---
author:
- Fangzhao Alex An
- 'Eric J. Meier'
- 'Jackson Ang’ong’a'
- Bryce Gadway
title: 'Supplementary materials for “Correlated dynamics in a synthetic lattice of momentum states”'
---

[^1]

[^2]

Fitting and simulation procedure
================================

We determine the tunneling energy $t$, average mean-field shift $U$, and initial condensate momentum $p_i$ by performing a combined fit of all experimental data in Figs.  2(a-d) of the main text to numerical simulations.  Specifically, we minimize the combined residuals of the two Bragg spectroscopy data sets for the $0\rightarrow\pm 1$ transitions and the two (positive and negative) sweep data sets as a function of the simulation parameters $t$, $U$, and $p_i$.

For each data set, we generated simulation curves by solving the nonlinear Schrödinger equation of Eq.  2 with appropriate detuning/well imbalance $\Delta$ (fixed $\Delta$ for the Bragg data and swept $\Delta$ for the sweep data).  The simulations assume a given tunneling energy $t$, initial condensate momentum $p_i$ (relating to an added intersite bias), and peak mean-field energy $U_0$. In the spirit of a local density approximation, we take into account the inhomogeneous density and inhomogeneous mean-field energy of our trapped atomic condensates by assuming a Thomas-Fermi density distribution and taking a population-weighted sum of simulation curves based on a homogeneous mean-field energy $U$, ranging from $0$ to $U_0$.

We took the residual sum of squares for each data set, and weighted each of these by the number of data points in the respective data set to give equal weighting to all data sets before summing up all of the values to get one fitting metric.  We then minimized this weighted sum with respect to the input values $t$, $U_0$, and $p_i$, obtaining tunneling energy $t/\hbar \approx 2\pi\times 390$ Hz, average mean-field energy $U/\hbar \approx 2\pi\times 1810$ Hz relating to an inhomogeneous distribution with peak mean-field energy $U_0/\hbar \approx 2\pi\times 3170$ Hz, and an initial condensate momentum $p_i \approx -0.018 \, \hbar k$.

Calculation of adiabatic energy levels
======================================

Even ignoring quantum fluctuations and nonclassical correlations, the nonlinear momentum-space interactions can result in interesting behavior of the two-mode condensate field, as described in the main text.  Here, we briefly describe the calculations (following the work of Ref.  [@GORDON]) used to determine the adiabatic energy levels and their model projections, as shown in Fig.  2 of the main text.  The identification of cusps, or swallow-tails, in the adiabatic energy level structure provides insight into the mechanism underlying the observed dynamical phenomenon of nonlinear self-trapping. 

We approximate the state of our two-mode Bose–Einstein condensate (with possible single particle states represented in terms of superpositions of the left ($|L\rangle \equiv |p/2\hbar k = 0\rangle$) and right well ($|R\rangle \equiv |p/2\hbar k = 1\rangle$) orbitals) in terms of the field $|\psi (\tau) \rangle = \sqrt{N} [\phi_L (\tau) |L\rangle +\phi_R (\tau) |R\rangle]$, where $\tau$ is the time variable, and the complex amplitudes $\phi_L$ and $\phi_R$ relate to populations $N_L = N|\phi_L|^2$ and $N_R = N|\phi_R|^2$ in the left and right well.  As described in the main text, the nonlinear Schrödinger equation describing the evolution of the condensate wave function results in the two coupled differential equations $$\begin{aligned}
i\hbar \dot{\phi}_L &= U [2 - |\phi_L|^2]\phi_L - t \phi_R \\
i\hbar \dot{\phi}_R &= U [2 - |\phi_L|^2]\phi_R - t \phi_L + \Delta (\tau)\phi_R \ ,\end{aligned}$$ where $U$ is the homogeneous mean-field energy relating to direct interactions, $t$ is the real-valued tunneling energy, and $\Delta$ is the site-to-site energy bias.  In experiment, the bias is swept in time as $\Delta (\tau) = \Delta_i - 2\Delta_i (\tau/T)$ for a sweep duration $T = 1$ ms. 

Intuition into the dynamical response of this system is gained by first considering the steady-state solutions to these coupled equations for various conditions of $U$, $t$, and $\Delta$. We determine the adiabatic energy levels relating to steady state solutions of the form $|\psi (\tau) \rangle = e^{-i\mu\tau} |\psi (0) \rangle$ by solving $$\begin{aligned}
\mu \phi_L &= [U(2 - |\phi_L|^2)]\phi_L - t \phi_R
\label{eee} \\
\mu \phi_R &= [\Delta + U(2 - |\phi_R|^2)]\phi_R - t \phi_L \ ,
\label{fff}\end{aligned}$$ where $\mu$ is the chemical potential. 

