Hyperfine structure in the microwave spectra of ultracold polar molecules

We investigate the microwave spectra of ultracold alkali metal dimers in magnetic, electric and combined fields, taking account of the hyperfine structure due to the nuclear spins. We consider the molecules 41K87Rb and 7Li133Cs, which are the targets of current experiments and demonstrate two extremes of large and small nuclear quadrupole coupling. We calculate the frequencies and intensities of transitions that may be used to transfer ultracold molecules between hyperfine states in a magnetic field, employing different polarizations of microwave radiation. In an electric field, the hyperfine levels display narrow avoided crossings at specific fields that we explain in terms of molecular alignment. The hyperfine splittings that arise in electric fields may hinder individual addressing in schemes to use ultracold molecules in quantum computation, but the structure of the spectra is suppressed in combined fields.


Introduction
Ultracold molecules have potential applications in many areas of physics ranging from precision measurement to quantum computing [1]. It has recently become possible to create ultracold alkali metal dimers in their ground rovibrational state in ultracold atomic gases, by magnetoassociation followed by state transfer using stimulated Raman adiabatic passage (STIRAP). This has been achieved for both polar 40 K 87 Rb [2] and nonpolar Cs 2 [3] and triplet 87 Rb 2 [4], and several other systems are being pursued [5][6][7]. Ground-state KRb molecules are now being used in experiments to probe ultracold chemical reactions in a fully quantal regime [8]. It is also possible to produce low-lying molecular states by direct photoassociation [9][10][11][12][13].
Alkali metal dimers have a rich spin structure because of the presence of two nuclei with nonzero spin. The nuclear spins interact with one another and with the molecular rotation through several different terms in the hyperfine Hamiltonian. In previous work, we have explored the hyperfine structure, and the effect of applied electric and magnetic fields, for both polar [14] and nonpolar [15] alkali metal dimers.
The microwave spectra of alkali metal dimers are important in a variety of contexts. They provide opportunities to transfer polar molecules between spin states and thus to control the nature of the molecular sample [16]. They are important in proposals to use ultracold molecules in the design of quantum computers [17][18][19], to control intermolecular interactions and create novel quantum phases [20][21][22], in condensed-phase models such as the molecular Hubbard Hamiltonian [19,23], to trap polar molecules using microwave radiation [24] and to use cavityassisted microwave cooling [25]. However, work on these topics has almost invariably neglected the effects of hyperfine structure.
We have previously given a brief report of the microwave spectrum of 40 K 87 Rb taking into account the hyperfine structure and the effects of electric and magnetic fields [26]. The purpose of the present paper is to give a more complete account of the spectra and to extend our predictions to other systems of current experimental interest. In the present paper, we compare and contrast the predicted spectra for 41 K 87 Rb and 7 Li 133 Cs, which are the targets of current experiments [5,12] and which demonstrate two extremes of large and small nuclear quadrupole couplings. We concentrate on spectra originating in the rotationless ground state, which is the principal focus of the experimental efforts. Rotational constant (B), rotational g-factor (g r ), nuclear g-factors (g 1 , g 2 ) and nuclear quadrupole ((eQq) 1 , (eQq) 2 ), shielding (σ 1 , σ 2 ), spin-rotation (c 1 , c 2 ), tensor spin-spin (c 3 ) and scalar spin-spin (c 4 ) coupling constants. Subscripts 1 and 2 refer to the less electronegative atom (K or Li) and to the more electronegative one (Rb or Cs), respectively. See [14] for further explanation of the parameters. 41  The structure of the paper is as follows. Section 2 describes the theoretical methods used, while sections 3, 4 and 5 discuss the spectra in the presence of magnetic, electric and combined fields, respectively.

