Phase diagram of two-component bosons on an optical lattice

We present a theoretical analysis of the phase diagram of two--component bosons on an optical lattice. A new formalism is developed which treats the effective spin interactions in the Mott and superfluid phases on the same footing. Using the new approach we chart the phase boundaries of the broken spin symmetry states up to the Mott to superfluid transition and beyond. Near the transition point, the magnitude of spin exchange can be very large, which facilitates the experimental realization of spin-ordered states. We find that spin and quantum fluctuations have a dramatic effect on the transition making it first order in extended regions of the phase diagram. For Mott states with even occupation we find that the competition between effective Heisenberg exchange and spin-dependent on--site interaction leads to an additional phase transition from a Mott insulator with no broken symmetries into a spin-ordered insulator.


I. INTRODUCTION
Recent observations of the superfluid to Mott insulator transition in a system of ultracold atoms in an optical lattice open fascinating prospects for studying many-body phenomena associated with strongly correlated systems in a highly controllable environment [1,2,3]. For instance, theoretical studies have shown that, with spinor bosonic or fermionic atoms in optical lattices, it may be possible to observe complex quantum phase transitions [4], to realize novel superfluidity mechanisms [5], and to probe onedimensional systems exhibiting spin charge separation [6].
Recently, Duan et al. [7] proposed a technique to implement interacting spin- 1 2 Hamiltonians using ultra-cold atoms, opening the door to controlled studies of quantum magnetism. In this approach the two-state bosonic or fermionic atoms are confined in an optical lattice where spin-dependent interactions and hopping are controlled by adjusting the intensity, frequency, and polarization of the trapping light. Deep in the Mott phase the motional degrees of freedom are frozen out and the remaining spin degrees of freedom are coupled by an effective Heisenberg exchange. In refs. [7,8], an effective spin hamiltonian was derived by perturbation theory for the case of a single atom per site and the limit of small tunneling. However, in practice Mott states with more than one atom per site are also of considerable interest and may exhibit richer phase diagrams. Furthermore, spin effects are expected to be important, and even stronger, at larger values of the tunnelling, where perturbation theory fails. For example, an important question that cannot be addressed by the perturbative treatments is how spin affects the transition into a superfluid phase and the properties of the superfluid phase itself.
In this paper we first extend the earlier approaches to the case of Mott states with general integer occupation. We find that at even fillings the competition between on-site interactions and nearest neighbor spin exchange leads to a transition from a spin ordered Mott state to one with no broken symmetries. Then we present a theoretical framework, which is non-perturbative in the tunneling and allows to describe both the superfluid and insulating phases in two-component systems. Using this approach we determine the phase diagram for a density of one atom per site. We find that the spin-ordered states persist up to the superfluid transition. In this region the critical temperature for spin ordering can be large, facilitating experimental realization of these phases. The z-antiferromagnetic state in particular, enjoys a negative zero-point energy which extends its domain far beyond the mean field prediction for the Mott phase. The transition between this state and the superfluid is found to be first order in contrast with the standard superfluid-insulator transition.
Before proceeding, we note that spin Hamiltonians can also be simulated by controlled collisions via frequent time-dependent shifts of the lattice potentials [9]. Compared with that method, the spin-dependent tunneling may have certain experimental advantages since it implements the desired Hamiltonian directly, thus circumventing imperfections and errors associated with rapid perturbations due to the lattice shifts. We also note the recent studies on quantum magnetism induced via magnetic dipole interactions of the condensed atoms [10]. The present approach results in much larger interaction strength per atom, and also allows for more flexible control over interaction properties.
The paper is organized as follows. In section II we describe the Hubbard model for two bosonic species on an optical lattice, which serves as our starting point. In section III, the perturbative approach of refs. [7,8] is extended to arbitrary integer filling and the insulating phase diagram is investigated. In section IV we present the mean-field description of the SF-MI transition in two component systems. The analytical predictions of a variational approach are compared to the results of a numerical mean-field analysis. In section V a theoretical framework is developed that incorporates the effect of quantum fluctuations and treats the magnetic interactions in the Mott and superfluid phases on an equal footing. In Section VI, this framework is used to analyze the full phase diagram for one atom per lattice site. The relevance of the present results in the light of realistic experiments is discussed in the concluding paragraphs of the paper.

