Crystallized and amorphous vortices in rotating atomic-molecular Bose-Einstein condensates

Vortex is a topological defect with a quantized winding number of the phase in superfluids and superconductors. Here, we investigate the crystallized (triangular, square, honeycomb) and amorphous vortices in rotating atomic-molecular Bose-Einstein condensates (BECs) by using the damped projected Gross-Pitaevskii equation. The amorphous vortices are the result of the considerable deviation induced by the interaction of atomic-molecular vortices. By changing the atom-molecule interaction from attractive to repulsive, the configuration of vortices can change from an overlapped atomic-molecular vortices to carbon-dioxide-type ones, then to atomic vortices with interstitial molecular vortices, and finally into independent separated ones. The Raman detuning can tune the ratio of the atomic vortex to the molecular vortex. We provide a phase diagram of vortices in rotating atomic-molecular BECs as a function of Raman detuning and the strength of atom-molecule interaction.

molecular vortices to the carbon-dioxide-type atomic-molecular vortices, then to the atomic vortices with interstitial molecular vortices, and finally to the completely separated atomic-molecular vortices. This result is in accordance with the predicted dissociation of the composite vortex lattice in the flux-flow of two-band superconductors 39 . The Raman detuning adjusts the population of atomicmolecular BECs and the corresponding vortices. This leads to the imbalance transition among vortex states. This study shows a full picture about the vortex state in rotating atomic-molecular BECs.

Results
The coupled Gross-Pitaevskii equations for characterizing atomicmolecular Bose-Einstein condensates. We ignore the molecular spontaneous emission and the light shift effect 28,[31][32][33] . According to the mean-field theory, the coupled equations of atomic-molecular BEC 17,33,40 can be written as In real experiment, it is observed that the coherent free-bound stimulated Raman transition can cause atomic BEC of 87 Rb to generate a molecular BEC of 87 Rb 27 . In numerical simulations, we use the parameters of atomic-molecular BECs of 87 Rb system with M m 5 2M a 5 2m (m 5 144.42 3 10 227 Kg), g m 5 2g a (a a 5 101.8a B , where a B is the Bohr radius), x 5 2 3 10 23 , and the trapped frequency v 5 100 3 2p. Note that if the change in energy in converting two atoms into one molecule (DU 5 2U Ta 2 U Tm ) 27 , not including internal energy, approaches zero, we can obtain the value 2g a 5 g m . The unit of length, time, and energy correspond to ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h= mv ð Þ p (<1.07 mm), v 21 (<1.6 3 10 23 s), and hv, respectively.
Crystallized and amorphous vortices in rotating atomic-molecular Bose-Einstein condensates. With rotation frequency V 5 0.8v, we study the influence of atom-molecule interaction on the formation of vortices. Figure 1 displays the densities and phases obtained under the equilibrium state with different atom-molecule interactions. The first and the second columns are the densities of the atomic BEC and the molecular BEC, respectively. The third column is the total density. The fourth and the fifth column are the corresponding phases of atomic and molecular BECs, respectively. The vortices can be identified in the phase image of BECs.
The composite of atomic and molecular vortices locates at the trap center to lower the system's energy. For the case of attractive atommolecule interaction (g am 5 20.87g a ), vortices form a square lattice [see Fig. 1(a)]. Interestingly, we can observe that the size of the molecular vortices can be divided into two types: one is big, and the other is small. Each atomic vortex has approximately the same size. Figure 2(a) further plots the vortices, where the atomic vortices overlap with a molecular one. The overlapping of atomic and Figure 1 | The densities and phases of atomic-molecular BECs of 87 Rb when the system reaches the equilibrium state. The rotation frequency is V 5 0.8v. The strength of atom-atom interaction is g a with the scattering length a a 5 101.8a B . The strength of molecule-molecule interaction g m is twice as much as that of atom-atom interaction. The strength of atom-molecule interaction and Raman detuning is set as (a) g am 5 20.87g a , e 5 14 hv, (b) g am 5 0, e 5 14 hv, (c) g am 5 0.87g a , e 5 14 hv and (d) g am 5 2g a , e 5 7 hv. Note that the fourth and fifth columns are the phases of atomic and molecular BECs, respectively. The unit of length is 1.07 mm.  41 ] of the BEC because within this distance, the order parameter 'heals' from zero up to its bulk value. The attractive interspecies interaction implies that the densities of the two BECs would have a similar trend to decrease and increase. It also causes some molecular vortices to overlap with atomic ones. In addition, the density of