Theoretical methods
The effective Hamiltonian of a 1 molecule in a specified vibrational state consists of rotational, hyperfine, Zeeman and Stark terms [27][28][29]. The hyperfine term includes contributions that represent the nuclear electric quadrupole interaction, the nuclear spin-rotation interaction, and the scalar and tensorial nuclear spin-spin interactions. The Zeeman term is dominated by the nuclear contribution describing the interaction of the nuclei with the magnetic fields. In the present work, we construct the Hamiltonian as described in [14] and diagonalize it to obtain the energy levels and wavefunctions.
The hyperfine coupling constants for 41 K 87 Rb and 7 Li 133 Cs have been evaluated as described by Aldegunde et al [14]. The resulting values are given in table 1. The calculations were carried out at the equilibrium geometries, R e = 4.07 Å for KRb [30] and R e = 3.67 Å for LiCs [31]. They are thus most suitable for the vibrational ground state but would also be a reasonable approximation for other deeply bound vibrational states.
There are three sources of angular momentum in a 1 molecule: the rotational angular momentum (N ) and the nuclear spins (I 1 and I 2 ). In the absence of a field, these couple together to give a total angular momentum F. However, in the presence of electric or magnetic fields, F is destroyed and the only conserved quantity is the total projection quantum number M F . It is most convenient to construct the Hamiltonian in an uncoupled basis set |N M N |I 1 M 1 |I 2 M 2 , 4 where M N , M 1 and M 2 are the projections of N , I 1 and I 2 onto the z-axis defined by the direction of the field.
We consider two different polarizations of the microwave field. When the polarization is parallel to the z-axis, the selection rules for dipole matrix elements between the uncoupled basis functions are N = ±1, M N = 0, M 1 = 0, M 2 = 0 and M F = 0. For any other axis of polarization, there are components that are circularly polarized with respect to z; the selection rules for these components are the same except for M N = ±1 and M F = ±1.

Microwave spectrum in a magnetic field
The experiments that produce ultracold 40 K 87 Rb [2] and 133 Cs 2 [3] in their ground rovibrational state involve two main steps. Pairs of ultracold atoms are first associated by sweeping the magnetic field across a zero-energy Feshbach resonance [32,33]. The resulting Feshbach molecules are then transferred to the ground rovibrational state by stimulated Raman adiabatic passage (STIRAP). The molecules remain in a magnetic field. It is possible, in principle, to populate a variety of hyperfine levels. However, the current experiments produce dimers with a particular value of M F determined by the atomic states used for magnetoassociation, as described below.
An important goal is to control the hyperfine state in which molecules are produced. In particular, the absolute ground state is stable to non-reactive collisions with other ground-state species (although reactive collisions with either atoms or molecules may still occur [8]). This section explores how microwave transitions can be used to transfer molecules between hyperfine states, as has now been achieved experimentally for 40 K 87 Rb [16]. We consider the use of (i) z-polarized microwave radiation to transfer population between N = 0 states of the same M F and (ii) radiation with a circularly polarized component to transfer population into the ground hyperfine state by changing M F . We discuss the mechanism of the transfer and the experimental conditions that favor or hinder it. The zero-field energy level patterns for the N = 0 and N = 1 levels of 41 K 87 Rb and 7 Li 133 Cs are shown in figure 1. The levels are labeled by the total angular momentum F, obtained by coupling the rotational angular momentum N and the nuclear spins I 1 and I 2 . The nuclear spin is 3/2 for 41 K, 87 Rb and 7 Li and 7/2 for 133 Cs. The splitting of the N = 0 states is due to the scalar spin-spin interaction and amounts to just a few kHz. The splitting of the N = 1 levels is caused mainly by the nuclear quadrupole interaction and amounts to 800 kHz for 41 K 87 Rb and 100 kHz for 7 Li 133 Cs. The difference stems from the very small nuclear quadrupole coupling constants for the 7 Li 133 Cs molecule, attributable to the particularly small values of the electric quadrupoles for the 7 Li and 133 Cs nuclei [34].
