External field control of collective spin excitations in an optical lattice of $^2\Sigma$ molecules

We show that an ensemble of $^2\Sigma$ molecules in the rotationally ground state trapped on an optical lattice exhibits collective spin excitations that can be controlled by applying superimposed electric and magnetic fields. In particular, we show that the lowest energy excitation of the molecular ensemble at certain combinations of electric and magnetic fields leads to the formation of a magnetic Frenkel exciton. The exciton bandwidth can be tuned by varying the electric or magnetic fields. We show that the exciton states can be localized by creating vacancies in the optical lattice. The localization patterns of the magnetic exciton states are sensitive to the number and distribution of vacancies, which can be exploited for engineering many-body entangled spin states. We consider the dynamics of magnetic exciton wavepackets and show that the spin excitation transfer between molecules in an optical lattice can be accelerated or slowed down by tuning an external magnetic or electric field.


INTRODUCTION
A major thrust of current experimental research is to create ultracold ensembles of polar molecules trapped on an optical lattice [1,2]. Optical lattices of ultracold molecules are predicted to be ideally suited for quantum simulation of complex many-body quantum systems [3][4][5] and the development of new schemes for quantum information storage and processing [6,7]. These applications exploit (i) the possibility to produce molecular ensembles in a Mott-insulator state, i.e. an ordered array with one molecule per lattice site [8,9]; (ii) the presence of long-range dipole -dipole interactions that can be used to couple molecules in different lattice sites [10]; (iii) the rotational structure of molecules; (iv) the possibility to address spectroscopically molecules in specific sites of an optical lattice [6]. For example, DeMille proposed to use rotational states of polar molecules confined in a one-dimensional optical lattice as qubits of a quantum computer entangled by the dipole -dipole interactions [6]. Recently, Micheli and coworkers showed that ultracold molecules in the 2 Σ electronic state trapped in a two-dimensional optical lattice can be used for engineering lattice-spin models that give rise to topologically ordered states [3]. This stimulated experimental work on the creation of ultracold molecules in electronic states with unpaired electrons [1].
Ultracold 2 Σ molecules can be produced by photoassociation of ultracold alkali metal atoms with ultracold alkaline earth [11] or closed shell lanthanide atoms [12,13], buffer gas loading [14,15] or direct laser cooling [16], as was recently demonstrated for the molecule SrF [17]. The presence of the unpaired electron in a 2 Σ molecule allows for new applications of ultracold molecules exploiting weak couplings of the electron spin with the rotational angular momentum of the molecule [1][2][3]. In particular, the electron spin of 2 Σ molecules can be used for encoding quantum information as in other spin-1/2 particles [18] and for quantum simulation of many-body spin dynamics [3]. In order to realize these applications, it is necessary to develop techniques for entangling the spin degrees of freedom and controlled preparation of many-body spin-dependent states of ultracold molecules on an optical lattice. Micheli and coworkers showed that the electric dipole -dipole interaction between 2 Σ molecules on an optical lattice leads to spin-dependent binary interactions whose parameters can be tuned by a combination of dc electric and microwave fields [3]. Here, we extend the work of Micheli and coworkers to explore the possibility of tuning the dynamics of collective spin excitations in an ensemble of 2 Σ molecules on an optical lattice by external electric and magnetic fields.
We consider SrF( 2 Σ) molecules in the ro-vibrationally ground state confined in a Mottinsulator state in an optical lattice. In the presence of a weak magnetic field, the lowest energy excitation of the molecular crystal corresponds to the spin-down to spin-up transition in an isolated molecule. We show that for certain combinations of superimposed electric and magnetic fields, the lowest energy excitation of the molecular crystal leads to the formation of a magnetic Frenkel exciton. The magnetic exciton is a many-body entangled state of the molecules. If some of the molecules are removed from the optical lattice to produce vacancies, which can be achieved by a recently demonstrated technique [19], the exciton undergoes coherent localization due to scattering by the impurities. Using a series of examples for SrF molecules in an optical lattice, we show that the localization of the magnetic exciton can be controlled by varying the concentration and distribution of vacancies as well as the external magnetic or electric fields. The system proposed here can be used for quantum simulation of spin excitation transfer in many-body crystals without phonons. We show that the spin excitation transfer between molecules can be accelerated or slowed down by tuning an external magnetic field. The Hamiltonian of an isolated 2 Σ molecule in the presence of superimposed electric and magnetic fields can be written as [20][21][22] where the first term determines the ro-vibrational structure of the molecule, γ SR is the constant of the spin-rotation interaction between the rotational angular momentum N and the spin angular momentum S of the molecule, E and B are the vectors of the electric and magnetic fields, d is the dipole moment of the molecule and µ B is the Bohr magneton.
