How is the temperature of the center of mass of a trapped particle (e.g. in a Paul or Penning trap with laser cooling) defined? I assume it has something to do with the equipartition theorem and Brownian motion, but there are so many "temperatures" that I'm having doubts of the standard temperature measurements in this case.
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Strictly speaking, the thermodynamic temperature is only defined when the particle is in thermodynamic equilibrium, where the quantum state is of the Gibbs form $$ \rho = \frac{e^{-H/k_B T}}{\mathrm{Tr} \left(e^{-H/k_BT}\right)}, $$ where $H$ is the Hamiltonian and $T$ is the temperature. However, generally a cold trapped atom will not be in thermodynamic equilibrium, so that the temperature is not strictly defined.
Usually in cold atom experiments people want to know how many vibrational excitations there are on average (often denoted by $\bar{n}$), since this is relatively easy to measure. When the atom is close to the potential minimum of a deep trap, the oscillations can be treated as approximately harmonic. In that case, the average number of vibrational excitations at thermal equilibrium is given by the Bose-Einstein distribution $$ \bar{n}_\mathrm{eq} = \left ( e^{\hbar \omega/k_B T} - 1 \right)^{-1}, $$ where $\omega/2\pi$ is the frequency of harmonic oscillations. In the limit of high temperature this reduces to the equipartition expression $$ E = \hbar\omega\bar{n}_\mathrm{eq} \approx k_B T,$$ where $E$ is the average energy.
Given some experimentally determined $\bar{n}$, one can determine an effective temperature according to the above formulae, whether or not the system is truly at equilibrium. However, in practice my friends doing trapped ion experiments seem much more interested in the quantity $\bar{n}$.
Mark Mitchison
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