A spin-orbit coupled Bose-Einstein condensate

1 System preparation

Our experiments began with nearly pure $\approx 1.8\times 10^{5}$ atom ${}^{87}\text{Rb}$ BECs in the $\mathinner{|{F=1,m_{F}=-1}\rangle}$ state confined in a crossed optical dipole trap. The trap consisted of a pair of $1064{\ \text{nm}}$ laser beams propagating along $\hat{x}-\hat{y}$ ($1/e^{2}$ radii of $w_{\hat{x}+\hat{y}}\approx 120{\ \mu\text{m}}$ and $w_{\hat{z}}\approx 50{\ \mu\text{m}}$) and $-\hat{x}-\hat{y}$ ($1/e^{2}$ radii of $w_{\hat{x}-\hat{y}}\approx w_{\hat{z}}\approx 65{\ \mu\text{m}}$).

We prepared equal mixtures of $\mathinner{|{F=1,m_{F}=-1}\rangle}$ and $\mathinner{|{1,0}\rangle}$ using an initially off resonant rf magnetic field $B_{\text{rf}}(t)\hat{x}$. We adiabatically ramped $\delta$ to $\delta\approx 0$ in $15{\ \text{ms}}$, decreased the rf coupling strength $\Omega_{\text{rf}}$ to about $150{\ \text{Hz}}\ll\hbar\omega_{q}$ in $6{\ \text{ms}}$, and suddenly turned off $\Omega_{\text{rf}}$, projecting the BEC into an equal superposition of $\mathinner{|{m_{F}=-1}\rangle}$ and $\mathinner{|{m_{F}=0}\rangle}$. We subsequently ramped $\delta$ to its desired value in $6{\ \text{ms}}$ and then linearly increased the intensity of the Raman lasers from zero to the final coupling $\Omega$ in $70{\ \text{ms}}$.

2 Magnetic fields

Three pairs of Helmholtz coils, orthogonally aligned along $\hat{x}+\hat{y}$, $\hat{x}-\hat{y}$ and $\hat{z}$, provided bias fields $(B_{x+y},B_{x-y},\text{and }B_{z})$. By monitoring the $\mathinner{|{F=1,m_{F}=-1}\rangle}$ and $\mathinner{|{1,0}\rangle}$ populations in a nominally resonant rf dressed state, prepared as above, we observed a short-time (below $\approx 10$ minutes) RMS field stability $g\mu_{\rm B}B_{\rm RMS}/h\lesssim 80{\ \text{Hz}}$. The field drifted slowly on longer time scales (but changed abruptly when unwary colleagues entered through our laboratory’s ferromagnetic doors). We compensated for the drift by tracking the rf and Raman resonance conditions.

Due to the small energy scales involved in the experiment, it was crucial to minimize magnetic field gradients. We detected stray gradients by monitoring the spatial distribution of $\mathinner{|{m_{F}=-1}\rangle}$-$\mathinner{|{m_{F}=0}\rangle}$ spin mixtures after TOF. Small magnetic field gradients caused this otherwise miscible mixture to phase separate along the direction of the gradient. We canceled the gradients in the $\hat{x}\!-\!\hat{y}$ plane with two pairs of anti-Helmholtz coils, aligned along $\hat{x}\!+\!\hat{y}$ and $\hat{x}\!-\!\hat{y}$, to $g\mu_{B}B^{\prime}/h\lesssim 0.7{\ \text{Hz}}/\mu\text{m}$.

References

3 SO coupled Hamiltonian

Our system consisted of a $F=1$ BEC with a bias magnetic field along $\hat{y}$ at the intersection of two Raman laser beams propagating along $\hat{x}+\hat{y}$ and $-\hat{x}+\hat{y}$ with angular frequencies $\omega_{L}$ and $\omega_{L}+\Delta\omega_{L}$, respectively. The rank-1 tensor light shift of these beams produced an effective Zeeman magnetic field along the $z$ direction with Hamiltonian $\hat{H}_{R}=\Omega_{R}\check{\sigma}_{3,z}\cos(2k_{L}\hat{x}+\Delta\omega_{L}t)$, where $\check{\sigma}_{3,x,y,z}$ are the $3\times 3$ Pauli matrices and we define $\check{1}_{3}$ as the $3\times 3$ identity matrix. If we take $\hat{y}$ as the natural quantization axis (by expressing the Pauli matrices in a rotated basis $\check{\sigma}_{3,y}\rightarrow\check{\sigma}_{3,z}$, $\check{\sigma}_{3,x}\rightarrow\check{\sigma}_{3,y}$, and $\check{\sigma}_{3,z}\rightarrow\check{\sigma}_{3,x}$) and make the rotating wave approximation, the Hamiltonian for spin states $\left\{\mathinner{|{m_{F}=+1}\rangle},\mathinner{|{0}\rangle},\mathinner{|{-1}\rangle}\right\}$ in the frame rotating at $\Delta\omega_{L}$ is

