Protocols for a many-body phase microscope: From coherences and d-wave superconductivity to Green's functions

TL;DR

The paper introduces a matter-wave microscope framework with Fourier-space manipulation to access long-range off-diagonal correlators in quantum-gas systems, enabling a practical “many-body phase microscope.” It lays out concrete protocols to measure equal-time coherences $g^{(1)}(\boldsymbol{d})$, four-point pairing correlators $C_{\mu,\nu}(\mathbf{d})$ for $d$-wave order, and non-equal-time Green's functions $G(\mathbf{k}_0,t)$ that map to ARPES-like spectral functions $A(\mathbf{k},\omega)$. It further shows how to detect hidden off-diagonal order of composite bosons in fractional Chern insulators by combining localized phase-coherence measurements with density information, realized via a local Raman pulse and microscope imaging. The work discusses detailed experimental considerations for Cs and Li atoms, including band-mapping, interaction management, and phase-stable Raman control, and outlines extensions to OTOCs, Majorana-edge signatures, and continuous-system implementations. Overall, the proposed protocols expand the experimental toolkit for characterizing exotic quantum many-body states and their dynamics with direct access to phases and coherences at nonlocal scales.

Abstract

Quantum gas microscopes probe quantum many-body lattice states via projective measurements in the occupation basis, enabling access to various density and spin correlations. Phase information, however, cannot be directly obtained in these setups. Recent experiments went beyond this by measuring local current operators and local phase fluctuations. Here we propose how Fourier-space manipulation in a matter-wave microscope allows access to various long-range off-diagonal correlators in experimentally realistic settings, realizing a many-body phase microscope. We demonstrate in particular how the fermionic d-wave superconducting order parameter in arbitrary Hubbard-type models, the non-equal time Green's function yielding the spectral function, or the hidden order of composite bosons in a fractional Chern insulator can be directly measured. Our results show the great potential of matter-wave microscopy for accessing exotic correlators including phases and coherences and characterizing intriguing quantum many-body states.

Figures (5)

• Figure 1: Proposed protocol for measuring equal-time Green's functions and off-diagonal long-range order. (a) Sketch of the general protocol using a matter-wave microscope. Details are described in the main text. 1) Atoms are initially in an optical lattice in a strongly-correlated regime, all in spin state $\uparrow$ (blue) ($t=0$). 2) After instantaneously switching off the lattice and interactions, a $T/4$ matter-wave pulse transforms the system into Fourier space, where directly afterwards (at $t=T/4$) a $\pi/2$ Raman pulse brings the system into superposition with an auxiliary spin state $\downarrow$ (yellow) that acquires a momentum kick. 3) After another $T/4$ step, a second $\pi/2$ Raman pulse in the matter-wave image plane (at $t=T/2$) but without momentum transfer makes lattice sites with a given distance $\mathbf{d}=\mathbf{i}-\mathbf{j}$ interfere (note that the matter-wave protocol inverts the image). 4) Spin-resolved single-atom imaging. (b) Example results of snapshot measurements. (c) If there is coherence between sites $\mathbf{i}$ and $\mathbf{j}$, one expects to see clear fringes in the measured density $\langle \hat{n}_{\mathbf{x},\uparrow}(4) \rangle_\varphi$ of the initial spin state $\uparrow$ as a function of the phase $\varphi$ of the second Raman pulse. Their amplitude corresponds directly to the $g^{(1)}(\mathbf{d})$ correlation function. The phase offset $\varphi_0$ of the fringe (e.g. due to time-reversal symmetry breaking) encodes the complex phase of $g^{(1)}(\mathbf{d})$. (d) Illustration of the protocol in phase space, where $T/4$ pulses swap the roles of $x$ and $p$, indicating the interferometer scheme. (e) The blue and yellow arrows indicate the utilized Raman transition between the two spin states.
• Figure 2: Proposed protocol for measuring the superconducting orders with arbitrary pairing symmetry. (a) Spinful version of the protocol in Fig. \ref{['fig:1']} using two physical spins (dark and light blue) and two auxiliary spins (dark and light yellow). Superconducting correlations can be detected from suitable four-point correlators $C_{\mu,\nu}(\mathbf{d})$ (see main text). (b) The four spin states are connected by separate Raman transitions. (c) Protocol for distinguishing between different pairing symmetries. The Raman transition for the spin-up state gives a kick that yields a translation by $\mathbf{d}_1=\mathbf{i}-\mathbf{j}$ (dark blue arrow), while the spin-down state is translated by $\mathbf{d}_2=\mathbf{i}-\mathbf{j}+\mathbf{e}_\mu-\mathbf{e}_\nu$ (light blue arrow) -- shown here for the combination $\mu=x$, $\nu=y$. This interferes the wavefunction of a pair of atoms, each displaced along a different direction, with itself at a controlled separation. (d) Illustration of the distinction between $s$-wave and $d$-wave superconducting order.
• Figure 3: Proposed protocol for measuring non-equal time correlators. (a) 1) A strongly correlated system is initialized, interactions and lattice potentials are switched off, and a $T/4$ pulse takes it into the Fourier plane. 2) A $\pi/2$ Ramsey pulse with one focused Raman beam at $\mathbf{x}_0$ extracts a particle in a selected momentum mode $\mathbf{k}_0(\mathbf{x}_0)$, see Eq. \ref{['eqkx']}. 3) In the image plane (at $T/2$) this mode is shifted and therefore isolated from the remaining many-body system, which 4) undergoes its intrinsic many-body dynamics after reloading into the original lattice and switching on interactions. The extracted momentum mode is indicated by the white hole in the system. 5) In a second Fourier plane (after applying another $T/4$ pulse), the selected momentum mode is brought to interference with the system via 6) a second $\pi/2$ Raman pulse giving access to the correlator $\langle \hat{a}^\dagger_{\mathbf{k}_0\uparrow}(t)\hat{a}_{\mathbf{k}_0\uparrow}(0) \rangle$ (see main text). (b) Illustration of the protocol in phase space in order to showcase the interferometer sequence.
• Figure 4: Measuring hidden off-diagonal long-range order in fractional quantum Hall systems. (a) 2D lattice with magnetic flux $\alpha$ and on-site interaction $U$, which gives rise to fractional quantum Hall phases for suitable parameters (left). Such systems can be described by composite bosons by attaching magnetic flux tubes (right). (b) Proposed protocol for measuring off-diagonal long-range correlations of composite bosons. 1)-4) the steps are as in Fig. \ref{['fig:1']}, but the second Raman pulse in step 3) is only locally applied via tweezers through a microscope. (c) For the evaluation, the measurements of the auxiliary spin (yellow) are shifted back to obtain the total density (each row is a cut through the 2D system), except for the two sites $\mathbf{i}, \mathbf{j}$ connected by the focused Raman beam (red box), where the coherence is evaluated from the spin contrast. The data in the dashed red boxes does not enter in the final evaluation.
• Figure 5: Protocol of Fig. \ref{['fig:3']} but for a spinful initial system, realized with $^6$Li atoms. Appropriate spin flips ensure weak interactions during the matter-wave protocol, but strong interactions in the initial system and during the time evolution of the recaptured system with one momentum mode isolated. Immediately after switching off the lattice, a Raman transfer ($\pi$ pulse) without momentum kick into the upper hyperfine states is applied and reduces the interaction strength during the matter-wave protocol. The same idea can be used in the protocol of Fig. \ref{['fig:2']} and combined with a ramp down of the magnetic field after the first $\pi$ pulse.