Tests for nonrandomness in quantum jumps

In a fundamental test of quantum mechanics, we have observed $\mathrm{228\hspace{0.17em}000}$ quantum jumps of a single trapped and laser cooled ${}^{88}\mathrm{Sr}^{+}$ ion. This represents a statistical increase of two orders of magnitude over previous similar analyses of quantum jumps. Compared to other searches for nonrandomness in quantum-mechanical processes, using quantum jumps simplifies the interpretation of data by eliminated multiparticle effects and providing near-unit detection efficiency of transitions. We measure the fractional reduction in the entropy of information to be $<6.5\times {10}^{-4}$ when the value of any interval between quantum jumps is known. We also find that the number of runs of successively increasing or decreasing interval times agrees with the theoretically expected values. Furthermore, we analyze $\mathrm{238\hspace{0.17em}000}$ quantum jumps from two simultaneously confined ions and find that the number of apparently coincidental transitions is as expected. Finally, we observe 8400 spontaneous decays of two simultaneously trapped ions and find that the number of apparently coincidental decays from the metastable state agrees with the expected value. We find no evidence for short- or long-term correlations in the intervals of the quantum jumps or in the decay of the quantum states, in agreement with quantum theory.

Figure 1

Partial energy-level diagram, transitions, and lifetimes in ${}^{88}\mathrm{Sr}^{+}$. We drive the $422\text{−}\text{nm}$ transition to Doppler cool and detect the ions, and the $1092\text{−}\text{nm}$ transition to prevent optical pumping into the $D_{3∕2}$ state. A laser of bandwidth $<10\text{kHz}$ drives the $674\text{−}\text{nm}$ transition to produce quantum jumps to and from $D_{5∕2}$ states.

Figure 2

Number of $422\text{−}\text{nm}$ photons counted by the detector over the time when all of the transitions shown in Fig. 1 are driven simultaneously.

Figure 3

The joint functions $U\left(x\mid y\right)$ (solid lines) and $U\left(y\mid x\right)$ (dashed lines) from Eqs. (5, 6) as functions of the number of intervals between pairs for bright and dark intervals and intervals between emissions and absorptions of 674-nm photons. Heavy dashed lines indicate the average values for each interval type.

Figure 4

The average values of $U\left(x\mid y\right)$ (solid boxes) and $U\left(y\mid x\right)$ (hollow boxes) from Fig. 3.

Figure 5

Differences between the expected and measured values of the number of runs up (squares) and down (triangles) for the four interval types. The values of the number of runs of length $\ell ⩾6$ have been magnified, and the runs up and down have been slightly offset along the $x$ axis for clarity. In total, each data set has $\sim \mathrm{57\hspace{0.17em}000}$ runs of various lengths both up and down.