⇦Chinese Physics Letters, 2016, Vol. 33, No. 4, Article code 040302⇨ Coherence of Disordered Bosonic Gas with Two- and Three-Body Interactions * Xin Zhang(张欣), Zi-Fa Yu(鱼自发), Ju-Kui Xue(薛具奎)** Affiliations Key Laboratory of Atomic and Molecular Physics and Functional Materials of Gansu Province, College of Physics and Electronics Engineering, Northwest Normal University, Lanzhou 730070 Received 7 January 2016 ^*Supported by the National Natural Science Foundation of China under Grant Nos 11475027, 11274255 and 11305132, the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grand No 20136203110001, the Natural Science Foundation of Gansu Province under Grant No 1506RJYA131, and the Creation of Science and Technology of Northwest Normal University under Grant Nos NWNU-KJCXGC-03-48 and NWNU-LKQN-12-12.
^**Corresponding author. Email: xuejk@nwnu.edu.cn Citation Text: Zhang X, Yu Z F and Xue J K 2016 Chin. Phys. Lett. 33 040302 Abstract We theoretically and numerically investigate the coherence of disordered bosonic gas with effective two- and three-body interactions within a two-site Bose–Hubbard model. By properly adjusting the two- and three-body interactions and the disorder, the coherence of the system exhibits new and interesting phenomena, including the resonance character of coherence against the disorder in the purely two- or three-body interactions system. More interestingly, the disorder and three-body interactions together can suppress the coherence of the purely three-body interactions system, which is different from the case in which the disorder and two-body interactions together can enhance the coherence in certain values of two-body interaction. Furthermore, when two- or three-body interactions are attractive or repulsive, the phase coherence exhibits completely different phenomena. In particular, if two- or three-body interactions are attractive, the coherence of the system can be significantly enhanced in certain regions. Correspondingly, the phase coherence of the system is strongly related to the effective interaction energy. The results provide a possible way for studying the coherence of bosonic gas with multi-atoms' interactions in the presence of the disorder. DOI:10.1088/0256-307X/33/4/040302 PACS:03.75.Lm, 03.75.Kk, 67.85.Hj © 2016 Chinese Physics Society Article Text In recent years, disorder in both bosonic and fermionic systems has attracted great interest.[1] Meanwhile, disorder effects in both noninteracting and interacting bosonic systems are also widely investigated. In particular, clarifying the interplay of interactions and disorder is essential for understanding many body quantum systems, including superfluid helium in porous media,[2] granular and thin-film superconductors[3,4] and light propagating in disordered media.[5] By properly adjusting the interaction and disorder, the interplay between the interaction and disorder in the ultracold Bose gas has become a particularly significant issue.[6-10] One important discovery is that, despite the fact that either disorder or interaction can suppress the superfluid density or phase coherence, their coupling can also enhance the phase coherence of the system.[11-13] Experimentally, using laser speckles[14] or incommensurable lattices[15,16] to control bosons in a disordered optical lattice can provide a proper physical system to investigate this issue. Up to now, theoretical studies of phase coherence of disordered bosonic gas are mainly focused on two-body interactions in general, which are most readily realizable experimentally.[17-19] If the atom density is high enough, i.e., in the case of the miniaturization of the devices in the integrated atom optics and deep lattices, three-body interactions induce many interesting phenomena.