Atomtronic protocol designs for NOON states

In this protocol we employ breaking of integrability through an applied field to subsystem $B$ and a measurement process. The protocol consists of three sequential steps, schematically depicted in Fig. 1:

1. (i)

   $\ket{\Psi_{1}^{\text{I}}}=\mathcal{U}(t_{\text{m}}-t_{\mu},0,0)\ket{\Psi_{0}}$;

2. (ii)

   $\ket{\Psi_{2}^{\text{I}}}=\mathcal{U}(t_{\mu},\mu,0)\ket{\Psi_{1}^{\text{I}}}$;

3. (iii)

   $\ket{\Psi_{3}^{\text{I}}}=\mathcal{M}\ket{\Psi_{2}^{\text{I}}}$,

where $t_{\text{m}}=\hbar\pi/(2\Omega)$ (see Methods) and $\mathcal{M}$ represents a projective measurement of the number of bosons at site 3 (which could be implemented, in principle, through Faraday rotation detection^{33, 34}). A measurement outcome of 0 or $M$ heralds a high-fidelity NOON state in subsystem $B$. For other measurement outcomes, the output is discarded and the process repeated (post-selection).

Now we specify an alternative protocol that does not involve measurements, so post-selection is not required. Employing the same initial state $\ket{\Psi_{0}}$, the following sequence of steps are implemented to arrive at a NOON state in subsystem $B$ (illustrated in Fig. 1):

• (i)

  $\ket{\Psi_{1}^{\text{II}}}=\mathcal{U}(t_{\text{m}}-t_{\nu},0,0)\ket{\Psi_{0}}$;

• (ii)

  $\ket{\Psi_{2}^{\text{II}}}=\mathcal{U}(t_{\nu},0,\nu)\ket{\Psi_{1}^{\text{II}}}$;

• (iii)

  $\ket{\Psi_{3}^{\text{II}}}=\mathcal{U}(t_{\text{m}}-t_{\mu},0,0)\ket{\Psi_{2}^{\text{II}}}$;

• (iv)

  $\ket{\Psi_{4}^{\text{II}}}=\mathcal{U}(t_{\mu},\mu,0)\ket{\Psi_{3}^{\text{II}}}$.

The analytic results provided above are obtained by employing the effective Hamiltonian in an extreme limit, with divergent applied fields acting for infinitesimally small times. Below we give numerical simulations of the protocols to show that, for physically realistic settings where the fields are applied for finite times, high-fidelity outcomes for NOON state production persist.

Throughout this section, we use $\ket{\Psi}$ to denote an analytic state, obtained in an idealized limit. We adopt $\ket{\Phi}$ to denote a numerically calculated state, obtained by time evolution with the EBHM Hamiltonian (2). Two sets of parameters are chosen to illustrate the results (expressed in Hz):

Set 1: {$U/\hbar=75.876$, $J/\hbar=24.886$, $\mu/\hbar=20.870$};

Set 2: {$U/\hbar=76.519$, $J/\hbar=73.219$, $\mu/\hbar=15.168$}.

The fidelities of Protocols I and II are defined as³⁵ $F_{\text{I}}=|\braket{\Psi_{3}^{\text{I}}}{\Phi_{3}^{\text{I}}}|$ and $F_{\text{II}}=|\braket{\Psi_{4}^{\text{II}}}{\Phi_{4}^{\text{II}}}|$, respectively. This is computed for $P\theta$ ranging from $0$ to $\pi$, achieved by varying $t_{\mu}$. In the case of Protocol II, we use $\nu=\mu$ for both sets of parameters. The systems considered here can, in principle, be implemented using existing hardware – see Physical proposal.