atomic BEC forms local nonzero minima at the region of the left molecular vortex [see Fig. 2(a)]. Here, the size of atomic vortices is obvious bigger than that of molecular vortices. Therefore, the local density of molecular vortices follows that of atomic vortices and the size becomes big when molecular vortices overlap with atomic vortices. When g am 5 0, atomic vortex lattices are triangular and the molecular vortices are amorphous state [see Fig. 1(b) and Fig. 2(d)]. Meanwhile, the total density (the third column) indicates that the molecular vortices and the atomic vortices form some structure like the carbon dioxide, which is also observed by a different method 17 . Figure 2(b) shows an enlarged configuration of the carbon dioxide vortices. Here, the size of atomic vortices is much larger than that of the molecular one. With repulsive interaction (g am 5 0.87g a ), vortex lattices are approximately hexagonal with a little deviation [see Fig. 1(c)]. Increasing atom-molecule interaction up to g am 5 2g a [see Fig. 1(d)], atomic-molecular BECs separate into two parts, molecular BEC locating at the center and atomic BEC rounding it. The results are understandable, since the mass of a molecule is twice as that of atom, molecular BEC tends to locate at the center. This is much different from that of the normal two-component BECs, where the same mass and intraspecies interactions are considered 5 . We have plotted a green circle to differentiate these vortices as two parts. The atomic vortices construct an approximately quadrangle lattice, especially near the center region, the atomic vortices overlap with a molecular one locating among four adjacent molecular vortices. Thus, vortex position indicates that vortices density of atomic BEC is half of that of molecular vortices. The atomic vortices expand over to the outskirts of the lattice where no overlapped molecular vortices appear [see Fig. 2(c)]. Figure 2(d) indicates that the carbon dioxide structure is not fixed in the same orientation. Similarly to Fig. 2(c), the carbon dioxide structure only exists at the center. However, the deviation of molecular vortices from the red lines d, e, and f is so large that we have to view the molecular vortices as an amorphous state. Vortex position in Fig. 2(e) shows that atomic vortices form the triangle lattice. All molecular vortices are distributed among atomic vortices, forming the hexagonal lattices without overlapping. Certainly, atomic vortices and molecular vortices are separated in Fig. 2(f) according to the immiscibility of atomic-molecular BECs with strong g am . We can conclude that the strength of atom-molecule interaction can adjust the composite degrees of vortices, and cause the overlapping composite, carbon-dioxide-type composite, interstitial composite and separation.
Furthermore, we find that the lattice configuration of vortices is very complex when atomic vortices and interstitial molecular vortices coexist. In Fig. 1(c), atomic vortices form the triangular lattice and interstitial molecular vortices display the honeycomb lattice. We further plot the densities of atomic BEC and molecular BEC at various cases in Fig. 3. When the number of atoms is much more than that of molecules, vortices in atomic BEC tend to form the triangular lattice, and vice versa. The lattice configurations are triangular in Figs. 3(a2), (b2), (e1) and (f1). Atomic vortices display square lattice in Figs. 3(a1)-3(c1). In all other subplots, the lattices are irregular and can be viewed as the amorphous state. For example, the number of adjacent molecular vortices which form bubbles 4 around some atomic vortices is not six but five in Figs. 3(d2)-(f2). In fact, the regular structures imply that both long-range order and short-range order should be remained. Thus, the observed random configuration is really amorphous.
The phase diagram of rotating atomic-molecular Bose-Einstein condensates. To explore the phase diagram of atomic-molecular vortices, we firstly show the modulation effect of Raman detuning . When the repulsive interaction is up to g am 5 2g a , the single molecular BEC or the single atomic BEC can be obtained by adjusting Raman detuning from {4 hv to 14 hv. The particle numbers can characterize the possible regions for the existence of atomic-molecular vortices. Figure 5 plots the phase diagram of atomic-molecular BECs. The stable atomic-molecular BECs system exists only when atom-molecule interaction is larger than { ffiffiffiffiffiffiffiffiffi g a g m p . When the Raman detuning is large enough, single atomic BEC occurs. Oppositely, if the Raman detuning is low enough, production changes into pure molecular BEC. Between these two regions, it is atomic-molecular BECs, where AMBEC(I) denotes the miscible mixture and AMBEC(II) stands for the phase separated mixture. Therefore, to explore atomic-molecular vortices, we mainly focus on AMBEC(I) region.
