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Bohmian solution to the measurement of many-body systems: S Sequential ti l currentt in i mesoscopic i electron l t devices d i X.Oriols, F.L.Traversa, G.Albareda. Universitat Autònoma de Barcelona E.mail: [email protected] C ll b t Collaborators: X C t i à A.Alarcón, X.Cartoixà, A Al ó A.Benali, A B li M.Yago, MY TN T.Norsen. What is the relation between electronics and Bohmian solution to the meas measurement rement of many-body man bod ssystems stems Electron devices 1.000.000.000 transistors per CPU !!! Bohmian mechanics Bohmian solution to the measurement of many-body systems: S Sequential ti l currentt in i mesoscopic i electron l t devices d i 1.- Introduction: 2.- The use of quantum (Bohmian) trajectories 3 Numerical results with the BITLLES simulator 3.- 4.- Conclusions and Future work Bohmian solution to the measurement of many-body systems: S Sequential ti l currentt in i mesoscopic i electron l t devices d i 1.- Introduction: 1.1.- Why we have to worry about many-body systems ? 1.2.- Why we have to worry about sequential measurement ? 2.- The use of quantum (Bohmian) trajectories 3.- Numerical results with the BITLLES simulator 4.- Conclusions and Future work 1.1.- Why we have to worry about many-body systems ? What we measure when we measure the current ? Amperimeter Ammeter I(t) S Source Drain W Wire W DUT I(t) measured in the ammeter VP(t) Wire RIN RL Integral (surface) conservation s J T r , t ds 0 VIN(t) I(t) computed t d in the DUT Differential (local) conservation J T ( r , t ) 0 Th many The Which magnitude accomplish the differential (local) conservation ? body problem d Jc 0 Continuity equation dE ( r , t ) dt J c (r , t ) 0 dt Poisson (Coulomb) E (r , t ) (r , t ) / q equation The total current measured at the ammeter is equal to that computed at the DUT 5 1.1.- Why we have to worry about many-body systems ? P.A.M. Dirac, 1929 The many body problem “The general theory of quantum mechanics is now almost complete. The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.” Max Born, 1960 ”It would indeed be remarkable if Nature fortified herself against further advances in knowledge behind the analytical difficulties of the many-body problem problem.” 6 Multi-time measurement and displacement current i time-dependent in ti d d t quantum t transport t t 1.- Introduction: 1.1.- Why we have to worry about many-body systems ? 1.2.- Why we have to worry about sequential measurement ? 2.- The use of quantum (Bohmian) trajectories 3.- Numerical results with the BITLLES simulator 4.- Conclusions and Future work 1.2.- Why we have to worry about sequential measurement ? Two “orthodox” dynamical law for describing the evolution of a quantum system: Unitary-evolution (when no measure): Schrodinger equation 1-law: d i x (t ) H x (t ) dt (t ) e iHt / (0) U (0) Non-unitary-evolution (when measure): “Collapse” 2-law: x (0) x ui (t ) (t ) ui ui (0) Example: a single-time single time (ensemble) measurement I I i P ( I i ) I i ui ui i i P ( I i ) ui ui ui Iˆ ui ui 2 i Iˆ ui I i ui Iˆ 1 ui ui i For ergodic g system y Time-average g is equal q to ensemble average g The prediction of DC current can be computed without worrying about “collapse” 8 1.2.