Phase-space microscopes for quantum gases: Measuring conjugate variables and momentum-weighted densities

Abstract

Quantum gas microscopes offer unprecedented insights into quantum many-body states of cold atomic gases. Here we introduce concrete protocols for extending quantum gas microscopes to measure in phase space, by mapping momentum onto auxiliary degrees of freedom and using positive operator-valued measures. We distinguish between two distinct operational modes. In the Husimi-Q phase space microscope, position and momentum are jointly measured; in this mode the fundamental quantum noise appears in the position measurement. Conversely, the averaged-mode phase space microscope extracts the spatial dependence of averages of the momentum density (and its moments); these averages can be retrieved with arbitrary spatial resolution. We illustrate the utility of these techniques in diverse physical settings.

Figures (4)

• Figure 1: Sketch of the protocol for mapping momentum to an auxiliary dimension to realize a 1D Husimi-Q phase-space microscope. (a) Two atoms with initial positions $x_1$ and $x_2$ and momenta $p_1$ and $p_2$ in the physical $x$ dimension. (b) A $T/4$ pulse is applied to move to the Fourier plane in the $x$ direction: the horizontal position of a particle now encodes its initial momentum so is labelled by $p_{1,2}$. (c) A spatially dependent impulse maps this momentum to a momentum in the auxiliary $z$ dimension. (d) A second $T/4$ pulse along both directions creates an image in the $x$ direction and maps the acquired $z$ momentum to a $z$ displacement. The protocol gives a choice of how to distribute uncertainty between $x$ and $p_x$.
• Figure 2: Sketches of protocols to map moments of the momentum density to an auxiliary spin state in averaged-mode phase-space microscopes. These are applied in the Fourier plane, so the position is labelled by momentum ${\bm p}$. (a) Spin-rotation-(i) transforms the initial $|S,S\rangle$ spin state (aligned vectors of lower layer) using a resonant Rabi pulse that circulates as $\vec{\Omega}\propto (-p_y,p_x,0)$ (orange central layer) to map the 2D momentum space onto the Bloch sphere (vectors of upper layer). (b) Spin-rotation-(ii) uses a uniform Rabi pulse $\vec{\Omega}\propto (1,0,0)$ in the presence of a quadratically varying Zeeman shift to imprint $|{\bm p}|^2$ in $S_z$. In both cases, the measured spin encodes moments of the momentum distribution.
• Figure 3: Measurement of a sharp step in the potential, with edge thickness $\delta$ mimicked by a state $\psi_{\rm edge}(x) = [1-\tanh(x/\delta)]/2$. (a) Density distribution as measured by a conventional QGM with point spread function $\propto {\rm e}^{-x^2/w^2}$. (b) Distribution of particles with momentum $p_{\rm m}=3\hbar/\sigma_{\rm c}$ as measured by a Husimi-Q phase space microscope with $\sigma_c = w$. The density measurement (a) becomes insensitive to $\delta$ for $\delta/w\lesssim 0.5$, while the phase-space measurement (b) continues to discriminate via the high-momentum tail.
• Figure 4: Particle density, kinetic energy density, and quartic momentum density around a 2D vortex of healing length $\xi$, computed for the ansatz FetterRMP$\psi(r,\theta) =\sqrt{\bar{\rho}} \,r\, {\rm e}^{{\rm i}\theta} /\sqrt{r^2+2\xi^2}$. The averaged-mode phase-space microscope also allows measurements of the latter two quantities.