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Taking the steady state of the reservoir rate equation shows that nres and n are proportional through \u03b3resnres=\u03b3inn. This energy renormalization modifies the spectrum of collective excitations as well.","type":"other"},{"key":"whgn-6889Cc24R1","unstructured":"We use \u210fgn\/m\u22c6 as a proxy for the amplitude of the polariton field. This quantity has the dimension of a velocity and corresponds to the Bogoliubov hydrodynamic speed of sound in the case \u03b4(v)=gn+gresnres, where the Bogoliubov dispersion is linear at low k. However, as discussed in [25] and [19], the Bogoliubov field becomes massive when gn+gresnres>\u03b4, which is the case here. In this regime, the notion of a hydrodynamic speed of sound no longer makes sense.","type":"other"},{"key":"whgn-6889Cc25R1","doi-asserted-by":"publisher","DOI":"10.1103\/t5dh-rx6w","type":"journal-article"},{"key":"whgn-6889Cc26R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.101.064505","type":"journal-article"},{"key":"whgn-6889Cc27R1","first-page":"23","type":"journal-article","volume":"11","author":"N.\u2009N. Bogolyubov","year":"1947","journal-title":"J. Phys. (USSR)","ISSN":"https:\/\/id.crossref.org\/issn\/0368-3400","issn-type":"print"},{"key":"whgn-6889Cc28R1","doi-asserted-by":"publisher","DOI":"10.1088\/0034-4885\/35\/1\/306","type":"journal-article"},{"key":"whgn-6889Cc29R1","unstructured":"Note that here we redefine \u03b4 from \u03b4(kp)=\u03c9p\u2212\u03c90\u2212\u210fkp2\/2m\u22c6 to \u03b4(v)=\u03c9p\u2212\u03c90\u2212m\u22c6v2\/2\u210f. In the plane wave pumping case, due to strict phase locking, v\u221dkp, so this change is purely conventional. However, in the general case where the polariton phase is not strictly determined by the pump phase, the effective detuning should depend on v rather than \u2207\u03d5p [19].","type":"other"},{"key":"whgn-6889Cc30R1","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6382\/abf1fc","type":"journal-article"},{"key":"whgn-6889Cc31R1","unstructured":"We observe the \u03c9B component of the mode, i.e., u(r)ei(m\u03b8\u2212\u03c9Bt) because OAI can only resolve optical fields of the same frequency as the probe. We cannot observe the four-wave mixing signal v(r)e\u2212i(m\u03b8\u2212\u03c9Bt) as in [25] because it exits the cavity at 2\u03c9p\u2212\u03c9pr and therefore does not interfere with the reference beam at \u03c9pr.","type":"other"},{"key":"whgn-6889Cc32R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.125.011301","type":"journal-article"},{"key":"whgn-6889Cc33R1","doi-asserted-by":"publisher","DOI":"10.1038\/nphys4151","type":"journal-article"},{"key":"whgn-6889Cc34R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.128.013901","type":"journal-article"},{"key":"whgn-6889Cc35R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevD.109.105024","type":"journal-article"},{"key":"whgn-6889Cc36R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-018-07882-8","type":"journal-article"},{"key":"whgn-6889Cc37R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-019-14243-6","type":"journal-article"},{"key":"whgn-6889Cc38R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.126.075301","type":"journal-article"},{"key":"whgn-6889Cc39R1","doi-asserted-by":"publisher","DOI":"10.5281\/zenodo.17699486","type":"software"}],"container-title":["Physical Review Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.aps.org\/article\/10.1103\/whgn-6889","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/harvest.aps.org\/v2\/journals\/articles\/10.1103\/whgn-6889\/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,9]],"date-time":"2025-12-09T12:14:57Z","timestamp":1765282497000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.aps.org\/doi\/10.1103\/whgn-6889"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,9]]},"references-count":39,"journal-issue":{"issue":"24","published-print":{"date-parts":[[2025,12]]}},"URL":"https:\/\/doi.org\/10.1103\/whgn-6889","relation":{},"ISSN":["0031-9007","1079-7114"],"issn-type":[{"value":"0031-9007","type":"print"},{"value":"1079-7114","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,9]]},"assertion":[{"value":"2025-07-25","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"\u00a9 2025 American Physical Society. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright_statement","label":"Copyright statement","group":{"name":"copyright","label":"copyright"}},{"value":"2025","name":"copyright_year","label":"Copyright year","group":{"name":"copyright","label":"copyright"}},{"value":"American Physical Society","name":"copyright_holder","label":"Copyright holder","group":{"name":"copyright","label":"copyright"}}],"article-number":"243801"}}