Entangled Quantum Dynamics of Many-Body Systems using Bohmian Trajectories

Transcription

1 Receved: 5 May 8 Acceped: 3 Augus 8 Publshed: xx xx xxxx OPEN Eagled Quaum Dyamcs of May-Body Sysems usg Bohma Trajecores Tare A. Elsayed,, Klaus Mølmer & Lars Bojer Madse Bohma mechacs s a erpreao of quaum mechacs ha descrbes he moo of quaum parcles wh a esemble of deermsc rajecores. Several aemps have bee made o ulze Bohma rajecores as a compuaoal ool o smulae quaum sysems cossg of may parcles, a very demadg compuaoal as. I hs paper, we prese a ovel ab-o approach o solve he may-body problem for bosoc sysems by evolvg a sysem of oe-parcle wavefucos represeg plo waves ha gude he Bohma rajecores of he quaum parcles. I hs approach, quaum eagleme effecs arse due o he eracos bewee dffere cofguraos of Bohma parcles evolvg smulaeously. The mehod s used o sudy he breahg dyamcs ad groud sae properes a sysem of eracg bosos. Numercal smulao of he quaum dyamcs of may-body sysems s plagued by he dmeso of he Hlber space whch creases expoeally wh he umber of parcles. Much of he progress heorecal codesed maer, aomc ad molecular physcs he pas few decades has bee acheved by fdg ew ways o crcumve hs problem. Some of he mos powerful approaches are desy fucoal heory, quaum Moe Carlo, desy marx reormalzao group 3 ad he mul-cofgurao me-depede Harree mehod 4. Recely, ocoveoal approaches based o mache learg 5, Bohma mechacs 6 9 ad wavele rasforms have bee proposed as well. Several mehods am o reduce he complexy of he umercal smulao of may-body sysems by resorg o low dmesoal objecs such as desy fucos, ad aural orbals or by usg mxed classcal-quaum dyamcs such as he Ehrefes approach 3 ad he surface hoppg mehod 4,5. I hs paper, we use aoher class of low-dmesoal objecs, amely sgle-parcle plo waves evolved cocurrely wh Bohma rajecores o exrac all he physcal formao abou he sysem. Wh he Bohma erpreao of quaum mechacs 6 8, he quaum mechacal wavefuco s a plo wave ha gudes he moo of he parcle he physcal space. Whle hs erpreao does o allevae he eed for dealg wh may-dmesoal fucos, he prospec of replacg he full may-parcle wavefuco by sgle-parcle wavefucos ha gude he Bohma parcles he physcal space was recely explored 9. However, hs dea was oly appled whe he eagleme bewee he parcles could be egleced 9,, hus rulg ou s applcao o srogly correlaed sysems. Oher approaches o rea may-body wavefucos wh rajecores volve approxmaos such as he mea-feld approxmao, or he sem-classcal approaches 3,4 or assume a wavefuco of a cera form 5. I he de Brogle-Bohm erpreao, quaum effecs are capured by a so-called quaum poeal whch ogeher wh he classcal poeal govers he moo of he parcle, see, e.g. 7. I has bee show recely ha hs erm ca be compued by modelg quaum pheomea by may eracg classcal worlds 9,6. I hs paper, we roduce a ovel approach o model quaum pheomea usg eracg cofguraos of quaum parcles guded by plo waves. Ths approach smulaes he mul-parcle quaum dyamcs a o-perurbave maer whou eglecg he eagleme or relyg o parcular assumpos abou he uderlyg quaum sae. We apply our approach o sudy he breahg dyamcs of few-boso sysems a rap wh log- ad shor-rage eracos ad compue he groud sae eergy for a exacly solvable sysem. The proposed approach s curre sage does o supersede esablshed umercal mehods or overcomes he scalg problem of smulag-may body sysems, bu offers a ew way forward ha may be furher developed o a full-fledged mehod. Zewal Cy of Scece ad Techology, 6h of Ocober Cy, Gza, 578, Egyp. Deparme of Physcs ad Asroomy, Aarhus Uversy, 8, Aarhus C, Demar. Correspodece ad requess for maerals should be addressed o T.A.E. (emal: are.ahmed.elsayed@gmal.com) Scefc REPOrTS (8) 8:74 DOI:.38/s

