Synthetic dimensions in ultracold polar molecules

Synthetic dimensions alter one of the most fundamental properties in nature, the dimension of space. They allow, for example, a real three-dimensional system to act as effectively four-dimensional. Driven by such possibilities, synthetic dimensions have been engineered in ongoing experiments with ultracold matter. We show that rotational states of ultracold molecules can be used as synthetic dimensions extending to many – potentially hundreds of – synthetic lattice sites. Microwaves coupling rotational states drive fully controllable synthetic inter-site tunnelings, enabling, for example, topological band structures. Interactions leads to even richer behavior: when molecules are frozen in a real space lattice with uniform synthetic tunnelings, dipole interactions cause the molecules to aggregate to a narrow strip in the synthetic direction beyond a critical interaction strength, resulting in a quantum string or a membrane, with an emergent condensate that lives on this string or membrane. All these phases can be detected using local measurements of rotational state populations.

In the lab frame, our system is described by the Hamiltonian where N j is the rotational angular momentum operator for the j th molecule, and B ∼ × O (GHz) is the rotational constant. The eigenstates |n, m of the first term in Eq. (S1) are described by two quantum numbers: the angular momentum quantum number n and its projection m on the z axis. Equation (S1) assumes that the molecules are rigid rotors, i.e all molecules are in the lowest vibrational state, and coupling to excited vibrational states is small. This assumption is valid up to n that approximately lies in a range between 40 and 60 in a typical diatomic molecule. (For example, for RbCs whose rotational constant is ∼ h×500 MHz and vibrational excitation energy is ∼ h×1.5 THz, the rotational excitation above the ground state is smaller than the vibrational excitation for n 55. For NaRb, whose rotational constant and vibrational excitation energy are ∼ h × 2 GHz and ∼ h × 3.2 THz, the rotational excitation above the ground state is smaller than the vibrational excitation for n 40). For larger n, the rotational states get dressed by rovibrational coupling. The matrix elements for the dipole moment operator between rotational states in a rigid rotor are [1,2] n, m|d p |n , m =(−1) p d 4π 3 Y n m (Θ, Φ)Y * nm (Θ, Φ)Y 1p (Θ, Φ) sin ΘdΘdΦ =(−1) p d 2n + 1 2n + 1 n, m; 1, p|n , m n, 0; 1, 0|n , 0 , whered p are the spherical components of the dipole operator, d is the molecule's permanent electric dipole moment, and p takes the values 0, ±1. Using the properties of Clebsch-Gordan coefficients, Eq. (S2) simplifies to n, m|d 0 |n − 1, m = dδ mm For larger n where the rigid rotor assumption breaks down, the polarization of the molecules typically reduces due to a larger internuclear separation, so the matrix elements are smaller than Eq. (S3). In this limit, matrix elements can be obtained from techniques in earlier works [3] which calculate dipole moments as a function of the internuclear separation. We find from Eqs. (S1) and (S3) that when the molecules are illuminated by a resonantẑ-polarized microwave of amplitude E (0) n , the synthetic tunneling strength from |n − 1, 0 to |n, 0 is J n = dE (0) n n √ 4n 2 −1 . When the molecules are illuminated by a σ + -polarized microwave, the tunneling strength from |n − 1, n − 1 to |n, n is J n = dE (0) n n 2n+1 . We choose the microwave fields such that J n = J is uniform in n: E for the m = 0 states, and E for m = n. Even when the dipole matrix elements are smaller than Eq. (S3) due to rovibrational mixing, the synthetic tunnelings can still be made uniform by compensating with stronger microwaves. To tune this, experiments can perform Rabi spectroscopy on the relevant two level system, and adjust the microwave amplitude until the desired tunneling rate is achieved. Dipole interactions induce an angular momentum exchange with an amplitude V ij n , between two molecules in adjacent rotational manifolds. When the two molecules are in m = 0 states, V ij n = n − 1, 0| i n, 0| jĤ dd |n, 0 i |n − 1, 0 j = 1−3(rij ·ẑ) 2 8π 0r 3 ij d 2 n 2 4n 2 −1 , whereĤ dd is the third term in Eq. (S1). For the m = n states, V ij n = 3(rij ·ẑ) 2 −1 8π 0r 3 ij d 2 n 2n+1 . These expressions are modified at large n, when rovibrational mixing is relevant [3].
Interactions between the rotational and nuclear spin angular momenta introduce some mixing between the molecule's rotational and hyperfine spin quantum numbers. However, because we add a small electric field ∼ O(10) V/cm, the energies of the different m states within a rotational manifold split by ∼ 10-100 MHz. This splitting is typically larger than the interaction between nuclear and rotational angular momentum. As a result, the rotational angular momentum decouples from the nuclear spins, so m is a good quantum number [4]. The expressions for the matrix elements for the dipole moment operator [Eq. (S3)] are valid in this limit. Additionally, the splitting of the m states within a rotational manifold is also larger than V ij n . Therefore, all dipole interaction-induced transitions to rotational states outside the Hilbert space in Eq. (1) are off-resonant and suppressed. The electric field does mix states between different rotational manifolds n, but the resulting additions to the Hamiltonian in Eq. (1) are insignificant for these electric field values.

