Interference, spectral momentum correlations, entanglement, and Bell inequality for a trapped interacting ultracold atomic dimer: Analogies with biphoton interferometry

Aiming at elucidating similarities and differences between quantum-optics biphoton interference phenomena and the quantum physics of quasi-one-dimensional double-well optically trapped ultracold neutral bosonic or fermionic atoms, we show that the analog of the optical biphoton joint-coincidence spectral correlations, studied with massless noninteracting biphotons emanating from Einstein-Podolsky-Rosen Bell-Bohm single-occupancy sources, corresponds to a distinct contribution in the total second-order momentum correlations of the massive, interacting, and time-evolving ultracold atoms. This single-occupancy contribution can be extracted from the total second-order momentum correlation function measured in time-of-flight experiments, which for the trapped atomic system contains, in general, a double-occupancy, NOON component. The dynamics of the two-particle system are modeled by a Hubbard Hamiltonian. The general form of this partial coincidence spectrum is a cosine-square quantum beating dependent on the difference of the momenta of the two particles, while the corresponding coincidence probability proper, familiar from its role in describing the Hong-Ou-Mandel coincidence dip of overlapping photons, results from an integration over the particle momenta. Because the second-order momentum correlations are mirrored in the time-of-flight spectra in space, our theoretical findings provide impetus for time-of-flight experimental protocols for emulating with (massive) ultracold atoms venerable optical interferometries that use two space-time separated and entangled (massless) photons or double-slit optical sources. The implementation of such developments will facilitate testing of fundamental aspects and enable applications of quantum physics with trapped massive ultracold atoms, that is, investigations of nonlocality and violation of Bell inequalities, entanglement, and quantum information science.

Figure 1

The Hubbard-dimer energy levels for all three cases of (i) spinless bosons, (ii) spin-1/2 bosons, and (iii) spin-1/2 fermions given by Eq. (15). See text for an explanation of the symbols $\phi$'s denoting the four Hubbard stationary wave functions, as well as their limiting $\mathrm{\Phi }$ forms at $\mathcal{U}\to \pm \infty$.

Figure 2

Total momentum correlation ${\mathcal{G}}_S(k_1,k_2)$ maps [top panels with blue background denoted as (a), (b), (d), (e), and (j)–(o)] and left-right joint-coincidence interferograms $p_{\mathrm{soc}}^S(k_1,k_2)$ [bottom panels with brown background denoted as (f), (g), (h), (i), and (p)–(u)] for the space-symmetric two-particle wave packet defined in Eqs. (34) and (35) for $\gamma =1/2$ and for two different times $t=0$ and $t=1\hslash /J$ at five different values of the Hubbard interaction parameter $\mathcal{U}=0,\pm 3,\pm 15$. (c) Integrated coincidence probability $P_{11}^S$ as a function of $\mathcal{U}$ for two values of time $t=0$ and $t=1\hslash /J$. Here and in all figures (unless explicitly stated otherwise): interwell distance $2d=2\mu \mathrm{m}$ and width $s=0.25\mu \mathrm{m}$. Momenta $k_1$ and $k_2$ in units of $1/\mu \mathrm{m}$. ${\mathcal{G}}_S(k_1,k_2)$ and $p_{\mathrm{soc}}^S(k_1,k_2)$ in units of $\mu {\mathrm{m}}^2$. See text for details.

Figure 3

Second-order total momentum correlation maps $\mathrm{[}{\mathcal{G}}_S(k_1,k_2)$, upper row, blue panels] and corresponding joint-coincidence spectral maps $\mathrm{[}{p}_{\mathrm{soc}}^S(k_1,k_2)$, lower row, brown panels] for the space-symmetric two-particle wave packet defined in Eqs. (34) and (35) and for $\gamma =1/2$. Three different interwell separations, [(a), (d)] $2d=2\mu \mathrm{m}$, [(b), (e)] $2d=3\mu \mathrm{m}$, and [(c), (f)] $2d=4\mu \mathrm{m}$, are considered. Note that the number of visible fringes increases with increasing $2d$. Remaining parameters $t=1\hslash /J$ and $\mathcal{U}=0$. Momenta $k_1$ and $k_2$ in units of $1/\mu \mathrm{m}$. ${\mathcal{G}}_S(k_1,k_2)$ and $p_{\mathrm{soc}}^S(k_1,k_2)$ in units of $\mu {\mathrm{m}}^2$.

