Spectrum of Two-Level Systems with Discrete Frequency Fluctuations

We study, theoretically and experimentally, an ensemble of two-level systems coupled to an environment which induces random jumps in their resonant frequency. We present a closed-form formula for the spectrum in terms of the resonant frequency distribution and the Poisson rate constant. For a normal distribution the spectrum deviates from a generalized Gumbel function, a well-known result for continuous stochastic Gaussian processes. We perform experiments with optically trapped cold ${}^{87}\mathrm{Rb}$ atoms and show that the predictions of our theory for a 3D harmonic trap match the measured spectra without fitting parameters.

Figure 1

The spectrum of an ensemble with a Gaussian detuning distribution as calculated by the solution of the discrete fluctuation model given in Eq. (2) [solid (black) line], and by the Kubo’s solution for a Gaussian process given in Eq. (3) [dashed (red) line]. The markers are the results of simulations without correlations between subsequent detunings [(black) squares and upper part of the schematics], and with correlations [(red) circles and lower part of the schematics].

Figure 2

A plot of the decay of coherence in the absence of fluctuations, $R_0(t/\tau )$, for a 3D harmonic trap [solid (black) line] compared to the decay with fluctuations, $R(t/\tau )$, with $\Gamma /\tau =6$ calculated by the solution of the discrete model for a 3D harmonic trap [dashed-dotted (blue) line] and for a Gaussian detuning distribution [dashed (red) line]. The inset shows the corresponding spectrums. Also plotted is the result of a Monte Carlo simulation with the same $\Gamma$ and $\tau$ [solid grey (green) line]. The simulations were done with 4000 atoms colliding at an average rate ${\Gamma }_{\mathrm{col}}=2.7\Gamma$ [18].

Figure 3

(a) A schematics of the experimental setup. (b) Experimental level scheme.

Figure 4

The envelopes, $R(t)$, of two Ramsey experiments with the same inhomogeneous time scale $\tau =42\mathrm{ms}$ and different $\Gamma$ [solid (black) lines], and theoretical calculations using Eqs. (4, 2) without fitting parameters [dashed (red) line]. The inset shows the decay of atomic-cloud waist oscillations fitted by an exponentially decaying cosine $e^{-\Gamma t}\cos (2\pi f_{0}t)+c$, with $\Gamma =130s^{-1}$, and $f_0$, which agrees well with the value measured by parametric excitation $f_0={\omega }_r/\pi$.