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The miscibility of quantum liquids is an interesting topic in many-body physics, which has been intensively investigated in 3He-4He superfluids and the mixtures of ultracold atoms. In the context of dual species Bose-Einstein condensates, the mean-field description has been well established, according to which, the miscibility condition is density independent and determined only by the ratio of inter- and intra-species interaction strength. Recently, Nadion and Petrov proposed that [ Phys. Rev. Lett. 126 115301 ], in the vicinity of the mixing-demixing threshold, quantum fluctuations play an important role to affect the equilibrium stability, and as a result, the partially miscible state emerges. This new phase of quantum matter opens up new perspectives to explore the beyond mean-field effect in ultracold atomic gases.In this work, according to the equation of state taking the Lee-Huang-Yang correction into consideration, we investigate the ground state phase diagram of repulsive binary Bose mixtures in the interacting regime suffering a weak mean-field instability. Under the thermodynamic balance conditions, the phase boundaries between the immiscible state, partially miscible state and the homogenous state are determined. For the equal-mass case, these phase transitions only take place on condition that intra-species interactions are in an asymmetric form. In terms of interaction parameters, we explicitly derive analytical expressions of the phase boundaries, which are appropriate to describe the transitions in sufficiently dilute atomic gases. At the quantum critical point, where the partially miscible state terminates, the susceptibility tensor of the density response exhibits a divergent behavior. For the unequal-mass case, the beyond-mean-field equation of state cannot be written in a compact form, thus the determination of the phase boundaries is more involved. By expanding the Lee-Huang-Yang energy expression to the terms linear in the concentration of the minority species, we analytically obtain the threshold density for the partially miscible transition. We also propose a discriminant function, from which the configuration of the partially miscible state can be identified for the given mass ratio and interaction strength. Applications of these theoretical results to experimental systems, such as sodium, potassium, and rubidium gases, are presented. -
Keywords:
- Bose-Einstein condensate /
- Bose mixture /
- miscibility /
- quantum fluctuation
[1] Graf E H, Lee D M, Reppy J D 1967 Phys. Rev. Lett. 19 417
Google Scholar
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[16] Ospelkaus S, Ospelkaus C, Humbert L, Sengstock K, Bongs K 2006 Phys. Rev. Lett. 97 120403
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[38] Petrov D S 2015 Phys. Rev. Lett. 115 155302
Google Scholar
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Google Scholar
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Google Scholar
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图 1 (a) 双组分BEC的不混溶态(IM)和两种部分混溶态(PM1, PM2)示意； (b), (c) 等质量玻色混合气体在平均场弱失稳区间的基态相图. HM表示均匀混溶相, 点线是本文给出的密度准至
$\varDelta_g^2$ 阶的解析相边界. 旋节线(虚线)与部分混溶态的相边界相交于★标记的临界点. 插图是在$(n_+, n_-)$ 平面绘出的相图, 其中绿色条纹填充的部分表示$n_i<0$ 的非物理区间. (b)图和(c)图的$\alpha$ 分别取3和1/3, 二者的$\varDelta_g$ 均为0.01Figure 1. (a) Illustrations of immiscible state (IM) and two kinds of partially miscible states (PM1 and PM2) of dual-species BEC; (b), (c) ground state phase diagram of equal-mass Bose mixtures in the regime of a weak mean-field-instability. HM represents the homogenous miscible phase, dotted-lines are the analytical phase boundaries given in the text. The spinodal line (dashed) meets the boundary of partially miscible state at the critical point denoted by ★. Inset: phase diagram in the
$n_+$ -$n_-$ plane. The area filled by green stripes is the unphysical region with$n_i<0$ . In plot (b),$\alpha = 3$ ; in plot (c),$\alpha = 1/3$ , for both plots,$\varDelta_g =0.01$ .
图 2 在不同
$n_1$ 取值下, 静态极化率$\chi_{11}$ 和$\chi_{12}$ 随2组分原子浓度的变化曲线. 两图中的点划线分别对应于$n_1=n_1^{\rm{C}}$ 时(19)式和(20)式给出的解析结果. 相互作用参数:$\alpha=3$ ,$\varDelta_g=0.01$ Figure 2. Static susceptibilities
$\chi_{11}$ and$\chi_{12}$ as functions of the concentration of species 2 with$n_1$ fixed at different values. The dash-dotted lines in two plots correspond to the analytical results given by Eqs. (19) and (20), respectively, with$n_1=n_1^{\rm{C}}$ . Interaction parameters:$\alpha = 3$ , and$\varDelta_g=0.01$ .
