, Y. Murakami, N. Bittner, A. Herrmann, H. U. Strand, P. Werner, and M. Eckstein, nessi: The non-equilibrium systems simulation package, Comput. Phys. Commun. 257, 107484 (2020).