2 Resuls Le us llusrae he usage of plo waves a D sysem cossg of parcles. The coordaes are deoed by x ad x, he poeal by V(x, x ) ad he wavefuco descrbg he full sysem s Ψ(x, x, ). I order o evolve he Bohma rajecores X () ad X () for he wo parcles (we deoe he Bohma rajecores hroughou hs paper by uppercase leers), we eed o evaluae he plo waves ψ (x, ). The plo waves are he full wavefuco projeced o he coordaes of all he parcles excep oe,.e., ψ( x, ) Ψ ( x, x, ) xj = Xj(), j ; hece, hey are also called codoal wavefucos (CWs) 7. I he absece of gauge felds, he Bohma veloces are compued erms of he plo waves as dx d xψ( x, ) = Im m ψ( x, ) x = X() where m s he mass of parcle. I s guaraeed ha he desy of he Bohma parcles evolved by Eq. () follows he evoluo of he desy fuco as compued by Schrödger s equao 8. I order o evolve he CWs whou havg o solve he me-depede Schrödger equao (TDSE) for Ψ(x, x, ), we roduce a geeralzed se of codoal wavefucos ψ ( x, ) defed as ψ Ψ ( x, x, ) ( x, ) x j xj= Xj (), j where he plo waves correspod o ψ. The equao of moo for ψ ( x, ) s gve by 8 ψ ( x, ) ψ ( x, ) = + m x = () dx () ψ j ( x, ) + ψ m d j ψ + +, Vx (, xj) ( x, ) x ( x, ) j xj= Xj(), j We see from hs equao ha he plo waves correspodg o dffere parcles erac drecly hrough he las hree erms of Eq. (3). I 9, a smlar equao of moo s derved erms of olocal poeals. I order o evolve he plo waves a exac maer usg Eq. (3) sead of evolvg he mul-dmesoal full wavefuco, we eed o evolve he whole herarchy of { ψ }. We llusrae he Mehods seco ha rucag hs herarchy a a fe order N s o a effce mehod o oba he correc dyamcs of a eagled sysem as he rucao errors propagae very qucly o ψ. Is here a way o avod he errors orgag from he rucaed orders? I urs ou ha he aswer s yes! To hs ed, we assume a asaz for he full wavefuco ha allows he calculao of he frs ad secod CWs, ψ ad ψ, whch we subsequely use o evolve he eracg plo waves ψ. () () (3) Ieracg Plo Waves. The mos geeral form for he wavefuco of a -parcle sysem s Ψ ( x, y, ) = c () φ( x) φ( y), j, j j where {φ } s a complee bass for he oe-body Hlber space, also referred o as orbals laer. Le us assume ha a fe umber of bass saes M s suffce o capure all he mpora feaures of he wavefuco. The codoal wavefucos of he frs parcle codoed o he secod parcle locaed a y = Y are expressed as (he me varable ad he parcle dex are omed o smplfy he oao) where a = j cjφj ( Y), b = c form as: ψ ( x) Ψ ( x, y) = aφ( x) ψ ( x) ψ ( x) j j φ ( y) y= Y Ψ ( x, y) y= Y Ψ ( x, y) j ad c = c = = y= Y y= Y j j φ ( y) j y= Y bφ( x) cφ( x) (4). These relaos ca be wre vecor a( Y) = Cφ( Y), b( Y) = Cφ( Y)ad c ( Y) = Cφ ( Y) (5) where a ( Y) = { a( Y), a ( Y), }, φ ( Y) = { φ( Y), φ ( Y), }, ec. Scefc REPOrTS (8) 8:74 DOI:.38/s