II. THE MODEL
We consider a system with two species of atoms or equivalently, atoms with two relevant internal states. The two species shall be denoted by the second quantized bosonic operators a and b. We assume that the two species are trapped by independent standing wave laser beams through polarization (or frequency) selection. Each laser beam creates a periodic potential in a certain direction v ασ sin 2 ( k α r), where k α is the wavevector of the light and σ = a, b is the species index. Throughout this work, we assume that the laser beams are orthogonal, creating either a square lattice in two dimensions or a cubic lattice in three dimensions. For sufficiently strong periodic potential and low temperatures the atoms will be confined to the lowest Bloch band. The low-energy Hamiltonian is then given by the Bose-Hubbard model for two boson species: Here i, j denotes the near neighbor sites, a i , b i are bosonic annihilation operators respectively for bosonic atoms of different spin states localized on site i, n ia = a † iσ a iσ , n ib = b † iσ b iσ . For the cubic lattice, using a harmonic approximation around the minima of the potential [3], the spin-dependent tunneling energies and the on-site interaction energies are given by Here v a,b is the depth of the optical potential for species a and b, v ab = 4v is the spin average potential in each direction, E R = 2 k 2 /2m is the atomic recoil energy, and a ab is the scattering length between the atoms of different spins. The intra-species interaction is given by (a a(b) are the corresponding scattering lengths). Furthermore, the magnitude of the interspecies interaction U can be additionally controlled by shifting the two lattices away from each other, which opens a wide range of U/V α to exploration. Note that spin-dependent tunnelling t µσ can be easily introduced by varying the potential depth v a and v b with control of the intensity of the trapping laser. We should also point out that the two atomic states generally have different energies (µ a = µ b in 1). In the spin language, this translates to a magnetic field in the z direction. However since there is essentially no transfer between the two populations, the experiment is performed with fixed magnetization and the chemical potentials can be set to fix this magnetization.
In this paper, we address primarily the case in which the total filling is commensurate with the lattice and the two species have equal density. A transition from a superfluid to a Mott insulator is expected, as in the usual case of a single species. However, in this system, magnetic order, associated with the pseudospin degrees of freedom (boson components), may occur as well.

III. DEEP MOTT PHASE: EFFECTIVE SPIN HAMILTONIAN
To illustrate the magnetic orders that can arise it is instructive to begin deep in the Mott insulator in the limit t a,b << U, V a,b , where the hamiltonian (1) can be simplified considerably. The low energy Hilbert space in this case contains states with a particular integer occupation on every site. However there is a remaining degeneracy associated with the spin (boson component) degrees of freedom. The degeneracy can be removed by an effective hamiltonian acting within the low energy subspace. This was done previously in Refs [7,8] for the case of a single atom per site, using second order perturbation theory in the hopping parameters. The result is Here | ↑ and | ↓ represent sites occupied by the a and b atoms respectively and the couplings are given by: We assume that the induced ordering field h can be cancelled by an externally applied field h ext . In this case the model obviously exhibits a transition between a x − y ferromagnet for J ⊥ > J z > 0 to an Ising antiferromagnet with z-Neel order (Fig. 1). We now extend the discussion to the case of any integer filling of N atoms per site. To see how things may become qualitatively different from the singly occupied case, consider first a Mott state with two atoms per site. The low-energy Hilbert space of a lattice site consists of the three states If V a,b >> U , the state a † b † | 0 has much lower energy than the other two. This implies a simple Mott state i a † i b † i | 0 which, unlike the Mott states in Fig. 1, does not break any symmetries. On the other hand, when V a,b is of the same order as U , all three states should be taken into account and more interesting phases may be possible. Therefore in the general case of N atoms per site we consider the The low-energy Hilbert space of a lattice site with N atoms per site can be constructed in a similar way. It contains the N + 1 states: where S ≡ N/2 and m = −S, . . . , S. Obviously a † and b † act as Schwinger bosons creating a multiplet of pseudospin S. The spin magnitude depends on the site occupancy. It is integer for even N and half integer for odd N . Now the effective hamiltonian within the spin S subspace can be derived by second order perturbation theory in a straightforward generalization of Ref. [7]. The result is where the interactions are given by: Note that if we take S→ 1 2 the parameters are identical to those of the effective spin-1 2 hamiltonian (2) in the case U ≈ V a = V b . Note also that additional terms such as (S z i S z j ) 2 do not arise. The (S z ) 2 term is of course just a constant for S = 1 2 and therefore has no effect in this case. However, it plays an important role at larger value of the spin, namely for occupations N > 1.
