NOON states via a quantum walk of bound particles

Tight-binding lattice models allow the creation of bound composite objects which, in the strong-interacting regime, are protected against dissociation. We show that a local impurity in the lattice potential can generate a coherent split of an incoming bound particle wave packet which consequently produces a NOON state between the endpoints. This is nontrivial because, when finite lattices are involved, edge-localization effects render challenging their use for nonclassical state generation and information transfer. We derive an effective model to describe the propagation of bound particles in a Bose–Hubbard chain. We introduce local impurities in the lattice potential to inhibit localization effects and to split the propagating bound particle, thus enabling the generation of distant NOON states. We analyze how minimal engineering transfer schemes improve the transfer fidelity and we quantify the robustness to typical decoherence effects in optical lattice implementations. Our scheme potentially has an impact on quantum-enhanced atomic interferometry in a lattice.

Figure 1

Scheme of the model: when a bound state is initially in the site 1 of a finite lattice, a suitably introduced local impurity (orange) in the chemical potential ${\mu }_j=-\beta {\delta }_{j,L/2+1}$ triggers a wave-packet splitting in a reflected $R$ and a transmitted $T$ component. If appropriately tuned it generates a NOON state between the endpoints (site 1 and $L$) at the transfer time. The two green peaks are local fields which realize a finite lattice model. The red arrow represents the natural direction of propagation of the bound particle once lowered the lattice depth at time $t=0$. We also add local fields in the first and last sites of the chain to inhibit the edge-localization effect and then to delocalize the bound particle from the edges.

Figure 2

Edge delocalization for a two-particle bound state: plot of the probability to have a bound particle in the first $P_{11}\left(t\right)$ and in the last $P_{LL}\left(t\right)$ site as a function of the time $t$, scaled for the transfer time $t^{*}\sim L/J_{\mathrm{eff}}$ for a uniform chain with $L=5$ and $U/J=5$ in the initial state $|\psi \left(0\right)\rangle \propto {\left(a_1^{†}\right)}^2|0\rangle$. We add a local field ${\mu }_j=-{\beta }^{\prime }({\delta }_{j,1}+{\delta }_{j,L})$ with strength ${\beta }^{\prime }=0$ (top), and ${\beta }^{\prime }={\beta }_{\mathrm{opt}}^{\prime }=J^2/4U$ (bottom). To compare the results with the single-particle case we plot, with a dashed black line, the probability $P_L\left(t\right)=|\langle 0|a_L{\left|\psi \left(t\right)\right|}^2$ for a single particle initially in $a_1^{†}|0\rangle$ (here $t^{*}\sim L/J$).

Figure 3

Optimal transfer of a two-particle bound state: (top) Analysis of the transfer fidelity $P_{LL}\left(t^{*}\right)$ for the initial state $|\psi \left(0\right)\rangle \propto {\left(a_1^{†}\right)}^2|0\rangle$ as a function of the onsite interaction $U/J$ when the optimal transfer scheme $J_1=J_L=J_0$ and $J_j=J$ for $j\ne (1,L)$ is included in the model (1). Here we also add two local impurities ${\mu }_j=-{\beta }_1({\delta }_{j,1}+{\delta }_{j,L})$ and ${\mu }_j=-{\beta }_2({\delta }_{j,2}+{\delta }_{j,L-1})$ where ${\beta }_1=(J_0^2-2J^2)/2U$ and ${\beta }_2=(J_0^2-J^2)/2U$ to eliminate the edge-locking effect. The $J_0$ value is chosen by numerically maximizing the transfer fidelity in a single-particle manifold [61]. To compare the difference between a single particle and a bound state, we plot (with a dashed line) also the single-particle transfer fidelity $P_L\left(t^{*}\right)=|\langle 0|a_L|\psi \left(t^{*}\right){\rangle |}^2$ obtained for a system initially in $a_1^{†}|0\rangle$. (bottom) Probability $P_{jj}\left(t\right)$ to have a two-particle bound state in site $j$ at time $t/t^{*}$ for a minimal engineered chain with $L=21$ and $U/J=8$.

Figure 4

Decoherence effects for a two-particle bound state: (top) Relative variation $\mathrm{\Delta }{P}_{LL}/P_{LL}(t^{*},\mathrm{\Gamma }=0)$ of the transfer fidelity $P_{LL}\left(t^{*}\right)$ with respect to the case in absence of decoherence, for a uniform chain with $U/J=3$ and length $L$. Here $\mathrm{\Delta }{P}_{LL}=|P_{LL}(t^{*},\mathrm{\Gamma })-P_{LL}(t^{*},\mathrm{\Gamma }=0)|$ and the dashed gray line is a threshold of a relative variation of the $5\%$. Several chain lengths $L$ are considered. (bottom) Relative variation $\mathrm{\Delta }{P}_L/P_L(t^{*},\mathrm{\Gamma }=0)$ for a single-particle state initially in $a_1^{†}|0\rangle$.

