Two-state Bogoliubov theory of a molecular Bose gas
We present an analytic Bogoliubov description of a BEC of polar molecules trapped in a quasi-2D geometry and interacting via internal state-dependent dipole-dipole interactions. We derive the mean-field ground-state energy functional, and we derive analytic expressions for the dispersion relations, Bogoliubov amplitudes, and dynamic structure factors. This method can be applied to any homogeneous, two-component system with linear coupling, and direct, momentum-dependent interactions. The properties of the mean-field ground state, including polarization and stability, are investigated, and we identify three distinct instabilities: a density-wave rotonization that occurs when the gas is fully polarized, a spin-wave rotonization that occurs near zero polarization, and a mixed instability at intermediate fields. These instabilities are clarified by means of the real-space density-density correlation functions, which characterize the spontaneous fluctuations of the ground state, and the momentum-space structure factors, which characterize the response of the system to external perturbations. We find that the gas is susceptible to both density-wave and spin-wave response in the polarized limit but only a spin-wave response in the zero-polarization limit. These results are relevant for experiments with rigid rotor molecules such as RbCs, -doublet molecules such as ThO that have an anomalously small zero-field splitting, and doublet- molecules such as SrF where two low-lying opposite-parity states can be tuned to zero splitting by an external magnetic field.
The experimental realization of Bose-Einstein condensation Anderson et al. (1995); Bradley et al. (1995); Davis et al. (1995) and Fermi degeneracy DeMarco and Jin (1999); O’Hara et al. (2002) in dilute samples of alkali atoms enabled many new discoveries and advances in the field of ultracold degenerate gases including the demonstration of the crossover from a Bose-Einstein condensate (BEC) to a Bardeen-Cooper-Schrieffer superfluid state Greiner et al. (2003); Bourdel et al. (2004) and the formation of self-assembled vortex lattices Abo-Shaeer et al. (2001); Schweikhard et al. (2004). Additionally, the microscopic “spin” degrees of freedom in these atomic systems have been used to explore more unconventional states of quantum matter, such as spin-orbit coupled Bose gases Lin et al. (2011); Galitski and Spielman (2013) and high-spin Bose gases Kawaguchi and Ueda (2012); Stamper-Kurn and Ueda (2013), which are host to a variety of novel quantum phases and phase transitions Stenger et al. (1998); Schmaljohann et al. (2004); Sadler et al. (2006). In all of these systems, ultracold temperatures have permitted the observation of coherent phenomena in the presence of relatively weak interactions.
Currently, promising candidates for realizing strong interactions are diatomic, heteronuclear molecules which can possess large electric dipole moments and interact strongly, even in very dilute molecular samples. Further, dipole-dipole interactions (ddi) are inherently long-range () and anisotropic Brown and Carrington (2003). In recent years, experimental groups have made remarkable progress toward cooling molecular samples to quantum degeneracy Doyle et al. (1995); Bethlem et al. (1999); Nikolov et al. (2000); Jochim et al. (2003); Kerman et al. (2004); Haimberger et al. (2004); Sage et al. (2005); Ni et al. (2008); Deiglmayr et al. (2008); Danzl et al. (2010); Stuhl et al. (2012); Takekoshi et al. (2014); Molony et al. (2014); Ospelkaus et al. (2008); Lang et al. (2008); Kuznetsova et al. (2009); Aikawa et al. (2010). Thus, ultracold molecules are among the most exciting prospects for future studies of strongly interacting quantum many-body systems Carr et al. (2009); Lemeshko et al. (2013).
