Experimental test of Bohr's complementarity principle with single neutral atoms

An experimental test of the quantum complementarity principle based on single neutral atoms trapped in a blue detuned bottle trap was here performed. A Ramsey interferometer was used to assess the wavelike behavior or particlelike behavior with second $\pi /2$ rotation on or off. The wavelike behavior or particlelike behavior is characterized by the visibility $V$ of the interference or the predictability $P$ of which-path information, respectively. The measured results fulfill the complementarity relation $P^2+V^2\le 1$. Imbalance losses were deliberately introduced to the system and we find the complementarity relation is then formally “violated.” All the experimental results can be completely explained theoretically by quantum mechanics without considering the interference between wave and particle behaviors. This observation complements existing information concerning Bohr's complementarity principle based on wave-particle duality of a massive quantum system.

Figure 1

(a) Schematic of BCP test with single photon in the conventional MZI. A single-photon state was transformed into a superposition of two paths, 1 and 2, in space by the first beam splitter BS1. The phase shift $\theta$ of two arms was tuned by scanning the voltage of the piezoelectric transducer (PZT). The presence or absence of the second beam splitter BS2 helped us observe the wavelike or particlelike behaviors of single photons. Two detectors, APD1 and APD2, were used to count photons from two output ports. (b) Schematic of the Ramsey interferometer system. A single atom state was split by the first microwave pulse MW1 into superposition states of $\left|0\right.〉$ and $\left|1\right.〉$, which were similar to the two paths and the state evolution in the time domain. After a certain time, the second microwave pulse MW2 was present (or absent) and the final state of the atom could be detected by regular probe-fluorescence measurement.

Figure 2

(a) Typical signal from single atoms: fluorescence recorded by the single-photon counting module (inset) and the corresponding histogram. (b) The average lifetime of a single atom in the blue trap was about 78 s, and the internal state lifetime for the trapped atoms in $\left|0\right.〉$ and $\left|1\right.〉$ was about 1.4 s (inset).

Figure 3

Test of BCP with single cesium atom. (a) Schematic of BCP relation test with single atom Ramsey interferometer. A single atom initially in state $\left|1\right.〉$ was subjected to two rotation operations created by a microwave pulse analog of the beam splitters in the usual MZI. In the presence of the second microwave pulse, the probability of measuring the qubit in a certain state depends on the relative phase $\theta$, thus exhibiting wavelike behavior, while in the absence of the pulse the probability is independent of $\theta$ and particlelike behavior appears. (b) Wavelike information $V^2$, which-path information $P^2$ and $P^2+V^2$ as functions of $R_1$, which is determined by the length of the first microwave pulse. The red dots, black squares, and blue diamonds stand for the measured values of $V^2,P^2$, and $P^2+V^2$ without losses, and the red solid, black dashed, and blue dotted lines are theoretical fittings according to Eqs. (2, 3, 4).

Figure 4

The results of BCP relation test when losses were introduced. (a) The results for losses occurring inside the interferometer. (b) The results for losses occurring outside the interferometer. The red dots, black squares, and blue diamonds stand for the measured values of $V^2,P^2$, and $P^2+V^2$ with losses ($L_1=0,L_2=0.5$) in path 1 and path 2, respectively. The red solid, black dashed, and blue dotted lines are theoretical fittings according to Eqs. (5)–(10).

Figure 5

Test of BCP relation of the second configuration. (a) Schematic of BCP relation test with Ramsey interferometer where the second pulse was varied. (b) Experiment results without losses. (c,d) Experiment results with losses occurring inside (c) and outside (d) the Ramsey interferometer. The red dots, black squares, and blue diamonds stand for the measured values of $V^2,P^2$, and $P^2+V^2$ and the red solid, black dashed, and blue dotted lines are theoretical fittings according to Eqs. (11, 12, 13, 14, 15, 16).

Figure 6

Elimination of the influence of the imbalance loss for the configuration with loss outside the Ramsey interferometer [Fig. 4]. The violation is eliminated by switching the input state and then averaging the results. The red dots, black squares, and blue diamonds stand for the measured values of $V^2,P^2$, and $P^2+V^2$, respectively, and the red solid, black dashed, and blue dotted lines are theoretical fittings according to Eqs. (2, 3, 4).