The results are presented in Fig. 2, where it is seen that $F_{\text{II}}$ is lower than $F_{\text{I}}$. This can be attributed to two primary causes. The first is that, while Protocol I takes $\tau_{I}\sim t_{\text{m}}$ to produce the final state, Protocol II requires double the evolution time $\tau_{II}\sim 2t_{\text{m}}$. The longer evolution time contributes to a loss in fidelity. The second reason is that, the measurement occurring in the final step of Protocol I has the effect of renormalizing the quantum state after collapse, which increases the fidelity of the resulting NOON state when a measurement of $r=0$ or $r=M$ is obtained. However, there is a finite probability that the measurement outcome is neither $r=0$ nor $r=M$ (see Supplementary Note 2).

In summary, both protocols display high fidelity results greater than 0.9. For Protocol I the outcomes are probabilistic (See Supplementary Note 2 for data). By contrast, the slightly lower fidelity results of Protocol II are deterministic.

A means to test the reliability of the system, through a statistical analysis of local measurement outcomes, is directly built into the design. This results from the system’s capacity to function as an interferometer³². For both protocols, once the output state has been attained we can continue to let the system evolve under $\mathcal{U}(t_{\text{m}},0,0)$. This yields the readout states, denoted as $\ket{\Psi_{\text{RO}}^{\text{I}}}$, $\ket{\Psi_{\text{RO}}^{\text{II}}}$ respectively for protocols I and II. In the idealized limits these are

where $c(\theta)\equiv\cos\left(P\theta/2\right)$ and $s(\theta)\equiv\sin\left(P\theta/2\right)$. For $\ket{\Psi_{\text{RO}}^{I}}$, the measurement probabilities at site 3 are $\mathcal{P}(0)=\cos^{2}\left({P\theta}/{2}\right)$ and $\mathcal{P}(M)=\sin^{2}\left({P\theta}/{2}\right)$. Combined with the probability of measuring $r=0,M$ in step (iii), we obtain four possibilities for the total probabilities as $\mathcal{P}_{\text{I}}(0,0)=\mathcal{P}_{\text{I}}(M,0)=0.5\cos^{2}\left({P\theta}/{2}\right)$ and $\mathcal{P}_{\text{I}}(0,M)=\mathcal{P}_{\text{I}}(M,M)=0.5\sin^{2}\left({P\theta}/{2}\right)$. Meanwhile, for $\ket{\Psi_{\text{RO}}^{II}}$, the measurement probabilities at site 3 are $\mathcal{P}_{\text{II}}(0)=\sin^{2}\left({P\theta}/{2}-{\pi}/{4}\right)$ and $\mathcal{P}_{\text{II}}(M)=\cos^{2}\left({P\theta}/{2}-{\pi}/{4}\right)$. As a numerical check, we consider the same sets of parameters from previous section. Then, we numerically calculate the above probabilities using the Hamiltonian (2), comparing the predicted analytic results with the numerical ones, as shown in Fig. 3. See Supplementary Note 2 for numerical probabilities of Protocol I, and related fidelity data. For results with Set 2, see Supplementary Note 3.

The Hamiltonian (2) has large energy degeneracies when $J=0$. Through numerical diagonalization of the intergable Hamiltonian for sufficiently small values of $J$, it is seen that the levels coalesce into well-defined bands, similar to that observed in an analogous integrable three-site model^{37, 39}. By examination of second-order tunneling processes (see Supplementary Note 1) In this regime, an effective Hamiltonian $H_{\rm eff}$ is obtained for this regime.

A very significant feature is that, for time evolution under $H_{\rm eff}$, both $N_{1}+N_{3}=M$ and $N_{2}+N_{4}=P$ are constant. The respective $(M+1)$-dimensional subspace associated with sites 1 and 3 and $(P+1)$-dimensional subspace associated with sites 2 and 4 provide the state space for the relevant energy band (see Supplementary Note 1). Restricting to these subspaces and using the effective Hamiltonian (6) yields a robust approximation for (2).

