Non-Markovian dynamics of an open quantum system in fermionic environments

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1 Non-Marovian dynamics of an open quanum sysem in fermionic environmens J. Q. You Deparmen of Physics, Fudan Universiy, Shanghai, and Beijing Compuaional Science Research Cener, Beijing Mi Chen (PhD suden a Fudan) Suppored by he Naional Naural Science Foundaion of China and he Naional Basic Research Program of China (973 Program) The Worshop on Correlaions and Coherence in Quanum Sysems Evora, Porugal, 8-1 Ocober, 1

2 Ouline Open quanum sysems and he maser equaion formalism: Non-Marov vs. Marov Non-Marovian quanum sae diffusion (QSD) in a bosonic bah: Some exacly solvable models Non-Marovian QSD in fermionic environmens Summary and ouloo

3 Wha is an open quanum sysem? How o describe is dynamics? Sysem plus environmen framewor Open sysem Environmen Closed sysem Considered sysem Noise Non-Marov or Marov Non-Marov is he rule, Marov is he excepion (N. G. van Kampen) Reduced densiy operaor: ρ () = Tr { ρ () } sys env o

4 Non-Marov vs. Marov Non-Marovian dynamics (memory environmen): The ime evoluion of he sysem s sae depends on is hisory. ρ() i = [ Hsys, ρ( )] + κ(, s) ρ( s) ds h Marov approximaion (memoryless environmen): Replace ρ( s) by ρ( ) Lindblad form: ρ i Γ = H + L L + L L h [ sys, ρ] ([, ρ ] [ ρ, ]) The reservoir is assumed o be of a broadband specrum, so as o have a correlaion ime (memory ime) much shorer han he dynamical (evoluion) ime of he considered sysem.

5 Maser equaion under Born-Marov approximaion Born approximaion (nd-order approximaion): Wea ineracion Typical assumpion: ρo () = ρsys () ρenv() ρ() i 1 = [ Hsys, ρ( )] d' [ H in ( ),[ Hin ( '), ρin ( ')]] h h env One can use i o derive a non-marovian maser equaion, bu his mehod is perurbaive (i.e., i applies o a wea sysem-bah coupling). Marov approximaion: ρin ρin Replace ( ') by ( ) ρ() i 1 = [ Hsys, ρ( )] d' [ H in ( ),[ Hin ( '), ρin ( )]] h h env H. Carmichael, An open sysems approach o quanum opics (Springer 1993)

6 Two non-perubaive mehods for non-marovian dynamics Feynman-Vernon influence funcional approach R. P. Feynman and F. L. Vernon, Ann. Phys. (N.Y.) 4, 118 (1963) Non-Marovian maser equaion for quanum Brown moion model : B. L. Hu, J. P. Paz, and Y. Zhang, Phys. Rev. D 45, 843 (199) Non-Marovian maser equaion of a double-quanum-do sysem: M. W. Y. Tu and W. M. Zhang, Phys. Rev. B 78, (8). Non-marovian quanum rajecories Non-Marovian quanum sae diffusion (NMQSD) L. Diosi, N. Gisin, and W. T. Srunz, Phys. Rev. A 58, 1699 (1998); W. T. Srunz, L. Diosi, and N. Gisin, Phys. Rev. Le. 8, 181 (1999); T. Yu, L. Diosi, N. Gisin, and W. T. Srunz, Phys. Rev. A 6, 91 (1999). Quanum hree-level sysem: J. Jing and T. Yu, Phys. Rev. Le. 15, 443 (1) Exension o he fermionic-bah case: Mi Chen and J.Q. You, arxiv:13.17; Wufu Shi, Xinyu Zhao and Ting Yu, arxiv:13.19.

7 Ouline Open quanum sysems and he maser equaion formalism: Non-Marov vs. Marov Non-Marovian quanum sae diffusion (QSD) in a bosonic bah: Some exacly solvable models Non-Marovian QSD in fermionic environmens Summary and ouloo

8 Some exacly solvable models Toal Hamilonian: H H ( g L = + o sys b g ) Lb + + ω bb Sochasic Schrödinger equaion (Diosi-Gisin-Srunz equaion) ψ + ( z ) = [ ihsys + Lz + L O(, z)] ψ ( z) (, ) = α( ) (,, ) Oz sosz ds ρ ψ ψ Μ{ ( z ) ( z)} iω ( s ) α ( s ) J ( ω ) e d ω == Spin-boson pure dephasing model ω H sys σ z = L = σ z Osz Spin-boson dissipaive model (,, ) = L ω = σ L = σ H sys z Osz (,, ) = (, ) f sσ f (, s) = [ iω+ α( s') f(, s') ds'] f(, s) f(,) = 1