MICROWAVE IMPLEMENTATION
Here, we briefly discuss the implementation of microwaves for the realization of the Hamiltonian in Eq. (1), and associated challenges. As a concrete example, we consider RbCs, whose rotational constant is B ≈ h × 500 MHz. Cold and dense samples of RbCs have been produced by both the Innsbruck [5,6] and Durham [7] groups. The microwave frequency required to excite a molecule from the |n − 1 to the |n rotational manifold is ω n = 2nB/ ≈ 2πn GHz. We consider the scenario where only the m = 0 rotational sublevels are populated, such that the number of synthetic lattice sites will be equal to the number of coupled rotational manifolds. In this case, an implementation of a 1D synthetic lattice with N rot synthetic lattice sites and open boundary conditions requires N rot −1 microwaves, spanning (sparsely) the range from 1 GHz to (N rot − 1) GHz.
The generation, amplification, and projection of microwaves to coherently couple a large number of synthetic sites can be straightforwardly accomplished using standard and commercially available microwave equipment. The required microwave waveform itself, n cos(ω n t), can be generated with lab-developed or commercially available arbitrary waveform generators presently capable of reaching up to at least 20 GHz [8]. Additionally, since only a finite number of discrete frequencies are actually required from the large frequency span, versatile frequency sources generated by direct digital synthesis may be used to stroboscopically sample the various ω n . Standard broadband amplifiers and waveguide horns can then be used to project these signals onto the molecular samples.
At least for some molecules like RbCs, dozens of coupled synthetic sites should be straightforward to realize with present microwave technology. The number of coupled synthetic sites is fewer for molecules which have a larger rotational constant; for example, the synthetic lattice will contain roughly ten sites for NaRb or KRb. In addition to being limited by the frequency range of waveform generators, the number of coupled synthetic sites is also limited by the lifetime of excited rotational states. The decay rate of a molecule with typical parameters (B = h × 500 MHz and d = 1D), from n = 20 due to stimulated emission by a microwave field at room temperature, is ∼ 1/ sec. The decay rate is larger at larger n. Therefore for decay to be insignificant on an experimental timescale∼ s, the number of coupled synthetic sites in this molecule is n 20. Even so, the synthetic dimension obtained here is significantly larger than can be obtained with alkali atoms.
Furthermore, a synthetic dimension of even larger sizes, up to hundreds of synthetic lattice sites, can be obtained using essentially the same equipment and approach, if we remove the restriction of using only the m = 0 or m = n rotational sublevels; unique spectral selectivity of state-to-state transitions is generally still provided by rotationalhyperfine energy shifts at the scale of ∼ 100 kHz.
The biggest challenge in the implementation of our scheme arises from the fact that different rotational states may experience different real space lattice potentials due to having different ac polarizabilities. This has two consequences: molecules in different rotational states have a) different zero point energies, which shift the microwaves off resonance, and b) different real space Wannier functions, which affect dipole matrix elements. The offsets in the zero point energies can be cancelled by tuning the microwave frequencies appropriately, and/or choosing "magic angles" for the polarization of the lattice lasers [9,10]. It may be more practical to tune the frequencies to resonance, because or under the single-particle terms in Eq. (S1) (red) with 100 microwaves. The black and red curves are indistinguishable by eye at the scale shown, demonstrating that the tight binding model accurately describes the molecule's evolution when it is driven by 100 microwaves. The microwave amplitudes were chosen such that J = B/200. This is a pessimistically large tunneling; the tight-binding model becomes more accurate as J/B is decreased. We expect that the typical synthetic tunneling will be J/B ∼ 10 −6 , as explained in the text. Inset: Difference between black and red curves. the polarizabilities are very sensitive to small deviations in the polarization angles. The effect on the dipole matrix elements can be compensated by adjusting the respective microwave amplitude until the desired matrix elements for a uniform synthetic tunneling are achieved. The only remaining challenge is then due to inhomogeneities in the real space optical lattice, and a harmonic trap in real space, which will lead to a spatially varying zero point energy. This challenge can be mitigated by using a flat trap, or by observing only a small region of the molecular cloud in a harmonic trap where the optical lattice is relatively uniform in real space. In such a small region, all microwaves can be tuned to resonance with the desired rotational transitions.
One might worry that the illumination of the cloud with a large number of microwaves presents an additional challenge: that the numerous off-resonant driving frequencies lead the molecules to decohere, and that the tight binding model Eq. (1) is no longer valid. However, there is no such decoherence, due to a large separation of energy scales between the microwave frequencies and microwave couplings. For example, a typical microwave frequency used is ω n ∼ O(GHz), and a typical synthetic tunneling, for moderate microwave amplitude, is J ∼ × O(kHz). For these values, the dynamics of the molecules are modeled accurately by the tight-binding model in Eq. (1). We illustrate this in Fig. S1, by comparing the dynamics of a molecule governed by [Eq. (1)], and one governed by the single particle terms in Eq. (S1) which include the 100 microwaves required to create a synthetic dimension with 101 sites. We find that even for very large synthetic tunnelings J ∼ ωn 200 , the dynamics in the two cases agree to better than 1% accuracy.