- Why we have to worry about sequential measurement ? Sequential S i l measurement Example: noise (current fluctuations) measurement Tootal Current (mA A) Engineers do not like noise, it makes errors in the device. Physicist Ph i i t enjoy j noise, i bbecause it shows h phenomena h that th t are not present in DC. I (t ) I ( t ) I ( t ) I R ( ) S( f ) Time (ps) fluctuations Autocorrelation Fourier transform S(f)= R( ) e -j2 f d ( )I(t+ ( ) t 2 t1 R(( ))= I(t) R( )= I(t 2 )I(t1) IDC - 2 We need the measurement of I(t2) after the measurement I(t1). 9 1.2.- Why we have to worry about multi-time measurement ? Sequential q measurement How we can compute the quantum two-times correlations ? I(t 2 )I(t1 ) j I (t j 2 )·I i (t1 )·P I j (t 2 ),I i (t1 ) i P I j (t 2 ),I i (t1 ) (0) ui ui Uˆ † ( ) u j u j Uˆ ( ) ui ui (0) How we can compute the probability ? t1 Measuring at t1 With the modulus of the wavefunction 2-law: (t1 ) ui ui (0) (t2 ) Uˆ ( ) (t1 ) Time Time-evolution 1-law: t2 Measuring g at t2 2-law: Final state (t2 ) u j u j (t2 ) (t2 ) u j u j Uˆ ( ) ui ui (0) 10 Bohmian solution to the measurement of many-body systems: S Sequential ti l currentt in i mesoscopic i electron l t devices d i 1.- Introduction: 2.- The use of quantum (Bohmian) trajectories 2.1.. . Bohmian o mechanics ec cs (quantum (qu u hydrodynamics) yd ody cs) 2.2.- The conditional wave function: a successful tool 2.3.- The computation of the displacement current 2.4- The computation of multi-time measurement 3.- Numerical results with the BITLLES simulator 4.- Conclusions and Future work 2.1.- Bohmian mechanics (quantum hydrodynamics) L. De Broglie 1924 D. Bohm J.S. Bell 1952 1960 t1 “Playing” with the many-particle Schrodinger equation.... ( r1,,..,, rN , t ) N 2 2 i rk U ( r1,.., rN , t ) · ( r1,.., rN , t ) t 2·m k 1 Look for a continuity equation: Define a velocity: 2 ( r1 ,.., r N , t ) t N k 1 t2 J k r1 ,.., r N , t 0 2 v k ( r1,.., r N , t ) J k ( r1,.., r N , t ) / ( r1,.., r N , t ) Define a Bohmian trajectory: r1 [ t ] r1 [ t 0 ] t to d t '·v k ( r1 [ t '],.., r N [ t '], t ') The computation of many many-particle particle trajectories exactly reproduce the evolution of the modulus of the wavefunction. 12 4 2.1.- Bohmian mechanics (quantum hydrodynamics) Example: A wave packet impinging upon a double barrier structure 1.- First, 2 - Second, 2.Second 3.- Third: ( x, t ) J ( x, t ) v( x, t ) | ( x , t )|2 t x i [t ] xoi v ( x i [t '], t ')·dt ' 0 Main criticism against Bohmian formalism: “…In any case, the basic reason for not paying attention to the Bohm approach is not some sort of ideological rigidity, but much simpler…It is just that we are all too busy with our own work to spend time on something that doesn’t seem likely to help us make progress with our real problems”. Steven Weinberg (private comunication with Shelly Goldstein) 13 Bohmian solution to the measurement of many-body systems: S Sequential ti l currentt in i mesoscopic i electron l t devices d i 1.- Introduction: 2.- The use of quantum (Bohmian) trajectories 2.1.. . Bohmian o mechanics ec cs (quantum (qu u hydrodynamics) yd ody cs) 2.2.- The conditional wave function: a successful tool 2.3.- The computation of the displacement current 2.4- The computation of multi-time measurement 3.- Numerical results with the BITLLES simulator 4.- Conclusions and Future work The many body problem 2.2.