{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/31105"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/31105","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"BEC-BCS crossover with Feshbach resonance for three-hyperfine-species model","abstract":"The BEC-BCS crossover problem has been intensively studied both theoretically and experimentally largely thanks to Feshbach resonances which allow us to tune the effective interaction between alkali atoms. In a Feshbach resonance, the effective s-wave scattering length grows when one moves toward the resonance point, and eventually diverges at this point. There is one characteristic energy scale, $\\delta_c$, defined as, in the negative side of the resonance point, the detuning energy at which the weight of the bound state shifts from predominatedly in the open-channel to predominated in the closed-channel. When the many-body energy scale (e.g. the Fermi energy, $E_{F}$) is larger than $\\delta_c$, the closed-channel weight is significant and has to be included in the many-body theory. Furthermore, when two channels share a hyperfine species, the Pauli exclusion between fermions from two channels also needs to be taken into consideration in the many-body theory. The current thesis addresses the above problem in detail. A set of gap equations and number equations are derived at the mean-field level. The fermionic and bosonic excitation spectra are then derived. Assuming that the uncoupled bound-state of the closed-channel in resonance is much smaller than the inter-particle distance, as well as the s-wave scattering length, $a_s$, we find that the basic equations in the single-channel crossover model are still valid. The correction first comes from the existing of the finite chemical potential and additional counting complication due to the closed-channel. These two corrections need to be included into the mean-field equations, i.e. the gap equations and the number equations, and be solved self-consistently. Then the correction due to the inter-channel Pauli exclusion is in the order of the ratio of the Fermi energy and the Zeeman energy difference between two channels, $E_F/\\eta$, which can be analyzed perturbatively over the previous corrections. Fermionic and bosonic excitation modes are studied. Similarly as the mean-field result, the basic structure follows that of the single-channel model, and the correction due to the inter-channel Pauli exclusion can be treated perturbatively with expansion parameter in the order of $E_F/\\eta$. In the bosonic excitation, a new out-of-sync phase mode emerges for the two-component order parameters. It is nevertheless gapped at the the pair-breaking energy.","abstract_html":"The BEC-BCS crossover problem has been intensively studied both theoretically and experimentally largely thanks to Feshbach resonances which allow us to tune the effective interaction between alkali atoms. In a Feshbach resonance, the effective s-wave scattering length grows when one moves toward the resonance point, and eventually diverges at this point. There is one characteristic energy scale, <span class=\"etd-inline-math\">&delta;<sub>c</sub></span>, defined as, in the negative side of the resonance point, the detuning energy at which the weight of the bound state shifts from predominatedly in the open-channel to predominated in the closed-channel. When the many-body energy scale (e.g. the Fermi energy, <span class=\"etd-inline-math\">E<sub>F</sub></span>) is larger than <span class=\"etd-inline-math\">&delta;<sub>c</sub></span>, the closed-channel weight is significant and has to be included in the many-body theory. Furthermore, when two channels share a hyperfine species, the Pauli exclusion between fermions from two channels also needs to be taken into consideration in the many-body theory. The current thesis addresses the above problem in detail. A set of gap equations and number equations are derived at the mean-field level. The fermionic and bosonic excitation spectra are then derived. Assuming that the uncoupled bound-state of the closed-channel in resonance is much smaller than the inter-particle distance, as well as the s-wave scattering length, <span class=\"etd-inline-math\">a<sub>s</sub></span>, we find that the basic equations in the single-channel crossover model are still valid. The correction first comes from the existing of the finite chemical potential and additional counting complication due to the closed-channel. These two corrections need to be included into the mean-field equations, i.e. the gap equations and the number equations, and be solved self-consistently. Then the correction due to the inter-channel Pauli exclusion is in the order of the ratio of the Fermi energy and the Zeeman energy difference between two channels, <span class=\"etd-inline-math\">E<sub>F</sub>/\\eta</span>, which can be analyzed perturbatively over the previous corrections. Fermionic and bosonic excitation modes are studied. Similarly as the mean-field result, the basic structure follows that of the single-channel model, and the correction due to the inter-channel Pauli exclusion can be treated perturbatively with expansion parameter in the order of <span class=\"etd-inline-math\">E<sub>F</sub>/\\eta</span>. In the bosonic excitation, a new out-of-sync phase mode emerges for the two-component order parameters. It is nevertheless gapped at the the pair-breaking energy.","abstract_has_math":true,"creators":["Zhu, Guojun"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Physics","degree_department":null,"school":null,"contributors":["Leggett, Anthony J.","Baym, Gordon A.","DeMarco, Brian L.","Willenbrock, Scott S."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-05-22T00:28:20Z","date_published":"2012-05-22T00:28:20Z","updated_at":"2026-07-22T22:25:30Z","subjects":["superconductivity","crossover","Bose-Einstein condensation","alkali gas"],"languages":["en"],"rights":["Copyright 2011 Guojun Zhu"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/31105","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Leggett, Anthony J.","Baym, Gordon A.","DeMarco, Brian L.","Willenbrock, Scott S."]