As we justify below, $\mathinner{|{m_{F}=+1}\rangle}$ can be neglected for large enough $\hbar\omega_{q}$, which gives the effective two-level Hamiltonian

for the pseudo-spin $\mathinner{|{\uparrow}\rangle}=\mathinner{|{m_{F}=0}\rangle}$ and $\mathinner{|{\downarrow}\rangle}=\mathinner{|{-1}\rangle}$ where $\Omega=\Omega_{R}/\sqrt{2}$. After a local pseudo-spin rotation by $\theta(\hat{x})=2k_{L}\hat{x}$ about the pseudo-spin $\hat{z}$ axis followed by a global pseudo-spin rotation $\check{\sigma}_{z}\rightarrow\check{\sigma}_{y}$, $\check{\sigma}_{y}\rightarrow\check{\sigma}_{x}$, and $\check{\sigma}_{x}\rightarrow\check{\sigma}_{z}$, the $2\times 2$ Hamiltonian takes the SO coupled form

The SO term linear in $\hat{k}_{x}$ results from the non-commutation of the spatially-dependent rotation about the pseudo-spin $z$ axis and the kinetic energy.

4 Effective two-level system

For atoms in $\mathinner{|{m_{F}=-1}\rangle}$ and $\mathinner{|{m_{F}=0}\rangle}$ with velocities $\hbar{\mathpzc k}_{x}/m\approx 0$ and Raman-coupled near resonance, $\delta\approx 0$, the $\mathinner{|{m_{F}=+1}\rangle}$ state is detuned from resonance owing to the $\hbar\omega_{q}=3.8E_{L}$ quadratic Zeeman shift. For $\delta/4E_{L}\ll 1$ and $\Omega<4E_{L}$, $\Delta(\Omega,\delta)\approx\delta[1-(\Omega/4E_{L})^{2}]^{1/2}$.

5 Effect of the neglected state

In our experiment, we focused on the two level system formed by the $\mathinner{|{m_{F}=-1}\rangle}$ and $\mathinner{|{m_{F}=0}\rangle}$ states. We verified the validity of this assumption by adiabatically eliminating the $\mathinner{|{m_{F}=+1}\rangle}$ state from the full three level problem. To second order in $\Omega$, this procedure modifies the detuning $\delta$ and SO coupling strength $\alpha$ in Eq. 1 by

In these expressions, we have retained only largest term in a $1/\omega_{q}$ expansion. In our experiment, where $\hbar\omega_{q}=3.8E_{L}$, $\delta$ is substantially changed at our largest coupling $\Omega=7E_{L}$. To maintain the desired detuning $\delta$ in the simple 2-level model (i.e., $\Delta\approx\delta+\delta^{(2)}=0$ in Fig. 1c), we changed $g\mu_{\rm B}B_{0}$ by as much as $3E_{L}$ to compensate for $\delta^{(2)}$. We did not correct for the always small change to $\alpha$.

Although both terms are small at the $\Omega=0.2E_{L}$ transition from miscible to immiscible, slow drifts in $B_{0}$ prompted us to locate $\Delta=0$ empirically from the equal population condition, $N_{\rm T\uparrow^{\prime}}=N_{\rm T\downarrow^{\prime}}$. As a result, $\delta$ in Eq. 1 implicitly includes the perturbative correction $\delta^{(2)}$.

