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As early as the 1950s, Prof. Yang and his collaborators realized that the most important interaction effects in a dilute quantum gas can be described by the s-wave scattering length between particles. This insight leads to universal descriptions of the interaction effects without the detailed knowledge of the interaction potential. They derived a formula expanding the energy density in terms of the gas parameter. This formula is later known as the Lee-Huang-Yang correction. However, it took forty years for the experimentalists to overcome several challenges and finally achieve degenerate quantum gases of atoms in 1995. The developments after 1995 have led to an exciting field known as “quantum gases” or “ultracold atomic gases”. The ultracold atom system has flexible tunability, allowing both the scattering length and the dimensionality to vary. The Lee-Huang-Yang corrections were observed from several experiments on ultracold atoms by increasing the scattering length. In addition, by reducing the dimensionality to one-dimension, several experiments on ultracold atoms have confirmed the Yang-Yang thermodynamics for one-dimensional bosons that Prof. Yang obtained in the 1960s and the large-N limit of one-dimensional fermions that Prof. Yang obtained around 2010. By increasing the dimensionality through using the idea of synthetic dimension, the experiment on ultracold atoms has also demonstrated the Yang monopole in the SU(2) non-abelian gauge field proposed by Prof. Yang in the 1970s. All of these experiments show the long-lasting impact of Prof. Yang’s theoretical work over several decades.
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Keywords:
- ultracold atomic physics
[1] Lee T D, Yang C N 1956 Phys. Rev. 104 254
Google Scholar
[2] Lee T D, Yang C N 1956 Phys. Rev. 105 1119
Google Scholar
[3] Lee T D, Huang K, Yang C N 1957 Phys. Rev. 106 1135
Google Scholar
[4] Anderson M H, Ensher J R, Matthews M R, Wieman C E, Cornell E A 1995 Science 269 198
Google Scholar
[5] Davis K B, Mewes M O, Andrews M R, van Druten N J, Durfee D S, Kurn D M, Ketterle W 1995 Phys. Rev. Lett. 75 3969
Google Scholar
[6] Huang K, Yang C N 1957 Phys. Rev. 105 767
Google Scholar
[7] Cornell E A, Wieman C E 2001 Rev. Mod. Phys. 74 875
Google Scholar
[8] Zhai H 2021 Ultracold Atomic Physics (Cambridge: Cambridge Universit Press)
[9] Chou T T, Yang C N, Yu L H 1996 Phys. Rev. A 53 4257
Google Scholar
[10] Chou T T, Yang C N, Yu L H 1997 Phys. Rev. A 55 1179
Google Scholar
[11] Altmeyer A, Riedl S, Kohstall C, Wright M J, Geursen R, Bartenstein M, Chin C, Denschlag J H, Grimm R 2007 Phys. Rev. Lett. 98 040401
Google Scholar
[12] Shin Y I, Schirotzek A, Schunck C H, Ketterle W 2008 Phys. Rev. Lett. 101 070404
Google Scholar
[13] Papp S B, Pino J M, Wild R J, Ronen S, Wieman C E, Jin D S, Cornell E A 2008 Phys. Rev. Lett. 101 135301
Google Scholar
[14] Navon N, Nascimbene S, Chevy F, Salomon C 2010 Science 328 729
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[15] Navon N, Piatecki S, Günter K, Rem B, Nguyen T C, Chevy F, Krauth W, Salomon C 2011 Phys. Rev. Lett. 107 135301
Google Scholar
[16] Skov T G, Skou M G, Jorgensen N B, Arlt J J 2021 Phys. Rev. Lett. 126 230404
Google Scholar
[17] Bethe H A 1931 Z. Physik 71 205
Google Scholar
[18] Yang C N 1967 Phys. Rev. Lett. 19 1312
Google Scholar
[19] Yang C N, Yang C P 1969 J. Math. Phys. 10 1115