Figure 5

Two-particle NOON state: plot of the probabilities $P_{ij}\left(t\right)$ to have one particle in site $i$ and the other in $j$, as a function of time, in units of the transfer time $t^{*}$, for two particles initially in $|\psi \left(0\right)\rangle \propto \left(a_1^{†}\right)|0\rangle$. Here we consider a uniform chain with an impurity ${\mu }_j=-\beta {\delta }_{j,L/2+1}$ where $\beta =0.789(J^2/2U)$ in a chain with $U/J=5$ and length $L=5$. The absence of the $P_{1L}\left(t^{*}\right)$ term is evidence that the output state at $t=t^{*}\simeq UL/J^2$ is the NOON state with two particles. The gray dashed line represents the results for an ideal lossless NOON state generation.

Figure 6

Finite-size effects in the two-particle NOON state generation: analysis of the scaling factor $\alpha$ where ${\beta }^{50-50}=\alpha J^2/U$, as a function of the chain length $L$ for generating a two-particle NOON state. The gray line represents the theoretical results from the effective Hamiltonian theory, which holds for $L\gg 1$. (inset) Analysis of the optimal value of $\beta$ of the local field ${\mu }_j=-\beta {\delta }_{j,L/2+1}$ which produces the NOON state with two particles as a function of $U/J$ for $L=5$. The red line is the fit $\beta ={\beta }^{50-50}=0.395J^2/U$.

Figure 7

Decoherence effects for a three-particle bound state: relative variation $\mathrm{\Delta }{P}_{LL}/P_{LL}(t^{*},\mathrm{\Gamma }=0)$ with respect to decoherence-free case, for a uniform chain with $U/J=2$. Here $\mathrm{\Delta }{P}_{LL}=|P_{LL}(t^{*},\mathrm{\Gamma })-P_{LL}(t^{*},\mathrm{\Gamma }=0)|$ and the dashed gray line is a threshold of a relative variation of the $5\%$. Several chain lengths $L$ are considered.

Figure 8

Three-particle NOON state: Joint probabilities $P_{ijk}\left(t\right)$ as a function of the time, in units of the transfer time $t^{*}$, for three particles initially in $|\psi \left(0\right)\rangle \propto \left(a_1^{†}\right)|0\rangle$ for a uniform chain with an impurity $\beta =0.099J^3/U^2$ and ${\beta }^{\prime }=J^2/8U$ in the middle of the chain with $U/J=5$ and length $L=5$. The absence of the $P_{1LL}\left(t\right)$ term is evidence that the output state at $t=t^{*}$ is the NOON state with two particles. The gray dashed line represents the results for an ideal lossless NOON state generation. We found also that $P_{1LL}\left(t\right)=P_{11L}\left(t\right)$.

Figure 9

Finite-size effects in the three-particle NOON state generation: Analysis of the scaling factor $\alpha$ as a function of chain length $L$ for a three-particle bound state, where ${\beta }^{50-50}=\alpha J^3/U^2$. The dashed gray line represents the theoretical value from the effective Hamiltonian description. (inset) Analysis of the optimal value of $\beta$ to produce the NOON state with three particles as a function of $U/J$ in a uniform chain with length $L=5$. The red line is the fit ${\beta }^{50-50}=\alpha J^3/U^2$ where $\alpha =0.099$.

Figure 10

NOON state detection with bound particles: Interference fringes in a Mach–Zehnder scheme for a two-particle bound state for a chain with $L=5$ and $U/J=5$. We plot the probability $P_{11}\left(t^{*}\right)$ to have the bound-state particle in the first site after a time evolution for $t^{*}\simeq LU/J^2$, when a phase factor $\varphi$ is introduced in the system. The line data represent the result for an ideal lossless Mach–Zehnder transformation.

Figure 11

NOON state detection after quenched interparticle interaction: interference fringes after the quench to $U/J=0$. The chain has length $L=5$ and we set $U/J=5$ for generating the two-particle NOON state in the edges at time $t^{*}\simeq LU/J^2$. Once the NOON state is generated the dynamics is frozen by increasing the lattice depth, then a controllable phase factor $\varphi$ is added by tilting the lattice. Once the interparticle interaction is quenched to $U/J=0$ we let the system evolve and we measure the probability $P_{1L}\left(t^{\prime \prime }\right)$ where $t^{\prime \prime }\simeq L/J$ is the transfer time of the free chain.

Figure 12

Phase estimation precision $\mathrm{\Delta }\varphi =1/\sqrt{F_Q}$, where $F_Q$ is the quantum Fisher information for the estimator $n_L$ in a uniform chain with $L=5$ respectively for a two-particle bound state and for a three-particle bound state (inset) as a function of the onsite interaction $U/J$. The red dashed (gray) dotted lines represent the ideal quantum (classical) lower bound $\mathrm{\Delta }{\varphi }_{\mathrm{quant}}=1/M$ ($\mathrm{\Delta }{\varphi }_{\mathrm{cl}}=1/\sqrt{M}$).