Each of the zero-field levels in figure 1 is (2F + 1) fold degenerate. This degeneracy can be lifted by applying a magnetic field. The energy pattern can be quite complicated as it consists of (2N + 1)(2I 1 + 1)(2I 2 + 1) states for each rotational level N . Figure 2 shows the Zeeman splitting for the N = 0 and 1 hyperfine levels of 41 K 87 Rb and includes all the 16 N = 0 and the 48 N = 1 hyperfine states. For 7 Li 133 Cs, the number of levels is even larger and they will not be shown.
We will first analyze transitions driven by a microwave field polarized parallel to the z-axis, which conserves the projection of the total angular momentum M F . Figure 3 shows the Zeeman splitting and avoided crossings of the 41 K 87 Rb (M F = +2) and 7        M F and is usually carried out using atoms in their absolute ground state in the magnetic field, which is ( f = I − 1/2, m f = I − 1/2) for all alkali metal atoms except 40 K. This is ( f = 1, m f = +1) for 7 Li, 41 K and 87 Rb and ( f = 3, m f = +3) for 133 Cs. Feshbach molecules are thus likely to be formed initially in M F = +2 and +4 for 41 K 87 Rb and 7 Li 133 Cs, respectively. In current experimental configurations, the STIRAP process used to transfer the molecules to the rovibrational ground state also conserve M F . The hyperfine state formed in this way is not generally the absolute ground state, which has M F = I 1 + I 2 .
There are important differences between dimers that contain 40 K and those that do not. 40 K is the only stable isotope of any alkali metal to have a negative g-factor and an inverted hyperfine structure. Because of this, the atomic ground state in a magnetic field is ( f = I K + 1/2 = 9/2, m f = −9/2) and heteronuclear Feshbach molecules formed from ground-state atoms have M F = I 1 − 5 [35]. The lowest molecular hyperfine state at high magnetic field has M F = I 1 − I K because of the negative value of g K for 40 K.
As the magnetic field increases from zero, the levels of each F split into components labeled by M F , as shown in figures 2 and 3. Avoided crossings occur between states with the same value of M F . While for the N = 0 levels the crossings occur at small magnetic fields proportional to |c 4 /(g 1 − g 2 )|, for the N = 1 states they take place at larger fields because of the larger zero-field splittings. Above the crossings, F is completely destroyed. For both molecules the N = 0 avoided crossings occur at fields below 20 G and are hard to see in figures 2 and 3. The N = 1 crossings take place at fields that depend on zero-field splittings E 0 divided by factors involving the two nuclear g-factors. Because of the very small zero-field splittings between some pairs of states, some of the avoided crossings can occur at fields as low as a few Gauss, although others are at much higher fields.
The nuclear Zeeman term, which dominates the Hamiltonian at high field, is diagonal in the uncoupled basis set. Because of this, the individual projections of the nuclear spins M 1 and M 2 become nearly good quantum numbers at sufficiently high field. When the g-factors for the two nuclei are substantially different, as for 41 K 87 Rb, the energy levels for each rotational state separate into distinct groups as the field increases (see figure 2). The g-factor for 87 Rb is much larger than that for 41 K, so that M Rb becomes a nearly good quantum number at much smaller fields than M K and the states gather together in groups characterized by a well-defined value of M Rb : the lowest set corresponds to M Rb = +3/2 and the top set to M Rb = −3/2. Similar behavior was observed for 40 K 87 Rb in [26].
For 41 K 87 Rb with M F = +2 the levels within each group display shallow avoided crossings and M K does not become well defined until fields above about 3000 G. For 7 Li 133 Cs, by contrast, the avoided crossings are complete by 40 G. In general, the number of levels in each group is N + I − + 1 − |M F − M + |, where I ± and M ± are quantum numbers for the nucleus with the larger/smaller g-factor. When there is more than one level in a group, there may be high-field crossings between them at a field of approximately E 0 /g − µ N , where g − is the smaller of the two g-factors and µ N is the nuclear magneton. For 41 K 87 Rb (N = 1), setting E 0 to the full range of zero-field energies gives an upper bound of about 5500 G, although the highest crossings in figure 3 actually occur around 2000 G. For 7 Li 133 Cs, the corresponding upper bound is about 140 G.
