Measurement Protocol for the Entanglement Spectrum of Cold Atoms

Entanglement, and, in particular, the entanglement spectrum, plays a major role in characterizing many-body quantum systems. While there has been a surge of theoretical works on the subject, no experimental measurement has been performed to date because of the lack of an implementable measurement scheme. Here, we propose a measurement protocol to access the entanglement spectrum of many-body states in experiments with cold atoms in optical lattices. Our scheme effectively performs a Ramsey spectroscopy of the entanglement Hamiltonian and is based on the ability to produce several copies of the state under investigation, together with the possibility to perform a global swap gate between two copies conditioned on the state of an auxiliary qubit. We show how the required conditional swap gate can be implemented with cold atoms, either by using Rydberg interactions or coupling the atoms to a cavity mode. We illustrate these ideas on a simple (extended) Bose-Hubbard model where such a measurement protocol reveals topological features of the Haldane phase.

Popular Summary

Entanglement is fundamental to our understanding of many-body quantum systems, and it is a key concept underlying a plethora of phenomena such as topological properties. Entanglement is also responsible for the complexity of simulating quantum many-body physics on a classical computer. One of the most powerful theoretical tools to characterize entanglement in a quantum many-body system is the so-called entanglement spectrum, which characterizes the statistical properties of a (reduced) quantum state. Thus far, however, because of the lack of an experimentally applicable measurement scheme, it has been used only as a theoretical concept. Here, we build on versatile tools to control and manipulate cold atoms available in current experiments, and we develop a protocol to measure the entanglement spectrum.

We propose to use Ramsey-type spectroscopy to access the spectrum of a many-body quantum state. Our protocol is based on the ability to produce several copies of the state being investigated and to couple these copies between one other and with an auxiliary atom. We show how these required ingredients can be implemented with cold atoms held in an optical lattice using Rydberg interactions and tunnel coupling. In particular, we make use of recently developed techniques to address atoms in optical lattices in a site-resolved way. We show how these tools can be combined to access the largest eigenvalues of the quantum state, and we find that the spectral resolution of our measurement scheme is determined by the total number of copies used.

We expect that our findings will extend the tools available in quantum gas experiments and elevate the entanglement spectrum from a fundamental but purely theoretical concept to a quantity that is measurable in the laboratory.

Figure 1

(a) Circuit representation of the protocol to determine the spectrum of a density operator $\rho$. It consists of $n$ stroboscopic steps (operations) that each involve an ancilla system and two copies of the state under investigation. (b) Each step can be constructed from three basic operations: tunnel coupling between neighboring copies (blue), a controlled (dispersive) phase shift for atoms in the lattice based on the state of the ancilla system (green), and rotations of the ancilla qubit (red). For a description of these operations, we refer to the main text [Eqs. (4), (5), and (7); see also Fig. 2]. (c) Construction of the stroboscopic step $U_{\mathrm{step}}$ from these elementary operations. Note that it differs from the one in Eq. (2) by an additional swap, $U_{\mathrm{step}}=S{U}_{\mathrm{step}}^{\prime }$. This additional swap ensures that all processes involve only neighboring copies (see Appendix pp1).

Figure 2

Realization of the elementary operations of the circuit in Fig. 1 with cold atoms. (a) The beam splitter $U_{\mathrm{BS}}$ [cf. Eq. (4)] can be realized by lowering the barrier between two copies in order to allow for tunneling between them. (b) To realize the controlled phase shift, an off-resonant laser to a Rydberg state $|s_b⟩$ is focused on the lattice sites in subsystem $\mathcal{A}$ of copy $k$. If the control atom is in the state $|0⟩$, the ac-Stark shift leads to an acquired phase (left); if it is in the Rydberg state $|1⟩$, the Rydberg blockade mechanism leads to a shift of the state $|s_b⟩$, suppressing the ac-Stark shift (right). (c) Illustration of the basic operations in step $k$ to determine the spectrum of the reduced density operator of only the first three lattice sites. Note that the blockade radius $r_B$ exceeds the size of the lattice. The rotations of the control atom $U_{\epsilon }$ correspond to Rabi pulses on a Rydberg transition. (d) Analogous setup for 2D systems. Several copies are created in different layers of a 3D lattice, with the ancillary atom trapped nearby. All steps in the protocol are the same as in the 1D case. We note that the subsystem $\mathcal{A}$ does not need to be contiguous.