Solution of these coupled equations determines the adiabatic eigenstate solutions, their energy values, and their projections onto the two available momentum modes (wells).  For a given combination of $U$, $t$, and $\Delta$ values, between two and four solutions with distinct $\mu$ and $|\phi_L|$ combinations (and with $|\phi_R|^2 = 1 - |\phi_L|^2$ fixing $|\phi_R|$) can be found, with the presence of more than two solutions relating to cusp or swallow-tail structures in the energy band structures.  The projected probabilities onto the left and right well states $|L\rangle$ and $|R\rangle$ are given by $$|\phi_L|^2 = 1 - |\phi_R|^2 = \frac{\Delta + U - \mu}{\Delta + 2 (U-\mu)}  \ .$$

The eigenstate solutions, having different $\mu$ values in general, are expressed in terms of these projected probabilities as $|\mathrm{I}\rangle = \sqrt{N} [ |\phi_L| |L\rangle - |\phi_R| |R\rangle ]$ (the ground state) and $|\mathrm{II}\rangle = \sqrt{N} [ |\phi_L| |L\rangle + |\phi_R| |R\rangle ]$ (the excited state).  The relevant $\mu$ values may be determined by solving the determinant $$\Bigg | \left(
   \begin{array}{cc}
     U(2-|\phi_L|^2) - \mu & -t \\
     -t & \Delta + U(2-|\phi_R|^2) - \mu \\
   \end{array}
 \right) \Bigg| = 0$$ in combination with solution of Eq.  \[eee\] and Eq.  \[fff\].  We note that in the main text (Fig.  2), these $\mu$ values are labeled as the energies $E$ of the adiabatic energy levels. 

Effect of interactions on quantum correlations in the momentum-space double well
================================================================================

The role of quantum fluctuations of the momentum-space double well site populations has been ignored in the main text, yet significant quantum correlations can develop in such a system as a result of interactions.  In particular, the momentum-space interaction shown to lead to the onset of nonlinear self-trapping in the main text can also be harnessed to generate squeezing of the momentum-space distribution.  Specifically, a generic two-mode quantum system featuring mode-dependent interactions can be mapped onto an effective collective spin Hamiltonian featuring a one-axis twisting term that can generate squeezed collective spin states [@RaghavanBigelow; @Esteve-Squeezing; @Jo-squeezing; @Gross-Squeezing]. 

We consider a system of particles distributed between two momentum modes (labeled mode $0$ and mode $1$) with a fixed total particle number $N = N_0 + N_1$. In the main text, when simulating the experimental data for the case of $N \sim 10^5$ atoms and only two sites, we ignored quantum fluctuations of the atomic populations.  This assumed that for a given site $n$ with particle number $N_n$, $\langle \hat{c}_n \rangle \sim \langle \hat{c}_n^\dagger \rangle \sim \sqrt{N_n}$, where $\hat{c}_n$ ($\hat{c}^\dagger_n$) is the annihilation (creation) operator for the mode $n$. This assumption allowed us to describe the system by a greatly simplified multimode Gross-Pitaevskii equation [@Trippenbach-FWM-theory].  More generally, following the treatment of Ref.  [@RaghavanBigelow], we may include the effect of quantum fluctuations by considering instead a second quantized Hamiltonian $H = H_\text{sp} + H_\text{int}$ that describes two-body interactions of atoms in the same and different modes ($H_\text{int}$), as well as the engineered synthetic lattice tunneling and site energies ($H_\text{sp}$).  For the double well, these contributions may be written as $$H_\text{sp} = \varepsilon_0 \hat{c}^\dagger_0 \hat{c}_0 + \varepsilon_1 \hat{c}^\dagger_1 \hat{c}_1 - t (\hat{c}^\dagger_0 \hat{c}_1 + \hat{c}^\dagger_1 \hat{c}_0)$$ and $$H_\text{int} = \frac{u_{00}}{2}(\hat{c}^\dagger_0 \hat{c}^\dagger_0 \hat{c}_0 \hat{c}_0) + \frac{u_{11}}{2}(\hat{c}^\dagger_1 \hat{c}^\dagger_1 \hat{c}_1 \hat{c}_1) + u_{01}(\hat{c}^\dagger_0 \hat{c}_0 \hat{c}^\dagger_1  \hat{c}_1) \ .$$ Here, $t$ is the tunneling energy, which we assume to be real, and $\varepsilon_0$ and $\varepsilon_1$ are the site energies (with energy bias $\Delta = \varepsilon_1 - \varepsilon_0$), which may be controlled through the Bragg frequency detuning.  The interaction coefficients are given by $u_{00} = u_{11} = U/N$ (for the assumed uniform mean-field energy $U$ and total particle number $N$) and $u_{01} = 2U/N$, with the difference arising from the added exchange interactions [@Trippenbach-FWM-theory]. 

