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10 Aug. 2010 Kyoto Yukawa Reduction of the system dynamics from the total system including the environments The University of Tokyo Seiji Miyashita Projection operator method For the master equation Real part Phonon Bottleneck phenomena in V15 Plateau induced by thermal effect sample Heat flow Heat bath Chiorescu, W. Wernsdorfer, A. Mueller, H. Boegge, B. Barbara, Phys. Rev. Lett. 84 (2000) 3454. Field sweeping with thermal bath Fast sweeping vAD v LZS Slow sweeping vAD vTH v vAD Magnetic Foehn Effect K. Saito & SM. JPSJ (2001) 3385. Fe2 Fe-rings Y. Shapira, et al PRB59 (1999) 1046 dM dH dM dH Y. Ajiro & Y. Inagaki Y. Narumi & K. Kindo H. Nakano & SM, JPSJ 70(2001) 2151 Fast Magnetization Tunneling in Tetranicke(II) SMM [Ni(hmp)(dmb)Cl]4 d iH , z X , R X , R dt V=0.002, ..... , 0.28T/s En-Che Yang,et al: Inorg. Chem. 45 (2006) 529 v=0.0512, ...., 0.0002 Boson system Spin-boson system from QMEnote (SM and T. Mori) Relation between the equation of motion and its steady solution Equation of motion up to the second order 1 L O 4 , L H s , 2 ( ) t i (a situational solution) L st 0 we may add any traceless W L st 2W O 4 if T rW 0 The diagonal elements are arbitrary in the order of O 2 0 Master equation leads the system to the equilibrium of the system O 2 The off-diagonal elements aredetermined in the order of O T. Mori and SM: JPSJ 77 (2008) 124005 (1-9). Complex admittance C. Uchiyama, M. Aihara, M. Saeki and S. Miyashita: PRE 80 (2009) 021128 (1-18). M. Saeki, C. Uchiyama, T. Mori and S. Miyashita: PRE 81, (2010) 031131 (1-33) 10 Aug. 2010 Kyoto Yukawa Study on the line shapes of the response function --Origins of the Width-The University of Tokyo Seiji Miyashita ESR line shape in strongly interacting spin systems Temperature-dependence of the shift and width in lowdimensional quantum spin systems Spin trimer: 3CuCl2 ・2Dioxane g B H F F AF (S=1/2)x3 paramagnetic EPR S=3/2 correlated state Y. Ajiro, et al: JPSJ 63 (1994) 859. Shift and width of the line shape • Intrinsic width due to assembly of the delta-functions Dmn e Em e En x mM n 2 , ( En Em ) Z • Quantum broadening due to quantum fluctuation of the field H JC 0 2 g b b b b i i N i 1 N z i i 1 • Transmission spectrum (input-output modes) C E T E / AE • Broadening width due to the interaction with the thermal bath 2 i 1 A 0 2 TrS B i iLS Microscopic expression of the line shape from the Hamiltonian of the system R. Kubo & K.Tomita JPSJ (1954) 888. R. Kubo: JPSJ 12 (1957) 570. Kubo formula H 2 J ijS i S j H 0 Siz H1 cost S ix ij i i 1 xx " ( ) (1 e ) M x (0) M x (t ) eit dt 2 Isotropic models (Paramagnetic Resonance) 1 g 2 E / R H , Perturbation H perturbation 2 [ J ( Six S xj Siy S jy ) J z Siz S zj ] ij D mn S S m n 3S m rmn S n rmn D 5 r3 rmn mn S S i ij j Expression of the admittance 1 xx " ( ) (1 e ) M x (0) M x (t ) eit dt 2 Eigenvalue and eigenvectors of the Hamiltonian H | m Em | m " D(mn ) ( ( En Em )) mn Dmn e Em e En x 2 mM n Z , ( En Em ) M x (0) M x (t ) m 2e i0t t / " 0 2 0 R width shift Nagata-Tazuke effect K.Nagata and Y.Tazuke, JPSJ 32(1972)337. (J. Kanamori & M.Tachiki : JPSJ 48 (1962) 50) One-dimensional Heisenberg antiferromagnet H0 c axis H0 // c axis Demonstration of the Nagata-Tazuke effects " D(mn ) ( ( En Em )) N=8 mn SM, T. Yoshino, A. Ogasahara: JPSJ 68 (1999) 655. R.E. Dietz, et al. PRL 26 (1971) 1186. T.T. Cheung, et al. PRB 17 (1978) 1266 Line shape of a spin chane with a staggered DM interaction Diz 1i D S. El Shawish, O. Cepas, and SM: PRB81, 224421 (2010). Line shape of a spin chain with a staggered DM interaction Diz 1i D T cf. D0 S xx h/ 0 all T S. El Shawish, O. Cepas, and SM: PRB81, 224421 (2010). Models Staggered DM model XXZ model Equivalence Difference Consideration on the line shape F (t ) ie iHt [ H , S x ]e iHt relaxation time moments of (t ) Memory function (short time) Memory function (long time) Double peak structure " ( ) it S , ( ) i ( t ) e dt ' ( ) i" ( ) 2 2 0 ( ' ( )) (" ( )) xx ( ) i (t )e it dt ' ( ) i" ( ) 0 (0) i (t ) dt 0 i C ' (0) 0, " (0) 1 C , t d 1 d d ( ) | 0 t(t ) dt t ' ( 0) , " (0) 0, 0 d C d C d t2 t d2 1 d2 d2 2 2 ( ) | 0 i t (t )dt 2i t ' ( 0) , " (0) 2 , 0 C C d 2 C d 2 d 2 d d d d2 4 " ( ) ( ' ( ))1 ' ( ) " ( ) " ( ) " ( ) d d d d d 2 S xx 2 2 2 2 2 2 d ( ' ( )) (" ( )) ( ' ( )) (" ( )) 2 2" ( ) ( ' ( )) 8" ( ) ( ' ( )) 2 t (" ( )) 2 2 2 2 C 2 C (" ( )) 2 3 1 t C 2 3 2 2 2 2 1 d ' ( ) ( ' ( )) d ' ( ) " ( ) d " ( ) d " ( ) d d 2 d 2 d d d ( ' ( ))1 ' ( ) " ( ) " ( ) d d 2 2 2 2 t 2 2 3 1 t 2 3 t 1 t C C C 2 C C C 2 t2 t d2 1 S xx 0 2 2 C C d 2 2 t2 C2 2 Estimated line shape in infinite chain Exact short range + spin diffusion long time tail with various cut-off times (tau_0,tau_c) Shift and width of the line shape • Intrinsic width due to assembly of the delta-functions Dmn e Em e En x mM n 2 , ( En Em ) Z • Quantum broadening due to quantum fluctuation of the field H JC 0 2 g b b b b i i N i 1 N z i i 1 • Transmission spectrum (input-output modes) C E T E / AE • Broadening width due to the interaction with the thermal bath 2 i 1 A 0 2 TrS B i iLS Coupling between spin system and cavity phonon system Spin system N m H spin H Siz , H spin m m m , H spin G G G , spin m G i 1 Cavity photon system Hcavity cavitybb Coupling N H couple g ( Sib Sib ), i 1 Transmission H spin H cavity H couple k g 0 trans Ek k m spin H trans Ek EG cavity H Coupling between spin system and cavity phonon system Spin system N m H spin H Siz , H spin m m m , H spin G G G , spin m G i 1 Cavity photon system Hcavity cavitybb Coupling N H couple g ( Sib Sib ), i 1 Transmission H spin H cavity H couple k Ek k g 0 m spin H trans Ek EG Enhancement of Rabi-oscillation and the vacuum-field Rabi splitting Enhancement of Rabi-oscillation Y. Kaluzny, P. G. , M. Gross, J. M. Raimond and S. Haroche, PRL 51, 1175 (1983) The vacuum-field Rabi splitting in the transmission spectrum G. S. Agarwal:, PRL 53, 1732 (1984). Splitting of PMR of DPPH The vacuum-field Rabi-splitting G. S. Agarwal:, PRL 53, 1732 (1984). DPPH I. Chiorescu, N. Groll, S. Bertaina, T. Mori and SM: PRB (2010) in press. (1004.3605) N-diamond arXiv 1006.0251 arXiv 1006.0242 Rubby S=3/2 Cr3+ Multi-photon effect N=nmax Super-radiance? ( ... , 0) ( ... , 1) ( ... , N) nmax: number of cavity photons in the ground state of spin system At N=nmax, a wide distribution of the Rabi frequences Eigenvalues and the transmission spectrum The vacuum-field Rabi-splitting G. S. Agarwal:, PRL 53, 1732 (1984). Photon emission spectrum d 1 H , a a a a 2a a ' aa aa 2aa dt i a a, , a (0) a, eq T ' ' T ra a i 0.1, ' 0.0, 0 1.0, g 0.2 Shift and width of the line shape • Intrinsic width due to assembly of the delta-functions Dmn e Em e En x mM n 2 , ( En Em ) Z • Quantum broadening due to quantum fluctuation of the field H JC 0 2 g b b b b i i N i 1 N z i i 1 • Transmission spectrum (input-output modes) C E T E / AE • Broadening width due to the interaction with the thermal bath 2 i 1 A 0 2 TrS B i iLS Line shape of the transmission g1 g2 c vac Thermal bath method Transmission in a steady state Aeikx it Ce ikx it Beikxit j-1 j j+1 j=0 i d H dt H E Input-output formulation Shift and width of the line shape • Intrinsic width due to assembly of the delta-functions Dmn e Em e En x mM n 2 , ( En Em ) Z • Quantum broadening due to quantum fluctuation of the field H JC 0 2 g b b b b i i N i 1 N z i i 1 • Transmission spectrum (input-output modes) C E T E / AE • Broadening width due to the interaction with the thermal bath 2 i 1 A 0 2 TrS B i iLS C. Uchiyama, M. Aihara, M. Saeki and S. Miyashita: PRE 80 (2009) 021128. Summary • Explicit expression of the the spectrum line shape: Line shale of a ring Heisenberg model with DM interaction Dmn e Em e En mM x 2 n / Z , ( En Em ) • Quantum broadening due to quantum fluctuation of the field Coupling of spin system and cavity photons H JC 0 2 g b b b b i i N i 1 N z i i 1 • Transmission spectrum (steady flow method) vs Broadening width due to the interaction with the thermal bath C E T E / AE i 1 A 0 2 TrS B i iLS 2 Thank you very much