Ultrafast preparation and detection of entangled atoms

Abstract

Atoms can form a molecule by sharing their electrons in binding orbitals. These electrons are entangled. Is there a way to break a molecular bond and obtain atoms in their ground state that are spatially separated and still entangled? Here, we show that it is possible to prepare these spatially separated, entangled atoms on femtosecond time scales from single oxygen molecules. The two neutral atoms are entangled in the magnetic quantum number of their valence electrons. In a time-delayed probe step, we use nonadiabatic tunneling, which is a magnetic quantum number-sensitive ionization mechanism. We find a fingerprint of entanglement in the measured ionization probability as a function of the angle between the light's quantization axis and the molecular axis. This establishes a platform for further experiments that harness the time resolution of strong-field experiments to investigate spatially separated, entangled atoms on femtosecond time scales.

Fig. 1.. Ultrafast preparation of an atomic Bell-like state.

(A) Oxygen molecule in its ground state X³Σ_g is excited by three-photon absorption from a circularly polarized laser pulse (photons are labeled with ℏω) at a central wavelength of 390 nm and an intensity of 1.0 × 10¹⁴ W/cm² at the time t = 0 fs. This triggers dissociation of the oxygen molecule via the 1³Π_u state and results in two neutral oxygen atoms. Panels (B to D) show the same as (A) but in more detail. (B) Oxygen molecules in their ground state X³Σ_g contain one electron with m = −1 and one electron with m = +1 in the ${\mathrm{\pi }}_{2\mathrm{p}}^{\ast }$ orbital. (C) Circularly polarized pump pulse with an anticlockwise-rotating electric field [indicated by “(L)” in (B)] vector excites the electron from ${\mathrm{\pi }}_{2\mathrm{p}}^{\ast }$ (with m = +1) to ${\mathrm{\sigma }}_{2\mathrm{p}}^{\ast }$ (with m = 0) by three-photon absorption. This leads to the 1³Π_u state that contains an excess electron with a magnetic quantum number of m = −1. (D) Upon dissociation, this produces two oxygen atoms in their ground state, which have defined magnetic quantum numbers [see blue filled circles in (B) to (D) and note that the indicated electron spins illustrate exemplarily cases only] (see Materials and Methods and Supplementary Materials for details). There is no way to tell which atom has which magnetic quantum number. This results in an entangled, Bell-like state, which is formed by one oxygen atom in “site A” and one oxygen atom in “site B.” A pump pulse with anticlockwise-rotating electric field predominantly prepares an entangled state, which can be written as $\mid {\text{Ψ}}_{-00-}^{+}⟩=\frac{1}{\sqrt{2}}({\mid m_{-1}⟩}_{\mathrm{A}}{\mid m_0⟩}_{\mathrm{B}}+{\mid m_0⟩}_{\mathrm{A}}{\mid m_{-1}⟩}_{\mathrm{B}})$.

Fig. 2.. Projection of the prepared Bell-like state to a new basis and tunnel ionization.

(A) Case in which a molecule dissociates at an angle of γ with respect to the light’s propagation axis is illustrated. At t = 1.5 ps, a circularly polarized probe pulse at an intensity of 4.5 × 10¹⁴ W/cm² hits the dissociating molecule. The laser pulse has a clockwise-rotating electric field [indicated by “(R)”] and projects the magnetic quantum number m in the molecular frame onto a new basis m′, as illustrated by the arrows within the blue shaded area. The fourth electron that is indicated by orange arrows is in a superposition of all three m′ states and entangled with the corresponding electron at the other site. The entanglement of these two electrons in the basis m is indicated by $\frac{1}{\sqrt{2}}({\mid m_{-1}⟩}_{\mathrm{A}}{\mid m_0⟩}_{\mathrm{B}}+{\mid m_0⟩}_{\mathrm{A}}{\mid m_{-1}⟩}_{\mathrm{B}})$. (B) At site A, nonadiabatic tunnel ionization occurs, which strongly prefers electrons with m′ = −1 and acts like a polarizer that projects the wave function on eigenstates that are defined by the new quantization axis. Single ionization at site A instantaneously affects the wave function at site B (ocher colored area). (C) With a certain probability, a second electron with m′ = −1 is liberated at site B by a sequential tunneling process, such that both oxygen atoms are singly ionized.

Fig. 3.. Strong-field ionization of two different Bell-like states.

(A) Experimental result for the ionization of one of the two atoms. The blue curve shows the ionization probability as a function of γ after predominantly preparing the entangled state $\mid {\text{Ψ}}_{-00-}^{+}⟩=\frac{1}{\sqrt{2}}({\mid m_{-1}⟩}_{\mathrm{A}}{\mid m_0⟩}_{\mathrm{B}}+{\mid m_0⟩}_{\mathrm{A}}{\mid m_{-1}⟩}_{\mathrm{B}})$. The red curve shows the same for the entangled state $\mid {\text{Ψ}}_{+00+}^{+}⟩=\frac{1}{\sqrt{2}}({\mid m_{+1}⟩}_{\mathrm{A}}{\mid m_0⟩}_{\mathrm{B}}+{\mid m_0⟩}_{\mathrm{A}}{\mid m_{+1}⟩}_{\mathrm{B}})$. (B) Fit of our model that uses the entangled states $\mid {\text{Ψ}}_{-00-}^{+}⟩$ and $\mid {\text{Ψ}}_{+00+}^{+}⟩$ to the data from (A) is indicated with solid lines. The dotted lines show the fit using a classical state, which is described by $\mid {\text{Ψ}}_{-0}^{+}⟩={\mid m_{-1}⟩}_{\mathrm{A}}{\mid m_0⟩}_{\mathrm{B}}$ in 50% of the cases and otherwise by $\mid {\text{Ψ}}_{0-}^{+}⟩={\mid m_0⟩}_{\mathrm{A}}{\mid m_{-1}⟩}_{\mathrm{B}}$ (the notations for $\mid {\text{Ψ}}_{+0}^{+}⟩$ and $\mid {\text{Ψ}}_{0+}^{+}⟩$ are analogous). (C and D) Same as (A) and (B) but for the single ionization of both of the two atoms. The results in (D) are obtained by using the models’ parameters that were found for (B) (see "Observables that shows a fingerprint of entanglement in strong field ionization"). The model that uses entangled states shows superior agreement with the experiment compared to the model that uses classical states. Intensity per volume element indicates that the measured yield has been divided by sin(γ). The experimental data have been symmetrized. Error bars show the SD of the statistical error only.