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arXiv:quant-ph/9801077 · Physics of the Solid State

Squeezed states of a particle in magnetic field

M. Ozana, A. L. Shelankov

Published Aug 1, 1998• 11 citations
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Abstract

For a charged particle in a homogeneous magnetic field, we construct stationary squeezed states which are eigenfunctions of the Hamiltonian and the non-Hermitian operator % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0dc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGybGbaK% aadaWgaaWcbaGaeuOPdyeabeaakiabg2da9iqadIfagaqcaiGacoga% caGGVbGaai4CaiabfA6agjabgUcaRiqadMfagaqcaiGacohacaGGPb% GaaiOBaiabfA6agbaa!4894! $$\hat X_\Phi = \hat X\cos \Phi + \hat Y\sin \Phi $$ , % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0dc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGybGbaK% aaaaa!3A82! $$\hat X$$ and Ŷ being the coordinates of the Larmor circle center and Φ is a complex parameter. In the family of the squeezed states, the quantum uncertainty in the Larmor circle position is minimal. The wave functions of the squeezed states in the coordinate representation are found and their properties are discussed. Besides, for arbitrary gauge of the vector potential we derive the symmetry operators of translations and rotations.

DOI
10.1134/1.1130543

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For a charged particle in a homogeneous magnetic field, we construct stationary squeezed states which are eigenfunctions of the Hamiltonian and the non-Hermitian operator % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0dc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGybGbaK% aadaWgaaWcbaGaeuOPdyeabeaakiabg2da9iqadIfagaqcaiGacoga% caGGVbGaai4CaiabfA6agjabgUcaRiqadMfagaqcaiGacohacaGGPb% GaaiOBaiabfA6agbaa!4894! X^Φ=X^cos⁡Φ+Y^sin⁡Φ\hat X_\Phi = \hat X\cos \Phi + \hat Y\sin \Phi X^Φ​=X^cosΦ+Y^sinΦ , % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0dc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGybGbaK% aaaaa!3A82! X^\hat XX^ and Ŷ being the coordinates of the Larmor circle center and Φ is a complex parameter. In the family of the squeezed states, the quantum uncertainty in the Larmor circle position is minimal. The wave functions of the squeezed states in the coordinate representation are found and their properties are discussed. Besides, for arbitrary gauge of the vector potential we derive the symmetry operators of translations and rotations.
The exchange interaction and effective mass of fermionic excitation in a low-density (r S ≫ 1) system of two-dimensional electrons are estimated from simple considerations. For the ratio of effective (renormalized due to interaction) to band mass, the dependence % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0xc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcgaqaai% aad2gadaahaaWcbeqaaiaacQcaaaaakeaacaWGTbaaaiabg2da9iaa% cIcadaWcgaqaaiaadgeaaeaadaGcaaqaaiaadkhadaWgaaWcbaGaam% 4uaaqabaaabeaaaaGccaGGPaGaciyzaiaacIhacaGGWbGaaiikaGGa% aiab-f7aHnaakaaabaGaamOCamaaBaaaleaacaWGtbaabeaaaeqaaO% Gaaiykaaaa!49B6! $${{m^* } \mathord{\left/ {\vphantom {{m^* } m}} \right. \kern-\nulldelimiterspace} m} = ({A \mathord{\left/ {\vphantom {A {\sqrt {r_S } }}} \right. \kern-\nulldelimiterspace} {\sqrt {r_S } }})\exp (\alpha \sqrt {r_S } )$$ is obtained, where A and α are constants on the order of unity. The effective g factor is independent of r S and is larger than its bare value in the two-valley case (silicon). Comparison with experimental data shows a qualitative agreement with silicon.
We study the kinetics of 2D Bose gas cooling provided Bose particles interact with 3D phonons. At low temperatures, phonon emission is prohibited by energy and momentum conservation. We show that both particle—particle scattering and impurity scattering assist Bose gas cooling. The temporal relaxation of temperature follows the law % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0Jd9sq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcKbay-l% qa-Xgagaafaaaa!38AA! $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} $$ above the Berezinski-Kosterlitz-Thouless phase transition point and T∼1/t after a Bose-Einstein 2D quasi-condensate develops.
Feller [1] and Kato [2] proved the existence of a unique minimal semigroup associated with the classical Fokker-Planck equation: (1.1) % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaeaada % WcaaqaaiaadsgacaWGWbWaaSbaaSqaaiaadMgacaWGQbaabeaaaOqa % aiaadsgacaWG0baaaiabg2da9maaqafabaGaamiCamaaBaaaleaaca % WGPbGaam4AaaqabaGccqqHPoWvdaWgaaWcbaGaam4AaiaadQgaaeqa % aOGaaiilaiaaywW7caWG0bGaeyyzImRaaGimaiaaywW7caWG3bGaam % yAaiaadshacaWGObGaaGPaVlaadMgacaWGUbGaamyAaiaadshacaWG % PbGaamyyaiaadYgacaaMc8Uaam4yaiaad+gacaWGUbGaamizaiaadM % gacaWG0bGaamyAaiaad+gacaWGUbGaaGPaVlaadchadaWgaaWcbaGa % amyAaiaadQgaaeqaaOGaaiikaiaaicdacaGGPaGaeyypa0JaeqiTdq % 2aaSbaaSqaaiaadMgacaWGQbaabeaaaeaacaWGRbaabeqdcqGHris5 % aaGccaGL9baaaaa!703E! dpijdt=∑kpikΩkj,t⩾0with initial condition pij(0)=δij} \left. {\frac{{d{p_{ij}}}}{{dt}} = \sum\limits_k {{p_{ik}}{\Omega _{kj}},\quad t \geqslant 0\quad with\,initial\,condition\,{p_{ij}}(0) = {\delta _{ij}}} } \right\} dtdpij​​=k∑​pik​Ωkj​,t⩾0withinitialconditionpij​(0)=δij​} and subject to the Markov condition: (1.2) % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaqUbcqqHPo % WvdaWgaaWcbaGaam4AaiaadQgaaeqaaOGaeyyzImRaaGimaiaadAga % caWGVbGaamOCaiaaykW7caWGRbGaeyiyIKRaamOAaiaaykW7caWGHb % GaamOBaiaadsgacaaMc8UaeyOhIu6aaabuaeaacqqHPoWvdaWgaaWc % baGaam4AaiaadQgaaeqaaOGaeyypa0JaeyOeI0IaeuyQdC1aaSbaaS % qaaiaadUgacaWGRbaabeaatCvAUfeBSn0BKvguHDwzZbqegeezVjwz % GyuyUD2CV52zGmfDKbIuaGqbaOGaa8hpaiabg6HiLcWcbaGaamOAai % abgcMi5kaadUgaaeqaniabggHiLdGccaGGUaaaaa!6B15! Ωkj⩾0for k≠j and ∞∑j≠kΩkj=−Ωkk<∞. {\Omega _{kj}} \geqslant 0for\,k \ne j\,and\,\infty \sum\limits_{j \ne k} {{\Omega _{kj}} = - {\Omega _{kk}} < \infty } . Ωkj​⩾0fork=jand∞j=k∑​Ωkj​=−Ωkk​<∞.