Figure 3

Entanglement spectrum of the extended 1D Hubbard model supporting symmetry-protected topological order in the ground state. (a) Optical lattice with a two-site unit cell corresponding to the Hamiltonian (8). The Hubbard parameter alternate between even and odd sites $t_{2j}\equiv t$, $t_{2j-1}\equiv t^{\prime }$, and analogously for ${ϵ}_j$ and $V_j$. (b) In the hard-core boson regime, this can be mapped to an alternating-bond spin-$1/2$ Heisenberg model (9), with $J=t/2=V/4$ and $J^{\prime }=t^{\prime }/2=V^{\prime }/4$ (see Appendix pp3). The strong antiferromagnetic bond (red) favors the formation of a spin-singlet dimer in the small $J^{\prime }/J$ limit. (c) ES ${\lambda }_{\alpha }$ of a bipartite split of regions $\mathcal{A}$ and $\mathcal{B}$, and corresponding entanglement energies ${\xi }_{\alpha }=-\ln ({\lambda }_{\alpha })$ (inset) ($J^{\prime }/J=0.1$, on $N=8$ sites), for one of the four degenerate ground states of Eq. (9).

Figure 4

Analysis of the protocol. (a) Ramsey signal as a function of the stroboscopic evolution time $t_n$ for a density operator with four degenerate eigenvalues ${\lambda }_{\alpha }=1/4$ $(\alpha =1,\dots ,4)$, using different sizes of the time step $\epsilon$. The finite size of the time step leads to a damping of the signal over a time $\tau \sim \mathcal{O}({\epsilon }^{-1})$ (cf. Appendix pp1). We indicate the measurement uncertainty due to shot noise $\mathrm{\Delta }{Z}_n=\sqrt{(1-⟨Z{⟩}_n^2)/N_{\mathrm{shot}}}$ for $N_{\mathrm{shot}}=200$ measurements per point. (b) Fourier-transformed Ramsey signal $s(\nu )=\sum _n\epsilon ⟨Z{⟩}_n\cos (2t_n\nu )$, with the uncertainty $\mathrm{\Delta }s\sim \epsilon \sqrt{\sum _n\mathrm{\Delta }{Z}_n^2}$. Each eigenvalue of the density operator gives rise to a Lorentzian whose width is determined by the size of the stroboscopic step. The product of the height and the width of the peak reveals that it stems from $d=4$ eigenvalues at $\lambda =0.25$. This is depicted in the inset by the circles, where the horizontal bars correspond to the width of the peak and indicate the spectral resolution. To observe the signature of the fourfold degeneracy, around $n_{\max }\sim 150$ copies are necessary ($\epsilon =0.1$). The position of the maximum, together with the height and the width of the resonance, allows one to determine the weight of the peak, i.e., the degeneracy ($d$) of the corresponding eigenvalue (see text). The inset shows the position of the resonance and its degeneracy $d$ for the peak at $\lambda \approx 0.25$ for the different values of $\epsilon =0.1$, 0.05, 0.025, 0.01. In all cases, one finds $d\sim 4$, as expected for fourfold degenerate eigenvalues. The horizontal bar indicates the width of the peak, ${\sigma }_{\alpha }$.

Figure 5

Effects of imperfect implementations of the individual operations on the Ramsey signal, for the same system parameters as in Figs. 3 and 4 (with $N=4$ sites in each copy). We separately show the effects of (a) residual interactions in the beam-splitter steps, (b) uncertainty of the tunnel times in the beam-splitter steps, (c) fluctuations in the phase acquired during the controlled phase gates, and (d) fluctuations in the rotation angle for single-qubit rotations (see text). For (b) and (c), we perform an average over different error realizations. The statistical uncertainties are indicated by the dashed lines. The size of the stroboscopic time step is $\epsilon =0.07$ in all cases.