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Using the BAW model of Ref.\u00a0[44], we can crudely estimate \u03bae for the \u03c9r\/2\u03c0=520\u2009\u2009MHz resonator. Taking \u03f5r\u22484.6 for ST quartz and an electrode length \u2113=12\u2009\u2009\u03bcm gives a static capacitance C0\u2248\u03f50(\u03f5r+1)\u2113\/2\u22480.3\u2009\u2009fF, leading to \u03bae=(8K2\/\u03c02)\u03c9r2C0Z0\u22482\u03c0\u00d730\u2009\u2009Hz. At T0=25\u2009\u2009mK, the internal loss rate is \u03bai=\u03c9rF\u03b40tanh(\u210f\u03c9r\/2kBT0)\u22482\u03c0\u00d72.4\u2009\u2009kHz for a typical TLS loss tangent F\u03b40=1.0\u00d710\u22125, yielding \u03bai\/\u03bae\u224880.","type":"other"},{"key":"q9g9-xbh2Cc57R1","unstructured":"In superconducting circuits, the most common Kerr-like nonlinearity arises from the power-dependent kinetic inductance of the thin superconducting films Refs. [49,50,80] and is characterized by a softening response (negative resonance shift with power). Mechanical systems such as MEMS based on doubly clamped beams can feature nonlinearities of purely geometric origin with both softening and hardening behaviors Refs. [112\u2013114]. We estimate this geometric effect to be negligible for our PCRs.","type":"other"},{"key":"q9g9-xbh2Cc58R1","unstructured":"This is true for upward frequency sweeps; however, for downward sweeps, the probe tone is only approximately resonant at the applied frequency where |S11| exhibits its minimum Ref. [67].","type":"other"},{"key":"q9g9-xbh2Cc59R1","doi-asserted-by":"publisher","DOI":"10.1088\/1367-2630\/ab97e8","type":"journal-article"},{"key":"q9g9-xbh2Cc60R1","unstructured":"A similar expression can be worked out for a two-port network. For the common case of a short-circuited \u03bb\/4 resonator notch-port coupled to a transmission line (the so-called \u201changer\u201d style), Refs.\u00a0[59,111] derive the following expressions: S11=S22=\u2212(Q\/Qe)\/(1+2jQx) and S12=S21=1+S11=1\u2212(Q\/Qe)\/(1+2jQx). The dissipated power is then Pd=(1\u2212|S11|2\u2212|S21|2)Ps=2(Q2\/QeQi)Ps\/(1+4Q2x2), a factor of 2 smaller than the expression for a one-port resonator in Eq.\u00a0(6).","type":"other"},{"key":"q9g9-xbh2Cc61R1","unstructured":"Equations\u00a0(1) and (2) describe a one-port resonator measured in reflection via a lossless circuit. As in Ref.\u00a0[59], we assume a power-independent coupling quality factor Qe and that nonlinearities arise solely through x(f,Pd) and Q(Pd), corresponding to a fixed circuit topology with power-dependent component values.","type":"other"},{"key":"q9g9-xbh2Cc62R1","unstructured":"Using that \u03a8(z)\u223cz\u2192\u221eln(z)+O(1\/z), one can directly verify from Eq.\u00a0(3) that limT\u21920\u0394fr(T)=0.","type":"other"},{"key":"q9g9-xbh2Cc63R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.103.224209","type":"journal-article"},{"key":"q9g9-xbh2Cc64R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevApplied.21.054044","type":"journal-article"},{"key":"q9g9-xbh2Cc65R1","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.86.1391","type":"journal-article"},{"key":"q9g9-xbh2Cc66R1","unstructured":"The state of near-resonant TLSs is also affected by the average phonon number n\u00af; however, their contribution to the overall resonance shift is negligible compared with the\u00a0one from the thermally populated, off-resonant TLS continuum.","type":"other"},{"key":"q9g9-xbh2Cc67R1","unstructured":"See Supplemental Materials at http:\/\/link.aps.org\/supplemental\/10.1103\/q9g9-xbh2 for details about S1 for the scalar model of the thermal conductance of the PCR tethers, S3 for the analysis of Swenson\u2019s equation and the nonlinear frequency response of the driven resonator, S4 for the derivation of the bistability criterion and the definitions of the characteristic phonon number scales, and S5 for the effect of external coupling rate on the onset of bistability, which includes Refs. [68\u201377].","type":"other"},{"key":"q9g9-xbh2Cc68R1","doi-asserted-by":"publisher","DOI":"10.1006\/spmi.1997.0561","type":"journal-article"},{"key":"q9g9-xbh2Cc69R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.81.232","type":"journal-article"},{"key":"q9g9-xbh2Cc70R1","doi-asserted-by":"publisher","DOI":"10.1088\/0953-8984\/9\/27\/007","type":"journal-article"},{"key":"q9g9-xbh2Cc71R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.71.205308","type":"journal-article"},{"key":"q9g9-xbh2Cc72R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.70.125425","type":"journal-article"},{"key":"q9g9-xbh2Cc73R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.2748342","type":"journal-article"},{"key":"q9g9-xbh2Cc74R1","doi-asserted-by":"publisher","DOI":"10.1002\/zamm.19210010109","type":"journal-article"},{"key":"q9g9-xbh2Cc75R1","doi-asserted-by":"crossref","DOI":"10.1002\/9780470977859","type":"book","volume-title":"The Duffing Equation: Nonlinear Oscillators and Their Behavior","author":"I. 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[115,116].","type":"other"},{"key":"q9g9-xbh2Cc92R1","doi-asserted-by":"publisher","DOI":"10.1146\/annurev-conmatphys-020911-125022","type":"journal-article"},{"key":"q9g9-xbh2Cc93R1","unstructured":"This approximation for the crossover Tc temperature can be derived using a known inequality constraining the derivative of the digamma function [117], (x+1\/2)\/x2\u2264\u03a8\u2032(x)\u2264(x+1)\/x2 for x>0. Solving TCF(Tc)=0 numerically yields kBTc\/hfr\u22480.4408, which is within 2% of the approximate result from Eq.\u00a0(A10).","type":"other"},{"key":"q9g9-xbh2Cc94R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.3273372","type":"journal-article"},{"key":"q9g9-xbh2Cc95R1","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970944","type":"book","volume-title":"Iterative Methods for Linear and Nonlinear Equations","author":"C.\u2009T. 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Chung","year":"2004","type":"conference-proceedings"},{"key":"q9g9-xbh2Cc117R1","doi-asserted-by":"publisher","DOI":"10.1515\/ms-2015-0010","type":"journal-article"}],"container-title":["Physical Review Applied"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.aps.org\/accepted\/10.1103\/q9g9-xbh2","content-type":"application\/pdf","content-version":"am","intended-application":"unspecified"},{"URL":"https:\/\/link.aps.org\/article\/10.1103\/q9g9-xbh2","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/harvest.aps.org\/v2\/journals\/articles\/10.1103\/q9g9-xbh2\/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,26]],"date-time":"2026-10-26T14:46:56Z","timestamp":1793026016000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.aps.org\/doi\/10.1103\/q9g9-xbh2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,10,26]]},"references-count":116,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2026,10]]}},"URL":"https:\/\/doi.org\/10.1103\/q9g9-xbh2","relation":{},"ISSN":["2331-7019"],"issn-type":[{"value":"2331-7019","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,10,26]]},"assertion":[{"value":"2026-02-03","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"\u00a9 2026 American Physical Society. 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