3 Fgure. The breahg dyamcs of wo eracg bosos a harmoc rap. The wo bosos are ally he groud sae of he rap before he harmoc eraco bewee hem s suddely swched o a =. The value of he reduced oe-body desy fuco ρ(x) a he org s moored durg he evoluo. The sregh of he harmoc eraco s he same as he rap sregh. We plo ρ() compued by he Ieracg Plo Waves mehod (IPW) usg 6 orbals ad by he exac wo-body wavefuco (Exac). The IPW resuls were obaed by averagg over 5 Bohma cofguraos of he wo parcles. The wo paels depc dffere me slces o llusrae he accuracy he log-me regme. The problem of fdg ψ ad ψ bols dow o fdg he coeffces b ad c cosug he vecors b ad c. Ths s accomplshed by mag use of a esemble of Bohma pars of coordaes {X, Y} whch are seleced ally from he oe-parcle desy fuco ρ(x) a = (For a sysem of bosos, he Bohma coordae of each boso s pced accordg o he sgle parcle desy ρ(x) whch correspods o he dagoal fuco of he wo body desy marx ρ(x, x ), where, for a wo-body problem, ρ( x, x ) = ψ( x, y) ψ ( x, y ) dydy ). Each of hese pars s called a cofgurao. If we ca represe boh φ ad φ for a cera value of Y as a lear superposo of all { φ ( Y )} correspodg o all members of he esemble,.e., f φ ( Y) = αφ ( Y ) where φ ( Y ) Y correspods o he h member of he esemble ad φ = βφ ( ) he follows from he leary Eq. (5) ha b ( Y) = α a ( Y ) ad c ( Y) = β a ( Y ). Fdg he values of α ad β s equvale o solvg a sysem of lear equaos. I hs way, we ca oba ψ, ψ whou ever cosrucg he coeffce marx C. I should be oed ha ψ ad ψ ca be deermed whou expressg ψ erms of a bass a all, sce {α } ad {β } deped oly o he ampludes of ay complee bass a he locao of he Bohma parcles. Afer {α } ad {β } are obaed, we ca express ψ ad ψ as ψ ( x) = α ψ ( x; Y ) ψ ( x) = β ψ ( x; Y ) Wh he CWs a our dsposal, we use he equao of moo (3) for = o evolve he esemble of CWs for all Bohma parcles as descrbed he Mehods seco. We call hs scheme Ieracg Plo Waves (IPW). Some observables ca be compued by averagg over he esemble of Bohma cofguraos {X, Y} such as x. Sce we have access o he CWs, we ca devse a more accurae mehod ha approxmaes he exac expresso of he expecao value of a operaor Â, ˆ A = Ψ ( x, ya ) ˆΨ( x, y) dxdy, by performg he egral over oe varable as a Rema sum over s Bohma coordaes,.e., ˆ A Δ ψ ˆ w w( x) Awψw( x) dx, w where ψ w (x) s he codoal wavefuco of he frs parcle codoed o he coordae of he secod parcle belogg o he w h cofgurao of he esemble, Δ w s he dsace bewee adjace values of Y a he w h cofgurao ad  w s he operaor  codoed o Y w. For wo-body operaors such as he eraco poeal V(x, y), Âw ( x) s gve by V(x; Y w ). Smlarly, he reduced oe-body desy ca be approxmaed as ρ( x) Δ ψ w w w( x) ψw ( x). Le us apply hs mehod o sudy he breahg dyamcs of wo bosos a harmoc rap, Vx ( ) = x. The bosos are ally codesed he groud sae of he rap, ad sar a breahg moo whe a harmoc eraco Vx (, y) = ( x y) s suddely swched o. A fe fxed se of orbals are ae o be he lowes se of egefucos of he oe-body problem wh he effecve poeal fel by oe parcle due o he oher oe, amely Veff( x) = x + ρ( yvx ) (, y) dy. We llusrae Fg. he behavor of ρ() for = = compued by he IPW mehod wh 6 orbals compared wh he exac dyamcs (aomc us are used he res of hs paper). (6) Scefc REPOrTS (8) 8:74 DOI:.38/s