When u can be neglected relative to J z or J ⊥ , the remaining terms in (6) form an anisotropic Heisenberg model. A standard, coherent state mean field theory is then possible, which yields x − y ferromagnetic order. A large positive u, in the absence of an ordering field (h = 0), acts to reduce the S z component of the spins. At some point, the classical coherent states that represent fully polarized spins, become unsuitable descriptions of the system. In particular, at large enough u all spins will be essentially confined to their lowest possible S z states.
When the spin is half integer (odd filling), there are two active states, at large u, corresponding to S z = ± 1 2 . The hamiltonian (6) then reduces to a spin-1 2 model, but the spin interactions remain practically identical. Thus, the essential physics is unchanged. We expect ferromagnetic spin order, as for small u, only with a reduced effective moment.
For integer spin (even filling) we expect qualitatively different behavior. At large enough u only the S z = 0 state will be important. We then expect that the system is well described by | Ψ = i | S, m = 0 i , a Mott state with no broken symmetries. The transition to this state at large u from the x − y ferromagnet at small u is formally identical to the transition from a superfluid to a Mott phase in the Hubbard model of single component bosons. A direct correspondence exists between the boson number in the Hubbard model and S z in (6). The Mott state of bosons is characterized by vanishing particle number fluctuations on a site. Similarly the transition here is into a state with well defined S z on each site.
To describe the transition we note that only the three states with lowest S z play an important role in its vicinity. We therefore write a homogenous mean field ansatz: 5 The variational energy in this state is given by: and we see that the minimum occurs for η = 0, π. The x − y order parameter is S + ∝ sin θ ≡ ψ. Therefore, to find the transition to a Mott insulator we expand the energy up to quadratic order in ψ and minimize it with respect to χ. Note that the quartic term is always positive since J z < J ⊥ . We then obtain the critical value of J ⊥ as a function of h: For magnetic fields h > u, the description in terms of the states Instead, a similar scheme can be carried out, using the states { | 0 , | 1 , | 2 }. This yields another lobe corresponding to a phase with well defined S z = 1. A schematic phase diagram is plotted in Fig 2. As the ordering field is increased, we obtain lobes corresponding to larger values of S z up to S z = S, where the spin is fully polarized. In practice the number of particles in each spin state is conserved independently.
In other words the experiment is done with fixed z magnetization and h is used as a theoretical tool to set this magnetization in our model. Here we fix zero magnetiztion by setting h = 0. In summary, we found that the Mott phases of two-component bosons with even site filling are markedly different from those with odd filling. At odd filings, the Mott regions of the phase diagram are essentially the same as those found for single occupation (see Fig. 1). These are all broken symmetry phases, either a x − y ferromagnet or a z-Neel state. The Mott phases at even filling are sketched in Fig. 2. Most notably, another Mott-type transition occurs within the Mott phase, between a x − y ferromagnet and a non symmetry breaking Mott state. Related spin ordering transitions for spin 1 bosons have been discussed recently by Imembakov et al. [12] and by Snoek and Zhou [13].
The perturbative expansion leading to (2) breaks down as the transition to a superfluid is approached and t a,b become comparable to U . The question arises, whether the phases predicted by the effective spin Hamiltonian still hold in this regime. More importantly, how do the effective spin interactions affect the nature of the transition to a superfluid and the superfluid phase itself?
To answer these questions we shall develop in the next two sections a theory which captures the effective spin interactions while also able to describe the transition to a superfluid.

IV. MEAN FIELD THEORY OF THE SUPERFLUID-MOTT TRANSITION
The usual, single component, Mott transition of bosons is well described by mean field theory [11,14]. It is thus natural to start our treatment of the two component case with a mean field approach. In order to capture the superfluid phase we extend the regime considered in the previous section to allow for arbitrary ratios of t a,b /U . However we shall confine ourselves to the case of a single atom per site and to the limit U, t a,b << V a , V b . Later we shall consider corrections due to finite intra-species interactions.
In this limit, it is particularly advantageous to use a variational approach, which is equivalent to meanfield theory [15]. The idea is to assume a site factorizable wave function associated with hard core bosons, which in our case takes the form The enormous reduction in Hilbert space, made possible by neglecting double occupation, is what makes these states convenient to work with. Specifically, it is easy to calculate expectation values. In addition, we shall see that they facilitate a fluctuation expansion about the mean field theory. Generalization to include higher occupations is possible but would make the subsequent calculations much more complicated. Note that a more general mean-field ansatz would include complex weights, however it is easily verified that this would not improve the variational energy.