We now consider the microwave transitions that are available to transfer molecules between hyperfine states with the same value of M F . Figure 4 shows the spectra for transitions between the two N = 0 hyperfine states (levels a and b) and their N = 1 counterparts (levels 1-6) for 41 K 87 Rb (M F = +2) and 7 Li 133 Cs (M F = +4) as a function of the magnetic field. The microwave field is z-polarized. Both molecules show a multi-line spectrum at low field but the spectra become dominated by a single transition in the strong-field limit. However, while a 100 G field is enough to cause this change for 7 Li 133 Cs, more than 4000 G is necessary for 41 K 87 Rb.
The progression from multi-line spectra to single-line-dominated spectra arises because the selection rules for z-polarized light only permit transitions This takes place at much lower magnetic fields for N = 0 than for N = 1. In the strong-field limit the only permitted transition from each N = 0 hyperfine state is given by the selection rule (1). This occurs at lower field for 7 Li 133 Cs than for 41 K 87 Rb because M K is not a good quantum number until very high field in the latter case.  Enlargements of the 0-600 G region for 41 K 87 Rb are shown. The microwave radiation is polarized parallel to the magnetic field. The most intense transition for each molecule is assigned a peak intensity of 1.
These spectra are important because they clarify under what conditions the N = 0 ↔ N = 1 microwave transitions can be used to transfer population between different N = 0 hyperfine states corresponding to the same M F through a two-photon process that uses the N = 1 levels as intermediate states. This will be feasible if there is at least one N = 1 level that displays a significant intensity from both the N = 0 states involved in the transfer. This requires substantially smaller magnetic fields for 7 Li 133 Cs than for 41 K 87 Rb.
Microwave radiation whose polarization is not parallel to the external field can be used to transfer population between hyperfine states with different M F values. For 41 K 87 Rb and 7 Li 133 Cs, the absolute ground state corresponds to M F = +3 and +5, respectively, except at very low magnetic field (see figure 3). It is therefore possible to reach the absolute ground state from 41 K 87 Rb (M F = +2) and 7 Li 133 Cs (M F = +4) if one of the two photons is circularly polarized. Whether this photon is responsible for the first or the second transition gives rise to two possible experimental implementations.  with significant intensity at low field but a single line dominates at sufficiently high field. The changeover occurs at considerably higher fields for 41 K 87 Rb than for 7 Li 133 Cs. The propensity rules for the possible paths are summarized in figure 6 for 7 Li 133 Cs. Solid, dashed and dotted lines represent transitions where none, one or two nuclear spin projections change and correlate with strong, medium and weak lines in the spectra. For either initial state (a or b), there are several paths to the lowest hyperfine state that combine one strong transition (solid line) with one weaker transition (dashed line). However, it should be noted that the propensity rules become stronger (more like selection rules) as the magnetic field increases. The overall two-photon transfer will therefore be more difficult at high fields and forbidden in the strong-field limit.
The propensity rules are very similar for 41 K 87 Rb (N = 0, M F = +2) ↔ (N = 0, M F = +3). The number and ordering of the levels and the distribution of strong, medium and weak transitions are identical to those for 7 Li 133 Cs. However, in 41 K 87 Rb, M K is not a nearly good quantum number until very high field so the propensity rules are not as strong.  In summary, the transfer of population between different N = 0 hyperfine states using microwave transitions will be facilitated by a larger mixture between the |N = 1, M N , M 1 , M 2 basis functions, which can be achieved by carrying out the transfer at low magnetic fields. The actual fields required depend on the magnitudes of the nuclear quadrupole coupling constants and the nuclear g-factors. The transfer is not possible in the limiting strong-field region.

Microwave spectrum in an electric field
The microwave spectra of ultracold molecules in electric fields are important in many contexts ranging from quantum information processing [17][18][19] to the creation of novel quantum phases [20,21]. In this section, we explore the effect of hyperfine structure on these spectra, focusing mostly on its relevance for the implementation of a quantum computer.