4 Geeralzao o may-parcle sysems. Geeralzg our algorhm o a may-parcle problem cossg of N B bosos s sraghforward. Le us deoe he coordaes of he parcles by x, y, z, ec., whle, as before, we deoe he Bohma coordaes by upper case leers. A sgle cofgurao of Bohma walers s deoed by ( X, Y, Z, ). Le us deoe he codoal wavfucos by ψ ( x; Y, Z, ) Ψ ( x, y, z, ) ψ Ψ ( x, y, z, ( x; Y, Z, ) ) ψ Ψ ( x, y, z, ( z; X, Y, ) ) y= Y, z= Z, y= Y, z= Z, x= X, y= Y, ad so o. The equao of moo for ψ ( x; Y, Z, ) s a smple geeralzao of Eq. (3) for he case of may parcles: ψ ( x; Y, Z, ; ) = ψ + x Vx ( ; Y, Z, ) ( x; Y, Z, ; ) m dy + ψ ( x; Y, Z, ; ) ψ ( d m x ; Y, Z, ; ) dz + ψ ( x; Y, Z, ; ) ψ ( + d m x ; Y, Z, ; ) + ec. Smlar equaos ca be wre for all he CWs correspodg o all parcles every cofgurao. A geerc asaz for he may-body wavefuco smlar o Eq. (4) reads, (7) Ψ ( x, y, z,, ) = cj () φ( x) φj( y) φ( z) j,, (8) I order o compue ψ (x;y, Z, ; ) from ψ w( x; Yw, Zw, ; ) belogg o all cofguraos, we eed o express he esor [ φj ( Y) φ ( Z) ] as a lear superposos of all he [ φj( Yw) φ ( Zw ) ] esors belogg o all cofguraos;.e., [ φj ( Y) φ( Z) ] = w αw[ φj( Yw) φ( Zw ) ]. Ths ca be doe wh he exsg umercal echques by rearragg all he M N B erms of he esors, where M s he umber of orbals, vecor forms ad solvg a lear sysem of equaos. Sce he sze of he vecors ow becomes expoeally bgger as he umber of parcles becomes larger, he boleec of hs mehod would be o ae a suffcely large umber of cofguraos ha esures havg a complee lear sysem. Therefore, a mmedae room for mproveme here would be o fd smar accs o overcome hs problem. Compug observables from codoal wavefucos. I order o compue he expecao value of a operaor, we ca rea he colleco of ormalzed CWs belogg o all he parcles as f hey descrbe ormal sgle parcle wavefucos.  x ca he be compued as ˆ A ψ x Aˆ x ( ) ψ ( x) dx N w w x w, where w ψ w ( x) s he ormalzed CW of parcle x belogg o he w h cofgurao ad N w s he umber of cofguraos. If  s a wo-body operaor such as he eraco poeal bewee wo parcles, we frs compue a mea-feld operaor, ad he rea as a oe-body operaor. For example, he mea-feld eraco poeal fel by oe parcle s compued as Vx ( ) ψ ( yvx ) ( y) ψ ( ydy ) N w w w. The expecao value of Vx ( ) s he w compued as a oe-body operaor. From he ormalzed colleco of all he CWs, we ca also ge a approxmao for he reduced desy marx of oe parcle ρ( x, x) ψ ( x ) ψ ( x) N w w w. Ths marx ca be used o w compue a se of aural orbals erms of he fe bass se used he posulaed asaz as he egesaes of ρ(x, x). May parcles a harmoc rap. Le us apply hs geeralzao o he dyamcs of 3 bosos ad 5 bosos a harmoc rap Vx ( ) = x for wo cases of erparcle eracos: log-rage aracve harmoc eraco ad shor-rage repulsve eraco. As he wo-parcle case, all he bosos ally resde he groud sae of he harmoc rap before he eraco s suddely swched o a =. We sudy he breahg dyamcs by compug x as a fuco of me. For a harmoc eraco of he form Vx (, y) = ( x y) we cosder wo cases of 5 bosos wh wea eracos ( =.) ad 3 bosos wh srog eraco ( = ) ad we use 3 ad 4 orbals he wo cases, respecvely. I boh cases we compare he resuls wh he umercally exac smulao usg mulcofguraoal me-depede Harree mehod for bosos (MCTDHB) 9 33 ad wh he Herma lm (HL) of Eq. (7) (also referred o as small eagleme approxmao) where all he o-herma erms Eq. (7) are dropped ou. The Herma lm s equvale o he me-depede quaum Moe-Carlo (TDQMC) of ref. 5 whch Scefc REPOrTS (8) 8:74 DOI:.38/s