In the Mott state, where each site is occupied by exactly one atom, the variational state simplifies even more It can be viewed as a pseudospin-1 2 state with a † | 0 = | ↑ and b † | 0 = | ↓ . The onset of superfluidity is characterized by the development of an order parameter sin θ = 0. More precisely, the superfluid order parameters of the two species in the state | Φ are given by: Now a classical energy functional can be written, which is defined by the expectation value of (1) in | Φ . Allowing for two sub-lattice order the energy function is: where z is the lattice coordination number. In the superfluid phase this function is minimized when both cos((χ i − η i )/2) = 1 and sin((χ i − η i )/2) = 1, which implies χ i = η i = π/2. The remaining degree of freedom θ is uniform on the lattice and found by minimizing The result is where t c = U/2z. We thus find a transition to a Mott insulating state for t a + t b < t c as illustrated by the circles in Fig. 3. This constitutes a straightforward generalization of the standard transition for a single species. By assuming the variational state (11), we neglected contributions from states with multiply occupied a or b bosons. To determine effects arising from the finite magnitude of the intra-species interaction we use a numerical self-consistent mean field field theory of (1). As first proposed in [14], the kinetic energy terms in the Hamiltonian are decoupled: In the homogeneous phase this leads to a sum of identical single-site Hamiltonians where the decoupling fields have to be determined self-consistently according to We have solved the combined set of Eqs. (18) and (19) numerically by diagonalizingH MF within a finite-size Hilbert space where we allow for up to M=9 bosons per species. We show results in Fig. (3), where it can be seen that for a small ratio U/V a,b the phase diagram is identical to that determined variationally. As V a,b decrease and approach U the Mott domain shrinks. For V a,b < U there is an instability toward a z-ferromagnetic superfluid. Since the experiment is done at fixed magnetization this would lead to phase separation into domains occupied only by a or by b atoms.
Note that in the Mott state where the order parameters a and b vanish, the ground state of H MF has precisely one atom per site but is completely independent of the relative weights of a and b atoms. Similarly, the variational energy in the Mott state (12) is a constant (−U/4), independent of the individual spin orientation. Thus the simple mean field approaches are unable to resolve spin order in the Mott state. To obtain spin order we shall in the next section consider quantum fluctuations around the variational mean field solutions.

V. EFFECT OF FLUCTUATIONS: "MAGNETIC" STATES
The situation we encountered when attempting to treat the Mott phase with the variational states is similar to the basic problem of frustrated quantum magnets. The classical energy of such systems, i.e the expectation value of the hamiltonian in a basis of coherent spin states, often contains a macroscopic degeneracy (see for example the review Ref. [16]). A general mechanism that can lift the degeneracy is "quantum order by disorder", whereby broken symmetry configurations are selected by the zero-point energy due to spinwaves [17]. A spinwave expansion in magnetism includes the quadratic fluctuations around coherent-state mean field configurations. We formulate a similar expansion in fluctuations about the mean-field states (11). As a first step we define second quantized bosonic operators, that create the appropriate Hilbert space: where | Ω is the vacuum of the new bosons and | 0 is an empty site. The new operators are analogues of Schwinger bosons in spin systems. Like the Schwinger bosons, they obey a holonomic constraint, namely that their total filling on a site is one. Now, we apply an orthogonal change of basis: In the new basis the variational state (11) is simply a singly occupied Fock state of the ψ 0 boson The three remaining bosons, ψ † 1,2,3 , create orthogonal fluctuations about the variational state. In the constrained Hilbert space of no double occupancy by the same species, the hamiltonian (1) may be written in terms of the ψ bosons. Furthermore, ψ 0i can be eliminated using the hard core constraint so that the hamiltonian is a function of only the three fluctuation operators ψ 1,2,3 . Assuming the fluctuations are small, we expand it to quadratic order in these operators. The exact form of the quadratic hamiltonian depends on the variational starting point which fixes the rotation matrix (21). For a two sub-lattice variational state, the fluctuation hamiltonian has the general form: while F k and G k are 6 × 6 matrices which depend on the variational parameters. Finally H f luc is diagonalized by a Bogoliubov transformation to obtain the excitation frequencies ω αk and the correction to the ground state energy: With the Bogoliubov transformation at hand it should be straightforward to calculate the average occupation of the fluctuations. For consistency of our approach we require: Let us now focus on the Mott phase. Recall that the variational energy is independent of the individual spin orientations, i.e. the parameters in the state (12). The fluctuation hamiltonian on the other hand will depend on the spin configuration. Before we compare the zero point energies corresponding to possible spin orders let us note a few general properties of the fluctuations in this case. Since the bosons p † i and h † i which create an extra particle or hole, are unoccupied in (12), they constitute a pair of orthogonal fluctuations. The third orthogonal fluctuation is φ † = cos( which creates a pseudospin of opposite orientation. Since the classical energy is independent of the spin configuration we expect that φ † i will not appear in the quadratic fluctuation hamiltonian. This reflects the fact that a local spin flip does not cost energy.
