Consideratii teoretice

O particula cu spin 1/2 evolueaza intr-un camp magnetic cu gradient pe o anumita directie (dispozitiv Stern-Gerlach) conform cu ecuatia Pauli - Schr odinger:

$$\frac \hbar i\cdot \frac \partial {\partial t}\left\vert \Psi \ri... ...angle +V( \widehat{\stackrel{\rightarrow }{r}})\left\vert \Psi \right\rangle =0$$ (7.2.1)

Operatorul $V(\widehat{\stackrel{\rightarrow }{r}})$ energia potentiala este:

$$V(\widehat{\stackrel{\rightarrow }{r}})=-\widehat{\stackrel{\rightarrow }{ \mu }}\otimes \overrightarrow{B}(\widehat{\stackrel{\rightarrow }{r}})$$ (7.2.2)

unde:

$$\widehat{\stackrel{\rightarrow }{\mu }}=\mu _0\cdot \widehat{\stackrel{ \rightarrow }{\sigma }}$$ (7.2.3)

este momentul magnetic al particulei. Expresia pentru $\overrightarrow{B}( \widehat{\stackrel{\rightarrow }{r}})$ trebuie sa fie compatibila cu ecuatiile Maxwell, dar daca campul magnetic prezinta o componenta uniforma mult mai mare decat cea neuniforma se poate arata ca [9] influenta componentei ne-uniforme transversale poate fi neglijata, astfel ca se poate lucra cu expresia:

$$\widehat{\stackrel{\rightarrow }{B}}=(B_0\cdot \widehat{1}+b\cdot \widehat{z} )\cdot \overrightarrow{e_z}$$ (7.2.4)

unde componenta uniforma $B_0$ nu este esentiala, pentru ca se poate lucra intr-o imagine de interactie. Ecuatia (7.2.1) devine:

$$\frac \hbar i\cdot \frac \partial {\partial t}\left\vert \Psi \ri... ...\cdot b\cdot \hat{z}\otimes \widehat{\sigma }_z\left\vert \Psi \right\rangle =0$$ (7.2.5)

unde se pot decupla cele doua componente spinoriale:

$$\left\vert \Psi \right\rangle =\left\vert \begin{array}{l} \psi _{+} \\ \psi _{-} \end{array} \right\rangle$$ (7.2.6)

in doua ecuatii:

$$\frac \hbar i\cdot \frac \partial {\partial t}\left\vert \psi _{\... ...angle \mp \mu _0\cdot b\cdot \widehat{z}\left\vert \psi _{\pm }\right\rangle =0$$ (7.2.7)

care se pot rezolva in reprezentarea impulsului $(p_z=\hbar \cdot k)$ [109]:

$$\Psi _{\pm }(k,t)=\exp \{i\cdot [\frac{(\mu _0\cdot b)^2}{24\cdot... ...\cdot \hbar })^2]\}\cdot \Psi _{\pm }(k\mp \frac{\mu _0\cdot b\cdot t}\hbar ,0)$$ (7.2.8)

$$\left\vert \psi _{\pm }(t)\right\rangle =\exp (i\cdot \frac{(\mu ... ...t^3)\cdot \exp (\mp i\cdot \frac{\mu _0\cdot b\cdot t}\hbar \cdot \hat{z})\cdot$$ (7.2.9)

$$\cdot \exp (-i\cdot \frac{\hbar \cdot t}{2\cdot m}\hat{k}^2)\cdot... ...dot b\cdot t}\hbar \cdot \hat{z})\cdot \left\vert \psi _{\pm }(0)\right\rangle$$

In aceste ecuatii se poate efectua o a-dimensionalizare naturala. Intr-adevar, marimea:

$$t_0=[\frac{2\cdot m\cdot \hbar }{(\mu _0\cdot b))^2}]^{\frac 13}$$ (7.2.10)

are dimensiune de timp. Fie:

$$\tau = \frac t{t_0}$$ (7.2.11)

$$\zeta = z\cdot \sqrt{\frac{2\cdot m}{\hbar \cdot t_0}}$$

$$\kappa = k\cdot \sqrt{\frac{\hbar \cdot t_0}{2\cdot m}}$$ (7.2.12)

(7.2.7) si (7.2.9) devin:

$$\frac \partial {\partial \tau }\left\vert \psi _{\pm }\right\rang... ...ight\rangle \mp i\cdot \widehat{ \zeta }\left\vert \psi _{\pm }\right\rangle =0$$ (7.2.13)

$$\left\vert \psi _{\pm }(\tau )\right\rangle =\exp (i\frac{\tau ^3... ...ot \frac \tau 2\cdot \hat{\zeta} )\cdot \left\vert \psi _{\pm }(0)\right\rangle$$ (7.2.14)