5 Fgure. Breahg dyamcs of 5 bosos ad 3 bosos usg IPW compared wh umercally exac dyamcs. (a,b) 5 bosos ad 3 bosos are ally he groud sae of a harmoc rap before a harmoc eraco of sregh =. ad = respecvely s swched o a =. The breahg dyamcs s compued by he Ieracg Plo Wave (IPW) mehod for 5 ad 3 bosos usg 3 ad 4 orbals respecvely ad compared wh he umercally exac dyamcs compued by MCTDHB ad wh he Herma lm (HL) soluo of Eq. (7) (see ex). I (c) we depc he me evoluo for a esemble of Bohma rajecores represeg he frs parcle, ad (d) he 5 rajecores correspodg o a sgle cofgurao for he case (a). does o ae eagleme o cosderao. I was also employed recely 34,35 order o devse a approxmae soluo for elecro-uclear dyamcs molecular sysems. I Fg., we show he resuls of compug x by averagg over he Bohma coordaes of all he parcles usg a sgle esemble coag cofguraos. We oce ha he IPW mehod s more accurae he wea eraco regme ha he srog eraco regme. The Bohma rajecores of he frs parcle all cofguraos are show Fg. (c) for =. whle he Bohma rajecores for all he 5 parcles a sgle cofgurao are show Fg. (d). The few cosa rajecores appearg Fg. (c) correspod o he cases where we maually se he Bohma veloces o be zero whe he deomaor Eq. () s below a cera hreshold. ( x y) I Fg. 3, we plo x afer swchg o a gaussa eraco Vx ( y) = ( / πσ ) e σ wh =., σ =.5 ad compare he resuls wh MCTDHB smulao ad he HL of Eq. (7). I hs calculao, we compue x from he expecao value of x usg he codoal wavefucos raher ha from he Bohma rajecores. I order o compue he groud sae eergy for a eracg sysem of parcles usg he IPW scheme, we alze he CWs ad he Bohma rajecores he groud sae of he oeracg Hamloa. Aferwards, we swch o he eraco adabacally. Accordg o he adabac heory 36, he sysem remas he groud sae of he saaeous Hamloa. I Fg. 4, we plo he evoluo of he eergy of he saaeous Hamloa of a 5-parcle sysem as we swch o he harmoc eraco Vx (, y) = ( x y) ada- NB 37 bacally ad compare wh he exac groud sae eergy E = + NB +. 5 for =.. The groud sae eergy compued by MCTDHB 38 s more accurae ha he IPW calculao for he same umber of orbals by several sgfca dgs. Perhaps a beer mehod o compue groud sae eergy s o propagae Eq. (7) complex me, whle evolvg he Bohma rajecores real me 5. The opmal relao bewee he real ad complex me evoluo cosues a eresg opc of research. Scefc REPOrTS (8) 8:74 DOI:.38/s

6 ( x y) Fgure 3. Breahg dyamcs of 5 bosos wh shor-rage repulsve eracos. Same as Fg. wh erparcle gaussa eracos Vx ( y) = ( / πσ ) e σ wh =., σ =.5. Three orbals are used he IPW calculao. Fgure 4. Groud sae eergy compuao usg IPW. Groud sae eergy for a sysem of 5 eracg bosos a harmoc rap wh harmoc eraco sregh =. compued by he IPW mehod (sold le) usg 3 orbals ad compared wh he exac groud sae eergy (dashed). The sysem s alzed he groud sae of he rap before he eraco s swched o adabacally as =. ( e. ). The eergy s compued wh respec o he saaeous value of. Dscusso We have preseed a promsg approach o aalyze he dyamc