- The conditional wave function: a successful tool The “BIG” many-particle wave-function for N particles: ( x1 ,..., x N , t ) v1 ( x1[t ],.., ] x N [t ], ] t) v1 ( x1[t ],.., ] xN [t ]], t ) M grid points M N configuration point J 1 ( x1 ,..., x N , t ) ( x1 ,..., x N , t ) 2 x1 x1 [ t ],...., x N xN [ t ] J 1 (( x1 , x2 [t ],.., xN [t ], t ) 2 ( r1 , x2 [t ],.., xN [t ], t ) x1 x1 [ t ] wave functions for N particles: The “LITTLE” conditional wave-functions ( x1 , x2 [t ],.., x N [t ], t ) 1 ( x1 , t ) M ·N M grid points configuration point The computation of many-particle trajectories from one BIG or from N LITTLE conditional wave functions are exactly identical. What is the equation satisfied by the conditional wave-function ? 15 2.2.- The conditional wave function: a successful tool 1 ( x1 , t ) 2 2 i U ( x , x [ t ], ] t ) G ( x , x [ t ], ] t ) i · J ( x , x [ t ], ] t ) 1 ( x1 , t ) 1 1 b 1 1 b 1 1 b 2 t 2 m x1 Good points : [X. Oriols, Phys. Rev. Lett. 98, 066803 (2007)] An exact procedure for computing many-particle Bohmian trajectories The correlations are introduced into the time-dependent potentials 4th The interacting potential from (a classical-like) Bohmian trajectories 5th There is a real potential to account for “non-classical” correlations 6th There is a imaginary potential to account for non-conserving norms Bad points : The terms G and J depends on the many-particle wave-function This is exactly the same difficulty found in the DFT (or TD-DFT) 16 2.2.- The conditional wave function: a successful tool Example: two Coulomb interacting particles q2 1 ( r1 , t ) 2 2 i 1 1 ( r1 , t ) 4 0 r1 r2 [t ] t 2 m * q2 2 ( r2 , t ) 2 2 i 2 t 2 m * 4 0 r1[t ] r2 2 ( r2 , t ) ; ; t J 1 ( r1 , t ) r1[t ] r1[to ] dt 2 | ( r 1 1, t ) | to t J 2 ( r2 , t ) r2 [t ] r2 [to ] dt | 2 ( r2 , t ) |2 to r1 r1 [ t ] r2 r2 [ t ] 17 2.2.- The conditional wave function: a successful tool Example: two (Coulomb and Exchange) interacting particles Ensemble results associated to x1 are indistinguishable from those associated to x2. But, Bohmian trajectories associated to particle x1 or to x2 are distinguishable. x1[to] -40 x1,to) ~ Emitter x1[t] ~ x1,to) to 0 x x1,t) 2 x2,to) ~ x1[t] x2,,t)) -20 x2[to] -40 Emitter 0 20 40 ~ to 60 x1[t] 2D Bohm Area=A=900 nm ~ ~ x2,to) x2[t] 1D' Bohm 0 x2,t) 100 120 140 160 180 200 220 Time t (fs) Emitter 1D Bohm 1D' 2D Bohm -10 -20 x2[t] 2D Bohm B h t 80 Collector 10 x1[t] 1D' Bohm t (b) Collector 20 x1,t) 2 Area=A=900 nm -30 x'1[0]=-40.50 nm x'2[0]= -23.25 x 23 25 nm x1[0]=-23.25 nm x2[0]= -40.50 nm -30 30 -20 20 [X. Oriols, Phys. Rev. Lett. 98, 066803 (2007)] Collector x -20 ~ (c) Emitter Pos sition x1[t] and d x2[t] (nm) 0 ~ 20 2 Area=A=2.25 m Collector Position x2 (nm) (a) 20 -10 10 0 10 20 Position x1 (nm) 18 2.2.- The conditional wave function: a successful tool Is it possible to improve the estimations of G and J functions ? 1 ( x1 , t ) 2 2 i U1 ( x1 , xb [t ], t ) G1 ( x1 , xb [t ], t ) i·J 1 ( x1 , xb [t ], t ) 1 ( x1 , t ) 2 2 m x t 1 Taylor expansion of the many-particle wave function: ( x1 , x2 , t ) 1 2 ( x1 , x2 , t ) ( x1 , x2 , t ) ( x1 , x2 , t ) x x [ t ] ( x2 x2 [t ]) ( x2 x2 [t ]) 2 .... 2 2 2 x2 x2 2! x x [t ] x x [t ] 2 2 2 2 The “LITTLE” conditional wave-functions for the derivatives: ( x1 , x2 , t ) x x [ t ] 1 ( x1 , t ) 2 2 n ( x1 , x2 , t ) n ( x1 , t ) n 1 x2 x x [t ] 2 2 A co coupled pled system s stem of eq equations ations for the deri derivatives ati es conditional wave-functions a e f nctions 1n ( x1 , t ) 2 2 1n ( x1 , t ) 2 n 2 d x2 [t ] n n n iU ( x1 , x2 , t ) n 1 i 1 ( x1 , t ) i1 ( x1 , t ) 1i ( x1 , t ) 2 n i t 2m x1 2m dt x2 i 0 i x x [t ] 2 For n=0,1,2,3,4,5....... 2 [T.Norsen, Found. Phys 40, 1858 (2010)] 19 Bohmian solution to the measurement of many-body systems: S Sequential ti l currentt in i mesoscopic i electron l t devices d i 1.