Generating Multiparticle Entangled States by Self-Organization of Driven Ultracold Atoms

We describe a mechanism for guiding the dynamical evolution of ultracold atomic motional degrees of freedom toward multiparticle entangled Dicke-squeezed states, via nonlinear self-organization under external driving. Two examples of many-body models are investigated. In the first model, the external drive is a temporally oscillating magnetic field leading to self-organization by interatomic scattering. In the second model, the drive is a pump laser leading to transverse self-organization by photon-atom scattering in a ring cavity. We numerically demonstrate the generation of multiparticle entangled states of atomic motion and discuss prospective experimental realizations of the models. For the cavity case, the calculations with adiabatically eliminated photonic sidebands show significant momentum entanglement generation can occur even in the “bad cavity” regime. The results highlight the potential for using self-organization of atomic motion in quantum technological applications.

Figure 1

Nonlinear self-organization via four-wave mixing of momentum modes in a driven BEC. (a) The self-organization along the $x$ axis can be driven by applying a $B$-field $B(t)$ oscillating near a Feshbach resonance, leading to an oscillating scattering length $a(t)$ [31]. (b) Absorption of a quantum of energy $\hslash {\omega }_{\text{mod}}$ leads here to a momentum-conserving scattering of two atoms with zero transverse momentum into modes with transverse momenta $\pm \hslash k_f$, where $k_f=\sqrt{m{\omega }_{\text{mod}}/\hslash }$. (c) Transverse self-organization with spatial period ${\mathrm{\Lambda }}_c$ for laser pumped atoms in a ring cavity ($\eta$ pump rate) with photon leakage rate $\kappa$. (d) In this system, an atomic sideband with transverse momentum $-\hslash q_c$ ($q_c=2\pi /{\mathrm{\Lambda }}_c$) is excited by scattering of an on-axis photon into a vacuum mode with transverse wave number $+q_c$. A correlated atom with transverse momentum $+\hslash q_c$ can then be created if this sideband photon is scattered back into the on-axis mode, with a converse process for the $-q_c$ photon sideband (not shown).

Figure 2

Populating the sideband Dicke-like states by unitary evolution via $H_B$ and $H_{\mathrm{ad}}$. (a) Maximum $⟨J_{\mathrm{eff}}^2⟩$ reached during temporal evolution, denoted by $J_{\max }^2$ (blue dots $H_B$, red squares $H_{\mathrm{ad}}$ at ${\eta }_{\mathrm{eff}}=1.6{\eta }_c$, and green triangles $H_{\mathrm{ad}}$ at ${\eta }_{\mathrm{eff}}=3{\eta }_c$) against the total atom number $N$, with the ideal Dicke state value of $N(N+2)/4$ denoted by the dashed line. (b) The Dicke-squeezed state forms a band around the equator of the Bloch sphere characterized by a large radius $⟨J_{\mathrm{eff}}^2⟩=⟨J_x^2+J_y^2⟩$ and a vanishing variance of the “spin” distribution along the $z$ axis, $⟨(\mathrm{\Delta }{J}_z{)}^2⟩$. (c) For unitary evolution with $H_B$ and $H_{\mathrm{ad}}$ at $N=40$, $⟨J_{\mathrm{eff}}^2⟩$ (blue line $H_B$, red line $H_{\mathrm{ad}}$ at ${\eta }_{\mathrm{eff}}=1.6{\eta }_c$, green line $H_{\mathrm{ad}}$ at ${\eta }_{\mathrm{eff}}=3{\eta }_c$) performs oscillations in time, nearly reaching the limit value of $N(N+2)/4=420$ (dashed line) for $H_B$. For $H_B$, $g=g_{\text{mod}}$, while for $H_{\mathrm{ad}}$, $g={\omega }_R$.

Figure 3

Generating multiparticle entanglement in the Dicke-like states via transverse self-organization for the full system Hamiltonian $H_{\mathrm{cav}}$. The dissipative dynamical evolution of (a) $⟨(\mathrm{\Delta }{J}_z{)}^2⟩$, (b) $⟨J_{\mathrm{eff}}^2⟩$ for $\eta$ values of $25{\omega }_R$ (blue, dashed), $35{\omega }_R$ (green, dot-dashed), and $45{\omega }_R$ (red, solid). (c) The $\eta$ scan of ${\xi }_m^2$ given by the lowest ${\xi }_{\mathrm{gen}}^2$ reached during temporal evolution, for the unitary case (purple, dots), and $\kappa =5{\omega }_R$ (red, squares), $\kappa =15{\omega }_R$ (yellow, triangles). Crossing of the dashed line (${\xi }_{\mathrm{gen}}^2=1$) indicates the existence of many-particle entanglement [102]. Solid lines in (c) are results for the $H_{\mathrm{ad}}$ model with ${\eta }_{\mathrm{eff}}=0.56\eta$. Simulation parameters: (a), (b) ($({\overline{\mathrm{\Delta }}}_c,{\overline{\mathrm{\Delta }}}_c^{\prime },U_0,\kappa )=(110,-45,10,5){\omega }_R$, and (c) $({\overline{\mathrm{\Delta }}}_c,{\overline{\mathrm{\Delta }}}_c^{\prime },U_0)=(110,-45,10){\omega }_R$, with (a)–(c) $N=8$.

Figure 4

Entanglement generation for a dissipative cavity with adiabatically eliminated photonic fields described by the Lindblad master equation with $H_{\mathrm{ad}}$ and decay rate $\gamma$ (see text). (a) Cavity decay rate $\kappa$ scan of ${\xi }_m^2$ (see text) for fixed $({\overline{\mathrm{\Delta }}}_c,{\overline{\mathrm{\Delta }}}_c^{\prime })=(1400,-1200){\omega }_R$, and $N=20$ (blue dots), $N=40$ (red squares), and $N=120$ (green triangles). (b) Sideband detuning ${\overline{\mathrm{\Delta }}}_c^{\prime }$ scan of ${\xi }_m^2$ for $N$ values as in (a), and fixed $\kappa =3\times {10}^4{\omega }_R$, with ${\overline{\mathrm{\Delta }}}_c=|{\overline{\mathrm{\Delta }}}_c^{\prime }|+4\kappa$ for each point. The dashed line indicates ${\xi }_{\mathrm{gen}}^2=1$. Solid lines are guide to the eyes. Simulation parameters: $U_0={10}^{-3}{\omega }_R$, ${\eta }_{\mathrm{eff}}=1.6{\eta }_c$.