图 3 W函数在
$\alpha$ -$\beta$ 平面内的零值线. 在零值线两侧, 可实现的部分混溶态具有不同的构型(见正文). 圆点标出了几种实验上已制备出的双组分BEC在参数平面内的位置Figure 3. Contour plot of
$W=0$ in$\alpha$ -$\beta$ plane. In the two regions separated by the contour, the achievable partially miscible state has different configuration (see text). Big dots mark several experimentally realized mixtures at their positions in the parameters plane.表 1 冷原子玻色混合气体中可实现的部分混溶构型, 以及不混溶相边界的密度值
$n_{i}^{\rm{IM}}$ . 这里约定质量轻的元素为组分1, 质量重的元素为组分2. 散射长度为平均场失稳阈值处的数据(以Bohr半径$a_0$ 为单位), 取自所列的参考文献.$n_{i}^{\rm{IM}}$ 的预测值是根据(25)式和(26)式计算的,$\varDelta_g$ 均设为$0.01$ Table 1. Configuration of partially miscible state and
$n_i^{\rm{IM}}$ in various Bose mixtures of cold atoms. Here, the lighter elements are set as species 1, and the heavier elements are set as species 2. Intraspecies scattering lengths at the mean-field immiscibility threshold, measured in unit of Bohr radius$a_0$ , are taken from the listed references. The predicted$n_i^{\rm{IM}}$ are computed using Eqs. (25) and (26), with$\varDelta_g=0.01$ .玻色混合气体 $a_{11}/ $$ a_0$ $a_{22}/a_0$ 部分混溶构型 $n_ {1}^{\rm{IM} }/ (10^{14} {\rm{cm} }^{-3})$ $n_{2}^{\rm{IM} }/ $$ (10^{14} {\rm{cm} }^{-3})$ 文献 23Na + 39K 52 7.8 PM2 21.46 72.14 [9] 23Na + 87Rb 54.5 100.4 PM1 2.87 4.12 [8] 39K + 41K 470 65 PM2 0.026 0.073 [10] 39K + 87Rb 10.2 100.4 PM1 3.19 1.52 [7] 41K + 87Rb 65 100.4 PM1 16.14 18.96 [12] 
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[1] Graf E H, Lee D M, Reppy J D 1967 Phys. Rev. Lett. 19 417
Google Scholar
[2] Pricaupenko L, Treiner J 1995 Phys. Rev. Lett. 74 430
Google Scholar
[3] Maciolek A, Krech M, Dietrich S 2004 Phys. Rev. E 69 036117
Google Scholar
[4] Papp S B, Pino J M, Wieman C E 2008 Phys. Rev. Lett. 101 040402
Google Scholar
[5] Thalhammer G, Barontini G, De Sarlo L, Catani J, Minardi F, Inguscio M 2008 Phys. Rev. Lett. 100 210402
Google Scholar
[6] McCarron D J, Cho H W, Jenkin D L, Koppinger M P, Cornish S L 2011 Phys. Rev. A 84 011603
Google Scholar
[7] Wacker L, Jorgensen N B, Birkmose D, Horchani R, ErtmerW, Klempt C, Winter N, Sherson J, Arlt J J 2015 Phys. Rev. A 92 053602
Google Scholar
[8] Wang F, Li X, Xiong D, Wang D 2016 J. Phys. B: At. Mol. Opt. Phys. 49 015302
Google Scholar
[9] Schulze T A, Hartmann T, Voges K K, Gempel M W, Tiemann E, Zenesini A, Ospelkaus S 2018 Phys. Rev. A 97 023623
Google Scholar
[10] Tanzi L, Cabrera C R, Sanz J, Cheiney P, Tomza M, Tarruell L 2018 Phys. Rev. A 98 062712
Google Scholar
[11] Li W X, Chen Y D, Sun Y T, Tung S, Julienne P S 2022 Phys. Rev. A 106 023317
[12] Burchianti A, D'Errico C, Prevedelli M, Salasnich L, Ancilotto F, Modugno M, Minardi F, Fort C 2020 Condes. Matter 5 21
Google Scholar
[13] Castilho P C M, Pedrozo-Peñafiel E, Gutierrez E M, Mazo P L, Roati G, Farias K M, Bagnato V S 2019 Laser Phys. Lett. 16 035501
Google Scholar