- Introduction: 2.- The use of quantum (Bohmian) trajectories 2.1.. . Bohmian o mechanics ec cs (quantum (qu u hydrodynamics) yd ody cs) 2.2.- The conditional wave function: a successful tool 2.3.- The computation of the displacement current 2.4- The computation of multi-time measurement 3.- Numerical results with the BITLLES simulator 4.- Conclusions and Future work 2.3.- The computation of the displacement current The TOTAL timetime-dependent current: current: E (Er,(tr), t ) ·ds ·ds Quantum ensemble average value I (tI)(t ) JJcc((tt))··ds ds tt SSDD S DS D Ensemble over all position distributions weighted by the probability 2 E ( r , t ) dr1..... drN ( r1 ,..., rN , t ) E ( r , r1 ,..., rN , t ) Solving the N-particle Poisson equation N E ( r , r1 ,..., rN , t ) q ( r ri ) i 1 Solving equation ¿?? S l i the th N-particle N ti l Schrodinger S h di ti The many 2 N ( r1 ,..., rN , t ) i 2k U ( r1 ,,...,, rN , t ) ( r1 ,,...,, rN , t )body problem t k 1 2 m 21 2.3.- The computation of the displacement current The TOTAL timetime-dependent “Bohmian”current: Bohmian”current: I (t ) J c (t ) ·ds SD E ( r, t ) t SD ·ds 2 E ( r , t ) dr d 1..... dr d N ( r1 ,..., rN , t ) E ( r , r1 ,..., rN , t ) An ensemble of Bohmian trajectories reproduces the wavefunction modulus 1 2 ( r1 ,..., rN , t ) lim M M M j 1 ( r1 r1 j [t ])····· ( rN rNj [t ]) Th prediction The di i off AC current can be b computedd from f an ensemble bl over trajectories. j i j j r1 [t ] ; 1 ( r1 , t ) …. j j rN [t ] ; N ( rN , t ) N j j j j E ( r , r1 [t ],..., rN [t ], t ) q ( r ri [t ]) 1 E ( r , t ) lim M M i 1 M j 1 j j E ( r , r1 [t ],..., rNj [t ], t ) 22 2.3.- The computation of the displacement current The TOTAL timetime-dependent “Bohmian”current: Bohmian”current: I (t ) SD J c (t ) ·ds SD E ( r, t ) t ·ds [A. Alarcon, JSTM (2009)] dE ( r , t ) J c (r , t ) 0 dt Each trajectory of the ensemble fulfils the TOTAL current conservation law. 23 Bohmian solution to the measurement of many-body systems: S Sequential ti l currentt in i mesoscopic i electron l t devices d i 1.- Introduction: 2.- The use of quantum (Bohmian) trajectories 2.1.. . Bohmian o mechanics ec cs (quantum (qu u hydrodynamics) yd ody cs) 2.2.- The conditional wave function: a successful tool 2.3.- The computation of the displacement current 2.4- The computation of multi-time measurement 3.- Numerical results with the BITLLES simulator 4.- Conclusions and Future work 2.4- The computation of multi-time measurement A “simple” explanation on Bohmian measurement…… measurement…… The wavefunction: r1,,...,, rN , y, t The trajectories: j j j j r1 [t], r2 [t], r3 [t],..., rN [t], yj[t]= I(t) The conditional wavefunction: j r1,..., rN , y [t], ] t The many body problem Time (ps) [A.Alarcón et. al. Chapter 6, Applied Bohmian Mechanics (2012)] 25 2.1.3.- Multi-time Bohmian measuremen of the current Is the Bohmian solution of the measurement equal to the Orthodox solution ? Bohmian trajectory I(t)=y[t] Copenhagen operators I(t)= eigenvalue of an Operator What would happen if I(t) were not equal to I(t) ? Copenhagen supporters: The measuring apparatus is wrong, because Bohmian mechanics has to reproduce orthodox quantum mechanics. Bohmian supporters: The operator is wrong, because orthodox quantum mechanics has to reproduce Bohmian mechanics. “N ï realism “Naïve li about b t operators” t ” [D. Durr, S. Goldstein, N. Zanghi, J. Stat. Phys., 116, 959-1055, (2004)] 26 2.4- The computation of multi-time measurement Is the Bohmian solution of the measurement equal to the Orthodox solution ? 