[20-24] Particularly, systems with three-body interactions have been shown to exhibit exotic quantum phases in their ground states,[25-28] such as topological phases or spin liquids and a chiral phase. So far, most of the relative quantum coherence investigations are limited to the system with two-body interactions. A comprehensive research of disordered bosonic gas with both two- and three-body interactions in a double-well potential is still missing. In this Letter, we discuss the coherence of the disordered bosonic gas with both effective two- and three-body interactions by a Bose–Hubbard model, which was realized in a double-well potential. We focus on the coherence characters determined by the two- and three-body interactions and the disorder. We find that the coupled effects of two- and three-body interactions and disorder induce new and interesting phenomena including the resonance character of coherence against disorder. More interestingly, the disorder and three-body interactions together can suppress the coherence of the system, which is different from the case in which the disorder and two-body interactions together can enhance the coherence in certain values of two-body interaction. Furthermore, when two- or three-body interactions are repulsive, the coupled effects of the two- and three-body interactions and the disorder will suppress the coherence. On the contrary, when two- or three-body interactions are attractive, the coherence can be enhanced in two special cases, in which the phase coherence and effective interaction energy have extreme values, in good agreement with the analytical value. To discuss the coherence of the system, we consider $N$ identical bosons with both two- and three-body interactions in a periodically disordered double-well potential,[29-37] which can occupy only two quantum states. The tunneling dynamics of bosons can be described by the two-site Bose–Hubbard model[38-43] $$\begin{align} \hat{H}=\,&-J(\hat{a}_{l}^†\hat{a}_{r}+\hat{a}_{r}^†\hat{a}_{l}) +\frac{U_{2}}{2}[\hat{n}_{l}(\hat{n}_{l}-1)\\ &+\hat{n}_{r}(\hat{n}_{r}-1)]+\frac{U_{3}}{6}[\hat{n}_{l}(\hat{n}_{l}-1)(\hat{n}_{l}-2)\\ &+\hat{n}_{r}(\hat{n}_{r}-1)(\hat{n}_{r}-2)]+\frac{\varepsilon}{2}(\hat{n}_{l}-\hat{n}_{r}),~~ \tag1 \end{align} $$ where $\hat{a}_{j}^†$ and $\hat{a}_{j}$ are the creation and annihilation operators, and $n_{j}=\hat{a}_{j}^†\hat{a}_{j}$ is the number operator in the $j$th ($j=l,r$) well, $J$ is the tunneling constant, and $U_{2}$ and $U_{3}$ are effective two- and three-body interactions, respectively. Note that the energy loss of three-body recombination is not considered in effective three-body interactions. The three-body recombination expressed by the imaginary part of $U_{3}$ is negligible on timescales compared with the three-body elastic collisional duration. In experiment, $U_{2}$ and $U_{3}$ can be controlled by the Feshbach resonance technique. Here $\varepsilon$ is the energy level mismatch between the two wells, which is introduced into the model via the random variation of $\varepsilon$ and is assumed to be randomly and uniformly distributed within the interval $\varepsilon\in[-\triangle,{\it\Delta}]$, where ${\it\Delta}$ describes the strength of the disorder. On experiment, this can be realized by randomly tilting the double-well potential. To begin with, we consider $N$ bosons trapped in disordered double-well potential with fixed $\varepsilon$. The Schrödinger equation $H|\psi\rangle=E|\psi\rangle$ can be represented in the Fock space. We define $|n\rangle=|n,N-n\rangle$, where $n=0,1,2,\ldots,N-1,N$. Expanding $|\psi\rangle=\sum_{n}\phi_{n}|n\rangle$, we can obtain $$\begin{alignat}{1} \!\!\!\!\!\!