The features of the spectrum in the presence of an electric field will differ drastically from those found for a magnetic field. Electric fields do not conserve parity and cause strong mixing of rotational levels. The strength of the Stark effect depends on the molecular electric dipole moment µ, 0.57 D for KRb [2] and 5.5 D for LiCs [36].
DeMille has proposed a quantum computing implementation in which the qubits are ultracold alkali dimers oriented along (|0 ) or against (|1 ) an external electric field [17]. The field strength changes with the position in the device so as to allow individual addressing of the molecules, which is performed using microwave transitions. The resulting frequency step between adjacent molecules depends on the detailed specification of the device and the molecular properties, but typical values for an array of 10 4 molecules in DeMille's proposal would be 250 kHz for KRb and 1.7 GHz for LiCs. The |0 and |1 states are mixtures of the M N = 0 levels that correlate with N = 0 and 1, respectively, at zero field. The goal of this section is to ascertain whether the hyperfine structure will interfere with the addressing process.
The top panel of figure 7 shows the lowest Stark energy levels of a 1 diatomic molecule, neglecting nuclear spin, as a function of the reduced field in units of B/µ, where B is the rotational constant of the molecule. The electric field orients the molecule and produces a space-fixed dipole moment d = µ cos θ , where θ is the angle between the internuclear axis and the electric field. The central panel in figure 7 shows the degree of orientation, defined as the expectation value of cos θ , for the states |0 and |1 . For electric fields in the interval (2-5)B/µ, the operating range of fields in DeMille's proposal, the |0 (|1 ) state corresponds to an orientation of the permanent dipole parallel (antiparallel) to the direction of the field. In the strong-field limit, the expectation value approaches unity for both molecular states, corresponding to a perfectly parallel orientation of the dipole.
As in section 3, we begin by analyzing the behavior of the energy levels. Because of the choice for |0 and |1 , we focus on M N = 0 states and transitions driven by a z-polarized microwave field. Figure 8 shows the Stark splitting of the hyperfine levels correlating with N = 1 for 41 K 87 Rb and 7 Li 133 Cs. The field-dependent average energy has been subtracted so that the hyperfine structure can be appreciated. Small electric fields, 0.3 kV cm −1 for 41 K 87 Rb and 0.02 kV cm −1 for 7 Li 133 Cs, are enough to make M N a nearly good quantum number and to separate the |M N | = 0 and 1 energy levels.
The M N = 0 states display a striking feature as a function of the electric field. At a field of 18.8 kV cm −1 for 41 K 87 Rb or 9.95 kV cm −1 for 7 Li 133 Cs, all the levels come very close together and almost cross. At this point the hyperfine splitting reduces to that caused by the (very small) scalar spin-spin interaction, as shown in the inset for 41 K 87 Rb. The effects of the nuclear quadrupole, the tensorial spin-spin and the nuclear spin-rotation interactions vanish and the splitting simplifies and coincides with that found for the N = 0 levels in the absence of fields. At this point, the total nuclear spin obtained by coupling I 1 and I 2 is a good quantum number. In addition to their effect on the microwave spectra, such close avoided crossings may cause nonadiabatic transitions for molecules in time-varying electric fields.
Near-crossings like those shown in figure 8 will be a universal feature of polar molecules in 1 states. The nuclear quadrupole interaction depends on the electric field gradient created by the electrons at the positions of the nuclei. This is a cylindrically symmetric second-rank tensor V (2) q with only a q = 0 component in the molecule-fixed frame. However, at fields high enough that M N is a good quantum number, the quadrupole splittings are governed by the electric field gradient in the space-fixed frame. This has components V (2) p given by where D 2 p0 (φ, θ, 0) is a Wigner rotation matrix and φ is the azimuthal angle. For the M N = 0 levels, only the component with p = 0 has diagonal matrix elements, and D 2 00 (φ, θ, 0) = P 2 (cos θ), the second Legendre polynomial. The time-averaged field gradient is thus proportional to the degree of alignment, defined as P 2 (cos θ) . As shown in the bottom panel also proportional to P 2 (cos θ ) , so it vanishes at the same point, while the nuclear spin-rotation interaction has no diagonal matrix elements for M N = 0 at any value of the field. For the |0 state, which correlates with N = 0, P 2 (cos θ) increases monotonically with field, from zero at zero field to unity in the high-field limit. The nuclear quadrupole and tensorial spin-spin splittings thus also increase monotonically and there are no close avoided crossings [14].
