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arXiv:1606.08466 · Physical Review A

Information and backaction due to phase-contrast-imaging measurements of cold atomic gases: Beyond Gaussian states

Ebubechukwu O. Ilo-Okeke, Tim Byrnes

Published Jul 25, 2016• 15 citations
arXivHTMLPDF

Abstract

We further examine a theory of phase contrast imaging (PCI) of cold atomic gases, first introduced by us in Phys. Rev. Lett. 112, 233602 (2014). We model the PCI measurement by directly calculating the entangled state between the light and the atoms due to the ac Stark shift, which induces a conditional phase shift on the light depending upon the atomic state. By interfering the light that passes through the BEC with the original light, one can obtain information of the atomic state at a single-shot level. We derive an exact expression for a measurement operator that embodies the information obtained from PCI, as well as the backaction on the atomic state. By the use of exact expressions for the measurement process, we go beyond the continuous variables approximation such that the non-Gaussian regime can be accessed for both the measured state and the postmeasurement state. Features such as the photon probability density, signal, signal variance, Fisher information, error of the measurement, and the backaction are calculated by applying the measurement operator to an atomic two-spin-state system. For an atomic state that is initially in a spin-coherent state, we obtain analytical expression for these quantities. There is an optimal atom-light interaction time that scales inversely proportional to the number of atoms, which maximizes the information readout.

DOI
10.1103/physreva.94.013617

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We further examine a theory of phase contrast imaging (PCI) of cold atomic gases, first introduced by us in Phys. Rev. Lett. 112, 233602 (2014). We model the PCI measurement by directly calculating the entangled state between the light and the atoms due to the ac Stark shift, which induces a conditional phase shift on the light depending upon the atomic state. By interfering the light that passes through the BEC with the original light, one can obtain information of the atomic state at a single-shot level. We derive an exact expression for a measurement operator that embodies the information obtained from PCI, as well as the backaction on the atomic state. By the use of exact expressions for the measurement process, we go beyond the continuous variables approximation such that the non-Gaussian regime can be accessed for both the measured state and the postmeasurement state. Features such as the photon probability density, signal, signal variance, Fisher information, error of the measurement, and the backaction are calculated by applying the measurement operator to an atomic two-spin-state system. For an atomic state that is initially in a spin-coherent state, we obtain analytical expression for these quantities. There is an optimal atom-light interaction time that scales inversely proportional to the number of atoms, which maximizes the information readout.
We introduce a theoretical framework for single-shot phase contrast imaging (PCI) measurements of spinor Bose-Einstein condensates (BECs). Our model allows for the simple calculation of the quantum backaction resulting from the measurement, and the amount of information that is read out. We find that there is an optimum time Gτ ∼ 1/N for the light-matter interaction (G is the ac Stark shift frequency, N is the number of particles in the BEC), where the maximum amount of information can be read out from the BEC. A universal information-disturbance tradeoff law ε(F)ε(G) ∝ 1/N(2) is found where ε(F) is the amount of backaction and ε(G) is the estimation error. The PCI measurement can also be found to be a direct probe of the quantum fluctuations of the BEC, via the noise of the PCI signal.
Phase Contrast Imaging (PCI) is a non-destructive method of probing the state of Bose-Einstein Condensates (BECs). In a PCI measurement, an off-resonant laser beam is illuminated on a BEC and a homodyne measurement is performed on the light which passes through the BEC. The interaction between laser beam and the atomic BEC encodes information about the atomic BEC on the phase-shift accumulated by laser beam. Past theories of PCI measurement have focused on continuous measurement, and treat the problem by deriving a master equation which causes an effective measurement induced decoherence. This allows for calculation of the backaction on the BEC, but not the information that is gained by the measurement. We introduce a simple theory to capture the physics of PCI measurement, which allows quantifying the information-disturbance trade-off, and works in the single shot regime. We show a universal scaling law for the information-disturbance trade-off that is valid for a wide range of particle numbers in the BEC.
We demonstrate that phase-contrast imaging (PCI) can reliably reconstruct the in situ density profile even for highly spatially and optically dense samples. In our experiment, we achieve high spatial densities of up to 7.9 × 1013 atoms/cm3 and optical depths up to 64 in a dense cold atomic cloud of 88Sr atoms. The use of a spatial light modulator instead of a fixed phase plate in the PCI setup provides enhanced flexibility and control of imaging parameters, making this imaging technique robust against imaging artifacts and adaptable to varying experimental conditions. Moreover, we quantify the conditions under which the standard single-atom polarizability model remains accurate for PCI in a density regime where collective effects should modify the atomic response of the system. We experimentally validate our statements by showing excellent agreement with time-of-flight measurements even at the highest densities. Our results establish PCI as a reliable and versatile method for characterizing spatially dense atomic clouds.
Measurement plays a crucial role in a quantum system beyond just learning about the system state: it changes the post-measurement state and hence influences the subsequent time evolution; further, measurement can even create entanglement in the post-measurement conditional state. In this work, we study how careful choice of parameters for a typical measurement process on cold atoms systems -- phase contrast imaging -- has a strong impact on both what the experimentalist observes but also on the backaction the measurement has on the system, including the creation and diffusion of quasiparticles emerging from the quantum many-body dynamics. We focus on the case of a Bose-Einstein-condensate array, in the low-temperature and low-momentum limit. Our theoretical investigation reveals regimes where the imaging light probes either the bare particle or quasiparticle dynamics. Moreover, we find a path to selectively measuring quasiparticle modes directly, as well as controlling over the measurement-induced creation and diffusion of quasiparticles into different momentum states. This lays a foundation for understanding the effects of both experimental approaches for probing many-body systems, but also more speculative directions such as observable consequences of `spontaneous collapse' predictions from novel models of quantum gravity on aspects of the Standard Model.
The linewidth of an atom laser is limited by density fluctuations in the Bose-Einstein condensate (BEC) from which the atom laser beam is outcoupled. In this paper we show that a stable spatial mode for an interacting BEC can be generated using a realistic control scheme that includes the effects of the measurement backaction. This model extends the feedback theory, based on a phase-contrast imaging setup, presented by Szigeti, Hush, Carvalho, and Hope [Phys. Rev. A 80, 013614 (2009)]. In particular, it is applicable to a BEC with large interatomic interactions and solves the problem of inadequacy of the mean-field (coherent state) approximation by utilizing a fixed number state approximation. Our numerical analysis shows the control to be more effective for a condensate with a large nonlinearity.
We present a detailed theoretical description of a recently proposed atomic scanning microscope in a cavity QED setup [D. Yang et al., Phys. Rev. Lett. 120, 133601 (2018)]. The microscope continuously observes atomic densities with optical subwavelength resolution in a nondestructive way. The superresolution is achieved by engineering an internal atomic dark state with a sharp spatial variation of population of a ground level dispersively coupled to the cavity field. Thus, the atomic position encoded in the internal state is revealed as a phase shift of the light reflected from the cavity in a homodyne experiment. Our theoretical description of the microscope operation is based on the stochastic master equation describing the conditional time evolution of the atomic system under continuous observation as a competition between dynamics induced by the Hamiltonian of the system, decoherence effects due to atomic spontaneous decay, and the measurement backaction. Within our approach we relate the observed homodyne current with a local atomic density and discuss the emergence of a quantum nondemolition measurement regime allowing continuous observation of spatial densities of quantum motional eigenstates without measurement backaction in a single experimental run.