Introducing coherent state superpositions in non-linear optics

Superposition is a fundamental principle of quantum mechanics. It states that if a quantum system can exist in two distinct states (A or B), it can also be in any superposition of the two, leaving the system’s exact state (A or B) undefined. This principle, when applied to coherent states —those that closely mimic the behavior of classical light—, leads to quantum light. And it turns out that the superposition of coherent states offers profound opportunities for advancing quantum science and technology. In particular, the applicability of coherent state superpositions in the field of nonlinear optics holds great promise for addressing unanswered questions and uncovering novel phenomena in quantum physics, quantum optics and quantum information.

However, nonlinear processes require extremely strong laser fields with a high mean number of photons to occur. This poses two major challenges for the integration of these states in nonlinear optics. First, achieving such high intensities while maintaining the non-classical properties of light is exceedingly difficult. And second, standard methods for witnessing the quantum nature of light are only effective at low photon numbers.

Now, a team of researchers from IESL FORTH, Dr. Theocharis Lamprou and Dr. Paraskevas Tzallas, and ICFO, Dr. Javier Rivera-Dean, Philipp Stammer and ICREA Prof. Maciej Lewenstein, has successfully addressed both issues. They created generalized coherent state superpositions (GCSS), verified their quantum nature and employed them to drive a nonlinear optical process (which, per se, is indicative of the high photon number and intensity of the states). Their findings, demonstrated both theoretically and experimentally, have been reported in Physical Review Letters.

Using an infrared laser pulse, the team generated GCSS with the highest mean photon number recorded for such states. They then directed this pulse onto an optical crystal, which absorbed the light and re-emitted it as its second harmonic —a frequency-doubled version of the original pulse. This phenomenon was induced by a nonlinear process called second harmonic generation.

“The generation of GCSS using a strong-field process was first observed in an earlier study from 2021. While these results suggested the potential for high-photon numbers, direct evidence was still lacking”, explains Dr. Javier Rivera. “In this work, we close that gap by demonstrating the high-photon number nature of GCSS through their ability to drive nonlinear optical processes”, he adds.

Bridging quantum optics and strong field physics

The study in itself already represents a fundamental application. It shows that, by combining quantum optics with strong-field physics —the field that explores how matter interacts with highly intense laser fields—, one can create intense laser fields with non-classical photon statistics, capable of driving nonlinear optical processes.

But the integration of these two disciplines, long sought after by researchers in both fields, opens further exciting possibilities. Quantum information science, for instance, which already overlaps with quantum optics, could benefit from incorporating strong-field physics. Similarly, attosecond science, which relies on high-intensity laser fields, might advance by leveraging quantum features often overlooked in traditional approaches.

Moreover, the team developed an approach that was free from decoherence effects. Decoherence, a process where quantum states lose their quality due to interactions with the environment, is a major obstacle for the vast majority of quantum information tasks. Developing quantum states that are resilient to decoherence or techniques to mitigate its effects is thus essential for advancing practical quantum technologies. Therefore, the states generated in the work, as well as the methods used to create them, could play a key role in overcoming the decoherence problem.

Reference:

Th. Lamprou, J. Rivera-Dean, P. Stammer, M. Lewenstein, and P. Tzallas, Nonlinear Optics Using Intense Optical Coherent State Superpositions. Phys. Rev. Lett. 134, 013601 (2025).

DOI: https://doi.org/10.1103/PhysRevLett.134.013601

Acknowledgements:

The Hellenic Foundation for Research and Innovation (HFRI) and the General Secretariat for Research and Technology (GSRT) under grant agreement CO2toO2 Nr.:015922, the European Union’s HORIZON-MSCA-2023-DN-01 project QU-ATTO under the Marie Skłodowska-Curie grant agreement No 101168628, the LASERLABEUROPE V (H2020-EU.1.4.1.2 grant no.871124) and ELI–ALPS. ELI–ALPS is supported by the EU and co-financed by the European Regional Development Fund (GINOP No. 2.3.6-15-2015-00001). H2020-EU research and innovation program under the Marie Skłodowska-Curie (No. 847517). Government of Spain (Severo Ochoa CEX2019-000910-S and TRANQI), Fundació Cellex, Fundació Mir-Puig, Generalitat de Catalunya (CERCA program) and the ERC AdG CERQUTE. ERC AdG NOQIA; MCIN/AEI (PGC2018-0910.13039/501100011033, CEX2019-000910-S/10.13039/501100011033, Plan National FIDEUA PID2019-106901GB-I00, STAMEENA PID2022-139099NB, I00, project funded by MCIN/AEI/10.13039/501100011033 and by the EU Next Generation EU/PRTR (PRTRC17.I1), FPI). QUANTERA MAQS PCI2019-111828-2; QUANTERA

DYNAMITE PCI2022-132919, QuantERA II Programme co-funded by H2020-EU program (No 101017733); Ministry for Digital Transformation and of Civil Service of the Spanish Government through the QUANTUM ENIA project call-Quantum Spain project, and by the EU through the Recovery, Transformation and Resilience Plan—Next Generation EU within the framework of the Digital Spain 2026 Agenda; Fundació Cellex; Fundació Mir-Puig; Generalitat de Catalunya (European Social Fund FEDER and CERCA program, AGAUR Grant No. 2021 SGR 01452, QuantumCAT \U16-011424, co-funded by ERDF Operational Program of Catalonia 2014-2020); Barcelona Supercomputing Center MareNostrum (FI-2023-1-0013) funded by the EU. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the EU, European Commission, European Climate, Infrastructure and Environment Executive Agency (CINEA), or any other granting authority. Neither the EU nor any granting authority can be held responsible for them (EU Quantum Flagship PASQuanS2.1, 101113690, EU H2020 FET-OPEN OPTOlogic (No 899794), EU Horizon Europe Program (No 101080086-NeQST); ICFO Internal “QuantumGaudi” project; EU H2020 program under the Marie Sklodowska-Curie (No. 847648); “La Caixa” Junior Leaders fellowships, La Caixa” Foundation (ID 100010434): CF/BQ/PR23/11980043. P.S. acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 847517.