Close avoided crossings such as those shown in figure 8 might also cause nonadiabatic transitions between hyperfine states for molecules in time-varying electric fields. This might occur either when sweeping an electric field or for molecules moving in an inhomogeneous field.  In order to keep the spectrum on a single frequency scale as a function of the electric field, we plot the hyperfine frequency shift instead of the absolute frequency. This is obtained by subtracting the frequency for the transition in the absence of hyperfine splittings. The most intense transition is assigned a peak intensity of 1.
increases, |M N | becomes better defined and the transition intensity concentrates in the M N = 0 branch (at higher frequency). This happens at smaller electric fields for 7 Li 133 Cs due to its larger electric dipole moment. The right-hand panels of figures 9 and 10 show the hyperfine structure of the spectra at larger fields, where only transitions into the two (N = 1, M N = 0) levels are permitted. These figures include the range of fields where DeMille's quantum computing scheme would operate: 7.7-19.3 kV cm −1 for 41 K 87 Rb and 6.1-10.2 kV cm −1 for 7 Li 133 Cs.
If the alkali dimers are going to work as qubits, it must be possible to switch repeatedly between the |0 and |1 states without exciting any other state of the molecule and without altering the state of neighboring molecules. The hyperfine structure will complicate the operation by making these conditions more difficult to fulfill: figures 9 and 10 show that for both molecules the spectra display several transitions with significant intensity at each field in The insets of figure 9 show a blow-up of this region. The hyperfine structure will also hinder the operation of the device at electric fields below the crossings: the intensity of the weaker transition is significant, mostly 10 −4 of that of the main peak, and the two lines spread over a range of frequency shifts comparable to the frequency step used for addressing (250 kHz). The spectrum for 7 Li 133 Cs ( figure 10) has different features. It shows two intense lines for the whole range of electric fields. Both peaks are compressed in a 10 kHz interval of frequency shifts, well below the addressing step for this molecule (1.7 MHz). The different behavior in the relative intensities for 41 K 87 Rb and 7 Li 133 Cs can be rationalized in terms of the following rules (which apply only to M N = 0 states). For molecules where the nuclear quadrupole interaction dominates the other hyperfine terms and one of the nuclear quadrupole coupling constants is much larger than the other, the individual nuclear spin projections M 1 and M 2 are nearly good quantum numbers for any electric field large enough to make the |M N | = 0 contributions negligible. The electric field not only causes the state to be dominated by M N = 0, but also indirectly makes M 1 and M 2 well defined. This explains why for 41 K 87 Rb, one transition is much more intense than the other except in the low field and the crossing regions. 40 K 87 Rb [26] is similar in this respect. By contrast, if the nuclear quadrupole interaction is not dominant or if the two nuclear quadrupole coupling constants are similar, M 1 and M 2 do not become well defined as the field increases. In this case, more than one intense line will appear even at high fields, as in the case of 7 Li 133 Cs.
In summary, the individual addressing of alkali dimers in a quantum computing device such as that proposed by DeMille [17] may be hindered by the existence of hyperfine structure. For molecules with large nuclear quadrupole constants, the hyperfine spectra of adjacent molecules may overlap, introducing an additional complication to the operation of the device.