For a large class of molecules (the “rigid rotors” e.g. KRb, RbCs, etc.), the lowest-lying microscopic degrees of freedom are rotational in nature, with characteristic energy splittings on the order of Brown and Carrington (2003). A number of theoretical proposals have discussed how these rotational levels can be manipulated to behave like “spins,” and how the state-dependent dipole-dipole interactions can be tuned (using a combination of DC electric and microwave fields) to emulate a broad class of quantum spin models, and thus to study quantum magnetism in a completely new context Micheli et al. (2006); Sadler et al. (2006); Yao et al. (2013); Manmana et al. (2013); Hazzard et al. (2013); Barnett et al. (2006); Gorshkov et al. (2011a, b); Kuns et al. (2011); Herrera and Krems (2011); Xiang et al. (2012); Lemeshko et al. (2012); Yan et al. (2013). Other molecules, such as -doublet (e.g. ThO, TiO) or doublet- (e.g. SrF) molecules, possess a set of low-lying opposite-parity electronic states with anomalously small energy separations Brown and Carrington (2003); Bohn (2010); Vutha et al. (2011). Recently, a sample of -doublet OH molecules was Stark-decelerated and evaporatively cooled to temperatures , approaching the quantum degenerate regime Stuhl et al. (2012). Unlike the rigid-rotors, the ground state of these molecules forms an effective spin-1/2 manifold, which is energetically far-removed from the higher-lying rotational states. Even in a very dilute sample, the dipole-dipole interaction energy can approach the doublet splitting, resulting in interesting dielectric properties Wilson et al. (2014).
Motivated by the experimental progress in the cooling and trapping of heteronuclear polar molecules, there has been a great deal of theoretical interest in understanding the role that strong dipole-dipole interactions play in BECs of molecules that possess spatial degrees of freedom. Many predictions have been made, including the emergence of a roton-maxon quasiparticle spectrum Santos et al. (2003); Ronen et al. (2006), anisotropic superfluid flow Ticknor et al. (2011), and structured vortex excitations Yi and Pu (2006); Wilson et al. (2009). However, very little work has been done to understand the role that the microscopic molecular structure plays in such systems.
In this paper, we present a robust theoretical description of bosonic molecules cooled to quantum degeneracy in which the microscopic nature of the molecules plays an important role. We investigate the mean-field ground state and mesoscopic structure of low-energy excitations by way of Bogoliubov-de Gennes perturbation theory. We present a general, analytic procedure for diagonalizing the fluctuation Hamiltonian, which results in analytic expressions for both the dispersion relations and Bogoliubov amplitudes, in terms of which we can calculate important many-body quantities such as two-point correlation functions and response functions. This procedure generalizes the method developed in Ref. Tommasini et al. (2003) for the case of momentum-dependent couplings that arise as a consequence of the long-range nature of the interactions.
Using these methods, we investigate a quasi-2D BEC of polar molecules in the presence of an external electric field that couples two low-lying molecular states where the molecules interact via state-dependent dipole-dipole interactions. We investigate the properties of the mean-field ground state by way of a Gaussian ansatz for the axial wave functions. By carefully investigating the nature of the two-point density-density and spin-spin correlation functions, we arrive at a complete physical picture of the dynamical instabilities that arise at large densities. We identify three distinct mechanisms for these instabilities, and we conclude with a discussion of three different candidate molecules and the associated parameter regimes (zero-field splitting, field strength, and density) to which these results apply.
The paper is organized as follows. In Sec. II, we present the theory for the internal structure of the molecules, developing a two-state approximation that allows for a unified treatment of a variety of different molecules. In Sec. III, we present the many-body treatment of the system where the many-body Hamiltonian, the Bogoliubov-de Gennes expansion, and the ground state energy functional in a Gaussian approximation are developed. In addition, we present the full analytic diagonalization of the fluctuation Hamiltonian, and thereby derive analytic expressions for important quantities such static response functions. In Sec. IV, we analyze the ground state energy functional and the behavior of energy and polarization of the mean-field ground state, including important limits. In Sec. V, we present analyses of the dispersion relations, depletions, static structure factors, and correlation functions all in the context of understanding the nature of the instabilities that appear for large enough density in the low-, intermediate-, and high-field regimes. In Sec. VI, we develop a full physical picture of the instabilities seen these regimes through analysis of the static structure factors and correlation functions. Finally, in Sec. VII, we conclude with a discussion of the implications of these results, including how to experimentally access the behavior using the candidate molecules addressed in Sec. II.
Ii Single Molecule Theory
In this paper, we consider a gas of polar molecules interacting via the dipole-dipole interaction in the presence of an external electric field . In such systems, the net polarization of the gas and the external field together induce a dipole moment in a particular molecule, which in turn modifies the overall polarization. In a semi-classical treatment Wilson et al. (2012a), we solve for both and self-consistently. Here, we are explicitly interested in the role that the microscopic molecular structure plays in determining the many-body behavior of a quantum degenerate gas of polar molecules. We build in a microscopic, quantum mechanical description of polarizability in molecular systems by including two low-lying opposite-parity states of the molecule that are coupled by an external field. This two-state approximation is general enough to provide a unified treatment of a wide class of molecules, including -doublets, doublet-’s, and rigid rotor molecules. We note that this description provides a unified picture of both dielectric physics—by building in the microscopic description of molecule polarizability—and of spin- systems with long-range interactions.
