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arXiv:1405.6016 · Physical Review Letters

Theory of Single-Shot Phase Contrast Imaging in Spinor Bose-Einstein Condensates

Ebubechukwu O. Ilo-Okeke, Tim Byrnes

Published Jun 13, 2014• 44 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.112.233602

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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 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.
Nondestructive imaging of spinor Bose-Einstein condensates (BEC) presents new opportunities for observing and manipulating macroscopic quantum phenomena in real time. In this talk I theoretically study harmonically trapped two-component BECs subject to weak measurement via phase contrast imaging. I discuss the application of weak measurement theory to spatially extended condensates and present a stochastic Gross-Pitaevskii equation for the condensate order parameter. I then examine the stability of spin-polarized and spin-unpolarized phases of the condensate subject to continuous weak measurement. Additionally, I demonstrate how weak measurement can be used to track the position of a domain wall between spin-polarized regions. I quantify the effects of measurement backaction on the stochastic dynamics of the domain wall and discuss how domain wall dynamics could be observed in experiment.
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.
Phase contrast imaging is used to observe Bose-Einstein condensates (BECs) at finite temperature in situ. The imaging technique is used to accurately derive the absolute phase shift of a probe laser beam due to both the condensate and the thermal cloud. The accuracy of the method is enhanced by using the periodicity of the intensity signal as a function of the accumulated phase. The measured density profiles can be described using a two relevant parameter fit, in which only the chemical potential and the temperature are to be determined. This allows us to directly compare the measured density profiles to different mean-field models in which the interaction between the condensed and thermal atoms is taken into account to various degrees.
We develop theory for fluctuations in atom number and spin within finite-sized cells of a spinor Bose-Einstein condensate. This theory provides a model of measurements that can be performed in current experiments using finite resolution in situ imaging. We develop analytic results for quantum and thermodynamic limits of the fluctuations and apply our theory to the four equilibrium phases of a spin-1 condensate. We then validate these limits and examine the behavior over a wide parameter regime using numerical calculations specialized to the case of a spinor condensate confined to be quasi-two-dimensional.