6 Origin of the effective interaction term

The additional $c^{\prime}_{\uparrow\downarrow}$ term in the interaction Hamiltonian for dressed spins directly results from transforming into the basis of dressed spins, which are

where $\hbar K_{x}/m$ is the group velocity, $K_{x}=q-q_{\uparrow}$ for $\mathinner{|{\uparrow^{\prime}}\rangle}$ and $K_{x}=q-q_{\downarrow}$ for $\mathinner{|{\downarrow^{\prime}}\rangle}$, and $\epsilon=\Omega/8E_{L}\ll 1$. Thus, in second quantized notation, the dressed field operators transform according to

where $q_{\uparrow}\approx-\sqrt{1-4\epsilon^{2}}k_{L}\approx-k_{L}$ and $q_{\downarrow}\approx\sqrt{1-4\epsilon^{2}}k_{L}\approx k_{L}$. Inserting the transformed operators into

gives the interaction Hamiltonian for dressed spins which can be understood order-by-order (both $c_{2}/c_{0}$ and $\epsilon$ are treated as small parameters). In this analysis, the terms proportional to $c_{2}$ are unchanged to order $c_{2}/c_{0}$, and we only need to evaluate the transformation of the spin-independent term (proportional to $c_{0}$). At $O(\epsilon)$ and $O(\epsilon^{3})$ all the terms in the expansion include high spatial frequency $e^{\pm 2ik_{L}x}$ or $e^{\pm 4ik_{L}x}$ prefactors. For density distributions that vary slowly on the $\lambda/2$ length scale these average to zero. The $O(\epsilon^{2})$ term, however, has terms without these modulations, and is

giving rise to $c^{\prime}_{\uparrow\downarrow}=c_{0}\Omega^{2}/(8E_{L}^{2})$.

7 Mean field phase diagram

We compute the mean-field phase diagram for a ground state BEC composed of a mixture of dressed spins in an infinite homogeneous system. This applies to our atoms in a harmonic trap in the limit of $R\gg\xi_{s}$, where $R$ is the system size, $\xi_{s}=\sqrt{\hbar^{2}/2m|c_{2}+c^{\prime}_{\uparrow\downarrow}|n}$ is the spin healing length and $n$ is the density. We first minimize the interaction energy $\hat{H}_{\rm I}$ at fixed $N_{\uparrow^{\prime},\downarrow^{\prime}}$, with an effective interaction $c^{\prime}_{\uparrow\downarrow}$ as a function of $\Omega$. The two dressed spins are either phase-mixed, both fully occupying the system’s volume $V$, or phase-separated with a fixed total volume constraint $V=V_{\uparrow^{\prime}}+V_{\downarrow^{\prime}}$. For the phase-separated case, minimizing the free energy gives the volumes $V_{\uparrow^{\prime}}$ and $V_{\downarrow^{\prime}}$, determined by $N_{\uparrow^{\prime},\downarrow^{\prime}}$ and $V$. The interaction energy of a phase-mixed state is smaller than that of a phase-separated state for the miscibility condition $c_{0}+c_{2}+c^{\prime}_{\uparrow\downarrow}/2<\sqrt{c_{0}(c_{0}+c_{2})}$, corresponding to $\Omega<\Omega_{c}$. This condition is independent of $N_{\uparrow^{\prime},\downarrow^{\prime}}$: for any $N_{\uparrow^{\prime},\downarrow^{\prime}}$ the system is miscible at $\Omega<\Omega_{c}$. Then, at a given $\Omega$, we minimize the sum of the interaction energy and the single-particle energy from the Raman detuning, $(N_{\uparrow^{\prime}}-N_{\downarrow^{\prime}})\delta/2$, allowing $N_{\uparrow^{\prime},\downarrow^{\prime}}$ to vary. For the miscible case $(\Omega<\Omega_{c})$, the BEC is a mixture with fraction $N_{\downarrow^{\prime}}/(N_{\uparrow^{\prime}}+N_{\downarrow^{\prime}})\in(0,1)$ only in the range of detuning $\delta\in(\delta_{0}-W_{\delta},\delta_{0}+W_{\delta})$, where $\delta_{0}=c_{2}n/2$, $W_{\delta}=|\delta_{0}|(1-\Omega/\Omega_{c})^{1/2}$ and $n=(N_{\uparrow^{\prime}}+N_{\downarrow^{\prime}})/V$. For the immiscible case ($\Omega>\Omega_{c}$), $W_{\delta}=(c_{2}/8c_{0})c_{2}n$ is negligibly small compared to $c_{2}n$.