This description takes the form of a two-state molecular Hamiltonian, given by
where , , is the effective dipole moment of the molecule in the strong-field limit, is the strength of the applied electric field, and is the zero-field splitting between two low-lying molecular states, the nature of which we will discuss below in the context of specific molecules. This Hamiltonian is written in the basis that diagonalizes the , where is the matrix of the dipole operator that lies along the molecular axis restricted to the lowest two molecular eigenstates. In this case,
We interpret the basis states in the basis as dipole states show dipole moments align () or anti-align () with the external field.
This description is convenient for multiple reasons. It provides a clear physical picture of the emergent physics, and it eliminates exchange interactions between molecules in the many-body Hamiltonian, enabling a fully analytic solution of the problem within the Bogoliubov de-Gennes framework. In addition, this allows for a unified many-body description of a wide class of dipolar BEC’s.
We have identified three classes of polar molecules relevant to modern experiments that are good candidates for experimentally realizing the results in this paper. These candidate classes are the rigid rotor molecules, -doublets, and doublet-s. The specific candidate molecule in the class of rigid rotors is RbCs, which has been cooled by means of both STIRAP Takekoshi et al. (2014); Ni et al. (2008) and photoassociation Ji et al. (2012). In the class of -doublets, we consider ThO, which is a candidate for eEDM searches Vutha et al. (2011). In the class of doublet-’s, we consider SrF, which has been laser cooled Shuman et al. (2010); Barry et al. (2012) and is a candidate for realizing magnetic Frenkel excitons in an optical lattice filled with such molecules Pérez-Ríos et al. (2010).
The following discussions of rigid rotor and -doublet molecules closely follow the discussions in Ref. Bohn (2010). The discussion of the doublet- molecules closely follows the discussion in Ref. Pérez-Ríos et al. (2010). In Sec. VI, we give a detailed accounting of the parameter regimes relevant to realizing the results discussed later in this paper for a subset of the molecules described in the following subsections.
ii.1 Rigid rotors
The Hamiltonian of a rotating molecule in the presence of an external electric field is given by
where is the rotational constant, is the total spatial angular momentum of the molecule, and is the dipole moment operator in the body-fixed frame. In the basis of eigenstates of and , the Hamiltonian is given by
and the matrix elements of the dipole operator components can be compactly expressed in terms of symbols as
The symbols enforce the selection rules and are even. Assuming that the external field is homogeneous and point along the lab-frame -axis, the only term that survives is the term.
The external electric field acts to mix angular momentum states according to Eq. (5). By diagonalizing , we can systematically include the mixing in of higher rotational states while still treating the system in a two-state approximation. In Fig. 1, we have plotted the lowest nine eigenergies of . We keep the lowest eigenstates and of , which are adiabatically connected at zero field to the states and , respectively. In this truncated basis, the molecular Hamiltonian takes the diagonal form,
where the splitting is a function of the field strength ; we have subtracted off a constant, field-independent offset. An effective dipole operator can be written in this truncated basis as
where the matrix elements in the basis are calculated by constructing in the basis using Eq. (5), transforming to the eigenbasis of , and restricting to the two lowest states and .
As noted previously, it is convenient to work in the eigenbasis of the effective dipole moment operator, in which case the molecular Hamiltonian takes the form of Eq. (1) with
The dipole moments are given by
In Fig. 2, we have plotted the matrix elements of both and as a function of . For , and are constant, and varies linearly with the external field. We therefore interpret the state as a molecular state with dipole moment that is aligned (anti-aligned) with the external field. We interpret as a zero-field splitting of the molecule, and the energies of the dipole states display a linear Stark shift. This interpretation requires that we work in the low-field limit where the effects of higher-lying rotational states are minimized.
