Continuous-variable multipartite vibrational entanglement

A compact scheme for the preparation of macroscopic multipartite entanglement is proposed and analyzed. In this scheme, the vibrational modes of a mechanical resonator constitute continuous-variable (CV) subsystems that entangle with each other as a result of their interaction with a two-level system (TLS). By properly driving the TLS, we show that a selected set of modes can be activated and prepared in a multipartite entangled state. We first study the entanglement properties of a three-mode system by evaluating the genuine multipartite entanglement, and we investigate its usefulness as a quantum resource by computing the quantum Fisher information. Moreover, the robustness of the state against the qubit and thermal noises is studied, proving a long-lived entanglement. To examine the scalability and structural properties of the scheme, we derive an effective model for the multimode system through elimination of the TLS dynamics. This work provides a step toward a compact and versatile device for creating a multipartite noise-resilient entangled state in vibrational modes as a resource for CV quantum metrology, quantum communication, and quantum computation.

Figure 1

(a) A TLS coupled to multiple vibrational modes of a mechanical resonator can intermediate their mutual interactions, which allows for creating a multipartite entangled state among those modes. This can be implemented, e.g., in (b) an HBAR whose longitudinal modes couple to a superconducting qubit via a piezoelectric device, or (c) the flexural modes of a membrane interacting with an embedded-atom-like defect. (d) In a multimode scheme, the set of modes are activated (here 1–6) by employing properly modulated drives that form two-mode squeezing interactions. The interaction is shown as a graph for two different modulation schemes: at the mode frequencies (e) and the half-sum frequencies (f). The edge thickness represents the interaction strength, while the active modes are highlighted in yellow.

Figure 2

Time evolution of the TPE (a) for different coupling rates (the curve labels show $g_0/\mathrm{\Gamma }$) at the Rabi frequency ${\mathrm{\Omega }}_0=3\mathrm{\Gamma }$, and (b) for different Rabi frequencies at $g_0=0.5\mathrm{\Gamma }$. (c) Robustness of longtime $E^{1\left|2\right|3}$: Density plot showing the tripartite entangled and separable parameter regions vs reservoir temperature and TLS pure dephasing rate $\tilde{\mathrm{\Gamma }}$ evaluated at $t=1000{\tau }_{\mathrm{FSR}}$ for ${\mathrm{\Omega }}_0=7\mathrm{\Gamma }$ and $g_0=0.5\mathrm{\Gamma }$. (d) The maximum normalized QFI as a function of coupling strength and drive amplitude.

Figure 3

Genuine $N$-partite entanglement for a set of six modes as a function of the coupling strength $g_0$ at ${\mathrm{\Omega }}_0=3\mathrm{\Gamma }$ and $t=1000{\tau }_{\mathrm{FSR}}$: (a) $N=3$, (b) $N=4$, (c) $N=5$, and (d) $N=6$.

Figure 4

The bipartite entanglement between the fundamental mode and the $k\mathrm{th}$ mode $E^{1|k}$ as a function of coupling rate $g_0$: (a) Logarithmic negativity at $T=10$ mK. (b) The entangled regions at five different temperatures. In both plots, the values correspond to ${\mathrm{\Omega }}_0=3\mathrm{\Gamma }$ at $t=1000{\tau }_{\mathrm{FSR}}$.

Figure 5

Comparison of the expectations values for the TLS operators: the exact (solid lines) vs mean-field approximation (dashed lines), which are solutions to Eqs. (A3). See the text for details.

Figure 6

(a) When the spectrum is noncommensurate, any desired set of modes can be activated by simply modulating the TLS drive on their frequencies (red arrows) or at half of the sum frequencies (green arrows). The activated modes form a complete graph, though weighted, for the two-mode squeezing interaction in both modulation strategies as illustrated for (b) mode frequencies modulation and (c) half of the sum frequencies modulation.

Figure 7

Normalized matrix elements of the effective two-mode squeezing coupling matrices ${\mathcal{F}}_{k,l}$: (a) and (b) for the commensurate spectrum with modulations at half sum frequencies and mode frequencies, respectively. In (c) and (d) the same are shown for a noncommensurate spectrum. The green square highlights the target set of modes, whereas the blue square illustrates the set of activated modes that are directly coupled to the target set.

Figure 8

Comparison of the entanglement $E^{1|2}$ as an illustration of the consistency: the original Hamiltonian (dashed orange) vs the effective Hamiltonian (solid blue) for three different coupling strengths. Here, ${\mathrm{\Omega }}_0=1\mathrm{\Gamma }$ and the other parameters are the same as in the main text.

Figure 9

The non-Gaussianity of the triangle system vs coupling strength. The parameters are given in the main text. Here, we set ${\mathrm{\Omega }}_0=3\mathrm{\Gamma }$ and evaluate the measure of non-Gaussianity at $t=1000{\tau }_{\mathrm{FSR}}$.