According to above analysis about atomic-molecular vortices and the corresponding atomic-molecular BECs, we calculate lots of other results and finally give a vortex phase diagram to summarize the vortex structures in Fig. 5. In (1) region [{ ffiffiffiffiffiffiffiffiffi g a g m p vg am v {0:1g a ], atomic-molecular vortices form the square lattice where the overlapped atomic-molecular vortices and the molecular vortices interlacedly exist. The carbon-dioxide-type atomic-molecular vortices occur in (2) region [20.1g a # g am , 0.5g a ]. Atomic vortices with interstitial molecular vortices emerge in region (3). In the AMBEC(II) region, atomic vortices and molecular vortices are separated. Certainly, in the region of atomic BEC (molecular BEC), atomic vortices (molecular vortices) favor to form the triangular lattice. The green region indicates that the created atomic and molecular vortices fully match with each other by roughly the ratio 152. Above the green region, more atomic vortices occur. Below the green region, more molecular vortices appear. Table 1 shows a summary of the details of various vortices in rotating atomic-molecular BECs.
Interestingly, the vortex phase diagram indicates some exotic transitions. (i) Imbalance transition: The increase of Raman detuning causes more atomic BEC. Thus, the pure molecular vortices change into carbon-dioxide-type atomic-molecular vortices (atom vortices with interstitial molecular vortices, and separated atomic-molecular vortices), and finally into single atomic vortices in region of 0 , g am , 0.5g a (0:5g a ƒg am ƒ ffiffiffiffiffiffiffiffiffi g a g m p , and g am w ffiffiffiffiffiffiffiffiffi g a g m p , respectively). In the region of { ffiffiffiffiffiffiffiffiffi g a g m p vg am v{0:1g a {0:1g a ƒg am ƒ0 ð Þ , the interlaced-overlapped atomic-molecular vortices (the carbon-dioxidetype atomic-molecular vortices) become into atomic vortices under a very high detuning parameter. (ii) Dissociation transition: By changing the atom-molecule interaction from attractive to repulsive, the composite atomic-molecular vortices change from overlapped to carbon-dioxide-type and finally into the independent separated ones.

Discussion
In this report, we focus on the strength of atom-molecule interaction and the Raman detuning term. The form of Hamiltonian in this paper is like that in the Ref. 17. In fact, a real experiment would include lots of other factors such as the light shift effect 28,31-33 , decay due to spontaneous emission 31 . In Ref. 33, Gupta and Dastidar have considered a more complicated model when they study the dynamics of atomic and molecular BECs of 87 Rb in a spherically symmetric trap coupled by stimulated Raman photoassociation process. In fact, the light shift effect almost has the same function as the Raman detuning term. Thus, it can be contributed to the Raman detuning term. This is the reason why we do not consider the light shift term in Hamiltonian like that in Ref. 33, but follows the form in Ref. 17.
In real experiment, it is believed that the single molecular BEC would occur when the Raman detuning goes to zero 28,31 . However, the measure of the remaining fraction of atom does not reach the minimum when Raman detuning is zero 28 . With the adiabatic consideration, the dynamical study also agrees with this point 33 . In fact, they show the evolutionary process of creating a molecular BEC from a single atomic BEC. Thus, particle number of molecular BEC varies with time but not fixed. The resonance coupling would cause the atomic BEC to convert into a molecular one as much as possible, but the molecular BEC also will convert into the atomic one. Therefore, the results in Ref. 28, 33 only shows a temporary conversion of atoms into molecules. In fact, when we use single atomic BEC as the initial condition and set c j k B T~0 , the temporary conversion of atomic BEC into molecular BEC can be observed with current damped projected Gross-Pitaevskii equations. It is obvious that the Raman detuning term in the Hamiltonian behaves just like the chemical potential to control the system's energy. The external potential for atomic BEC is fixed to be V a (r) and molecular BEC experiences the trap potential V m (r) 1 e. Here, our method initially derives from the finite-temperature consideration: the system is divided into the coherent region with the energies of the state below E R and the noncoherent region with the energies of the state above E R 42,43 . So, our method will behavior just likes to catch the particles with a shallow trap and exchange particles with an external thermal reservoir. But ultimately we remove the external thermal reservoir to get system to the ground state. Raman detuning changes the depth of shallow trap to m m 2 e. The molecular BEC will be converted by atoms until the system reaches the equilibrium state. Therefore, a maximum of creating molecular BEC does not occur at the equilibrium state when Raman detuning varies. Instead, molecule number decreases monotonously when Raman detuning increases.
Why do atomic-molecular vortices display so rich lattice configurations? In fact, atomic vortices and molecular vortices tend to be attractive in region (1) and (2). Otherwise, the overlapped atomicmolecular vortices and the carbon-dioxide-type ones can not occur.
