Cold polar molecules in two-dimensional traps: Tailoring interactions with external fields for novel quantum phases

We discuss techniques to engineer effective long-range interactions between polar molecules using external static electric and microwave fields. We consider a setup where molecules are trapped in a two-dimensional pancake geometry by a far-off-resonance optical trap, which ensures the stability of the dipolar collisions. We detail how to modify the shape and the strength of the long-range part of interaction potentials, which can be utilized to realize interesting quantum phases in the context of cold molecular gases.

Figure 1

(Color online) System setup: Polar molecules are trapped in the $(x,y)$ plane by an optical lattice made of two counterpropagating laser beams with wave vectors $\pm {\mathbf{k}}_{\mathrm{L}}=\pm k_{\mathrm{L}}{\mathbf{e}}_z$ (arrows on the top and bottom). The dipoles ${\mathbf{d}}_j$ are aligned in the $z$ direction by a dc electric field ${\mathbf{E}}_{\mathrm{dc}}\equiv E_{\mathrm{dc}}{\mathbf{e}}_z$ (arrow on the left). An ac (microwave) field ${\mathbf{E}}_{\mathrm{ac}}$ is indicated (arrow on the right). Inset: Definition of polar $\left(\vartheta \right)$ and azimuthal $\left(\phi \right)$ angles for the relative orientation of the intermolecular collision axis $\mathbf{r}$ with respect to a space-fixed frame with axes $\{{\mathbf{e}}_x,{\mathbf{e}}_y,{\mathbf{e}}_z\}$.

Figure 2

(Color online) Qualitative sketch of effective 2D potentials $V_{\mathrm{eff}}^{2\mathrm{D}}\left(\mathit{\rho }\right)$ for polar molecules confined in a 2D (pancake) geometry. Here, $\mathit{\rho }=r\sin \vartheta (\cos \phi ,\sin \phi )$ is the 2D coordinate in the plane $z=0$ and $\rho =r\sin \vartheta$ (see inset of Fig. 1). Solid line: Repulsive dipolar potential $V_{\mathrm{eff}}^{2\mathrm{D}}\left(\mathit{\rho }\right)=C_3∕{\rho }^3$ induced by a dc electric field. Dash-dotted line: Steplike potential induced by a single ac (microwave) field and a weak dc field. Dashed line: Attractive potential induced by the combination of several ac (microwave) fields and a weak dc field. Here, the potentials $V_{\mathrm{eff}}^{2\mathrm{D}}\left(\mathit{\rho }\right)$ and the separation $\rho$ are given in arbitrary units. For the steplike case (dash-dotted line) $\rho =1$ corresponds to the Condon point $r_{\mathrm{C}}$ of Sec. 3C.

Figure 3

(Color online) Sketch of the phase diagram for a homogeneous 2D system of polar molecules interacting via the effective 2D repulsive potential $V_{\mathrm{eff}}^{2\mathrm{D}}\left(\mathit{\rho }\right)=C_3∕{\rho }^3$. $T$ is the temperature in units of $T_d\equiv C_3∕k_{\mathrm{B}}{a}^3$, with $a$ the average interparticle distance and $k_{\mathrm{B}}$ the Boltzmann constant. The symbol $r_d\equiv E_{\mathrm{int}}∕E_{\mathrm{kin}}=C_{3}m∕{\hslash }^{2}a$ is the interaction $(E_{\mathrm{int}}=C_3∕a^3)$ to kinetic energy $(E_{\mathrm{kin}}={\hslash }^2∕m{a}^2)$ ratio. A crystalline phase appears for large ratios $r_d>r_{\mathrm{QM}}$ and small temperatures $T<T_{\mathrm{m}}$. The critical ratio $r_{\mathrm{QM}}\approx 18\pm 4$ for the quantum melting to a superfluid phase has been determined in Ref. 31, while the classical melting temperature $T_{\mathrm{m}}$ (dashed line) to a normal gas phase has been calculated in Ref. 37. The finite-temperature superfluid to normal fluid phase transition is of the Berezinskii-Kosterlitz-Thouless type [38, 39] and it appears below the upper bound $T_{\mathrm{KT}}=\pi {\hslash }^2∕2k_{\mathrm{B}}m{a}^2$ (dash-dotted line). The crossover to an unstable regime for small repulsion and finite confinement in the $z$ direction (see Fig. 1) is indicated by a hatched region (see text, Sec. 3).

