Quantum mechanics allows events to happen with no definite causal order: this can be verified by measuring a causal witness, in the same way that an entanglement witness verifies entanglement. Here, we realize a photonic quantum switch, where two operations andB act in a quantum superposition of their two possible orders. The operations are on the transverse spatial mode of the photons; polarization coherently controls their order. Our implementation ensures that the operations cannot be distinguished by spatial or temporal position-further it allows qudit encoding in the target. We confirm our quantum switch has no definite causal order by constructing a causal witness and measuring its value to be 18 standard deviations beyond the definite-order bound. DOI: 10.1103/PhysRevLett.121.090503 In daily experience, it is natural to think of events happening in a fixed causal order. Strikingly, it has been proposed that quantum physics allows for nonclassical causal structures where the order of events is indefinite [1,2]. It has been theoretically shown that such a possibility provides an advantage for computation [3], communication complexity [4,5], and other information processing tasks [6][7][8]. Furthermore, investigations of indefinite causal orders suggest a promising route towards a theory that combines general relativity and quantum mechanics [9,10].Indefinite causal orders can be studied using a framework that distinguishes whether some experimental situationcalled a "process"-is compatible with a fixed causal order of the events or not. An example of a process with indefinite causal order is the "quantum switch" [1]. In the quantum switch, the order in which two quantum operation andBconsidered as "black box operations"-are performed on a target system is coherently controlled by a control quantum system (Fig. 1). This can also be seen as a particular case of "superposition of time evolution" [11]. The advantages provided by the quantum switch arise from the fact that it cannot be reproduced by an ordinary quantum circuit which uses the same number of black box operations [3][4][5][6][7].Here, we present an optical implementation of the quantum switch where the control system is the photon's polarization and the target is the transverse spatial mode. We verify indefinite causal order by introducing a causal witness [14,15], for which we obtain a value 18 standard deviations beyond the bound for definite ordering. One notable achievement of our experiment is that it opens the possibility of encoding more than two levels in the target system-transverse spatial mode can indeed be highdimensional and hence can act as a qudit.In previous implementations [12,13], the location of each black box-the spot where photons go through a set of wave plates-was different depending on the order, resulting in four distinct locations in space [ Fig. 1(c)]. Furthermore, the photons had a coherence length much shorter than the distance between the two sets of wave plates: in effect, the operations could also be distinct in...

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We describe a dispersive Faraday optical probe of atomic spin which performs a weak measurement of spin projection of a quantum gas continuously for more than one second. To date focusing bright far-off-resonance probes onto quantum gases has proved invasive, due to strong scalar and vector light shifts exerting dipole and Stern-Gerlach forces. We show that tuning the probe near the magiczero wavelength at 790 nm between the fine-structure doublet of 87 Rb cancels the scalar light shift, and careful control of polarization eliminates the vector light shift. Faraday rotations due to each fine-structure line reinforce at this wavelength, enhancing the signal-to-noise ratio for a fixed rate of probe-induced decoherence. Using this minimally-invasive spin probe we perform microscale atomic magnetometry at high temporal resolution. Spectrogram analysis of the Larmor precession signal of a single spinor Bose-Einstein condensate measures a time-varying magnetic field strength with 1 µG accuracy every 5 ms; or equivalently makes > 200 successive measurements each at 10 pT/ √ Hz sensitivity.

We continuously observe dynamical decoupling in a spin-1 quantum gas using a weak optical measurement of spin precession. Continuous dynamical decoupling aims to dramatically modify the character and energy spectrum of spin states to render them insensitive to parasitic fluctuations. Continuous observation measures this new spectrum in a single-preparation of the quantum gas. The measured time-series contains seven tones, which spectrogram analysis parses as splittings, coherences, and coupling strengths between the decoupled states in real-time. With this we locate a regime where a transition between two states is decoupled from magnetic field instabilities up to fourth order, complementary to the parallel work at higher fields by Trypogeorgos et al. (arXiv:1706.07876). The decoupled microscale quantum gas offers magnetic sensitivity in a tunable band, persistent over many milliseconds: the length scales, frequencies, and durations relevant to many applications, including sensing biomagnetic phenomena such as neural spike trains.From Hahn echoes to dynamical decoupling, pulse sequences have been used to protect spin superpositions from inhomogeneities and parasitic fluctuations, prolonging quantum coherence and circumventing deleterious energy shifts [1][2][3]. A complementary strategy is continuous dynamical decoupling: replacing the pulse sequence with an uninterrupted coupling of bare spin states yielding dressed spin eigenstates with new quantization direction, spectrum, and coupling. The new spectrum protects against low-frequency fluctuations, while the new couplings admit band-tunable sensing [4]. Continuous dynamical decoupling has been applied to nitrogen-vacancy centers [5][6][7][8] Here we demonstrate how a weak continuous measurement of spin precession can probe the spectrum of a continuously decoupled spin-1 quantum gas in a single experimental preparation ('shot'). Time-resolved Fourier spectroscopy of this measurement record reveals not only all dressed-state splittings and their relative immunity to noise but also dressed state coherences and coupling strengths. Estimating the eigenspectrum of the multi-level dressed system brings into view a higherorder decoupling than exists in dynamically-decoupled two-level systems. In this regime, one transition is only quartically-sensitive to noise, surviving much larger amplitude fluctuations than conventional quadratic decoupling. Further, a new transition arises between states which are otherwise uncoupled, completing a cyclic coupling of all dressed states. This low-frequency magnetic stability combined with continuous detection is immediately applicable to band-tunable magnetometry [5,6,12] and experiments preparing delicate spin-entangled manybody states [13]; whereas the unconventional cyclic coupling could be applied to emulation of frustrated quantum spin chains [14].Atomic Zeeman states |m z = −1, 0, 1 in a magnetic field B z e z can be decoupled from fluctuations in B z by applying a perpendicular radiofrequency (rf) field B rf e x cos ω rf t...

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In this article we present a full description of the quantum Kerr Ising model-a linear optical network of parametrically pumped Kerr non-linearities. We consider the non-dissapative Kerr Ising model and, using variational techniques, show that the energy spectrum is primarily determined by the adjacency matrix in the Ising model and exhibits highly non-classical cat like eigenstates. We then introduce dissipation to give a quantum mechanical treatment of the measurement process based on homodyne detection via the conditional stochastic Schrodinger equation. Finally we identify a quantum advantage in comparison to the classical analogue for the example of two anti-ferromagnetic cavities.

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