We assume that both E and B are directed along the quantization axis z. We consider molecules in the vibrationally ground state and use the rigid-rotor approximation for our calculations. It is convenient to use the basis of direct products [21] of the rotational |NM N and spin |SM S wave functions to evaluate the eigenvectors and eigenvalues of Hamiltonian (1). Here, M N and M S denote the projections of N and S, respectively, on the z axis. The diagonalization of Hamiltonian (1) yields the energy levels of the molecule in superimposed electric and magnetic fields, shown in Figure 1 for the particular example of SrF. The eigenvectors of Hamiltonian (1) are linear combinations The index f denotes the f -th excited state of the molecule. We neglect the hyperfine structure of the molecule. This is a good approximation for the magnetic fields considered in the present work.
At zero electric field, the magnitude of the rotational angular momentum N is conserved. that couples states of different parity. Therefore, the crossing between states β and γ is real in the absence of an electric field and becomes avoided in the presence of an electric field. The properties of 2 Σ molecules, such as alignment and orientation, are very sensitive to electric and magnetic fields near these avoided crossings [22][23][24]. In the present work, we show that the avoided crossings depicted in Figure 1 can be exploited for inducing and controlling collective spin excitations of 2 Σ molecules on an optical lattice.

Magnetic excitons induced by electric fields
We consider an optical lattice of SrF molecules prepared in the absolute (vibrational, rotational and Zeeman) ground state with one molecule per lattice site. We assume that the lattice sites are separated by 400 nm and the tunneling of molecules between lattice sites is suppressed, i.e. the molecules are in a Mott insulator state. This can be achieved by applying laser fields of high intensity [9]. With the current technology, it is possible to create optical lattices that provide harmonic confinement for ultracold atoms and molecules with the vibrational frequencies of the translational motion up to 100 -150 kHz [25]. We assume that the molecules populate the ground state of the lattice potential and neglect the center-of-mass motion of the molecules that broadens the spectral lines of the molecular crystal by about 5 % [26].
The Hamiltonian describing the optical lattice with identical 2 Σ molecules in the presence of superimposed electric and magnetic fields is where r n is the position of the n-th lattice site,Ĥ as,n is Hamiltonian (1) for molecule in site n,V dd is the electric dipole-dipole interaction between molecules in different lattice sites, and N mol is the total number of molecules. Hamiltonian (3) can be rewritten in terms of the operatorsB † n,f andB n,f that create and annihilate a molecular excitation f in site n as follows [27]: whereĤ ǫ f is the energy of the f -th excited state of the molecule and The eigenstates of Hamiltonian (5) can be written as where the wave functions describe the ensemble of N mol − 1 molecules in the ground state φ 0 and one molecule located at r i in the f -th excited state. Eq. (7) gives the wave function for a Frenkel exciton corresponding to an isolated f -th excitation of the molecule [27]. In general, different excitonic states are coupled by the terms J f,g n,m with f = g in Eq. (4) and the eigenstate of Hamiltonian (3) is a linear combination of the excitonic states (7) associated with different excitations f .

Hamiltonian (5) can be diagonalized by the unitary transformation [27]
where k is the exciton wavevector and L f (k) = n J f,f n,0 e ik·rn determines the exciton dispersion in the absence of inter-exciton couplings. If all off-diagonal couplings J f,g n,m with f = g are much smaller that the energy separation of the molecular state f from other molecular states, the coupling terms J f,g n,m in Eq. (4) can be neglected and Eq. (7) provides an accurate description for the wave function of exciton f . This corresponds to the two-level approximation [27,28].