ad sac properes of sysems cossg of several bosos by evolvg a sysem of ouary equaos ha goes beyod he small eagleme approxmao ad he mea-feld approxmao. Our mehod bulds o he formal expaso (Eq. 3), bu as we fd ha he rucao of hs se of equaos qucly leads o errors, we roduce ad apply a rucao-free mehod ha provdes he lowes order plo waves a self-cosse maer. The accuracy of hs ew approach s cofrmed bu also ouperformed by he sae-of-he-ar MCTDHB algorhm. I he MCTDHB mehod, creasg he umber of orbals (M) s cofroed wh he expoeally large umber of cofguraos of permaes ha eeds o be ae o accou. We have a smlar scalg M N B problem our approach; he umber of cofguraos of Bohma parcles has o be larger ha order o avod havg a udeermed lear sysem of equaos whe solvg for α w. So, he complexy of our approach sll creases expoeally wh he umber of parcles. I s worh meog also ha MCTDHB s much faser ha our algorhm. Whle a ypcal resul he prevous fgures aes a few hours o compue, aes much less me by he well developed MCTDHB. Improvemes o our mehod may come from: () sraeges ha mmze he umber of cofguraos, ad hece he compuaoal power, requred o evolve he plo waves whou havg a uderdeermed sysem of equaos, ad () opmal choce of he bass fucos (possbly a adapve se of orbals) ha lead o he mos compac represeao of he full wavefuco, ad hece he smalles umber of orbals o capure he dyamcs of he may-parcle wavefuco. I s sll a ope queso wheher he o-herma erms he equaos of moo ca be replaced by a effecve eagleme poeal ha maes he equaos uary ad a he same me capures he eagleme Scefc REPOrTS (8) 8:74 DOI:.38/s

7 he sysem. For sysems cossg of may parcles,.e., N B >, he eagleme of he groud sae s so small 39 ha eve he Herma lm 5 ca be effce for smulag he dyamcs volvg a small umber of exced saes. I prcple, geeralzg he IPW approach o fermoc sysems s sraghforward, as log as we choose he al sae wh he proper symmery requremes. However, for fermos wo problems arse. Frs, due o he Paul excluso prcple, we eed a large umber of orbals o descrbe a fermoc sae ad herefore, he umber of fermos ha ca be aalyzed s small compared o bosos. Secod, he codoal wavefucos for fermos wll have odes, ha complcae compug he velocy of he Bohma walers aroud hose odes. Sce he ode problem s a well ow problem for smulag quaum dyamcs wh Bohma rajecores 6, he mehods developed hs regard he leraure 4,4 may beef he soluo of hs problem. As a fal comme, we oe ha order o descrbe eagled dyamcs, we eed o cosder may eracg cofguraos of Bohma parcles he prese approach. A smlar suao arses 9,6 where he cocep of eracg classcal worlds was roduced. Ths smlary bewee he wo approaches may be worh furher aeo dscussos of he foudaos of quaum mechacs. Mehods Dervao of Equao (3). Le us derve Eq. (3) wh respec o ψ x we fd by he cha rule ha ( x, x, ) x (, ). Sce ψ Ψ ( x, ), x= X() ψ( x, ) = Ψ( x, x) dx() Ψ( x, x) + x d x x = X () x= X() By exchagg he me ad spaal dervaves he frs erm o he R.H.S. ad usg he TDSE:. (9) we oba afer subsug bac Eq. (9) ψ( x, ) Ψ = ( x, x) + Vx (, x) m m Ψ ( x, x), ψ x = + (, ) m x x + dx() + ψ ( x, ) + ψ ( x, ). m d ( Vx (, x ) Ψ( x, x )) x= X() By applyg he cha rule o he secod erm o he R.H.S. we oba ( Vx (, x ) Ψ ( x, x )) = ψ ( x, ) V ( x, x ) whch afer subsug Eq. () recovers Eq. (3). x () x= X() () x x = X () = Evolvg a herarchy of plo waves. Le us llusrae he effcecy of evolvg a rucaed herarchy of ψ usg Eq. (3) order o compue he dyamcs of a eagled sysem. We cosder he eagled dyamcs of wo parcles of masses m = ad m = subjec o he harmoc poeal Vx (, x ) = x + x wh =.. Le us ae he al sae o be he eagled groud sae of he Hamloa wh he poeal fuco Vx (, x ) = x + x + ( x x ) 3 wh = =., 3 =. ad he masses of he parcle m = ad m =. Ths s a eagled sae. We evolve he Bohma rajecores for he al codos X =, X =. We frs rucae he herarchy a N =, hus mag Eq. (3) uary. Ths case correspods o he Herma lm,.e., oerag plo waves. Fgure 5 shows ha he Bohma rajecory evolved by he correspodg plo wave devaes from he rajecory compued from he exac plo wave already a half a cycle of he oscllaory moo. Icreasg he deph of he herarchy o N = 7 oly exeds he rage of accurae dyamcs for aoher cycle. I hs example, we see clearly ha alhough he parcles are o-eracg, we eed o accou for he eraco bewee he plo waves of he wo parcles correcly hrough he hgher order CWs eve whe he rao of he parcles masses s :. Oherwse, he errors orgag from rucag he herarchy of { ψ } propagae very fas o ψ. Icreasg N beyod N wll o help prologg he rage of accuracy because of he umercal errors he calculao of he hgher-order dervaves of he wavefuco. Sce he errors afflc he plo waves hrough he las wo erms Eq. (3), hs mehod of evolvg he plo waves s mos suable whe we are eresed he dyamcs of a very lgh parcle eracg wh a much heaver oe over a very shor me scale. I hs case, we ca om he las wo erms for he heavy parcle whle reag hem for he lgh parcle,.e., do a semclasscal approxmao 4 for oe parcle oly. Ieracg Plo Waves for wo parcles. For he wo-parcle IPW smulaos Fg. we express all he CWs erms of he bass se {φ }, ad evolve he expaso coeffces {a }. Equao (3) s he expressed as Scefc REPOrTS (8) 8:74 DOI:.38/s

8 Fgure 5. Eagled quaum dyamcs of wo parcles a harmoc rap. (a) A caroo represeg he plo waves gudg Bohma parcles movg D. Alhough he wo parcles do o erac, her eagleme mples a couplg bewee he wo plo waves gudg he wo Bohma rajecores (he dashed le). (b) The rajecores of he lgh parcle compued from he exac plo waves (sold blue) ad usg plo waves evolved usg Eq. (3) (dashed). The case for N = 7 correspods o eracg plo waves evolved wh he help of a herarchy of codoal wavefucos { ψ, ψ } up o = 7 (see Eq. (3)), whle N = correspods o oerag plo waves (he Herma lm). We oce ha he former case s more accurae ha he laer. The wo parcles have mass rao :. ω s he rap frequecy of he lgh parcle. φ m a ( x) x () φ( ) = () + a x Vx (, Y) a( ) φ ( x) φ + φ. m c x dy () ( ) b() ( x) d By ag he er produc wh each of he orbals {φ (x)} we oba he me dervave of he expaso coeffces { a }. Ths sysem of equaos s he solved usg a fourh-order Ruge-Kua mehod. Propagag he codoal wavefucos. Each of equaos (3) ad (7) represes a sysem of coupled olear ad ouary dffereal equaos ha ca be cas he form ψ( x, ) = Hψ( x, ) + W( x, ), where he frs erm o he RHS represes he uary par of he equao ad W(x, ) represes he ouary par whch s a fuco of all oher CWs. If, e.g., H s a cosa Hamloa, a geeral soluo for hs equao aes he form: H H