Figure 4

(Color online) Solid lines: Energies $E_{J,M}$ (left) and states $\mid {\varphi }_{J,M}⟩$ (right) of Eq. (4) with $J=0,1$, for a molecule in a weak dc electric field ${\mathbf{E}}_{\mathrm{dc}}=E_{\mathrm{dc}}{\mathbf{e}}_0$ with $\beta \equiv d{E}_{\mathrm{dc}}∕B\ll 1$. The dc-field-induced splitting $\hslash \delta$ and the average energy separation $\hslash \overline{\omega }$ are $\hslash \delta =3B{\beta }^2∕20$ and $\hslash \overline{\omega }=2B+B{\beta }^2∕6$, respectively. Dashed and dotted lines: Energy levels for a molecule in combined dc and ac fields (The ac Stark shifts of the dressed states are not shown). Dashed line: The ac field is monochromatic, with frequency $\omega$, linear polarization $q=0$, and detuning $\Delta =\omega -(\overline{\omega }+2\delta ∕3)>0$. Dotted lines: Schematics of energy levels for an ac field with polarization $q=\pm 1$ and frequency ${\omega }^{\prime }\ne \omega$.

Figure 5

(Color online) BO potentials $E_{J;Y;\sigma }\left(\mathbf{r}\right)$ as a function of the distance $r$, for two molecules interacting in the absence of external fields, $E_{\mathrm{dc}}={\omega }_{\perp }=\beta =0$ (see text and Table ). Here, $J=J_1+J_2$, $Y$, and $\sigma$ are the total number of rotational excitations shared by the two molecules, the quantum number associated with the projection of the total internal angular momentum onto the collision axis ${\mathbf{e}}_r\bullet ({\mathbf{J}}_1+{\mathbf{J}}_2)$, and the permutation symmetry under the exchange of the particles, respectively. The solid and dashed curves correspond to symmetric $(\sigma =+)$ and antisymmetric $(\sigma =-)$ eigenstates, respectively. Each potential energy surface is labeled by the corresponding energy and eigenstate (see Table ). Here, $r_B\equiv (d^2∕B{)}^{1∕3}$, with $d$ the permanent dipole moment and $B$ the rotational constant of each molecule, respectively. Note that the $\Pi$ and $\Delta$ states are doubly degenerate.

Figure 6

(Color online) BO potentials $E_{J;M;\sigma }(r,\vartheta )$ for two molecules colliding in the presence of a dc field, with $\beta \equiv d{E}_{\mathrm{dc}}∕B=1∕5$ and $(J;M;\sigma )$ the quantum numbers of Table . The solid and dashed curves correspond to symmetric $(\sigma =+)$ and antisymmetric $(\sigma =-)$ eigenstates, respectively. (a) BO potentials for the 16 lowest-energy eigenstates $E_n(r,\vartheta =0)$. The molecular-core region is identified as the region $r<r_B=(d^2∕B{)}^{1∕3}$, while for $r\gg r_B$ the eigenstates group into manifolds separated by one quantum of rotational excitation $2B$. (b), (d) Blowups of the first excited energy manifold of (a) in the region $r\gtrsim r_B$ for $\vartheta =0$ and $\pi ∕2$, respectively. Note the electric-field-induced splitting $\hslash \delta \equiv 3B{\beta }^2∕20$ (see Sec. 2B1). The distance $r_{\delta }$ where the dipole-dipole interaction becomes comparable to $\hslash \delta$ is $r_{\delta }=(d^2∕\hslash \delta {)}^{1∕3}$. (c), (e) Blowups of the ground-state potential $E_{0,0;+}(r,\vartheta )$ of (a) in the region $r\gtrsim r_B$ for $\vartheta =0$ and $\pi ∕2$, respectively. The distance $r_{\star }$ [cf. Eq. (22)], where the dipole-dipole interaction becomes comparable to the van der Waals attraction is indicated. Note the attractive (repulsive) character of the potential for $\vartheta =0$ $(\vartheta =\pi ∕2)$ and $r>r_{\star }$.