In a certain class of molecules, there exist two low-lying states and of opposite parity whose splitting is much smaller than the rotational spitting . These two states are said to comprise a -doublet Bohn (2010). The Hamiltonian for such molecules is given by
where is an external electric field oriented along the molecular axis, is the zero-field splitting of the molecule, and is a function of the total electronic and rotational angular momentum quantum numbers of the states and .
In the basis that diagonalizes , the molecular Hamiltonian again takes the form of Eq. (1), where and , and the dipole moments are given by . We again interpret these two states as dipole states that either align or anti-align with the external field. These states are the strong-field states of the Hamiltonian, and since , we can interpolate between the weak-field limit and the strong field limits while still neglecting the effects of rotations of the molecule.
Following the discussion in Ref. Pérez-Ríos et al. (2010), the Hamiltonian for a molecule in the presence of external electric and magnetic fields can be written as
where the first term includes both vibrational and rotational terms, is the molecular spin, is the rotational angular momentum, is the molecular dipole moment and is the Bohr magneton. We assume that the molecules are in their vibrational ground states and can be approximated as rigid rotors. It is shown in Ref. Pérez-Ríos et al. (2010) that there are two low-lying states () and () of opposite parity that can be tuned to zero splitting via the external magnetic field. The electric field is the only term in the Hamiltonian that couples opposite-parity eigenstates, so this crossing is exact at zero-field. Restricting our attention to only these two states, we can write the Hamiltonian in the form of Eq. (10), where can now be tuned by an external magnetic field to be much smaller than the rotational splitting. We again work in the basis where the dipole moment operator is diagonal, and everything carries over from the -doublet section.
Iii Many-Body Hamiltonian and Bogoliubov-de Gennes Analysis
In this section, we present the general many-body treatment of a molecular BEC interacting via electric dipole interactions in the case where the molecules can be treated in a two-state approximation, as discussed in Sec. II. Using a Bogoliubov-de Gennes analysis, we derive both the ground-state energy functional and the second-order fluctuation Hamiltonian in the grand-canonical ensemble via a Gaussian ansatz for the axial (trap-axis) wave functions. We analytically diagonalize the fluctuation Hamiltonian. This procedure results in analytic expressions for the low-energy dispersion relations and Bogoliubov amplitudes, in terms of which we can write important many-body properties such as the quantum depletion and static structure factors.
iii.1 Many-body Hamiltonian in the dipole basis
The full many-body Hamiltonian is given by
where is the single-particle Hamiltonian for the center-of-mass motion of the molecule, and
where we have assumed that the induced dipoles lie along the -axis.
Expanding the field operator as a two-component spinor
the interaction Hamiltonian becomes
The rotation to the single-molecule strong-field basis removes any exchange interactions in the Hamiltonian, and we are left with only direct interaction terms.
The remaining terms in the Hamiltonian (Eq.(12)) can be expressed in terms of the field operators , and the result is
where , given by
is the single-molecule Hamiltonian, and , given by
is the Hamiltonian for the linear coupling between dipole states that arises as a consequence of the zero-field splitting. We introduced chemical potentials to work in the grand-canonical ensemble, and , given by
which is the number operator for the internal state . We consider a gas of polar molecules harmonically trapped in quasi-2D, in which case the Hamiltonian is given in cylindrical coordinates by
iii.2 Bogoliubov Theory
Here, we perform a Bogoliubov-de Gennes analysis of the full many-body Hamiltonian derived above. When the axial trapping is sufficiently tight, we can expand the field operator as the product of an axial wave function and a field operator for the in-plane motion. We further expand the field operator as a sum of condensate and fluctuation terms, yielding
where is the number of particles occupying molecular state , is an in-plane wave vector, is a normalized, state-dependent axial wave function, and is the in-plane area of the system. Neglecting the fourth-order terms in the expansion, Eq. (17) becomes
where , given by
is the ground-state energy functional, and , given by
is the second-order fluctuation Hamiltonian. These terms are expressed in terms of the single-particle parameters,
and the interaction parameters,
where is the oscillator length associated with the axial trapping potential.
