Complementarity stands at the heart of quantum mechanics(QM). It is especially important when describing interferometers. Clocks are a key tool for interrogating the effects of general relativity(GR) such as gravitational time dilation. The interface of QM and GR is still poorly understood. Here we prepare a clock interferometer, whereby a single clock traverses a spatial Mach-Zehnder type interferometer. The clock is thus in a spatial superposition of two distinct wave packets[1]. By inducing an effective time-lag difference, we theoretically derive and experimentally test a new complementarity relation for quantum clocks in the context of the gravitational red-shift[2]:

V^2 + (C · D)^2 ≤ 1. V is the inteferometric visibility, and D is the distinguishability arising from the time-lag difference. C indicates the clock’s capability of creating distinguishability and is determined by the internal population of clock spin states. We study in detail this complementary relation by independently measuring V , C and D and prove that the quantum clock complementarity rule is sound. These results demonstrate a simplified and efficient platform for studying the fundamental physics at the intersection of QM and GR, which may be applied, for example, to discussions about the Compton clock and gravitationally-induced decoherence.

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[1] Yair Margalit, Zhifan Zhou, Shimon Machluf, Daniel Rohrlich, Yonathan Japha and Ron Folman, A self-interfering clock as a “which path” witness, Science 349, 1205 (2015).

[2] Zhifan Zhou,Yair Margalit, Daniel Rohrlich, Yonathan Japha and Ron Folman, Quantum complementarity of clocks in the context of general relativity, Class. Quantum Grav. 35, 185003 (2018).