(E-E_{n})|n\rangle=-JM_{n+1}|n+1\rangle-JM_{n-1}|n-1\rangle,~~ \tag2 \end{alignat} $$ where $$\begin{align} E_{n}=\,&\frac{U_{2}}{2}(N^{2}+2n^{2}-2Nn-N)+\frac{U_{3}}{6}[N^{3}-n^{3}\\ &+3Nn(n-N+2) -3(N^{2}+n^{2})\\ &+2(N-n)]+\frac{\varepsilon}{2}(2n-N),\\ M_{n+1}=\,&\sqrt{(n+1)(N-n)},\\ M_{n-1}=\,&\sqrt{(n(N-n+1)}. \end{align} $$ From Eq. (2), we can easily evaluate the eigenenergies and eigenfunctions by exact diagonalization. At zero temperature, the phase coherence of the system can be defined as $C_{\varepsilon}=\frac{1}{N}\sum\langle\hat{a}_{l}^†\hat{a}_{r}\rangle_{\varepsilon} =\frac{1}{N}\sum_{n=0}^{N}(M_{n+1}\phi_{n}^{0}\phi_{n+1}^{0})$, where $\phi_{n}^{0}$ is the ground state wave function in the Fock space, and $\langle\hat{a}_{l}^†\hat{a}_{r}\rangle$ is the expectation value of $\hat{a}_{l}^†\hat{a}_{r}$ at fixed $\varepsilon$. The results of $C$ for the disordered case can be obtained by averaging those results for fixed $\varepsilon$, that is, $\bar{C}=\frac{1}{2{\it\Delta}}\int^{{\it\Delta}}_{-{\it\Delta}}Cd\varepsilon$, which is under the curves of $C$ at the area of $\varepsilon\in[-{\it\Delta},{\it\Delta}]$. Here $C>0$ (or $\bar{C}>0$) means that the system is in the coherent (quasi-coherent) state, whereas $C\rightarrow0$ (or $\bar{C}\rightarrow0$) indicates that the system is in the decoherent state. To investigate the phase coherence of the system, we choose $N=10$ and diagonalize Eq. (2) numerically. The results of $C$ versus the disorder $\varepsilon$ with different $U_{2}$ and $U_{3}$ are shown in Fig. 1. From Fig. 1(a), it can be seen that when $U_{2}=0$ and $U_{3}=0$, $C$ quickly decreases to zero with increasing $\varepsilon$, since noninteracting particles can only tunnel from one well to the other if $\varepsilon < J$. The disorder in the noninteracting system suppresses the phase coherence. Interestingly, with the increase of $U_{3}$, $C$ oscillates and resonates against $\varepsilon$, then quickly decreases when $\varepsilon^{\ast}=20U_{3}$. This is the coupled effect of disorder and three-body interactions, which results in complex characters. When $\varepsilon < \varepsilon^{\ast}$, $C$ oscillates with large amplitude due to the effect of the three-body interactions. When $\varepsilon>\varepsilon^{\ast}$, $C$ quickly decreases and approaches zero, since the effective interaction energy is less than the disordered potential, which results in the destruction of the coherence. We find that the emergence of $\varepsilon^{\ast}$ depends on $U_{3}$. However, when $U_{3}=0$, the situation becomes completely different (see Fig. 1(b)). Compared with Fig. 1(a), we find that, with the increase of $U_{2}$, $C$ quickly decreases when $\varepsilon^{\ast}=10U_{2}$. Specifically, when $U_{2}\leq1$, $C$ decreases with $\varepsilon$ monotonously, rather than oscillates. When $U_{2}>1$, $C$ oscillates with $\varepsilon$, but with small amplitude. Interestingly, the critical disorder $\varepsilon^{\ast}$ for the destruction of coherence in the purely three-body interaction system is twice as much as that in the purely two-body interaction system. However, when the coherence preserves oscillation, the value of $C$ in the system with purely two-body interaction is larger than that in the system with purely three-body interaction. That is, the coherence of the system with only two-body interaction can be more easily destructed by disorder (with small $\varepsilon$). In the purely two- or three-body interactions system, the critical disorder for destruction of coherence increases with the interactions. We can note that only two- or three-body interactions have different influence characters on the coherence.

Fig. 1. (Color online) The value of $C$ versus $\varepsilon$ for different $U_{3}$ ($U_{2}=0$) (a) and for different $U_{2}$ ($U_{3}=0$) (b).

Fig. 2. (Color online) The value of $C$ versus $\varepsilon$ for $U_{2}=5$ and $U_{3}=0$, 1, 2, 3, 4 and 5.