ψ( x, ) = e e Wx (, ) d + ψ( x, ) I order o propagae ψ(x, ) for a sgle me sep from = o = δ usg hs soluo, boh he operaor e Hδ ad e H are performed usg a spl-operaor mehod 4. The egral s performed usg he rapezodal δ H H δ rule e W( x, ) d [ e W( x, δ) + Wx (,)]. Mscellaeous umercal echques. We use he spl operaor echque 35 order o propagae he exac wo-parcle wavefuco magary me (o geerae he groud sae) ad real me (as Fg. ) or o geerae he exac Bohma rajecores (as Fg. 5). The evaluao of he codoal wavefucos a he poso of he Bohma parcles was performed by FFT-based erpolao. The spaal dervaves of he wavefuco o compue { ψ, ψ } are compued usg he Fas Fourer Trasform wh he FFTW pacage. Solvg he lear sysem of equaos Eq. (5) was performed by he LAPACK roue gelsd() whch uses sgular value decomposo ad a dvde ad coquer mehod o compue he mmum-orm soluo o a lear leas squares problem (Iel Mah Kerel Lbrary Developer Referece). () () Scefc REPOrTS (8) 8:74 DOI:.38/s

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Harmoc eraco model ad s applcaos Bose-Ese codesao. J. Sa. Phys. 3, 63 (3). 38. Lode, A. U., Sama, K., Alo, O. E., Cederbaum, L. S. & Srelsov, A. I. Numercally exac quaum dyamcs of bosos wh medepede eracos of harmoc ype. Phys. Rev. A 86, 6366 (). 39. Sperlg, J. & Walmsley, I. Eagleme macroscopc sysems. Phys. Rev. A 95, 66 (7). 4. Goldfarb, Y. & Taor, D. J. Ierferece bohma mechacs wh complex aco. J. Chem. Phys. 7, 6 (7). 4. Ber, E. R. e al. Adapve quaum moe carlo approach saes for hgh-dmesoal sysems. I Appled Bohma Mechacs, 33 (Pa Saford Publshg, ). 4. Kosloff, R. Tme-depede quaum-mechacal mehods for molecular dyamcs. J. Phy. Chem. 9, 87 (988). Acowledgemes T. A. Elsayed has D. A. Decer, H. Myag, A. I. Srelsov, L. F. Buchma, C. Leveque ad Adreas Cara for bref dscussos, ad I. Chrsov for hs commes o a earler verso of he mauscrp. The auhors acowledge facal suppor by Vllum Foudao. Par of hs paper was wre whle T. A. Elsayed was vsg Huer College of he Cy Uversy of New Yor. Auhor Corbuos T.A.E. coceved he research dea, desged he IPW algorhm for wo ad may parcles ad mplemeed o he compuer. L.B.M. ad K.M. provded helpful sghs ad feedbac durg he developme of he mehod. All auhors parcpaed he daa aalyss ad he roubleshoog. The mauscrp was wre by T.A.E. wh exesve feedbac from L.B.M. ad K.M. Addoal Iformao Compeg Ieress: The auhors declare o compeg eress. Publsher's oe: Sprger Naure remas eural wh regard o jursdcoal clams publshed maps ad suoal afflaos. Scefc REPOrTS (8) 8:74 DOI:.38/s

10 Ope Access Ths arcle s lcesed uder a Creave Commos Arbuo 4. Ieraoal Lcese, whch perms use, sharg, adapao, dsrbuo ad reproduco ay medum or forma, as log as you gve approprae cred o he orgal auhor(s) ad he source, provde a l o he Creave Commos lcese, ad dcae f chages were made. The mages or oher hrd pary maeral hs arcle are cluded he arcle s Creave Commos lcese, uless dcaed oherwse a cred le o he maeral. If maeral s o cluded he arcle s Creave Commos lcese ad your eded use s o permed by sauory regulao or exceeds he permed use, you wll eed o oba permsso drecly from he copyrgh holder. To vew a copy of hs lcese, vs hp://creavecommos.org/lceses/by/4./. The Auhor(s) 8 Scefc REPOrTS (8) 8:74 DOI:.38/s