![ \[FIG:figs1\] **Squeezing in a momentum-space double well.** (** a**) Visualization of many-particle ($N = 100$) spin states $|\Psi\rangle$ through their overlap with different coherent spin states $|\langle \theta,\varphi|\Psi\rangle|^2$. Shown are the cases of an initial coherent spin state $|\pi/2,\pi\rangle$ (upper plot), and the transformed state after evolution under $H_{\mathrm{sq}}$ for a time $\kappa \tau_{\mathrm{sq}}/\hbar = 0.0173 \pi$ (lower plot). (** b**) Squeezing along the $\hat{z}$-axis, $\xi_z$, for different evolution times $\kappa \tau_{\mathrm{sq}}/\hbar = \{0.1,0.25,0.5,0.75,1\}\times 0.0173 \pi$ (solid lines, with colors varying from blue to red) and for different angles of rotation $\varphi_{\mathrm{rot}}$ of the final distribution about $\hat{J}_x$. ](FigS1.pdf){width="\columnwidth"}

Analysis of this interacting two-mode system is simplified by considering the atomic distribution in terms of a collective spin.  Each atom is imbued with an effective spin-$1/2$ degree of freedom relating to the two momentum modes that may be occupied.  The total state of the system may be considered as an effectively large, collective spin with maximum length $N/2$. One may define effective angular momentum operators relating to the coherences and macroscopic occupations $N_0$ and $N_1$ of two possible momentum orders (ignoring thermal and quantum depletion), given by $$\hat{J}_x = (\hat{c}_0^\dag \hat{c}_1 + \hat{c}_1^\dag \hat{c}_0)/2$$ $$\hat{J}_y = i(\hat{c}_0^\dag \hat{c}_1 - \hat{c}_1^\dag \hat{c}_0)/2$$ $$\hat{J}_z = (\hat{c}_1^\dag \hat{c}_1 - \hat{c}_0^\dag \hat{c}_0)/2 \ .$$ In this modified description in terms of a collective spin, we will consider a basis of Dicke states $|j,m\rangle$ having total spin $j = N/2$ and $\hat{z}$-projection $m = (N_1 - N_0) / 2$, describing the collective two-mode number states.  Uncorrelated coherent spin states (CSSs) of the form $$|\theta , \varphi\rangle = \sum_{m = -j}^{j} f_{m}^j (\theta) e^{-i(j+m)\varphi}|j,m\rangle \ ,$$ for $f_{m}^j (\theta) = \binom{2j}{j+m}^{1/2}\cos(\theta/2)^{j-m}\sin(\theta/2)^{j+m}$, result from a global rotation of spin-polarized states $|j,j\rangle$ about the spin vector $\hat{n}_\varphi = \cos(\varphi) \hat{J}_x + \sin(\varphi) \hat{J}_y$ by an angle $\theta$ [@CSS-RMP].  In this description in terms of a collective spin, the momentum-space interaction gives rise to an effective nonlinear squeezing Hamiltonian $H_{\mathrm{sq}} = \kappa \hat{J}_z^2$ (after removing the contribution from mode-independent interactions), where $\kappa = (u_{00} + u_{11})/2 - u_{01} = -U/N$ [@RaghavanBigelow]. 

When combined with single particle manipulations of the effective spin degree of freedom through control of $H_{sp}$, the interactions can be used to generate correlations and entanglement in the double well system.  We examine the case of how interactions modify an initially prepared CSS $|\pi/2,\pi\rangle$ aligned along $-\hat{J}_x$. For short evolution times, the nonlinear Hamiltonian $H_{\mathrm{sq}}$ leads to a “shearing” of such coherent states.  This is depicted in Fig.  , for the initial CSS (upper plot) and the sheared, non-classical squeezed state after a time $\kappa\tau_{\mathrm{sq}}/\hbar = 0.0173 \pi$ (lower plot), through the overlap of these states with CSSs of varying $\theta$ and $\varphi$ values.  For ease of calculation, dynamics are shown for the case of only $N = 100$ atoms ($j = 50$). 

Figure  shows, for sheared distributions relating to various evolution times $\tau_{\mathrm{sq}}$, the $\hat{z}$-axis squeezing parameter $\xi_z = 2 j \frac{\langle \Delta\hat{J}_z^2\rangle}{j^2 - \langle\hat{J}_z\rangle^2}$ as a function of rotation angle $\varphi_\text{rot}$ about the $\hat{J}_x$ spin axis.  For typical experimental parameter values ($N = 10^5$ atoms, $U / \hbar = 2 \pi \times 1.5$ kHz), an optimal squeezing of $\xi_z^\text{min} \approx (3j^2)^{-1/3}$, relating to $-33$ dB (after appropriate rotation of the sheared distribution about the $\hat{J}_x$ spin axis), would be expected after a total duration $\tau_{\mathrm{sq}} \approx 1.13 j^{1/3}(\hbar/U) \approx 6.6$ ms [@RaghavanBigelow].  The same squeezing could be achieved in a shorter time through a temporary modification of the atomic scattering length, such as through a magnetically tunable Feshbach resonance.  Shorter squeezing timescales would help to mitigate effects owing to spatial separation of the momentum orders, which would in practice degrade and limit the available squeezing.  Additionally, refocusing $\pi$ (echo) pulses or twist-and-turn squeezing schemes [@Muessel-TWIST] can be used to maintain spatial overlap of the momentum states. 

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[^1]: These authors contributed equally to this work

[^2]: These authors contributed equally to this work