Figure 7

(Color online) Contour plot of the effective potential $V(\rho ,z)$ of Eq. (25), for two polar molecules interacting in the presence of a dc field $\beta >0$, and a confining harmonic potential in the $z$ direction, with trapping frequency ${\omega }_{\perp }={\omega }_{\mathrm{c}}∕10$, where ${\omega }_{\mathrm{c}}\equiv (12C_{3;0}∕m{r}_{\star }^5{)}^{1∕2}$ of Eq. (24) and $r_{\star }=(2\mid C_{6;0}\mid ∕C_{3;0}{)}^{1∕3}$ of Eq. (22). The contour lines are shown for $V(\rho ,z)∕V_{\star }\ge 0$, with $V_{\star }=B{\beta }^4∕54$. Darker regions represent stronger repulsive interactions. The combination of the dipole-dipole interactions induced by the dc field and the harmonic confinement leads to realization of a 3D repulsive potential. The repulsion due to the dipole-dipole interaction and the harmonic confinement is distinguishable at $z\sim 0$ and $z∕r_{\star }\sim \pm 7$, respectively. Two saddle points (circles) located at $({\rho }_{\perp },\pm z_{\perp })$ separate the long-distance region where the potential is repulsive $\sim 1∕r^3$ from the attractive short-distance region. The gradients of the potential are indicated by dash-dotted lines. The thick dashed line indicates the instanton solution for the tunneling through the potential barrier.

Figure 8

(Color online) The Euclidean action $S_{\mathrm{E}}$ of Eq. (28) as a function of ${\omega }_{\perp }∕{\omega }_{\mathrm{c}}$ (solid line). For ${\omega }_{\perp }<{\omega }_{\mathrm{c}}^{\prime }\approx 0.88{\omega }_{\mathrm{c}}$ $({\omega }_{\perp }>{\omega }_{\mathrm{c}}^{\prime })$ the bounce occurs for $z\left(0\right)\ne 0$ [within the plane $z\left(0\right)=0$]; see text. The point ${\omega }_{\mathrm{c}}^{\prime }$ is signaled by a circle. The dashed line is the $C_{6;0}$-independent expression $S_{\mathrm{E}}\approx 7.01S_0({\omega }_{\perp }∕{\omega }_{\mathrm{c}}{)}^{1∕5}$ (see text), with $S_0=\sqrt{m\mid C_{6;0}\mid }∕r_{\star }^2$. For ${\omega }_{\perp }>{\omega }_{\mathrm{c}}^{\prime }$ the action is $S_{\mathrm{E}}\approx 5.78S_0$, which is ${\omega }_{\perp }$ independent, consistent with the bounce occurring in the $z=0$ plane.

Figure 9

(Color online) The ground-state effective 2D BO potentials $V_{\mathrm{eff}}^{2\mathrm{D}}\left(\mathit{\rho }\right)$ of Eq. (29) as a function of the molecular separation $\rho$ in the $z=0$ plane for various strengths of the dc electric field $\beta =0,0.1,0.15,0.2,0.3,0.4$. The quantity $a_{\perp }$ is the harmonic oscillator length in the $z$ direction. The molecular parameters are chosen as $(d^4{m}^{3}B∕{\hslash }^6{)}^{1∕2}=1.26\times {10}^6$ and the frequency of the harmonic potential in the $z$ direction is ${\omega }_{\perp }=15B∕{10}^6\hslash$. This corresponds to the case of $\mathrm{SrO}$ with a mass $m=104\mathrm{amu}$, a rotational constant $B\approx h\times 10\mathrm{GHz}$, and a permanent dipole moment $d\approx 8.9\mathrm{D}$ in a tight confining potential with ${\omega }_{\perp }=2\pi \times 150\mathrm{kHz}$, where $a_{\perp }\approx 25\mathrm{nm}$.