Figure 2b shows the mean field phase diagram as a function of $(\Omega,\delta)$, where $\delta/E_{L}$ is displayed with a quasi-logarithmic scaling, ${\rm sgn}(\delta/E_{L})\left[\log_{10}\left(|\delta/E_{L}|+|\delta_{\rm min}/E_{L}|\right)-\log_{10}|\delta_{\rm{min}}/E_{L}|\right]$, in order to display $\delta$ within the range of interest. This scaling function smoothly evolves from logarithmic for $|\delta|\gg\delta_{\rm{min}}$, $\approx{\rm sgn}(\delta/E_{L})\log_{10}|\delta/E_{L}|$, to linear for $|\delta|\ll\delta_{\rm min}$, $\approx\delta$, where $\delta_{\rm min}/E_{L}=0.001E_{L}=1.5~$Hz.

In our measurement of the dressed spin fraction $f_{\downarrow^{\prime}}$ (see Fig. 3a), $\delta=0$ is determined from the $N_{\rm T\uparrow^{\prime}}=N_{\rm T\downarrow^{\prime}}$ condition. We identify this condition as $\delta=\delta_{0}$ and apply it for all hold time $t_{h}$. Because $|\delta_{0}|\approx 3~$Hz is below our $\approx 80~$Hz RMS field noise, we are unable to distinguish $\delta_{0}$ from 0.

8 Recombining TOF images of dressed spins

To probe the dressed spin states (Eq. 6), each of which is a spin and momentum superposition, we adiabatically mapped them into bare spins, $\mathinner{|{\uparrow,{\mathpzc k}_{x}=q_{\uparrow}+k_{L}}\rangle}$ and $\mathinner{|{\downarrow,{\mathpzc k}_{x}=q_{\downarrow}-k_{L}}\rangle}$, respectively. Then, in each image outside a $\approx 90{\ \mu\text{m}}$ radius disk containing the condensate for each spin distribution, we fit $n_{\rm T{\uparrow^{\prime}},\rm T{\downarrow^{\prime}}}(x,y)$ to a gaussian modeling the thermal background and subtracted that fit from $n_{\rm T{\uparrow^{\prime}},\rm T{\downarrow^{\prime}}}(x,y)$ to obtain the condensate 2D density $n_{\uparrow^{\prime},\downarrow^{\prime}}(x,y)$. Thus, for each dressed spin we readily obtained the temperature, total number $N_{\rm T\uparrow^{\prime},\rm T\downarrow^{\prime}}$, and condensate densities $n_{\uparrow^{\prime},\downarrow^{\prime}}(x,y)$.

To analyze the miscibility from the TOF images where a Stern-Gerlach gradient separated individual spin states, we recentered the distributions to obtain $n_{\uparrow^{\prime}}(x,y)$ and $n_{\downarrow^{\prime}}(x,y)$. This took into account the displacement due to the Stern-Gerlach gradient and the nonzero velocities $\hbar{\mathpzc k}_{x}/m$ of each spin state (after the adiabatic mapping). The two origins were determined by the following: we loaded the dressed states at a desired coupling $\Omega$ but with detuning $\delta$ chosen to put all atoms in either $\mathinner{|{\downarrow^{\prime}}\rangle}$ or $\mathinner{|{\uparrow^{\prime}}\rangle}$. Since $q_{\uparrow,\downarrow}=\mp(1-\Omega^{2}/32E_{L}^{2})k_{L}$ (see Fig. 1c), these velocities $\hbar{\mathpzc k}_{x}/m=\hbar(q_{\uparrow}+k_{L})/m,\hbar(q_{\downarrow}-k_{L})/m$ depend slightly on $\Omega$, and our technique to determine the distributions’ origin accounts for this effect.

9 Calibration of Raman Coupling

Both Raman lasers were derived from the same Ti:Sapphire laser at $\lambda\approx 804.1$ nm, and were offset from each other by a pair of AOMs driven by two phase locked frequency synthesizers near $80{\ \text{MHz}}$. We calibrated the Raman coupling strength $\Omega$ by fitting the three-level Rabi oscillations between the $m_{F}=-1,0,\ {\rm and}\ +1$ states driven by the Raman coupling to the expected behavior.