We propose a physical construction, consisting of dysprosium ¹⁶⁴Dy atoms trapped in an optical lattice, to test the theoretical results. The trapping is accomplished by employing two sets of counterpropagating laser beams with wavelength $\lambda=0.532$ $\mu$m and waist $w_{0}$, with $w_{0}\gg\lambda$. We consider each set of beams to cross with the other at an angle of $90^{\circ}$ (cyan beams in Fig. 5) , generating a square, two-dimensional optical lattice, in which the distance between nearest wells is $l={\lambda}/{2}=266$ nm. We also consider a Gaussian beam propagating towards the z-direction (blue beam in Fig. 5), with $\lambda=0.532$ $\mu$m and waist $w_{1}=1.0$ $\mu$m, aligned to the center of a four-site square plaquette, to isolate it from the rest of the lattice. Then, to achieve a pancake-shaped trap, it is necessary to include a set of two beams with $\lambda=0.532$ $\mu$m and waist $w_{2}\sim w_{0}$, whose orientations are disposed at an angle of $\alpha=60^{\circ}$ from each other (orange beams in Fig. 5), inducing a trapping aspect ratio of $\kappa^{2}\equiv\omega_{z}/\omega_{\text{r}}=1.464$. Together, they generate the potential $V(\textbf{r})$:

where $m$ is the atom’s mass, $\displaystyle\omega_{\text{r}}=\sqrt{\frac{2}{m}\left(V_{0}k^{2}+\frac{2V_{1}}{w_{1}^{2}}\right)}$ and

are, respectively, the radial and transverse trapping frequencies. Above, $\omega=\sqrt{{4V_{1}}/({mw_{1}^{2}})}$ arises due to the isolation of the four-well system from the optical lattice. The values $V_{1}=V_{0}$ and $V_{2}=9V_{0}$ are, respectively, the central beam’s and the x-z crossing beam’s potential depths, $V_{0}$ is the 2D lattice potential depth, $l$ is the distance between nearest sites, $k=2\pi/\lambda$ is the wave number, and $d_{\text{sw}}=\lambda/\left(2\sin\left({\alpha}/{2}\right)\right)$ is the distance between nearest wells along the z-axis. Since we are considering $\alpha=60\degree$, the minimum distance between the system’s horizontal layer and the next upper (or lower) layer is $d_{\text{sw}}=2l=\lambda$, which makes irrelevant the tunneling contributions between different horizontal layers.

To establish equivalency between $V(\textbf{r})$ and the Hamiltonian of Eq. (2), we employ the standard second-quantization procedure. From this, we calculate the on-site interaction parameter $U_{0}$ as:

where $\kappa$ is related to the trapping (pancake) shape aspect, $\eta\equiv m\omega_{\text{r}}/(2\hbar)$, $g\equiv 4\pi\hbar^{2}a/m$, with $a$ being the s-wave scattering length (tunable via Feshbach Resonance), $C_{\text{dd}}\equiv\mu_{0}\mu_{1}^{2}$ is the coupling constant, where $\mu_{0}$ is the vacuum magnetic permeability, $\mu_{1}$ is the atomic magnetic moment, and $f(\kappa)$ is a function that describes how the dipolar interaction behaves for different geometries (encoded in $\kappa$)³⁸. Taking site 1 as the “starting point’, the parameter $U_{1\text{j}}$, which accounts for the dipole-dipole interaction between atoms at sites 1 and j, is expressed as:

where $J_{0}$ is the Bessel function of first kind, $d_{1\text{j}}=l/\delta$, if $j=2,4$, and $d_{1\text{j}}=l\sqrt{2}/\delta$, if $j=3$. Here, the on-site dipolar interaction is given by $\displaystyle U_{\text{dip}}=\lim_{d_{ij}\to 0}U_{1\text{j}}\propto f(\kappa)$. The term $\delta=1+2V_{1}/(V_{0}k^{2}w_{1}^{2})$ arises when isolating the four-site region from the rest of the lattice, which causes the wells to slightly approach each other.