When both the two- and three-body interactions are considered, the situation exhibits new and interesting phenomena. Figure 2 shows the results of $C$ with $U_{2}=5$ for $U_{3}=0$, 1, 2, 3, 4 and 5. Compared with Fig. 1, we note that, when $U_{2}=5$ and $U_{3}=0$, the resonance character of curve $C$ against disorder becomes clear, and the local peaks or valleys of the curve emerge at certain values of $\varepsilon$. This nonmonotonic behavior of $C$ depends on the two- and three-body interactions. With the increase of $U_{3}$, the peaks or valleys of the curve shift to large values of $\varepsilon$ and the gaps between the two nearby peaks or valleys increase with $U_{3}$. The coherence of the system can be enhanced or suppressed by disorder (with large $\varepsilon$). That is, the coherence of the system depends on the coupled effects of the two- and three-body interactions and disorder. The resonance character observed in Figs. 1 and 2 is induced by the cooperation of disorder and interaction. If the disorder and interaction strength satisfy the condition that there should be no extra interaction energy cost for one particle tunneling from one well to the other, then the tunneling must be enhanced or suppressed. This condition can be obtained analytically by the variational analysis of the state energy $E_{n}$, i.e., by using $\partial E_{n}/\partial n=U_{2}(2n-N)+U_{3}(N-\frac{N^{2}}{2}-2n+Nn)+\varepsilon=0$, we obtain $$\begin{alignat}{1} \varepsilon^{\ast}=U_{2}(N-2n)+U_{3}\Big(\frac{N^{2}}{2}+2n-N-Nn\Big).~~ \tag {3} \end{alignat} $$ With $N=10$, we can obtain $$\begin{align} \varepsilon^{\ast}=(10-2n)(U_{2}+4U_{3}).~~ \tag {4} \end{align} $$ When $U_{2}=0$, $\varepsilon^{\ast}=4U_{3}(10-2n)$. When $U_{3}=0$, $\varepsilon^{\ast}=U_{2}(10-2n)$. Here $n=5, 4,\ldots, 0$. Then, we can find that the resonance character of the coherence observed in Figs. 1 and 2 can be understood by Eq. (4). When $\varepsilon=\varepsilon^{\ast}$, the interaction can screen the disordered potential and can smooth out the effective potential in the occupied sites. Hence the effective interaction energy has maximum/minimum values and then the corresponding phase coherence is suppressed/enhanced at the values of $\varepsilon^{\ast}$. The explicit couple among coherence $\bar{C}$, the two-body interaction $U_{2}$, the three-body interactions $U_{3}$ and the disorder strength ${\it\Delta}$ are further shown in Figs. 3 and 4. Figure 3(a) shows that when $U_{3}=0$ and ${\it\Delta}=0$, the variation of $\bar{C}$ against $U_{2}$ decreases monotonously. That is, the two-body interaction energy dominates over the tunneling energy, then the coherence is suppressed with the increase of $U_{2}$. Different from the clean system (i.e., ${\it\Delta}=0$), when ${\it\Delta}\neq0$, $\bar{C}$ is nonmonotonic and the local maxima of $\bar{C}$ emerges at a certain value of $U^{\ast}_{2}$. That is, without three-body interactions, when $U_{2} < U^{\ast}_{2}$, the two-body interaction energy is less than the disordered potential, the two-body interaction can screen the disordered potential and thus can smooth out the effective potential for remaining particles. The coherence is then enhanced with the increase of $U_{2}$. When $U_{2}>U^{\ast}_{2}$, the two-body interaction energy dominates over the disordered potential, and the coherence is suppressed by the single-particle localization with the increase of $U_{2}$. Here $U^{\ast}_{2}$ increases with the disorder. Figures 3(b)–3(d) show that the behavior of $\bar{C}$ is completely different when three-body interactions are present (i.e., $U_{3}\neq0$). When two- and three-body interactions are mutually repulsive (i.e., $U_{3}=2.5$), $\bar{C}$ decreases monotonically with $U_{2}$, as shown in Fig. 3(b). That is, energies of two- and three-body interactions dominate over the disorder potential and together suppress the coherence. Figure 3(c) shows the situation with attractive three-body interactions (i.e., $U_{3}/JN=-1$). Interestingly, when $U_{2}$ increases, the peaks of the curve located at certain values of $U_{2}$ (i.e., $U_{2}/JN=1,5$) emerge. This is different from the results of Fig. 3(b) where $\bar{C}$ decreases with increasing $U_{2}$ monotonically. We can also find that the nonmonotonic behavior of $\bar{C}$ can still exist even in the clean system. That is, the system with attractive three-body interactions can enhance the coherence in certain regions. Similar results are illustrated in Fig. 3(d), where variations of $\bar{C}$ against $U_{2}$ for $U_{3}=-5$ and ${\it\Delta}$ are shown.