[14] Mi C D, Nawaz K S, Wang P J, Chen L C, Meng Z M, Huang L H, Zhang J 2021 Chin. Phys. B 30 063401
Google Scholar
[15] Hadzibabic Z, Stan C A, Dieckmann K, Gupta S, Zwierlein M W, Gorlitz A, Ketterle W 2002 Phys. Rev. Lett. 88 160401
Google Scholar
[16] Ospelkaus S, Ospelkaus C, Humbert L, Sengstock K, Bongs K 2006 Phys. Rev. Lett. 97 120403
Google Scholar
[17] Ferrier-Barbut I, Delehaye M, Laurent S, Grier A T, Pierce M, Rem B S, Chevy F, Salomon C 2014 Science 345 1035
Google Scholar
[18] Yao X C, Chen H Z, Wu Y P, Liu X P, Wang X Q, Jiang X, Deng Y, Chen Y A, Pan J W 2016 Phys. Rev. Lett. 117 145301
Google Scholar
[19] Roy R, Green A, Bowler R, Gupta S 2017 Phys. Rev. Lett. 118 055301
Google Scholar
[20] Taglieber M, Voigt A C, Aoki T, Hansch T W, Dieckmann K 2008 Phys. Rev. Lett. 100 010401
Google Scholar
[21] Hara H, Takasu Y, Yamaoka Y, Doyle J M, Takahashi Y 2011 Phys. Rev. Lett. 106 205304
Google Scholar
[22] Ravensbergen C, Corre V, Soave E, Kreyer M, Kirilov E, Grimm R 2020 Phys. Rev. Lett. 124 203402
Google Scholar
[23] Ciamei A, Finelli S, Trenkwalder A, Inguscio M, Simoni A, Zaccanti M 2022 Phys. Rev. Lett. 129 093402
Google Scholar
[24] Pethick C J, Smith H 2008 Bose-Einstein Condensation In Dilute Gases (New York: Cambridge University Press) pp350–352
[25] Pitaevskii L, Stringari S 2016 Bose-Einstein Condensation and Superfluidity (New York: Oxford University Press) pp401–403
[26] Navarro R, Carretero-González R, Kevrekidis P G 2009 Phys. Rev. A 80 023613
Google Scholar
[27] Wen L, Liu W M, Cai Y, Zhang J M, Hu J 2012 Phys. Rev. A 85 043602
Google Scholar
[28] Bisset R N, Kevrekidis P G, Ticknor C 2018 Phys. Rev. A 97 023602
Google Scholar
[29] Pattinson R W, Billam T P, Gardiner S A, McCarron D J, Cho H W, Cornish S L, Parker N G, Proukakis N P 2013 Phys. Rev. A 87 013625
Google Scholar
[30] Lee K L, Jørgensen N B, Liu I K, Wacker L, Arlt J J, Proukakis N P 2016 Phys. Rev. A 94 013602
Google Scholar
[31] Cikojević V, Markić L V, Boronat J 2018 New J. Phys. 20 085002
Google Scholar
[32] Wen L, Guo H, Wang Y J, Hu A Y, Saito H, Dai C Q, Zhang X F 2020 Phys. Rev. A 101 033610
Google Scholar
[33] Gutierrez E M, Oliveira G A, Farias K M, Bagnato V S, Castilho P C M 2021 Appl. Sci. 11 9099
Google Scholar
[34] Schaeybroeck B V 2013 Physica A 392 3806
Google Scholar
[35] Ota M, Giorgini S, Stringari S 2019 Phys. Rev. Lett. 123 075301
Google Scholar
[36] Spada G, Parisi L, Pascual G, Parker N G, Billam T P, Pilati S, Boronat J, Giorgini S 2022 arXiv: 2211.09574
[37] Naidon P, Petrov D S 2021 Phys. Rev. Lett. 126 115301
Google Scholar
[38] Petrov D S 2015 Phys. Rev. Lett. 115 155302
Google Scholar
[39] Larsen D M 1963 Ann. Phys. (Berlin) 24 89
[40] Balabanyan G O 1986 Theor. Math. Phys. 66 81
Google Scholar
[41] Christensen R S, Levinsen J, Bruun G M 2015 Phys. Rev. Lett. 115 160401
Google Scholar
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