Figure 10

(Color online) Schematic representation of the effects of an ac microwave field on the interaction of two molecules. The solid and dashed lines are the bare $(E_{\mathrm{ac}}=0)$ potentials $E_n\left(r\right)\equiv E_{J;Y;\sigma }\left(r\right)$ of Sec. 3A1 for the symmetric $(\sigma =+)$ and antisymmetric $(\sigma =-)$ states, respectively. An ac field of frequency $\omega =2B+\Delta$ is blue detuned by $\Delta =3B∕{10}^6\hslash$ from the single-particle rotational spacing $2B$, with Rabi frequency $\Omega$. The dipole-dipole interaction splits the excited-state manifold, making the detuning position dependent. Eventually, the combined energy of the bare ground-state potential $E_{0;0;+}\left(r\right)$ and of an ac photon (vertical arrow) becomes degenerate with the energies of the bare symmetric $E_{1;\pm 1;+}\left(r\right)$ and antisymmetric $E_{1;0;-}\left(r\right)$ potentials. The corresponding resonant points are denoted as $r_{\mathrm{C}}=(d^2∕3\hslash \Delta {)}^{1∕3}$ (circles) and $r_{\mathrm{C}}^{\prime }=(2d^2∕3\hslash \Delta {)}^{1∕3}$, respectively. The resulting dressed ground-state potential is sketched by a thick solid line. For molecular parameters of $\mathrm{SrO}$ ($B\approx h\times 10\mathrm{GHz}$ and $d\approx 8.9\mathrm{D}$) the detuning corresponds to $\Delta ∕2\pi =30\mathrm{kHz}$, and the lengths $r_B$ and $r_{\mathrm{C}}$ are given by $r_B\approx 11\mathrm{nm}$ and $r_{\mathrm{C}}\approx 0.5\mathrm{\mu }\mathrm{m}$, respectively.

Figure 11

(Color online) The dressed adiabatic potentials ${\tilde{E}}_n(r,\vartheta )$ of Eq. (37) for two molecules interacting in an ac field. The setup is the same as in Fig. 10. The field polarization is linear $(q=0)$ in (a1) and (a2), while it is circular $(q=1)$ in (b1) and (b2). The solid (red) and dashed (blue) lines correspond to the potentials for the symmetric $(\sigma =+)$ and antisymmetric $(\sigma =-)$ states, respectively. The thick continuous (black) line is the adiabatic dressed ground-state potential ${\tilde{E}}_{1;0;+}\left(r\right)$. (a1) and (b1) show the potentials as a function of the separation $r$ for interactions in the $z=0$ plane $(\vartheta =\pi ∕2)$. The position of the resonant Condon point $r_{\mathrm{C}}$ is indicated by a vertical line. (a2) and (b2) show the angular dependence of the potentials at $r=r_{\mathrm{C}}$. Note that for $q=0$ (a2) ${\tilde{E}}_{0;0;+}\left(r\right)$ becomes degenerate with ${\tilde{E}}_{1;1_{\pm };+}(r_{\mathrm{C}},\vartheta )$ at $\vartheta =0,\pi$, while it is nondegenerate at all angles for $q=1$ (b2), suggesting better shielding. The potential ${\tilde{E}}_{1;0;-}(r_{\mathrm{C}},\vartheta )$ has an energy larger than ${\tilde{E}}_{1;0;+}(r_{\mathrm{C}},\vartheta )$ for all angles $\vartheta$, indicating a level crossing ar $r>r_{\mathrm{C}}$ (see text).