1 k N ˆ Î1 q p k kx Lx m k 1 Total current operator Mˆ I1 t px px px I1 t , I px Its associated ((degenerate) g ) pprojector=POVM j f t Mˆ I1 t 0 t Final (unnormalized) state after measurement is a total momentum eigenstate. A.- Following a collapsed wave function The probability of being transmitted at t2 and reflected at t3 is zero zero. B.- Following a trajectory The probability of being transmitted at t2 and reflected at t3 is zero. 27 2.4- The computation of multi-time measurement Partition “Noise” comparison for electron devices without (many-body) Coulomb ? [M.Buttiker PRL1990] SAMPLE (A) quantization approach 1.5 An electron “created” at one contact | 1 a 1 | 0 0.2 1.0 Fano Factor nT n R F=S(0)/2q(II) can not “emerge” at two different contacts 1 | b 1 b 1 b 2 b 2 | 1 0 0.3 0.1 0.5 0.0 0.0 2.0 Buttiker results 0.3 1.5 SAMPLE (B) mission coefficiient T(E) Transm 2nd Buttiker results 2.0 0.2 1.0 0.1 0.5 0.0 0.00 0.05 0.10 0.15 0.20 0.0 0.25 Applied bias V (Volts) 28 Bohmian solution to the measurement of many-body systems: S Sequential ti l currentt in i mesoscopic i electron l t devices d i 1.- Introduction: 2.- Theoretical discussion 3.- Numerical results with the BITLLES simulator 4.- Conclusions and Future work 3.2.- DC current beyond mean field [X.Oriols, APL(2005) ] Effect of Coulomb correlation on current and noise 6 4 (a) Our approach 5 # Particles emitter Esaki results [16] 2 EFE 1 ER Colector Emitter # Particles emitter x=0 # Particles collector EFC L x=L 4 Visual guided line 3 2 1 FROZEN POTENTIALS SELF-CONSISTENT POTENTIALS 0 0 1.4 Our approach 1.2 Our approach Fano o Factor Buttiker results [6] Fano F Factor Our approach # Particles collector Cu urrent (A) Cu urrent (A) 3 (a) 1.0 Visual guided line ER FROZEN 1.2 ER FROZEN 1.0 (b) 0.8 0 8 0.0 (b) FROZEN POTENTIALS 0.1 0.2 0.3 Applied bias (Volts) 0.4 0.8 0 8 0.0 SELF-CONSISTENT POTENTIALS 0.1 0.2 0.3 0.4 Applied bias (Volts) 30 Bohmian solution to the measurement of many-body systems: S Sequential ti l currentt in i mesoscopic i electron l t devices d i 1.- Introduction: 2.- Theoretical discussion 3.- Numerical results with the BITLLES simulator 4.- Conclusions and Future work 4.- Conclusions and future work There is a need for predictions of the dynamics properties of quantum devices….. 1.- FORMALISM: We have shown the ability of many-particle Bohmian trajectories (extracted from conditional wave functions) to accurately and efficiently compute time dependent transport (AC, noise, transients,…). 2.- SIMULATOR: Using these ideas, we have developed a user-friendly, handmade simulator for time-dependent electron transport, named: Bohmian Interacting Transport for non-equiLibrium eLEctronic Structures [G. Albareda et al. Phys. Rev. B 79, 075315 (2009)] [G. Albareda et al. Phys. Rev. B 82, 085301(2010)] [F. L. Traversa, et al. IEEE Trans. on Electron devices, 58(7) 2104 (2011) ] Freely available at http:\europe.uab.es\bitlles 32 4.- Conclusions and future work Acknowledgment g This work has been partially supported by the Ministerio de Ciencia e Innovación under Project No. TEC2009-06986 and by the DURSI of the Generalitat de Catalunya under d C Contract t t No. N 2009SGR783. 2009SGR783 Thank you very much for your attention BITLLES freelyy available at http:\europe.uab.es\bitlles X.Oriols and J.Mompart, A li d Bohmian Applied B h i M Mechanics: h i From Nanoscale Systems to Cosmology, (2011). ISBN: 978-981-4316-39-2 33