The attractive force makes atomic vortices and molecular vortices behave similarly. Thus, both atomic and molecular vortex lattices in region (1) are square. In region (2), atomic vortices display the triangular lattice. Molecular vortices seem to follow the triangular lattice but the interaction among vortices causes the considerable deviation. Obviously, the CO 2 -type structures do not follow the fixed direction, i.e., long-range order vanishes but there is still short-range order. Thus, we have to view molecular vortices as the amorphous state. In region (3), atomic vortices and molecular vortices can not form the carbon dioxide structure. Because the size of molecular vortices is smaller than that of atomic vortices, it tends to locate at the interval of the lattice of atomic vortices. When the number of one component is much more than that of the other, the vortices of this component dominate over the vortices of the other component. The former is easy to form the regular vortex lattice. The latter has to follow the interaction of the former and forms the vortex lattice. The amorphous state originates from the competition between atomic vortices and molecular vortices, especially when the number of atom and molecule has the considerable proportion [see Figs. 3(d1) and 3(d2)]. In that case, short-range order is only partly kept and ultimately long-range order is destroyed. Certainly, this also causes the distribution of vortices in one component is relatively regular and that in the other component is amorphous.
The structural phase transitions of vortex lattices are explored through tuning the atom-molecule coupling coefficient and the rotational frequency of the system 17 . Certainly, the Archimedean lattice of vortices in Ref. 17 is one of the interstitial-composite-structures. Here, we show the crystallized and amorphous vortices by the combined control of Raman detuning and atom-molecule interaction. In fact, when we increase the value of x, the CO 2 -type structure of vortices are easy to be created. Even the interstitial-composite structure we now obtain in Fig. 3 would transfer into the CO 2 -type structure if x is big enough. We have also considered the effect of rotation frequency. With the attractive interaction of atom-molecule (g am 5 20.87g a ), Figure 6 shows various rotation frequencies to produce the vortices. Figure 6(a) indicates that no vortex would occur with V 5 0. For V 5 0.2v, only one molecular vortex is induced. In atomic BEC, the phase indicates no vortex is created although there is a local minimum of density near the center. For V 5 0.4v, the phase indicates that there is an atomic vortex. In fact, we find the atomic vortex is overlapped with a molecular vortex. Undoubtedly, more and more vortices emerge when rotation frequency increases. When the rotation frequency is up to V 5 0.8v, we can obtain a regular square vortex lattice. Meanwhile, each atomic vortex is overlapped with a corresponding molecular vortex. Obviously, vortices and vortex lattice may not be induced with a slow rotation. This is the reason why we favor to investigate the vortices with a fast rotation in Figs. 1-4.
We now show that ultracold Bose gases of 87 Rb atoms are a candidate for observing the predicted atomic-molecular vortices. By  In summary, we have observed various new atomic-molecular vortices and the lattices controlled by atom-molecule interaction and Raman detuning. Including the regular vortex lattices, we have displayed amorphous vortex state where vortices do not arrange regularly but like amorphous materials. We have obtained the vortex phase diagram as function of Raman detuning and atom-molecule interaction in the equilibrium state. Vortex configuration in atomicmolecular BECs includes the overlapped atomic-molecular vortices, the carbon-dioxide-type vortices, the atomic vortices with interstitial molecular vortices, and the completely separated atomic-molecular vortices. The lattice configuration of vortex mainly depends on atom-molecule interaction. For example, the overlapped atomicmolecular vortices display the square lattice. When the carbon-dioxide-type vortices occur, atomic vortices show the triangular lattice and molecular vortices show the amorphous state. Atomic vortices and interstitial molecular vortices can show several types of lattice, such as triangular, honeycomb, square and amorphous. And both atomic and molecular vortices show the triangular lattice in the incomposite region and in single BEC. Our results indicate that atom-molecule interaction can control the composite of atomic and molecular vortices, and can also cause novel dissociation transition of vortex state. Furthermore, the Raman detuning can control the numbers of particles in atomic-molecular BECs and approximately lead to the linear decrease of molecular vortices. This may induce the imbalance transition from atomic-molecular vortices to pure atomic (molecular) vortices. This study shows rich vortex states and exotic transitions in rotating atomic-molecular BECs.

Methods
We use the damped projected Gross-Pitaevskii equation (PGPE) 42 to obtain the ground state of atomic-molecular BEC. By neglecting the noise term according to the corresponding stochastic PGPE 43 , the damped PGPE is described as where,Ĥ j Y j~i h LY j Lt , T is the final temperature, k B is the Boltzmann constant, m j is the chemical potential, and c j is the growth rate for the jth component. The projection operator P is used to restrict the dynamics of atomic-molecular BEC in the coherent region. Meanwhile, we set the parameter c j k B T~0 :03. The initial state of each Y j is generated by sampling the grand canonical ensemble for a free ideal Bose gas with the chemical potential m m,0 5 2m a,0 5 8 hv. The final chemical potential of the noncondensate band are altered to the values m m 5 2m a 5 28 hv.