BOUND STATE FOR TWO MOLECULES
The Schrödinger equation for the relative wavefunction of two molecules, f (y = m − n) = f mn , is We substitute the ansatz that f (y) = e −λy for y > 0. The first line in Eq. (S4) tells us that the energy of this bound state is E = −4J cosh λ. When V ≤ 0, we find that the ground state satisfies f (y) = f (−y). Using the second line in Eq. (S4), we find that λ solves the implicit transcendental equation: The solution for λ in Eq. (S5) is positive for all V ≤ 0. When V > 0, we find that in the ground state, f (y) = −f (−y). In this case, λ solves the transcendental equation: The solution for λ in Eq. (S6) is positive only when V ≥ 2J. The positive solution for λ, and the ground state energy, are plotted in Fig. S2.

OTHER VARIATIONAL APPROXIMATIONS
In the main text we presented the results for the many-body phase diagram based on a cluster mean field theory. Here we consider two alternative approximations that corroborate the findings there.

Single-site mean field approximation
Our single-site mean field ansatz for a 1D chain of molecules is The variational energy per molecule in this approximation is Similar to the cluster mean field approximation in the main text, we found that this ansatz also predicts two binding transitions, which occur at V 0 and V 2.15J in the thermodynamic limit. The binding transitions again correspond to a spontaneous dimensional reduction to a fluctuating quantum string. The string hosts an emergent hardcore condensate at |V /J| = ∞.
The variational ansatz for a square lattice of molecules is obtained by a straightforward extension of the 1D ansatz in Eq. (S7). That is, bipartition the square lattice into two intercalated checkerboards in the conventional way, run the index i over one checkerboard, and j over the other checkerboard. Here again, we found that two binding transitions occur, similar to the cluster approximation. When V 0 or V 1.5J, the system undergoes a spontaneous dimensional reduction to a fluctuating quantum membrane. The membrane hosts an emergent hardcore condensate at |V /J| = ∞.
Fermionic-pair mean field approximation for a 1D chain In this approximation, we generalize our system to a grand canonical ensemble of slave fermions, by introducing a chemical potential. The modified Hamiltonian that describes our system iŝ We relax the constraint that exactly one molecule occupies each real lattice site, and fix the chemical potential µ such that this constraint is held on average: n ĉ † njĉnj = 1 ∀j. Motivated by our exact solution for two molecules, we assume mean fields for bound molecules in the synthetic dimension: To gain a simple understanding, we assume that V ij n = V ij and J n = J > 0 are uniform in n. Further, we consider only nearest-neighbor interactions, V ij = V δ |i−j|,1 , and periodic boundary conditions in both real and synthetic dimensions. Later we perform a numerical calculation, where we include nonuniform physical values V n in the lowlying rotational states, and open boundary conditions.