For a uniform state with χ i = χ the hamiltonian assumes a simple form where the couplings depend on χ The hamiltonian is diagonalized by a standard Bogoliubov transformation: which yields the excitation modes In addition, there is a zero mode ω 3 (k) = 0 corresponding to local spin flips, which reflects the macroscopic degeneracy at the classical level. Higher order terms in the fluctuations take into account the corrected potential landscape and generate a dispersion of the spin flip mode ω 3 (k). Since here we are interested in the zero point energy, we need not go beyond quadratic fluctuations. The quantum correction to the ground state energy is calculated from the prescription (26) where we have added the last term perturbatively in t α /V α . This is justified in the regime of interest t α , U << V α . The minimum of ∆E(χ) occurs for χ = π/2, which corresponds to pseudospins aligned on the x − y plane. Note that the dispersions of the particle and hole excitations (31) are degenerate in this case. Their gap vanishes when t a + t b = U/2z, which marks the transition to a superfluid in agreement with the variational result (16).
To check the consistency of our fluctuation expansion the local density of fluctuations in the x − y ferromagnet can be calculated using the Bogoliubov transformation (30): Fig. 4 plots the mean square fluctuation as a function of (t a + t b )/U at a constant ratio t a /t b . It can be verified that the mean square local fluctuation is smaller than 1/4 throughout the phase diagram. This constitutes a posteriori justification for our expansion which relied on the smallness of the fluctuations. We should comment though that the occupation of the zero mode cannot be calculated at this order. If the x − y state is indeed stable, interactions would generate a dispersion which would lead to a finite local ground state occupation. We now consider the canted state The angle θ parameterizes a continuous path from the z-Neel state (θ = 0) to the x − y ferromagnet (θ = π/2). Since | Ψ(θ) is not translationally invariant, neither will be the fluctuation hamiltonian derived from it. An elegant way to overcome this difficulty is to apply a unitary particle hole transformation on sub lattice B for i ∈ B. In the spin language this is equivalent to a π rotation of the spins in the B sub-lattice about their x axis. The rotation changes the hopping terms in the hamiltonian (1) but it also transforms | Ψ(χ) to a translationally invariant state Our procedure can now be carried out with the new hamiltonian and the transformed state. The fluctuation hamiltonian assumes the form which can be diagonalized by a Bogoliubov transformation. In the z-Neel state (θ = 0) the excitation energies assume a particularly simple form Note that contrary to the x − y state, the excitations are non degenerate. The gap in ω 1,2 (k) vanishes on the lines t a,b = U/2z respectively. Thus the z-Neel state is locally stable toward formation of a superfluid within these boundaries. There is however a dangerous zero mode ω 3 (k) = 0 which may be destabilized by higher order terms in the fluctuation hamiltonian. This mode corresponds to φ † which describes spin fluctuations (φ † k ) toward the x − y ferromagnetic state. In regions where the z-Neel state is ultimately stable these corrections would just generate a dispersion for ω 3 (k). The quantum zero-point energy of the fluctuations in the z-Neel state is given by: The mean local fluctuation can be calculated in the same way as before It is plotted in Fig. 4, which demonstrates that fluctuations about the z-Neel state are also small. Before we address the full Mott domain, it is instructive to evaluate the energy corrections (32) and (40) deep in the Mott phase, where we can compare the result with the effective spin hamiltonian (2). It is also much easier to evaluate the zero point energy in this limit. For t a , t b << U we can expand the square roots in (32) and (40), then perform the momentum sums exactly, with the result These are identical to the mean-field energies in the effective spin hamiltonian (2). Thus we see that our fluctuation analysis about the variational states captures the essential spin interactions. In the next section we address the stability of the spin states over the entire parameter regime to derive a phase diagram.