Figure 4

Total momentum correlation ${\mathcal{G}}_S(k_1,k_2)$ maps (top panels, blue background) and left-right joint-coincidence interferograms $p_{\mathrm{soc}}(k_1,k_2)$ (bottom panels, brown background) for the space-symmetric wave packet defined in Eqs. (34) and (35) for $\gamma =1/2$ and for two different times $t=0$ and $t=\pi /4\hslash /J$ at four different values of the Hubbard parameter $\mathcal{U}=-3,0,3,15$ (from top to bottom). The frames on the left display the integrated joint-coincidence probability $P_{11}^S$ as a function of time $t$ for these four values of $\mathcal{U}$. Momenta $k_1$ and $k_2$ in units of $1/\mu \mathrm{m}$. ${\mathcal{G}}_S(k_1,k_2)$ and $p_{\mathrm{soc}}^S(k_1,k_2)$ in units of $\mu {\mathrm{m}}^2$. See text for details.

Figure 5

Integrated joint-coincidence probability curves $P_{11}^S$ [see Eq. (40)] as a function of $\mathcal{U}$ for four characteristic values of time, (a) $t=\pi /4\hslash /J$, (b) $t=\pi /2\hslash /J$, (c) $t=3\pi /4\hslash /J$, and (d) $t=\pi \hslash /J$. This case corresponds to the space-symmetric wave packet defined in Eqs. (34) and (35), but for a variable $\gamma \left(\mathcal{U}\right)$ given by Eq. (38). The insets display corresponding total second-order momentum correlation ${\mathcal{G}}_S(k_1,k_2)$ maps (right panels, blue background) and left-right joint-coincidence interferograms $p_{\mathrm{soc}}(k_1,k_2)$ (left panels, brown background). Momenta $k_1$ and $k_2$ in units of $1/\mu \mathrm{m}$. ${\mathcal{G}}_S(k_1,k_2)$ and $p_{\mathrm{soc}}^S(k_1,k_2)$ in units of $\mu {\mathrm{m}}^2$. See text for details.

Figure 6

Diagrams of the four different amplitudes that contribute to the total second-order momentum correlations $\mathcal{G}(k_1,k_2)$ [the total fourth-order coincidence ${\tilde{P}}_2(x_1,x_2)$ in Ref. [28]; see Eq. (5) therein]. In these diagrams, the two-photons (particles) in each event (I)–(IV) are described by two lines of the same color. In the diagrams labeled as (I) and (II) (green and magenta colors, respectively), both particles (photons) originate from the same well (light source). In the diagrams labeled as (III) and (IV) (blue and red colors, respectively), each particle (photon) in the pair originates from a different well (light source). $L\left(A\right)$ and $R\left(B\right)$ denote the two trapping wells (light sources). $X_1$ and $X_2$ denote two positions in the time-of-flight expansion cloud (the optics far field). In the case of the optics far field, the symbols $x_1$ and $x_2$ (lowercase) are often used [28], instead of the uppercase $X_1$ and $X_2$. The positions of the two particles in the time-of-flight cloud are related to the single-particle momenta $k_1$ and $k_2$ at the double-well trap as $X_j=\hslash k_j{t}_{\mathrm{TOF}})/M,j=1,2$ [31], where $M$ is the mass of the atom and $t_{\mathrm{TOF}}$ is the experimental time of flight. Note that the use of the term “second-order” by us corresponds to the use of the term “fourth-order” in quantum optics.

Figure 7

[(a), (b)] von Neumann entropy for mode entanglement and (c) concurrence for the Hubbard-dimer eigenstates ${\phi }_i,i=1,...,4$ as a function of $\mathcal{U}$. In panel (a), the case of two spinless bosons with only three eigenstates is presented. The maximum value attained by the von Neumann entropy is $ln\left(3\right)=1.09861$ at $\mathcal{U}=-\sqrt{2}$ for the ${\phi }_1$ state or at $\mathcal{U}=\sqrt{2}$ for the ${\phi }_3$ state. The asymptotic values at $\mathcal{U}\to \pm \infty$ are zero or $ln\left(2\right)=0.693147$ (nonsymmetric curves). In panel (b), the case of two spin-1/2 bosons or fermions is presented. The maximum value attained (at $\mathcal{U}=0)$ is $2ln\left(2\right)=1.38629$, whereas the asymptotic values for $\mathcal{U}\to \pm \infty$ are $ln\left(2\right)=0.693147$ (symmetric curves). In panel (c), the concurrence for all three cases yields the same result (of course the space antisymmetric state ${\phi }_4$ is missing for the spinless bosons). The maximum asymptotic value attained by the concurrence (for $\mathcal{U}\to \pm \infty )$ is unity, whereas the minimum value (at $\mathcal{U}=0)$ is zero (symmetric curves).