9 Some exacly solvable models Damped harmonic oscillaor model Hsys = ωaa L= a Osz (,, ) = (, ) f sa f (, s) = [ iω+ α( s') f(, s') ds'] f(, s) f(,) = 1 Quanum Brown moion model p 1 H sys m q m = + ω L= q (,, ) (, ) (, ) (,, ') s' ' Osz = f sq+ gs p i jss zds Exac non-marovian maser equaion a () b () ρ ρ ρ ρ ρ ρ i i = i[ Hsys, ] + [ q, ] + [ q,{ p, }] + c()[ q,[ p, ]] d()[ q,[ q, ]] B. L. Hu e. al., Phys. Rev. D 45, 843 (199); W. T. Srunz and T. Yu, Phys. Rev. A 69, 5115 (4).

10 Ouline Open quanum sysems and he maser equaion formalism: Non-Marov vs. Marov Non-Marovian quanum sae diffusion (QSD) in a bosonic bah: Some exacly solvable models Non-Marovian QSD in fermionic environmens Solid-sae quanum circuis (see, e.g., JQY and Nori, Naure 474, (11) ; Xiang, Ashhab, JQY and Nori, arxiv: , o appear in Rev. Mod. Phys.) Fermionic bahs: Elecric leads, bacground charge flucuaions, Summary and ouloo

11 Quanum confined sysem coupled o fermionic reservoirs Quanum do Double quanum do The oal Hamilonian of a quanum confined sysem coupled o wo fermionic reservoirs H = H + H + H o sys in env H g c a g c a H c in = ( L L L + R R R +..) H a a a a env = ( ωl L L + ωr R R) In a quanum sae diffusion approach, environmens are required o be iniially a zero emperaure, so as o convenienly represen he environmenal degrees of freedom wih he coheren sae basis.

12 Bogoliubov ransformaion: Convering a nonzero- o a zero-emperaure problem As for environmens iniially wih a nonzero emperaure, one can map he nonzero-emperaure densiy operaor o a zero-emperaure densiy operaor using a Bogoliubov ransformaion [T. Yu, Phys. Rev. A 69, 617 (4)]. By including he par involving holes in he elecric leads, he oal Hamilonian can be wrien as H = H + ( g c a + H. c.) + ω a a + ω b b o sys Performing Bogoliubov ransformaion for fermionic operaors: a n d n e = 1 b n e n d = 1 + H = H + n g c d + H c + n g c e + H c + ' o sys 1 (..) (..) H H n g c d e n g c e e H c iω iω ' o ( ) = sys + ( ) ω d d + ω e e The fermionic bahs wih nonzero iniial emperaures are mapped o virual fermionic bahs wih zero iniial emperaure.

13 Fermionic coheren saes represenaion Iniial condiion Ψ = ψ = d e d = e = Fermionic coheren sae basis for he environmens z = z = e z d Grassmann variables z, w w e w = w = e zw = z w In he coheren sae represenaion Ψ = ih () Ψ o ψ z w zw Ψ ψ (, ) ( z, w) = i zw Ho( ) Ψ Use he following equaions: d zw = z zw d zw = zw z z () s δ = ds z z z () s δ e zw w zw w () s δ ds w w w () s = δ = (chain rules) e zw = zw w

14 Non-Marovian quanum sae diffusion (QSD) equaion Non-Marovian QSD equaion (sochasic Schrödinger equaion) Noise funcion ψ = ih ψ c z + sys [ ( ) δ cw ()] ψ c α 1( s) ψ ds c α ( s) ψ ds δz() s δw() s iω = iω z () i 1 n g z e w () = i n g w e Saisical properies of noise funcion M{ z ( )} = M{ z ( ) z ( s)} = M{ z ( ) z ( s)} = α ( s) = dω[1 n ( ω)] J ( ω) e ω i ( s) 1 ( ) M{ w( )} = M{ w( ) w( s)} = M{ w ( ) w ( s)} ( s) d n ( ) J ( ) e iω s = α = ω ω ω zz ww M{ L} e { L} d zd w δ zz= z z ww= w w d z = dzdz d w = dw dw The compleeness relaion for coheren saes: zz ww e zw zw d zd w = 1