These equations are purely general for the case where the particles interact exclusively via direct interactions via a state-independent central potential (see Eq. (15)). In the case of dipole-dipole interactions between dipoles aligned with the trap-axis, the interaction parameter can be written in the simple form (see Appendix A)
In order to calculate the parameters, we need the axial wave functions . We can minimize the ground state energy with respect to and the population-normalized axial wave functions , and enforce the normalization condition . Extremizing with respect to yields . Extremizing with respect to results in a set of coupled differential equations, given by
which is constrained by both the normalization condition above and the equilibrium condition, .
Here, we instead employ a Gaussian ansatz for the axial wave functions, given by
This allows us to find analytic expressions for the interaction parameters, and we find that this ansatz results in good qualitative agreement with the results obtained by using the numerical solutions of Eq. (30). The parameters in Eq. (24) can be evaluated analytically, and the ground state energy per particle can be written as
where is the relative phase between the two components, is the total 2D areal density scaled by , is the relative number of molecules occupying molecular state , and the interaction parameter is given by
In order to find the ground state energy, we minimize Eq. (32) with respect to and , yielding and
|which determines the relative population via the equilibrium condition, .|
|In light of the Gaussian ansatz and energy minimization procedure, the fluctuation Hamiltonian (Eq. (25)) can be written as|
where the interaction parameter is explicitly given by
and is the complementary error function.
iii.3 Diagonalization of the fluctuation Hamiltonian
In Ref. Tommasini et al. (2003), the Bogoliubov diagonalization procedure was generalized in order to deal with a coherently coupled two-state BEC whose atoms interact via contact interactions. Here, we generalize this procedure for the case of the momentum-dependent couplings (Eq. (36)) that arise when the particles interact via an anisotropic, long-range interaction. We note that this procedure can be applied to any BEC with a linear coupling term and state-dependent, momentum-dependent interaction couplings, as long as there are only direct interactions (see Eqs. (28) and (25)). The diagonalization procedure results in analytic expressions for the Bogoliubov amplitudes, and they can be combined with (if necessary) numerical values of the interaction parameters to yield important quantities such as response functions.
The diagonalization procedure consists of the following steps: (1) a canonical transformation of the plane-wave operators that removes the linear coupling between the two modes; (2) a transformation to a set of non-Hermitian coordinate operators scaled in such a way that the momentum terms are left invariant under a further rotation of the coordinate operators; (3) a rotation of the coordinate operators that decouples the two components; and (4) a transformation back to a set of bosonic operators in terms of which the fluctuation Hamiltonian is diagonal. Since in-plane center-of-mass momentum is conserved in this system, this procedure is identical for each block of the Hamiltonian corresponding to a particular momentum.
For purposes of clarity, in what follows, we make the replacement .
The first transformation is given by
and it can be easily shown that the new creation and annihilation operators and satisfy the canonical boson commutation relations. Under this transformation, the Hamiltonian takes the form
are the single-particle energies, and
are a set of dressed momentum dependent interaction parameters.
We next define a set of non-Hermitian coordinate operators via the transformation,
are the dispersion relations in the absence of interactions between anti-aligned dipoles. These coordinate operators are non-Hermitian—i.e. and —but they still satisfy the canonical position-momentum commutation relations. Under this transformation, the Hamiltonian is
The next step is to decouple modes 1 and 2 by defining a new set of coordinate operators,
and the angle functions are given by
|Again, these operators are not Hermitian, but they do satisfy the canonical position-momentum commutation relations. Under this transformation, the Hamiltonian takes the form|
The final transformation, given by
results in the diagonal Hamiltonian,
where the operators satisfy canonical bosonic commutation relations. The first term in Eq. (50) is a state-independent offset, due to quantum fluctuations, which can be absorbed into the ground state energy. From the second term, it is apparent that are the two branches of the dispersion relation for this system. Furthermore, in the case where the dressed coupling constant uniformly vanishes, these two dispersion relations reduce to . We will see in Sec. V that these two decoupled modes can be interpreted as spin-wave and density-wave modes. The presence of interactions between molecules in different dipole states couples these two modes, and this leads to a rich quasi-particle spectrum in which either density- or spin-wave behavior can dominate.
Since we have explicit expressions for the operator transformations, we can write the original plane-wave operators in terms of the quasi-particle operators as
where the ’s and ’s are known as Bogoliubov amplitudes. Many quantities that characterize the system—such as structure factors and correlation functions—can be written in terms of these amplitudes, and we therefore quote the results here. They are