In the present work, we consider the lowest energy excitation of the molecular crystal corresponding to the Zeeman transition from state α to state β depicted in Figure (7). To prove this we calculated the energy band of the lowest energy exciton by the direct diagonalization of Hamiltonian (4) with four coupled excitonic states and by the diagonalization of Hamiltonian (5), which corresponds to the two-level approximation [27]. Figure 2 demonstrates that the two-level approximation for the lowest energy exciton in the system considered here is very accurate.  apply an electric field to couple the molecular states of different parity; (ii) tune the magnetic field adiabatically to a value B > B 0 (see Figure 1), where the α → β exciton bandwidth is large; (iii) generate the lowest energy excitation; (iv) detune the magnetic field to a value B ≪ B 0 and turn off the electric field. As mentioned in the previous section, when the electric field is absent and the magnetic field is B ≪ B 0 , state α is a pure M S = −1/2 state and state β is a pure M S = +1/2 state. If carried out faster than the inherent time scale of the exciton dynamics (see next section), step (iv) must therefore project the excitonic wave function on the many-body state with The variation of the magnetic field in step (iv) must be generally faster than h/J to preserve the exciton state, but slow enough to preclude the non-adiabatic transitions to molecular state γ. This can be achieved because the splitting of the molecular states β and γ at the avoided crossing is much greater than J. We have confirmed that the magnetic field can be detuned to a value B ≪ B 0 without changing the magnitudes of the coefficients C i by timedependent calculations as described in the next section. Wave function (11) is a many-body analogue of the two-body EPR state with spin-1/2 particles a| ↑ | ↓ + b| ↓ | ↑ [18,29]. In the following section, we show that the expansion coefficients C i in Eq. (11) can be modified by creating vacancies in the molecular crystal.

Localization of magnetic excitons in the presence of vacancies
In an ideal, infinitely large crystal, the exciton wave function is completely delocalized and the probability to find any molecule in the excited state is the same. If the molecular crystal contains impurities in the form of other molecules or vacancies, the exciton wave function is modified and the excitons may undergo coherent localization [27]. The localization patterns depend on the dimensionality of the crystal, the concentration and the distribution of impurities as well as the exciton -impurity interaction strength. Wurtz and coworkers have recently presented the results of an experiment showing that atoms in specific sites of an optical lattice can be selectively evaporated by focusing an electron beam onto a par-ticular lattice site [19]. This technique is general and can be applied to molecules as well as atoms. This method can be used to create mesoscopic ensembles of ultracold particles with arbitrary spatial patterns. In the following two sections, we show that the interactions of magnetic excitons (11) with empty lattice sites are very strong and that the entangled spin states given by Eq. (11) can be effectively modified by evaporating molecules from the optical lattice to create vacancies. In this section, we consider a one-dimensional array of 2000 SrF molecules on an optical lattice with lattice spacing a = 400 nm. Since the exciton coupling constant J is negative (cf, Figure 2), the exciton has a positive effective mass, i.e.
the energy of the exciton increases with the wavevector [27]. An excitation of molecules by laser field produces excitons with small wavevector [27] so we focus on a state near the bottom of the energy spectrum.