},{"key":"dc:creator","label":"Author","values":["Zhu, Guojun"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2012-05-22T00:28:20Z","2012-05"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Physics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["superconductivity","crossover","Bose-Einstein condensation","alkali gas"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2011 Guojun Zhu"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/31105"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The BEC-BCS crossover problem has been intensively studied both theoretically and experimentally largely thanks to Feshbach resonances which allow us to tune the effective interaction between alkali atoms. In a Feshbach resonance, the effective s-wave scattering length grows when one moves toward the resonance point, and eventually diverges at this point. There is one characteristic energy scale, $\\delta_c$, defined as, in the negative side of the resonance point, the detuning energy at which the weight of the bound state shifts from predominatedly in the open-channel to predominated in the closed-channel. When the many-body energy scale (e.g. the Fermi energy, $E_{F}$) is larger than $\\delta_c$, the closed-channel weight is significant and has to be included in the many-body theory. Furthermore, when two channels share a hyperfine species, the Pauli exclusion between fermions from two channels also needs to be taken into consideration in the many-body theory. The current thesis addresses the above problem in detail. A set of gap equations and number equations are derived at the mean-field level. The fermionic and bosonic excitation spectra are then derived. Assuming that the uncoupled bound-state of the closed-channel in resonance is much smaller than the inter-particle distance, as well as the s-wave scattering length, $a_s$, we find that the basic equations in the single-channel crossover model are still valid. The correction first comes from the existing of the finite chemical potential and additional counting complication due to the closed-channel. These two corrections need to be included into the mean-field equations, i.e. the gap equations and the number equations, and be solved self-consistently. Then the correction due to the inter-channel Pauli exclusion is in the order of the ratio of the Fermi energy and the Zeeman energy difference between two channels, $E_F/\\eta$, which can be analyzed perturbatively over the previous corrections. Fermionic and bosonic excitation modes are studied. Similarly as the mean-field result, the basic structure follows that of the single-channel model, and the correction due to the inter-channel Pauli exclusion can be treated perturbatively with expansion parameter in the order of $E_F/\\eta$. In the bosonic excitation, a new out-of-sync phase mode emerges for the two-component order parameters. It is nevertheless gapped at the the pair-breaking energy.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2012-01-03T20:29:57Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 3 0_deposit.zip: 3365985 bytes, checksum: 08a59e223486064e4867352fa28e0765 (MD5) newSlide.pdf: 738090 bytes, checksum: c3d15e56dea617d6ba734097aeb64e16 (MD5) Zhu_Guojun.pdf: 1111819 bytes, checksum: 7e6b74f2ec58a24da2cc3b2a8a959df1 (MD5)","Made available in DSpace on 2012-05-22T00:28:20Z (GMT). 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There is one characteristic energy scale, $\\delta_c$, defined as, in the negative side of the resonance point, the detuning energy at which the weight of the bound state shifts from predominatedly in the open-channel to predominated in the closed-channel. When the many-body energy scale (e.g. the Fermi energy, $E_{F}$) is larger than $\\delta_c$, the closed-channel weight is significant and has to be included in the many-body theory. Furthermore, when two channels share a hyperfine species, the Pauli exclusion between fermions from two channels also needs to be taken into consideration in the many-body theory. The current thesis addresses the above problem in detail. A set of gap equations and number equations are derived at the mean-field level. The fermionic and bosonic excitation spectra are then derived. Assuming that the uncoupled bound-state of the closed-channel in resonance is much smaller than the inter-particle distance, as well as the s-wave scattering length, $a_s$, we find that the basic equations in the single-channel crossover model are still valid. The correction first comes from the existing of the finite chemical potential and additional counting complication due to the closed-channel. These two corrections need to be included into the mean-field equations, i.e. the gap equations and the number equations, and be solved self-consistently. Then the correction due to the inter-channel Pauli exclusion is in the order of the ratio of the Fermi energy and the Zeeman energy difference between two channels, $E_F/\\eta$, which can be analyzed perturbatively over the previous corrections. Fermionic and bosonic excitation modes are studied. Similarly as the mean-field result, the basic structure follows that of the single-channel model, and the correction due to the inter-channel Pauli exclusion can be treated perturbatively with expansion parameter in the order of $E_F/\\eta$. In the bosonic excitation, a new out-of-sync phase mode emerges for the two-component order parameters. It is nevertheless gapped at the the pair-breaking energy.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2012-01-03T20:29:57Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 3 0_deposit.zip: 3365985 bytes, checksum: 08a59e223486064e4867352fa28e0765 (MD5) newSlide.pdf: 738090 bytes, checksum: c3d15e56dea617d6ba734097aeb64e16 (MD5) Zhu_Guojun.pdf: 1111819 bytes, checksum: 7e6b74f2ec58a24da2cc3b2a8a959df1 (MD5)","Made available in DSpace on 2012-05-22T00:28:20Z (GMT). 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