In the fermion-pair approximation, the mean-field Hamiltonian for a 1D chain of molecules iŝ Here, ∆ = ∆ j,j+1 n and ∆ = ∆ j+1,j n . We self-consistently diagonalizeĤ MF to find ∆, ∆ and µ. We find different solutions when V < 0 and V > 0. In both regimes, the parameters ∆, ∆ and µ solve the implicit equations: (S13) When V < 0, ∆ = ∆ and g( k) = cos k Z sin k X is a self-consistent solution. When V > 0, ∆ = −∆ and g( k) = sin k Z cos k X is a self-consistent solution. The free energy per molecule is Here, X and Z refer to the real and synthetic directions (not the same as x and z), and k X and k Z are momenta conjugate to X and Z. The mean field ground states are achiral p-wave superfluids of slave fermions. The order parameter is even (odd) along the synthetic direction, and odd (even) along the real direction, when V < 0 (V > 0). This is related to the parity of the two-molecule ground state when V < 0 and V > 0. In the thermodynamic limit of N rot and N mol , the system is in the superfluid phase for V < 0 and V > 2J. For N rot 1 and V /J → ±∞, the mean field superfluids are identical to the two-molecule ground state in Eq. (2) when they are projected onto the physical sector of unit molecular filling. The system undergoes a phase transition to a normal Fermi liquid phase at V = 0 and V = 2J. V ij n = V δ |i−j|=1 , iŝ where ∆ A = ∆ j,j+Â n and ∆ A = ∆ j+Â,j n (Â =X orŶ). When the dipole interactions V n are equal and negative in both real directions, there are two allowed self-consistent solutions: (S16) The first solution in Eq. (S16) describes an achiral superfluid, while the second solution is chiral. When the dipole interactions V n are equal and positive in both real directions, there are two allowed self-consistent solutions: (S17) The first solution again describes an achiral superfluid, while the second solution is chiral. We calculate the energy of the chiral and achiral superfluids from Eq. (S14), and plot the superfluid phase with the least energy in Fig. S3. We find that the chiral phase is the most stable superfluid in a narrow region of V /J for small N rot . When projected to the physical subspace, the chiral superfluids correspond to non-Abelian Ising anyonic phases [11].

Comparison of energy in various approximations
In a 1D chain, the total variational energy per molecule in the cluster mean field approximation [Eq.
n f * mn f * m ,n+1 f m,n+1 f m ,n + f * nm f * n+1,m f n+1,m f n,m + c.c. (S18) The 1D variational energy in the single-site and fermionic-pair mean field approximations were given in Eqs. (S8) and (S14). For simplicity, in this section, we modify the expressions to include only nearest-neighbor dipole interactions, and plot the variational minimum in the three approximations in Fig. S4.The physics is unchanged when the  S4. The ground state energy for a 1D chain of molecules in three different approximations: a cluster mean field approximation (solid), single-site mean field (dashed), and fermion-pair mean field (dotted). The calculation was performed with only nearest-neighbor dipolar interactions and the physical values of Vn. The size of the synthetic dimension was chosen to be Nrot = 20. The cluster mean field state always has the lowest energy. The energies are similar when the full long-ranged dipolar interaction is included.
full dipole interaction is included. We find that the cluster mean field state always has the lowest energy. The singlesite mean field state has nearly the same energy as the cluster mean field state for V < 2J, and the fermionic-pair state displays similar behavior as the single-site and cluster mean field states. These observations suggest that all three approximations capture the essential features of the ground state, and the cluster mean field state is the closest approximation to the true ground state.
We also calculated the variational energies of the ground state of a square lattice of molecules in the three approximations. We again found that the cluster mean field state has the lowest energy, and the other approximations result in similar energies.