A fluctuation hamiltonian can be derived in a similar way for the superfluid phase where we find the three excitation modes: with t + = t a + t b and t − = t a − t b . The zero point energy correction in the superfluid phase is evaluated using the prescription (26). A discussion of the collective modes and of the nature of the superfluid phase is deferred to the next section.

VI. PHASE DIAGRAM FOR 1 ATOM PER SITE
In this section we combine the ingredients prepared in the last sections to present a phase diagram for a lattice with an average occupation of one atom per site. From the variational approach we found that the x − y ferromagnet becomes unstable towards a superfluid state when t a + t b > U/2z. The mean field phase diagram was sketched in Fig (3). However the boundaries of these phases with the z-Neel state remain undetermined. It is the quantum zero point energy of fluctuations that selects ordered magnetic states from a degenerate variational energy.
To analyze the stability of the spin states we need to calculate the derivatives with respect to θ of the zero point energies corresponding to these phases: The last term is added perturbatively in t α /V α and corrects for a large but finite intra-species interaction. It is easily seen that the first derivative of the modes ω α vanishes identically at the points θ = 0 and θ = π/2, corresponding to the z-Neel and x − y states. Consequently these states are either minima or maxima of the zero point energy. The second derivative at the z-Neel state is given by: where we have denoted τ α ≡ 2zt α /U and v α = V α /U . The domain of stability of the phase is obtained by numerically evaluating the momentum sum in (45). The resulting domain of stability is of the general shape illustrated in Fig. 5. Note that the phase boundaries deep in the Mott state (t a,b << U ) are linear and coincide with the result obtained from the effective hamiltonian (2). However we find in contrast with the effective spin hamiltonian, that even for true hard core interactions V a,b →∞, there is a finite x − y ferromagnetic domain. Note that the Mott z-Neel domain in Fig. (5) extends beyond the mean field transition to the superfluid which occurs at t a + t b = U/2z. As seen in Fig. 6, this is due to a lower ground state energy (including the quantum corrrection) than the superfluid. In the remainder of this section we shall examine the nature of the phases and transitions in Fig. 5.

A. Metastability and hysteresis
It is an interesting observation that over a significant parameter range, quantum fluctuations favor the z-Neel state even where its variational energy alone is higher than that of the superfluid. What kind of transition then, is marked by the lines t a,b = U/2z, where the z-Neel state finally becomes unstable? It could be one of the two: (i) A first order transition into the superfluid state or (ii) a second order transition into a supersolid, namely a superfluid that retains Ising order.
We shall see that the former indeed occurs, but this is not immediately obvious. Consider the excitation modes (39). Since only one of them becomes gapless on the transition lines t a,b = U/2z, one might guess that these lines mark the formation of a supersolid. However we now show that at the classical level the supersolid is unstable to formation of a uniform superfluid.
A variational state describing a supersolid is given by

VII. DISCUSSION AND CONCLUSIONS
It is interesting to consider the present results in light of the current experimental possibilities. First of all, we have shown that magnetic phases are robust in the sense that they persist up to the transition into the superfluid state. Specifically, spin-ordered states appear even near the boundary of this phase transition. Spin-exchange interactions in this regime are most easily accessible experimentally since the relevant energy scales are largest and comparable to on-site interaction. In this regime magnetic phases are therefore relatively insensitive to perturbations due to e.g. inhomogeneous magnetic field variations. Secondly, we note the existence of several metastable states in this regime, indicating that the system is likely to display interesting dynamics as the optical potential is lowered across the transition point. In particular, hysteresis and abrupt changes in the state of the system can be expected. At the same time, our results indicate that spin-ordered states are qualitatively different for odd and even numbers of particles per site. Both are likely to be observable in any realistic realization, since the inhomogeneous trapping potential typically leads to domains with different occupation.
Finally, it is important to note that detection of the complex states, of the type discussed in this paper, presents an interesting challenge in its own right. It turns out that the quantum nature of strongly correlated magnetic states can be revealed by spatial noise correlations in the image of the expanding gas [19]. Specifically, atoms released from a Mott-insulating state of the optical lattice display sharp (Bragg) peaks in the density-density correlation function as a consequence of quantum statistics and such peaks can be used to probe the spin ordered Mott states proposed for two component bosons.
In summary, we presented a theoretical analysis of the phase diagram of two component bosons on an optical lattice. We extended earlier treatments which were valid only deep in the Mott phase toward the MI-SF transition and beyond and were thus able to map a complete phase diagram. In addition we identified a transition into a Mott phase with no broken symmetries, which occurs only at even fillings.