Here we give an overview of the origin for the effective Hamiltonian. Recall that the integrability condition is $U_{13}=U_{24}=U_{0}$ and $U_{12}=U_{23}=U_{34}=U_{14}$. When $J=0$, the Fock state $\ket{M-l,P-k,l,k}$ is eigenstate of the Hamiltonian (2) with energy

where $C=(U_{0}+U_{12})N^{2}/{4}-{U_{0}}/{2}$. The result is independent of $l$ and $k$, indicating degeneracies. For small values of $J$, the degeneracies are broken and lead to energy levels in well-defined bands, each with $2(M+1)(P+1)$ energy levels, except for $N$ even, where the band with the highest energy, $M=P$, will have $(M+1)(P+1)$ levels. The level energy structure of the case we are analyzing, with $N=15$, is shown in Supplementary Figure 1. In it, we highlight in cyan the band with $M=4$ and $P=11$ (and vice versa), while the vertical lines marks the two sets of parameters pointed in the main text (repeated here, expressed in Hz):

Set 1: {$U/\hbar=75.876$, $J/\hbar=24.886$, $\mu/\hbar=20.870$};

Set 2: {$U/\hbar=76.519$, $J/\hbar=73.219$, $\mu/\hbar=15.168$}.

An effective Hamiltonian for each band is obtained by consideration of second-order processes. Associated to labels $M$ and $P$, such that $N=M+P$, we obtain

For a given initial Fock state, the resonant regime is achieved when the expectation energy lies in a region characterized by an energy band. There, the values of the integrability-breaking parameters $\mu$, $\nu$ may be as large as the band-separation allows, which is depicted in Supplementary Figure 2.

Supplementary Table 1 shows the measurement probabilities of Protocol I, as well as the fidelity of the resulting state with the respective NOON state, for $M=4$, $P=11$ and the two aforementioned sets of parameters. The resulting NOON state from Protocol I can be either symmetric ($r=0$) or antisymmetric ($r=M$). For intermediate values for the outcome of measuring $N_{3}$, we calculate the fidelity of the resulting state with the symmetric NOON state ($r=0,1$) or the antisymmetric state ($r=2,3,4$), respectively.

For less ideal choices of parameters, it is possible to perform a fitting on the readout probabilities amplitudes, such that

where $c_{00}$, $c_{MM}$, $c_{0}$ and $c_{M}$ are constants that are obtained by fitting the numerically-evaluated data with the analytic models. By choosing the parameters of Set 2, we obtain the following constants from a least-squares fitting: $c_{00}=0.938$, $c_{MM}=0.893$, $c_{0}=0.954$ and $c_{M}=0.909$. The results are shown in Supplementary Figure 3.

Here we analyze the system’s robustness in the presence of a perturbation parameter, and outline a method to enhance performance. Supposing that the integrability condition is subject to an error, denoted by $\xi$:

We find that the fidelities for the parameters Set 1 are above 0.9 for an error parameter $\xi/J$ up to $\sim 0.01\%$, while the parameter Set 2 is able to produce NOON state with fidelities above $0.9$ up to $\xi/J\sim 0.03\%$.

To enhance the fidelity, we propose a procedure that consists of both positive ($+\xi$) and negative ($-\xi$) deviations in Eq. (S.2). This can be done, for instance, by considering a sequence of pulses¹. This is appropriate when considering an error parameter in the physical setup: after fixing the desired (approximate) s-wave scattering length, the trapping frequency adjustment may not have the required precision, allowing for a minimum-error of $\pm\xi$.

Considering perturbations $H^{\pm}(\mu,\nu)$ of the Hamiltonian, with the form

set $\bar{U}$, $\bar{J}$ and $\bar{\mu}$ as the mean values for the two cases $+\xi$ and $-\xi$. We then calculate the times $t_{\text{m}}$ and $t_{\mu}$ from these mean values. Next, running a simulation that alternates $N_{\delta t}$ times between the extreme coupling values during the integrable time evolution over $t_{\text{m}}-t_{\mu}$ leads to an increase in the system’s tolerance to the error, as depicted in Supplementary Figure 4.