Editing Fluctuation–dissipation theorem (section)

=== Quantum version===

The fluctuation-dissipation theorem relates  the [[correlation function]] of the observable of interest <math>\langle \hat{x}(t)\hat{x}(0)\rangle</math> (a measure of fluctuation) to the imaginary part of the [[response function]] <math>\text{Im}\left[\chi(\omega)\right]=\left[\chi(\omega)-\chi^*(\omega)\right]/2i</math> in the frequency domain (a measure of dissipation).  A link between these quantities can be found through the so-called [[Kubo formula]]<ref>{{cite journal |title=The fluctuation-dissipation theorem |doi=10.1088/0034-4885/29/1/306 | volume=29 |year=1966 |journal=Reports on Progress in Physics |pages=255–284 |author=Kubo R |issue=1 |bibcode=1966RPPh...29..255K|s2cid=250892844 }}</ref>

<math display="block">\chi(t-t')=\frac{i}{\hbar}\theta(t-t')\langle [\hat{x}(t),\hat{x}(t')] \rangle</math>

which follows, under the assumptions of the [[Linear response function|linear response]] theory, from the time evolution of the [[ensemble average]] of the observable <math>\langle\hat{x}(t)\rangle</math> in the presence of a perturbing source. Once Fourier transformed, the Kubo formula allows writing the imaginary part of the response function as

<math display="block">\text{Im}\left[\chi(\omega)\right] = \frac{1}{2\hbar} \int_{-\infty}^{+\infty}\langle \hat{x}(t)\hat{x}(0) - \hat{x}(0)\hat{x}(t)\rangle e^{i\omega t}dt.</math>

In the [[canonical ensemble]], the second term can be re-expressed as

<math display="block">\langle \hat{x}(0) \hat{x}(t)\rangle = \operatorname{Tr} e^{-\beta \hat{H}}\hat{x}(0)\hat{x}(t) = \operatorname{Tr} \hat{x}(t) e^{-\beta \hat{H}}\hat{x}(0) = \operatorname{Tr} e^{-\beta \hat{H}}\underbrace{e^{\beta \hat{H}}\hat{x}(t) e^{-\beta \hat{H}}}_{\hat{x}(t-i\hbar\beta)}\hat{x}(0)=\langle \hat{x}(t-i\hbar\beta) \hat{x}(0)\rangle</math>

where in the second equality we re-positioned <math>\hat{x}(t)</math> using the cyclic property of trace. Next, in the third equality, we inserted <math>e^{-\beta \hat{H}}e^{\beta \hat{H}}</math> next to the trace and interpreted <math>e^{-\beta\hat{H}}</math> as a time evolution operator <math>e^{-\frac{i}{\hbar}\hat{H}\Delta t}</math>  with [[imaginary time]] interval <math>\Delta t = -i\hbar\beta</math>. The imaginary time shift turns into a <math>e^{-\beta\hbar\omega}</math> factor after Fourier transform

<math display="block">\int_{-\infty}^{+\infty}\langle \hat{x}(t-i\hbar\beta)\hat{x}(0)\rangle e^{i\omega t}dt = e^{-\beta\hbar\omega}\int_{-\infty}^{+\infty} \langle \hat{x}(t)\hat{x}(0)\rangle e^{i\omega t}dt</math>

and thus the expression for <math>\text{Im}\left[\chi(\omega)\right]</math> can be easily rewritten as the quantum fluctuation-dissipation relation <ref>{{cite journal |title=Fundamental aspects of quantum Brownian motion |doi=10.1063/1.1853631 | volume=15 |year=2005 |journal=Chaos: An Interdisciplinary Journal of Nonlinear Science |page=026105 |author=Hänggi Peter, Ingold Gert-Ludwig|issue=2 |pmid=16035907 |arxiv=quant-ph/0412052 |bibcode=2005Chaos..15b6105H |s2cid=9787833 |url=https://nbn-resolving.org/urn:nbn:de:bvb:384-opus4-301764 }}</ref>

<math display="block">S_{x}(\omega)=2\hbar\left[n_\text{BE}(\omega)+1\right]\text{Im}\left[\chi(\omega)\right]</math>

where the power spectral density <math>S_{x}(\omega)</math> is the Fourier transform of the auto-correlation <math>\langle \hat{x}(t) \hat{x}(0)\rangle</math> and <math>n_\text{BE}(\omega)=\left(e^{\beta\hbar\omega}-1\right)^{-1}</math> is the [[Bose-Einstein Statistic|Bose-Einstein]] distribution function. The same calculation also yields

<math display="block">S_{x}(-\omega) = e^{-\beta\hbar\omega}S_{x}(\omega) = 2\hbar\left[n_\text{BE}(\omega)\right] \text{Im}\left[\chi(\omega)\right]\neq S_{x}(+\omega)</math>

thus, differently from what obtained in the classical case, the power spectral density is not exactly frequency-symmetric in the quantum limit. Consistently, <math>\langle \hat{x}(t)\hat{x}(0)\rangle</math> has an imaginary part originating from the commutation rules of operators.<ref>{{cite journal |title=Introduction to Quantum Noise, Measurement and Amplification |doi=10.1103/RevModPhys.82.1155 |volume=82 |year=2010 |journal= Reviews of Modern Physics|page=1155 |arxiv=0810.4729 |last1=Clerk |first1=A. A. |last2=Devoret |first2=M. H. |last3=Girvin |first3=S. M. |last4=Marquardt |first4=Florian |last5=Schoelkopf |first5=R. J. |issue=2 |bibcode=2010RvMP...82.1155C |s2cid=119200464 }}</ref> The additional "<math>+1</math>" term in the expression of <math>S_x(\omega)</math> at positive frequencies can also be thought of as linked to [[spontaneous emission]]. An often cited result is also the symmetrized power spectral density

<math display="block">\frac{S_x(\omega)+S_x(-\omega)}{2} = 2\hbar\left[n_\text{BE}(\omega)+\frac{1}{2}\right] \text{Im}\left[\chi(\omega)\right] = \hbar\coth\left(\frac{\hbar\omega}{2k_BT}\right)\text{Im}\left[\chi(\omega)\right].</math>

The "<math>+1/2</math>" can be thought of as linked to [[quantum fluctuation]]s, or to [[Zero-point energy|zero-point motion]] of the observable <math>\hat{x}</math>. At high enough temperatures, <math>n_\text{BE}\approx (\beta\hbar\omega)^{-1}\gg 1</math>, i.e. the quantum contribution is negligible, and we recover the classical version.