15 The O-operaor and is equaion of moion Non-Marovian QSD equaion (sochasic Schrödinger equaion) ψ = ih ψ c z + sys [ ( ) δ cw ()] ψ c α 1( s) ψ ds c α ( s) ψ ds δz() s δw() s Lie he Diosi-Gisin-Srunz equaion in he bosonic-bah case, we inroduce he O-operaors: O δ δ z δ ψ (, ) z w = O (, s, z, w ) ψ( z, w ) δ w () s ψ (, ) z w = O1(, s, z, w ) ψ( z, w ) () s Equaion of moion for he O-operaors: δ O z w = c O (,,, ) 1 (,, z, w ) = c = [ ih ( c O + c O ), O ] + Q + ({ c, O } z ( ) + { c, O } w ( )) n + sys ' '1 ' ' n n ' n ' ' n ' ' ' O O O O Q c δ c δ c δ c δ L1 R1 L R n = L + R + L + R δλn δλn δλn δλn Λ = 1 z () s Λ = w () s

16 Non-Marovian maser equaion Sochasic reconsrucion of he reduced densiy operaor: ρ = Tr Ψ Ψ = e ψ ( z, w ) ψ ( z, w) d zd w M{ ψ ( z, w ) ψ ( z, w)} z z w w env Sochasic projecion operaor P z w z w ρ = M { P } ψ(, ) ψ(, ) Non-Marovian maser equaion ρ = ihsys ρ + c Μ PO 1 z w c Μ O1 z w P [ c, Μ{ O (, z, w ) P }] + [ c, Μ{ PO (, z, w)}] [, ] ([, { (,, )}] [, { (,, ) }] Generally, i is complex, bu can have a simple form for a given sysem. Marov limi α ( s) [1 n ] ( s) 1 δ Γ α ( s) n ( s) Γ δ Γ ρ ρ ρ ρ ρ ρ ρ ρ = ih [ sys, ] + [ n( c c cc cc ) + (1 n)( c c cc cc )]

17 Applicaions: (1) Single quanum do Single quanum do H = H + H + H o sys in env H = ( ω a a + ω a a ) env L L L R R R = L L L + R R R + H sys = ωcc L = R = Hin ( g c a g c a H. c.) c c c Srong Coulomb blocade: U Consrain: Only one elecron is allowed in he single quanum do O-operaors 1 = L + R (,,, ) = (, ) + L + R O (, s, z, w ) f (, s) c q (, s, s')[ w ( s') w ( s')] ds' O s z w f s c q (, s, s')[ z ( s') z ( s')] ds'

18 Applicaions: (1) Single quanum do Exac non-marovian maser equaion ih [ sys, ] 1( )[ c, c] ( )[ cc, ] 1( ) ρ = ρ +Γ ρ +Γ ρ Γ [ c, cρ ] Γ( ) [ c, ρc] wih ime-dependen raes: Γ () = [ α ( s) A (, s) α ( s) B (, s)] ds j 1 j j ( s iω ) Aj(, s) + β( s s') A (, ') ' (, ) j s ds = U s s ( s iω ) Bj( s, ) + β( s s') B(, ') ' (, ) j s ds = Vs s ( iω ) h(, s) β( s') h(, s') ds' = s s Final condiion a : s = 1 1 α ( s) = α ( s) + α ( s) j Lj Rj Us (, ) = α ( shs ) (, ') ds' Vs (, ) = α ( shs ) (, ') ds' 1 β ( s s') = α ( s' s) + α ( s s') 1 A (,) = B (,) = h(,) = 1 A(,) = B(,) =

19 Applicaions: (1) Single quanum do Exac non-marovian maser equaion ih [ sys, ] 1( )[ c, c] ( )[ cc, ] 1( ) ρ = ρ +Γ ρ +Γ ρ Γ [ c, cρ ] Γ( ) [ c, ρc]

20 Applicaions: (1) Single quanum do Marov limi: 1 1 Γ1() [1 nl( ω)] Γ L + [1 nr( ω)] Γ 1 1 Γ( ) nl( ω) ΓL nr( ω) Γ R R Maser equaion in he Lindblad form: ρ = ρ + Γ ρ ρ ρ + ρ ρ ρ ih [ sys, ] [ n( c c cc cc ) (1 n)( c c cc cc )] Large bias limi: μl ω μr Zero emperaure: > > nl ( ω ) = 1 nr ( ω ) = & ρ = Γ ρ + Γ ρ & ρ =Γ ρ Γ ρ L R L R 11 & ρ = [ iω +Γ +Γ ] ρ 1 L R 1 empy do sae 1 occupied do sae S. A. Gurviz and Ya. S. Prager, Phys. Rev. B 53, 1593 (1996)