Figure 12

(Color online) (a) Schematic representation of the effects of a dc and an ac microwave field on the interaction of two molecules. The solid and dashed lines are the bare potentials $E_n\left(\mathbf{r}\right)\equiv E_{J;M;\sigma }(r,\vartheta )$ of Sec. 3A2 with $\vartheta =\pi ∕2$ for interactions in the presence of the dc field only, for the symmetric $(\sigma =+)$ and antisymmetric $(\sigma =-)$ states, respectively. The dc field induces a splitting $\hslash \delta$ of the first excited manifold of the two-particle spectrum. A microwave field of frequency $\omega =\overline{\omega }+2\delta ∕3+\Delta$ is blue detuned by $\Delta >0$ from the single-particle rotational resonance. The dipole-dipole interaction further splits the excited-state manifold, making the detuning space dependent. Eventually, the combined energy of the bare ground-state potential $E_{0;0;+}\left(\mathbf{r}\right)$ and an ac photon (black arrow) becomes degenerate with the energy of the bare symmetric $E_{1;0;+}(r,\pi ∕2)$. The resonant point $r_{\mathrm{C}}=(d^2∕3\hslash \Delta {)}^{1∕3}$ occurs at $r\approx 46r_B$. A second resonant Condon point occurs at (much) shorter distances $r_{{\mathrm{C}}^{\prime }}\lesssim r_{\delta }=(d^2∕\hslash \delta {)}^{1∕3}$ with an antisymmetric potential (not shown). (b) Blowup of the potentials of (a) with $M=0$ (see text in Sec. 3C2). The dressed ground-state potential is sketched by a thick solid line.

Figure 13

(Color online) Dressed adiabatic potentials ${\tilde{E}}_{J;M;\sigma }\left(\mathbf{r}\right)$ of Eq. (47) for the interaction of two molecules polarized by a (weak) dc field ${\mathbf{E}}_{\mathrm{dc}}=E_{\mathrm{dc}}{\mathbf{e}}_z$ with $\beta \equiv d{E}_{\mathrm{dc}}∕B=1∕10$ and dressed by an ac field with linear polarization $q=0$. The ac-field detuning and Rabi frequency are $\Delta =3B∕{10}^6\hslash$ and $\Omega =\Delta ∕4$, respectively. For a typical rotational spacing of $B\sim h\times 10\mathrm{GHz}$ these numbers entail $\Delta ∕2\pi =30\mathrm{kHz}$ and $\Omega ∕2\pi =7.5\mathrm{kHz}$. (a) Dressed adiabatic potentials ${\tilde{E}}_n\left(\mathbf{r}\right)$ of Eq. (47) plotted as a function of $r$ for $z=0$ $(\vartheta =\pi ∕2)$. The potentials corresponding to symmetric (antisymmetric) states are given by solid (dashed) lines, and indicated by ${\tilde{E}}_{J;M;\sigma }\left(\mathbf{r}\right)$ for $J=0,1,2$, $M=0$, $\sigma =+$ ($J=1$, $M=0$, $\sigma =-$). The dressed ground-state potential ${\tilde{E}}_{0;0;+}\left(\mathbf{r}\right)$ has the highest energy (thick solid line). The other potentials are asymptotically detuned by multiples of $\Delta$. The ground-state potential ${\tilde{E}}_{0;0;+}\left(\mathbf{r}\right)$ shows an avoided crossing with the symmetric potential ${\tilde{E}}_{1;0;+}\left(\mathbf{r}\right)$ at $r=r_{\mathrm{C}}$. (b) Dressed adiabatic potentials ${\tilde{E}}_{J;M;\sigma }\left(\mathbf{r}\right)\equiv {\tilde{E}}_{J;M;\sigma }(\rho ,z)$ for the symmetric states $(\sigma =+)$ of Eq. (47) plotted as a function of $\rho =r\sin \vartheta$ and $z=r\cos \vartheta$. Darker regions (in the online version: darker red/darker blue) correspond to stronger repulsive/attraction. For $z=0$ ($\rho$ axis) we recognize the case of (a), where the symmetric ground-state potential has the largest energy. The position of the Condon point $r_{\mathrm{C}}$ is indicated by an arrow. Accordingly, the avoided crossing with the potential ${\tilde{E}}_{1;0;+}\left(\mathbf{r}\right)$ observed at $r=r_{\mathrm{C}}$ in (a) is now clearly visible in transparency, below the upper layer. For $\mid z\mid >0$ the potential ${\tilde{E}}_{1;0;+}$ becomes less and less repulsive. For $\mid z\mid >\rho ∕\sqrt{2}$ we have $1-3{\cos }^2\vartheta <0$, and thus ${\tilde{E}}_{1;0;+}\left(\mathbf{r}\right)$ is attractive and the Condon point vanishes since the two states are off resonant. (c) Dressed adiabatic potential as in (b), but also showing the antisymmetric states $(\sigma =-)$. We see that the dressed potential ${\tilde{E}}_{1;0;-}\left(\mathbf{r}\right)$ for the antisymmetric state with $(1;0;-)$ is strongly attractive in the plane, i.e., for $z=0$, which corresponds to the profile shown in (a). With increasingly separation $\mid z\mid ∕r>0$ the potential become less and less attractive. For $\mid z\mid ∕\rho >1∕\sqrt{2}$ we have $3{\cos }^2\vartheta -1>0$ and the potential becomes repulsive. Thereby a crossing between the asymmetric state and the ground state appears at a (second) Condon point ${\mathbf{r}}_{\mathrm{C}}^{\prime }$ (dashed line), as the two states are resonant, but due to the permutation symmetry do not couple.