Se aleg conditiile initiale astfel încât la momentul initial sa avem un puls gaussian:

$$\psi _{\pm }(\zeta ,0)=\left\langle \zeta \mid \psi _{\pm }(0)\ri... ...dot \zeta _0^{1/2}}\exp \left( -\frac 12\cdot (\frac \zeta {\zeta _0})^2\right)$$ (7.2.15)

(7.2.14) devine:

$$\psi _{\pm }(\zeta ,\tau )=\frac 1{\pi ^{1/4}\cdot \zeta _0^{1/2}... ...\tau }{ \zeta _0^2}\right) }\cdot (\frac{\zeta \mp \tau ^2}{\zeta _0})^2\right)$$ (7.2.16)

care este un puls gaussian centrat in $\pm \tau ^2$. Aceste rezultate corespund unei masuratori ideale cu dispozitivul Stern-Gerlach. Desigur, prezenta altor factori, cum ar fi: limitarea spatiala a campului magnetic, efectele de margine, delocalizarea pachetelor datorita propagarii libere, fluctuatii termice ale marimilor macroscopice implicate duc la grade de ne-idealitate ale masuratorii. Desi in manuale se argumenteaza ca practic suprapunerea celor doua componente poate fi facuta oricat de mica, o serie de lucrari arata ca factorii mentionati fac ca masuratoarea sa devina non-exacta [9,53].

In continuare studiem efectul prezentei unui termen stochastic perturbator in Hamiltonian:

$$\frac \hbar i\cdot \frac \partial {\partial t}\left\vert \psi _{\... ...ight\rangle +\hat{B}(t)\cdot \widehat{z}\left\vert \psi _{\pm }\right\rangle =0$$ (7.2.17)

Pentru operatorul $\hat{B}(t)$ consideram expresia [20]:

$$\hat{B}(t)=i\cdot \stackrel{n}{\sum }\kappa _n\cdot \sqrt{\frac 1... ...+\hat{a}_n^{\dagger }\cdot \exp \left( -i\cdot \omega _n\cdot t\right) \right)$$

unde $\left\{ \omega _n\right\}$ si $\left\{ \kappa _n\right\}$ sunt marimi constante care descriu oscilatorii ce alcatuiesc rezervorul termic. Luand $\kappa _n\varpropto \omega _n$, forma a-dimensionala a (7.2.17) este:

$$\frac \partial {\partial \tau }\left\vert \psi _{\pm }\right\rang... ...right\rangle \mp i\cdot \widehat{ \zeta }\left\vert \psi _{\pm }\right\rangle -$$ (7.2.18)

$$-\eta \cdot \sum_nf_n\left( -\hat{a}_n\cdot \exp \left( i\cdot \o... ...\cdot t\right) \right) \widehat{\zeta }\left\vert \psi _{\pm }\right\rangle =0$$

unde $f_n$ sunt factori numerici care descriu distributia frecventelor $\left\{ \omega _n\right\} _n$, iar $\eta$ este un factor a-dimensional, care va fi estimat mai tarziu. Operatorii campului actioneaza in spatiul Hilbert al dispozitivului, iar ecuatia este subinteleasa in spatiul produs tensorial; pentru a scrie ecuatia de evolutie pentru particula investigata, trebuie sa luam urma in ecuatia von Neumann - Liouville peste spatiul Hilbert al dispozitivului. Fie starea initiala a dispozitivului descrisa de distributia canonica:

$$\hat{\rho}_b=\frac 1{Z\left( \beta \right) }\exp (-\beta \hbar \sum_n\omega _n\hat{a}_n^{\dagger }\hat{a}_n)$$

iar cea a particulei $\hat{\rho}_s$. Intr-o imagine de interactie care ''ascunde'' evolutia ideala, ecuatia von Neumann - Liouville este:

$$\frac{\partial \hat{\rho}}{\partial \tau }=-\frac i\hbar \left[ \... ...ta }\otimes \hat{B}(\tau )) \widehat{U}_0\left( \tau \right) ,\hat{\rho}\right]$$ (7.2.19)

Avand in vedere stationaritatea starii dispozitivului, putem scrie solutia ( 7.2.19) ca:

$$\hat{\rho}=\widehat{U}\left( \tau \right) \hat{\rho}_s\otimes \hat{\rho}_b \widehat{U}^{\dagger }\left( \tau \right)$$

$$\widehat{U}\left( \tau \right) =\mathbb{T}\exp (-\frac i\hbar \in... ...idehat{\zeta }\left( \tau ^{\prime }\right) \otimes \hat{B}(\tau ^{\prime })))$$

in care $\mathbb{T}$ este operatorul de ordonare temporala, iar $\widehat{ \zeta }\left( \tau \right)$ este operatorul $\widehat{\zeta }$ in imaginea de interactie. Avem de estimat o urma de tipul:

$$\mathrm{tr}\left( \exp \left( A^{*}(\tau )\hat{a}^{\dagger }-A(\tau )\hat{a} \right) \exp (-\beta \hat{a}^{\dagger }\hat{a})\right)$$

in care, daca aplicam formula Campbell-Baker-Hausdorf, obtinem:

$$\sum_{n=0}^\infty \exp (-\beta \hbar \omega n-\frac 12\left\vert A(\tau )\right\vert ^2+2i\sqrt{n}\mathcal{I}m(A(\tau )))$$ (7.2.20)