21 Applicaions: () Double quanum do Double quanum do (a) U U, V = Consrain: A mos wo elecrons are allowed in he DQD. (b) , V H = ωcc+ ω cc +Ω ( cc+ cc) cl c1 sys O-operaors H a a a a H = H + H + H env = ( ωl L L + ωr R R) o sys in env Consrain: Only one elecron is allowed in he DQD. = cr = c 1 = L + 4 R O (, sz,, w) f (, sc ) f (, sc ) [ f (, ss, ') w( s') f (, ss, ') w( s')] ds' = L + 4 R H g c a g c a H c in = ( L L L + R R R +..) O (, sz,, w) m (, sc ) m (, sc ) [ m (, ss, ') w( s') m (, ss, ') w( s')] ds'

22 Applicaions: () Double quanum do Exac non-marovian maser equaion = [, ] +Γ ( )[, ] +Γ ( )[, ] +Γ ( )[, ] +Γ ( )[, ] i Hsys L1 c1 c1 L c1 c1 L3 c1 c L4 c1 c ρ ρ ρ ρ ρ ρ +Γ ()[ ()[ ()[ R1()[ c, ρc1] + Γ R c, c1 ρ] + Γ R3 c, ρc] + ΓR4 c, c ρ ] () Γ L 1 [ c1, ρc1] ΓL( ) [ c1, c1ρ ] ΓL3( ) [ c1, cρ] Γ L4( ) [ c1, ρc] Γ () R1 [ c, ρc1] ΓR( ) [ c, c1ρ ] ΓR3( ) [ c, cρ] Γ R4( ) [ c, ρc] wih ime-dependen raes A j (, s) B j (, s) Γ () = [ α ( s) A (, s) α ( s) B (, s)] ds j 1 j j saisfy a se of inegro-differenial equaions, wih final condiions A (,) = A (,) = B (,) = B (,) = 1 L1 R3 L R4 A (,) = B (,) = for oher, j j j

23 Applicaions: () Double quanum do Marov limi: 1 1 Γ L1() = [1 nl( ω1)] Γ Γ L L() = n ( 1) L ω Γ 1 1 Γ R3() = [1 nr ( ω)] Γ Γ R R4() = n ( ) R ω Γ L R Γ () =Γ () =Γ () =Γ () = R1 R L3 L4 Maser equaion in he Lindblad form: ρ = ρ + Γ ρ ρ ρ + ρ ρ ρ ih [ sys, ] [ n( c c cc cc ) (1 n)( c c cc cc )]

24 Applicaions: () Double quanum do Maser equaion in he Lindblad form: ρ = ρ + Γ ρ ρ ρ + ρ ρ ρ ih [ sys, ] [ n( c c cc cc ) (1 n)( c c cc cc )] Large bias limi and zero emperaure condiion μ ω ω μ L > 1, > R 1 nl( ω ) = 1 nr( ω ) = empy do 1, lef (righ)do occupied 3 boh dos occupied Srong Coulomb-blocade regime (a) Srong inerdo Coulomb repulsion & ρ = Γ ρ +Γ ρ L R & ρ ρ ρ ( ρ ρ ) 11 =Γ L +Γ R 33 + iω 1 1 & ρ ( ) ρ ( ρ ρ ) = Γ L +ΓR iω 1 1 & ρ33 =ΓLρ ΓRρ33 Γ +Γ & ρ = i( ω ω ) ρ + iω ( ρ ρ ) ρ S. A. Gurviz and Y. S. Prager, PRB 53,1593 (1996); L R M. W. Y. Tu and W. M. Zhang, PRB 78, (8). (b) Boh srong inra- and inerdo Coulomb repulsions & ρ = Γ ρ +Γ ρ L R & ρ =Γ ρ + iω ( ρ ρ ) 11 L 1 1 & ρ = ΓRρ iω( ρ1 ρ1) Γ & ρ = i( ω ω ) ρ + iω ( ρ ρ ) ρ R T. H. Soof and Y. V. Nazarov, PRB 53,15 (1996)

25 Summary Summary and ouloo Non-Marovian quanum sae diffusion (QSD) approach is exended o deal wih he fermionic environmens. Non-Marovian maser equaion is derived. The approach is applied o single quanum do and double quanum do, each coupled o wo elecric leads. Ouloo Connecion o Feynman-Vernon influence funcional approach? Non-Marovian QSD for a spin environmen? Non-Marovian QSD for he sysem nonlinearly coupled o eiher a bosonic or fermionic bah?