Figure 14

(Color online) Contour plot of the effective potential $V(\rho ,z)$ of Eq. (51) for two polar molecules interacting in the presence of a weak dc field and an ac field. The field parameters are the same as in Fig. 13. The frequency of the confining harmonic potential in the $z$ direction is ${\omega }_{\perp }=\Delta ∕5$. Darker regions represent stronger repulsive interactions. The white region for $\rho <1∕2$ indicates a potential $V(\rho ,z)<0$. The combination of the dipole-dipole interactions induced by the dc electric and ac (microwave) fields and of the quadratic confinement leads to realization of a 3D repulsive potential for $r\gg r_{\mathrm{C}}$. In particular, two saddle points located at $({\rho }_{\perp },\pm z_{\perp })$ (circles) separate the repulsive long-distance regime $r\gg r_{\mathrm{C}}$ from the short-distance regime $r<r_{\mathrm{C}}$ where diabatic losses occur. The dotted line signaled by ${\mathbf{r}}_{\mathrm{C}}^{\prime }$ indicates the location of the crossing between the ground-state potential and the energy ${\tilde{E}}_{1;0;-1}\left(\mathbf{r}\right)$ of the antisymmetric state (see text and Fig. 13).

Figure 15

(Color online) Effective 2D potentials ${\tilde{E}}_{J;M;\sigma ;k}^{2\mathrm{D}}\left(\mathit{\rho }\right)$ with $(J=0,1,2;M=0;\sigma =\pm 1;k=0,1,2,\dots )$ of Eq. (55) for the ac- and dc-field setups of Fig. 14. The strength of the (weak) dc field is $\beta =d{E}_{\mathrm{dc}}∕B=1∕10$. The ac field has polarization $q=0$, is blue detuned by $\Delta =3B∕{10}^6\hslash$, and the saturation amplitude is $\Omega ∕\Delta =1∕4$. (a), (b), and (c) correspond to a harmonic oscillator frequency of the confining potential ${\omega }_{\perp }∕\Delta =1∕2,2$, and 5, respectively. The thick dashed lines indicate the (single-band) effective potentials ${\tilde{E}}_{J;0;\sigma }^{2\mathrm{D}}\left(\mathit{\rho }\right)$ of Eq. (53). The solid lines are the nonperturbative potentials $E_{J;0;\sigma ;k}^{2\mathrm{D}}\left(\mathit{\rho }\right)$ of Eq. (55), where $k=0,1,2,\dots$ denote the $k$th transversal excitations in the $z$ direction. The ground-state potential approaches twice the single-particle Stark shifts, $2(E_{0;0}+E_{0;0}^{\prime })$ (see text), for $\rho \to \infty$ and it is indicated by arrow(s). The harmonic oscillator length $a_{\perp }$ for a typical mass $m\approx 100\mathrm{amu}$, a dipole moment $d\approx 8.9\mathrm{D}$, and a rotational constant $B\approx h\times 10\mathrm{GHz}$ is indicated on the $\rho$ axis. The corresponding detuning is $\Delta \approx 2\pi \times 30\mathrm{kHz}$ and the trapping frequency is ${\omega }_{\perp }∕2\pi \approx 15$, 60, and $150\mathrm{kHz}$ in (a), (b), and (c), respectively. The gray region corresponds to $r<r_{\delta }$.