4. THE DENSITY MATRIX

Transcription

1 4. THE DENSTY MATRX The desy marx or desy operaor s a alerae represeao of he sae of a quaum sysem for whch we have prevously used he wavefuco. Alhough descrbg a quaum sysem wh he desy marx s equvale o usg he wavefuco, oe gas sgfca praccal advaages usg he desy marx for cera me-depede problems parcularly relaxao ad olear specroscopy he codesed phase. The desy marx s defed as he ouer produc of he wavefuco wh s cojugae. (4.1) Ths mples ha f you specfy a sae χ, χ ρ χ gves he probably of fdg a parcle he sae χ. s ame derves from he observao ha plays he quaum role of a probably desy. f you hk of he sascal descrpo of a classcal observable obaed from momes of a probably dsrbuo P, he ρ plays he role of P he quaum case: A AP A da (4.2) A A Tr A (4.3) where Tr[ ] refers o racg over he dagoal elemes of he marx, Tr[ ] a a. a The las expresso s obaed as follows. f he wavefuco for he sysem s expaded as he expecao value of a operaor s c (4.4) Aˆ c c m Aˆ (4.5), m Also, from eq. (4.1) we oba he elemes of he desy marx as c c m,, m m (4.6) We see ha, he desy marx elemes, are made up of he me-evolvg expaso coeffces. Subsug o eq. (4.5) we see ha, A ˆ A m Tr Aˆ pracce hs makes evaluag expecao values as smple as racg over a produc of marces. (4.7) Adre Tokmakoff, 11/7/214

2 p. 4-2 Wha formao s he desy marx elemes, ρm? The dagoal elemes ( = m) gve he probably of occupyg a quaum sae: cc p (4.8) For hs reaso, dagoal elemes are referred o as populaos. The off-dagoal elemes ( m) are complex ad have a me-depede phase facor c c c c (4.9) m m e Sce hese descrbe he cohere oscllaory behavor of cohere superposos he sysem, hese are referred o as cohereces. So why would we eed he desy marx? becomes a parcularly mpora ool whe dealg wh mxed saes, whch we ake up laer. Mxed saes refer o sascal mxures whch we have mperfec formao abou he sysem, for whch me mus perform sascal averages order o descrbe quaum observables. For mxed saes, calculaos wh he desy marx are grealy smplfed. Gve ha you have a sascal mxure, ad ca descrbe pk, he probably of occupyg quaum sae k, evaluao of expecao values s smplfed wh a desy marx: ˆ Aˆ p A (4.1) k k k k p (4.11) k k k k Aˆ TrAˆ (4.12) Evaluag expecao value s he same for pure or mxed saes. Properes of he desy marx We ca ow summarze some properes of he desy marx, whch follow from he defos above: 1) s Herma sce m 2) Sce probably mus be ormalzed, Tr 1 3) We ca ascera he degree of pure-ess of a quaum sae from 2 1 for pure sae Tr < 1 for mxed sae

3 p. 4-3 addo, whe workg wh he desy marx s covee o make oe of hese race properes: 1) The race over a produc of marces s vara o cyclc permuao of he marces: Tr ABCTr CAB Tr BCA 2) From hs resul we see ha he race s vara o uary rasformao: Tme-evoluo of he desy marx 1 Tr S AS Tr S AS Tr A The equao of moo for he desy marx follows aurally from he defo of ad he me-depede Schrödger equao. H H H, (4.13) (4.14) Equao (4.14) s he Louvlle-Vo Neuma equao. s somorphc o he Heseberg equao of moo, sce ρ s also a operaor. The soluo o (4.14) s U U Ths ca be demosraed by frs egrag eq. (4.14) o oba (4.15) () d H, (4.16) f we expad eq. (4.16) by eravely subsug o self, he expresso s he same as whe we subsue U exp d H( ) (4.17) o eq. (4.15) ad collec erms by orders of H().

4 p. 4-4 Noe ha eq. (4.15) ad he cyclc varace of he race mply ha he me-depede expecao value of a operaor ca be calculaed eher by propagag he operaor (Heseberg) or he desy marx (Schrödger or eraco pcure): A ˆ Tr Aˆ Tr AU ˆ U Tr Aˆ (4.18) For a me-depede Hamloa s sraghforward o show ha he desy marx elemes evolve as m U U m (4.19) (4.2) e From hs we see ha populaos, () ( ), are me-vara, ad cohereces oscllae a he eergy splg. The desy marx he eraco pcure For he case whch we wsh o descrbe a maeral Hamloa H uder he fluece of a exeral poeal V(), H H V (4.21) we ca also formulae he desy operaor he eraco pcure,. From our orgal defo of he eraco pcure wavefucos U (4.22) We oba as U U (4.23) S S Smlar o he dscusso of he desy operaor he Schrödger equao, above, he equao of moo he eraco pcure s V, (4.24) V where, as before, UVU. Equao (4.24) ca be egraed o oba

5 p. 4-5 d V, (4.25) Repeaed subsuo of o self hs expresso gves a perurbao seres expaso d V 1 1, 2 2 d2 d 1 V 2, V 1, 2 d d 1 d 1 V, V 1,, V 1, 1 2 (4.26) (4.27) Here ad s he h -order expaso of he desy marx. Ths perurbave expaso wll play a mpora role laer he descrpo of olear specroscopy. A h order expaso erm wll be proporoal o he observed polarzao a h order olear specroscopy, ad he commuaors observed eq. (4.26) are proporoal o olear respose fucos. Smlar o eq. (4.15), equao (4.26) ca also be expressed as U U (4.28) Ths s he soluo o he Louvlle equao he eraco pcure. descrbg he me-evoluo of he desy marx, parcularly whe descrbg relaxao processes laer, s useful o use a superoperaor oao o smplfy he expressos above. The Louvlle equao ca be wre shorhad erms of he Lovlla superoperaor L where L s defed he Schrödger pcure as ˆ ˆ L (4.29) L Aˆ H, Aˆ (4.3) Smlarly, he me propagao descrbed by eq. (4.28) ca also be wre erms of a superoperaor G, he me-propagaor, as G s defed he eraco pcure as G (4.31)

6 p. 4-6 GA ˆ U Aˆ U (4.32) Gve he egesaes of H, he propagao for a parcular desy marx eleme s G a b H/ H/ ab e e ab e a b (4.33) Usg he Louvlle space me-propagaor, he evoluo of he desy marx o arbrary order eq. (4.26) ca be wre as 2 d d d Gˆ V Gˆ V Gˆ V (4.34)