Microwave spectrum in combined electric and magnetic fields
Combining electric and magnetic fields provides an additional degree of control over molecular properties and interactions. In this section, we explore the effects of applying parallel electric and magnetic fields on the microwave spectra. Once M 1 and M 2 become nearly good quantum numbers, the spectrum is dominated by a single line. Even a relatively small magnetic field is sufficient to achieve this for both molecules, and there are no shallow avoided crossings at high magnetic field such as those that appear for 41 K 87 Rb in the absence of an electric field, as discussed in section 3.
The subsidiary transitions are 6 or 7 orders of magnitude weaker for 41 K 87 Rb at 500 G or 7 Li 133 Cs at 100 G and their intensities continue to decrease at higher magnetic fields. Similar behavior is expected for all other alkali metal dimers. It is therefore possible to produce a singleline-dominated spectrum, with no crossings as a function of electric field, simply by applying an additional magnetic field. The use of combined electric and magnetic fields thus restores the simplicity needed to apply DeMille's quantum computing scheme.

Conclusions
We have explored the microwave spectra of ultracold alkali metal dimers in applied magnetic, electric and combined fields taking into account the hyperfine structure. We have compared the spectra for 41 K 87 Rb and 7 Li 133 Cs, which have large and small values, respectively, of the nuclear quadrupole coupling constants. Our results can therefore be extrapolated to understand the microwave spectra of any other alkali dimer.
The alkali dimers have a very complicated spin structure arising from the presence of two nonzero nuclear spins. Because of this, there are many possible spectra that might be considered. We have focused mostly on spectra in which the initial hyperfine state has N = 0 and the same value of M F as Feshbach molecules produced from atoms in their absolute ground states. However, our programs can readily be applied to other M F states and most of our conclusions can be extended straightforwardly to such states. The microwave field is polarized parallel to the external fields. In order to keep the spectrum on a single frequency scale as a function of the electric field, we plot the hyperfine frequency shift instead of the absolute frequency. The most intense transition for each value of the magnetic field is assigned a peak intensity of 1.
The zero-field splitting is much larger for N = 1 than for N = 0 because of the presence of nuclear quadrupole coupling. At low magnetic field and zero electric field, the microwave spectra display many lines of significant intensity. This persists until the nuclear Zeeman term dominates the zero-field splitting for N = 1. Some of the structure starts to disappear at relatively low magnetic fields, of as little as a few Gauss, but some of the structure persists to fields of thousands of Gauss in systems with large quadrupole coupling constants where one of the nuclear g-factors is small, such as in 41 K 87 Rb. At relatively low fields, it is possible to find two-photon paths that allow molecules to be transferred between N = 0 hyperfine states via N = 1 states. However, in the high-field limit, the spectra are single-line-dominated and no such paths exist.
An electric field mixes different rotational states and creates orientation and alignment. The oriented states have possible applications in quantum computing [17]. For oriented molecules, the levels correlating with N = 0 and 1 both have a substantial nuclear quadrupole structure. Even a very small electric field is sufficient to separate the N = 1 states with M N = 0 and ±1, in parallel electric and magnetic fields. The lowest electric field shown is 0.03 kV cm −1 . The microwave field is polarized parallel to the external fields. In order to keep the spectrum on a single frequency scale as a function of the electric field, we plot the hyperfine frequency shift instead of the absolute frequency. The most intense transition for each value of the magnetic field is assigned a peak intensity of 1.
enough to prevent significant mixing between them. This simplifies the spectrum to some extent. However, the hyperfine levels for N = 1 display close avoided crossings as a function of field, which arise at points where the oriented molecules have zero alignment. Even away from the crossings, there is more than one transition with significant intensity at each electric field, and this may cause complications in designing a quantum computer. Combined electric and magnetic fields provide a convenient way to restore simplicity in the microwave spectra and eliminate unwanted transitions. The magnetic field resolves the near degeneracy that exists when it is not present.
The calculations described in the present paper are also likely to be important in other applications of ultracold polar molecules, such as in the creation of novel quantum phases [20,21] and the development of quantum simulators for condensed-phase problems [19,23]. The spin structure must be taken into account in a full treatment of collisions involving alkali metal dimers.