Un calcul numeric arata ca la temperatura fixata, expresia de mai sus, este foarte oscilanta, depinzand dramatic de valoarea lui $A(\tau ),$ care la randul sau depinde de un factor $\exp \left( i\cdot \omega _n\cdot \tau \right) .$ Rezulta de aici faptul ca actiunea operatorilor de camp se poate traduce numeric in aparitia unor numere aleatoare complexe, a caror amplitudine o inglobam in factorii numerici $f_n$. Varianta adaptata numeric pentru (7.2.18) devine:

$$\frac \partial {\partial \tau }\left\vert \psi _{\pm }\right\rang... ...right\rangle \mp i\cdot \widehat{ \zeta }\left\vert \psi _{\pm }\right\rangle -$$ (7.2.21)

$$-\eta \cdot \sum_nf_n\cdot \sin (\omega _n\cdot \tau +\phi _n)\widehat{\zeta }\left\vert \psi _{\pm }\right\rangle =0$$

unde $\phi _n$ sunt numere aleatoare uniform distribuite in $\left[ 0,2\pi \right] ,$ iar $f_n$ numere aleatoare distribuite dupa componenta $\exp (-\beta \hbar \omega n)$ din (7.2.20). In reprezentarea pozitiei (7.2.21) devine:

$$\frac{\partial \psi _{\pm }}{\partial \tau }-i\cdot \frac{\partia... ...n (\omega _n\cdot \tau +\phi _n)\zeta \psi _{\pm }\left( \zeta ,\tau \right) =0$$ (7.2.22)

$$-\eta \cdot \sum_nf_n\cdot \sin (\omega _n\cdot \tau +\phi _n)\cdot \zeta \psi _{\pm }\left( \zeta ,\tau \right) =0$$

Forma ideala a (7.2.22) ($\eta =0$) este transformata Fourier a ecuatiei Airy. Daca

$$G_{\pm }(\zeta ,\zeta ^{\prime };\tau )=\frac 1{2\sqrt{\pi i\tau ... ...ilon ^3}{12}\mp \frac{\tau _\epsilon }2(\zeta -\zeta ^{\prime })\right) \right)$$ (7.2.23)

este functia Green a acestei ecuatii ( $\tau _\epsilon =\tau -i\epsilon$, $\epsilon >0$ pentru asigurarea convergentei), i.e.:

$$\psi _{\pm }\left( \zeta ,\tau \right) =\int d\zeta ^{\prime }\cd... ...zeta ,\zeta ^{\prime };\tau )\cdot \psi _{\pm }\left( \zeta ^{\prime },0\right)$$ (7.2.24)

se poate rezolva (7.2.22) folosind o metoda iterativa, cu conditia ca $\eta$ sa fie suficient de mic:

$$\psi _{\pm }\left( \zeta ,\tau \right) =\sum_m\psi _{\pm }^{\left( m\right) }\left( \zeta ,\tau \right) \cdot \eta ^m$$ (7.2.25)

$$\frac{\partial \psi _{\pm }^{\left( m\right) }}{\partial \tau }-i... ...i\cdot \zeta \cdot \psi _{\pm }^{\left( m\right) }\left( \zeta ,\tau \right) =$$

$$=\sum_nf_n\exp \left( i\cdot (\omega _n\cdot \tau +\phi _n)\right) \cdot \zeta \psi _{\pm }^{\left( m-1\right) }\left( \zeta ,\tau \right)$$ (7.2.26)

Folosind (7.2.23) scriem solutia iterata:

$$\psi _{\pm }^{\left( m\right) }\left( \zeta ,\tau \right) =\stack... ... ^{\prime }\cdot G_{\pm }(\zeta ,\zeta ^{\prime };\tau -\tau ^{\prime })\times$$

$$\times \sum_nf_n\cdot \exp \left( i\cdot (\omega _n\cdot \tau ^{\... ...si _{\pm }^{\left( m-1\right) }\left( \zeta ^{\prime },\tau ^{\prime }\right) =$$ (7.2.27)

$$=\stackrel{\tau }{\int_0}d\tau ^{\prime }\cdot \sum_nf_n\cdot \ex... ...psi _{\pm }^{\left( m-1\right) }\left( \zeta ^{\prime },\tau ^{\prime }\right)$$

Folosind solutia ideala (7.2.16) ca intrare in metoda iterativa se poate obtine solutia (7.2.25).