To find the eigenstates of a molecular crystal with vacancies, we use the two-level approximation and modify Hamiltonian (5) Figure 4.  This should produce a spin excitation localized on a single molecule. The spin excitation transfer can then be induced by removing the field gradient, tuning the magnetic field to a value B ∼ B 0 and applying an electric field that gives rise to significant couplings J f,f n,m in Eq. (5). The spin excitation of a single molecule is a localized exciton wave packet. When the coupling matrix elements J f,f n,m are non-zero, the wavepacket must exhibit dynamics of spin excitation transfer determined by the strength of the coupling matrix elements J f,f n,m . In order to explore the effects of electric and magnetic fields on the dynamics of the exciton wavepackets and determine the timescale of the spin excitation transfer, we consider a small ensemble of SrF molecules on a one-dimensional optical lattice. In a finite-size ensemble of molecules, the spin excitation should propagate to the edge of the molecular sample and return to the original molecule, leading to revivals of the exciton wavepacket. We expand the total wave function of the system in terms of direct products (12) with time-dependent expansion coefficients, The substitution of this expansion in the time-dependent Schrödinger equation with Hamiltonian (5) yields a system of first-order coupled differential equations for the expansion coefficients F n (t). We integrate these equations to obtain the time evolution of the quantities |F n (t)| 2 , which give the probability for the spin excitation to be found at time t in lattice site n. Figure 5 shows the autocorrelation function |A(t)| 2 = | Ψ(t = 0)|Ψ(t) | 2 , which describes the probability for the spin excitation to be on the initially excited molecule (molecule four in this example) in a system of seven molecules calculated as a function of time for four different magnitudes of the magnetic field near the avoided crossing shown in  Figure 7 demonstrates the interaction of the spin excitation with the potential barriers produced by two (panel a) and three (panel b) adjacent vacancies. This calculation shows that the excitation can tunnel through the vacancies and that the rate for tunneling through three empty lattice sites is almost one order of magnitude smaller than that for tunneling through two empty sites. Figure 7 demonstrates that the spin excitation in a mesoscopic ensemble of 2 Σ molecules in an optical lattice separated by three empty lattice sites can be considered isolated on the time scale of < 5 ms. SUMMARY We have shown that the unique energy-level structure of 2 Σ molecules can be exploited for controlled preparation of many-body entangled states (11) of non-interacting molecular spins We use SrF molecules for our illustrative calculations in this paper. SrF has a relatively large dipole moment, a relatively small rotational constant and a very weak spin-rotation interaction by comparison with other 2 Σ molecules [32]. The effects studied here should be more pronounced, making the experiments easier, in an ensemble of molecules with a larger dipole moment and a larger spin-rotation interaction constant or with an optical lattice with smaller lattice-site separations. The experimental work on the creation of ultracold 2 Σ molecules that should exhibit similar behavior as SrF is currently underway in several laboratories [15][16][17]33]. We note that molecules in the 3 Σ electronic state exhibit similar avoided crossings in combined electric and magnetic fields [34,35], which significantly widens the range of molecules that can be used for the experiments proposed here.
The absence of interactions between the spin states in the coherent superpositions (11) should make the entanglement robust against decoherence due to vibrational motion of molecules in the lattice. We note that this entanglement is created over distances spanning thousands of molecules in an optical lattice separated by 400 nm, which amounts to mm size scales. We showed that the exciton states can be localized by creating vacancies in the optical lattice. Our results demonstrate that the localization patterns of the magnetic exciton states are sensitive to the number and distribution of vacancies. This can be exploited for engineering entangled states with different patterns. Excitons determine the optical properties of crystals and it might also be interesting to explore the interaction of the magnetic excitations with electromagnetic radiation, which can be achieved either through coupling to a two-photon optical field [36] or to a single photon rf field [37]. The localization of excitons by vacancies in the presence of tunable dc electric and magnetic fields can then be used for controlled preparation of polaritons. The experimental techniques for producing optical lattices of ultracold molecules in the ro-vibrationally ground states [9] and for creating vacancies in optical lattices with controlled spatial arrangements [19] have been recently demonstrated.
The system proposed here can be used to study dynamics of spin excitation transfer in molecular aggregates. We have shown that localized magnetic excitations in finite-size arrays of molecules exhibit revivals in the presence of electric and magnetic fields near E 0 and B 0 .
The frequency of the revivals can be controlled by varying the magnitude of the electric and magnetic fields. Our calculations show that the magnetic excitation can tunnel through multiple vacancies in adjacent lattice sites. The tunneling rate is dramatically suppressed when the number of vacancies in adjacent lattice sites is increased. Our results indicate that the time scale for the magnetic excitation transfer through three adjacent vacancies is > 5 ms. This suggests a possibility of studying spin excitation transfer by measuring the exciton wavepacket revivals in small molecular aggregates. The localized spin excitations in an optical lattice of 2 Σ molecules can be created and observed by applying a gradient of a magnetic field and performing a spatially-resolved spectroscopic measurement [6,38].