Fig. 3. (Color online) The value of $\bar{C}$ versus $U_{2}$ for different ${\it\Delta}$ and $U_{3}$.

Fig. 4. (Color online) The value of $\bar{C}$ versus $U_{3}$ for different ${\it\Delta}$ and $U_{2}$.

From Fig. 4, we should note that, when the two-body interaction is absent (i.e., $U_{2}=0$), $\bar{C}$ decreases with the increase of $U_{3}$ monotonically for all values of ${\it\Delta}$ (see Fig. 4(a)). This is different from the results of Fig. 3(a) where the two-body interaction and disorder together can enhance the coherence. That is, three-body interaction energy dominates over the disordered potential and together they suppress the coherence. As shown in Fig. 4(b), when two- and three-body interactions are mutually repulsive (i.e., $U_{2}=5$), $\bar{C}$ decreases with the increase of $U_{3}$, which is the same as Fig. 3(b). If the two-body interaction is attractive (i.e., $U_{2}=-1, -5$) as shown in Figs. 4(c) and 4(d), with the increase of $U_{3}$, the peak of the curve located at certain values of $U_{3}$ emerges, which is also in good agreement with the analytical values. Figures 3 and 4 also show that, when $-U_{3} < U_{2} < -4U_{3}$, the coherence $\bar{C}$ is entirely suppressed. Disorder can enhance the coherence (i.e., $\bar{C}$ in the disordered system is higher than that in the clean system) in certain regions. In Figs. 3(c), 3(d), 4(c) and 4(d), the phase coherence of the system exhibits the interesting behavior. For example, Figs. 3(d) and 4(d) show that the coherence is enhanced at certain values of $U_{2}$ (i.e., $U_{2}/JN= 5,20$) and $U_{3}$ (i.e., $U_{3}/JN=1.25,5$). We can analyze this phenomenon by Eq. (4). Setting $\varepsilon=0$, Eq. (4) reduces to $(N-2n)(U_{2}+4U_{3})=0$, then we obtain two special cases of $N=2n$ and $U_{2}=-4U_{3}$. The case of $N=2n$ corresponds to $U_{2}/JN=5$ in Fig. 3(d) and $U_{3}/JN=5$ in Fig. 4(d), where particles distribute equally in two potential wells and $U_{2}=-U_{3}$ is satisfied. The case of $U_{2}=-4U_{3}$ corresponds to $U_{2}/JN=20$ in Fig. 3(d) and $U_{3}/JN=1.25$ in Fig. 4(d). For those two cases, due to the fact that either two- or three-body interactions are attractive, the attractive interaction energy and the repulsive interaction energy can be compensated for each other, which results in the fact that the total effective interaction energies have minimum values and the corresponding phase coherence is enhanced. We can note that the phase coherence of the system is strongly related to the effective interaction energy. When two- or three-body interactions are attractive, the phase coherence is enhanced and the effective interaction energies have extreme values in the two special cases of $N=2n$ and $U_{2}=-4U_{3}$. In conclusion, within a two-site Bose–Hubbard model, the interplay between the two- and three-body interactions and the disorder is investigated. We find that, with only disorder or interaction, the coherence is suppressed by single-particle localization, since the disordered potential or effective interaction energy dominates over tunneling energy. When both interaction (two- and three-body interactions) and disorder are present, the coherence is enhanced in the parameter regime where they are comparable in strength. Correspondingly, the phase coherence of the system is strongly related to the effective interaction energy. In particular, when two- or three-body interactions are attractive, the phase coherence is significantly enhanced and the total effective interaction energies have extreme values in two special cases. The results provide an in depth insight into the physics of the disordered bosonic gas and stimulate investigations into disordered multi-atoms' systems. 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