Figure 8

von Neumann entropy for mode entanglement for two spinless bosons described in space by the time-dependent wave packet $\mathrm{\Omega }\left(t\right)$ in Eq. (34). The four left frames [(a)–(d)] describe variations of the von Neumann entropy at given times $t=0$ [(a) and (c)] and $t=\pi /4\hslash /J$ [(b) and (d)] as a function of $\mathcal{U}$, and for two different cases of the mixing parameter $\gamma$ (see Sec. 3d), namely $\gamma =1/2$ [(a) and (b)] and $\gamma \left(\mathcal{U}\right)=\frac{1}{4}(\mathcal{U}-\sqrt{{\mathcal{U}}^2+16})$ [(c) and (d)]. The four right frames [(e)–(h)] describe variations of the von Neumann entropy at given values of $\mathcal{U}=2$ [(e) and (g)] and $\mathcal{U}=30$ [(f) and (h)] as a function of time $t$, and for the same two different cases of the mixing parameter $\gamma$, namely $\gamma =1/2$ [(e) and (f)] and $\gamma \left(\mathcal{U}\right)$ [(g) and (h)]. In agreement with the findings from the section on the von Neumann entropy for eigenstates [see Fig. 7], the range of oscillatory variations of $S_{\mathrm{vN}}$ extends from zero to a value of $ln\left(3\right)=1.09861$.

Figure 9

Von Neumann entropy for mode entanglement for two spin-1/2 bosons or fermions described by the time-dependent wave packet $\mathrm{\Omega }\left(t\right)$ in Eq. (34). The four left frames [(a)–(d)] describe variations of the von Neumann entropy at given times $t=0$ [(a) and (c)] and $t=\pi /4\hslash /J$ [(b) and (d)] as a function of $\mathcal{U}$ and for two different cases of the mixing parameter $\gamma$ (see Sec. 3d), namely $\gamma =1/2$ and $\gamma \left(\mathcal{U}\right)=\frac{1}{4}(\mathcal{U}-\sqrt{{\mathcal{U}}^2+16})$. The four right frames [(e)–(h)] describe variations of the von Neumann entropy at given values of $\mathcal{U}=2$ and $\mathcal{U}=30$ as a function of time $t$, and for the same two different cases of the mixing parameter $\gamma$, namely $\gamma =1/2$ and $\gamma \left(\mathcal{U}\right)$. In agreement with the findings from the section on the von Neumann entropy for eigenstates [see Fig. 7], the range of oscillatory variations of $S_{\mathrm{vN}}$ extends from $ln\left(2\right)=0.693147$ to a value of $2ln\left(2\right)=1.38629$.

Figure 10

Concurrence for all three cases of a pair of particles described in space by the time-dependent wave packet $\mathrm{\Omega }\left(t\right)$ in Eq. (34). The four top frames [(a)–(d)] describe variations of the concurrence at given times $t=0$ [(a) and (c)] and $t=\pi /4\hslash /J$ [(b) and (d)] as a function of $\mathcal{U}$, and for two different cases of the mixing parameter $\gamma$ (see Sec. 3d), namely $\gamma =1/2$ [(a) and (b)] and $\gamma \left(\mathcal{U}\right)=\frac{1}{4}(\mathcal{U}-\sqrt{{\mathcal{U}}^2+16})$ [(c) and (d)]. The four bottom frames [(e)–(h)] describe variations of the concurrence at given values of $\mathcal{U}=2$ [(e) and (g)] and $\mathcal{U}=30$ [(f) and (h)] as a function of time $t$, and for the same two different cases of the mixing parameter $\gamma$, namely $\gamma =1/2$ [(e) and (f)] and $\gamma \left(\mathcal{U}\right)$ [(g) and (h)]. In agreement with the findings from the section on the concurrence for eigenstates [see Fig. 7], the range of potential oscillatory variations of $\mathcal{C}C$ extends from zero to unity.