Google Scholar
[20] Lieb E, Liniger W 1963 Phys. Rev. 130 1605
Google Scholar
[21] van Amerongen A H, van Es J J P, Wicke P, Kheruntsyan K V, van Druten N J 2008 Phys. Rev. Lett. 100 090402
Google Scholar
[22] Yang C N, You Y Z 2011 Chin. Phys. Lett. 28 020503
Google Scholar
[23] Pagano G, Mancini M, Cappellini G, Lombardi P, Schafer F, Hu H, Liu X J, Catani J, Sias C, Inguscio M, Fallani L 2014 Nat. Phys. 10 198
Google Scholar
[24] Zhai H 2015 Rep. Prog. Phys. 78 026001
Google Scholar
[25] Yang C N, Mills R L 1954 Phys. Rev. 96 191
Google Scholar
[26] Yang C N 1978 J. Math. Phys. 19 320
Google Scholar
[27] Sugawa S, Salces-Carcoba F, Perry A R, Yue Y, Spielman I B 2018 Science 360 1429
Google Scholar
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图 1 “Lee-Huang-Yang修正”的实验证实. 纵轴压强
$ h $ 反映了系统的状态方程, 横轴是$ \mu {a}^{3}/g $ . 这里,$ a $ 就是文中的散射长度$ {a}_{\mathrm{s}} $ ;$ \mu $ 是巨正则系统的化学势;$ g=4\mathrm{\pi }{\hslash }^{2}a/m $ ,$ m $ 是原子的质量. 黑色点是实验结果, 红色实线是加入“Lee-Huang-Yang修正”的理论结果, 黑色实线是蒙特卡罗计算的结果, 红色虚线是没有加入“Lee-Huang-Yang修正”的平均场结果. 插图是弱相互作用区域的结果 [15]Figure 1. Experimental observation of the Lee-Huang-Yangcorrection. The equation-of-state, measured through the pressure h, is plotted as a function of
$ \mu {a}^{3}/g $ . Here$ a $ is the s-wave scattering length$ {a}_{\mathrm{s}} $ in the text.$ \mu $ is the chemical potential in the grand canonical ensemble.$ g=4\mathrm{\pi }{\hslash }^{2}a/m $ . The solid red line is theoretical result with the Lee-Huang-Yang corrections, and the solid black line is the result obtained by quantum Monte Carlo simulation. The red dashed line is the mean-field result without including the Lee-Huang-Yang corrections. The inset is a zoom-in plot of the weakly interacting regime[15].
图 2 Yang-Yang Thermodynamics的实验证实 (a)—(d) 原位成像的密度分布; (e)—(h) 动量分布. 实线是Yang-Yang Thermodynamics的理论预言, 虚线是无相互作用玻色子的结果. 黑色的点是实验数据点[21]
Figure 2. Experimental observation of Yang-Yang thermodynamics: (a)–(d) The in-situ density distribution; (e)–(h) the momentum distribution. The solid lines are obtained from the Yang-Yang thermodynamics and the dashed lines are the results of non-interacting bosons[21].
图 3 N分量一维费米气体的实验结果.
$ \beta ={\omega }_{\mathrm{B}}/{\omega }_{x} $ , 其中$ {\omega }_{\mathrm{B}} $ 是测得的沿$\hat{x}$ 方向系统呼吸模的频率,$ {\omega }_{x} $ 是沿$\hat{x}$ 方向谐振子势阱的频率. 横轴是费米子内态的数目N. 其中正方形方块是实验数据, 圆圈是理论预言. 上面的横线是无相互作用(N = 1)费米子的情况; 下面的横线是单分量玻色子的情况, 即$ N\to \infty $ 的极限情况[23]Figure 3. Experiments on N-component one-dimensional Fermi gas.
$ \beta ={\omega }_{\mathrm{B}}/{\omega }_{x} $ , where$ {\omega }_{\mathrm{B}} $ is the breathing mode frequency along$\hat{x}$ and$ {\omega }_{x} $ is the harmonic trap frequency along$\hat{x}$ . This frequency is plotted as a function of the number of fermion components N. The squares are experimental data and the circles are theoretical predictions. The upper horizontal line shows the theoretical value for non-interacting Fermi gas (N = 1) and the lower line shows the result for spinless bosons, as$N \to \infty$ limit[23].
图 4 实验观测到Yang monopole (a) 随着Yang monopole移入或移出球面, 系统拓扑的变化; (b) 第一类陈数(下图)和第二类陈数(上图). 第一类陈数总是零. 第二类陈数随着Yang monopole移出球面, 从1变成0[27]
Figure 4. Experimental observation of Yang monopole: (a) Illustration of the topological transition when the Yang monopole moves out of the sphere; (b) the first (lower) and the second (upper) Chern number. The first Chern number is constantly zero and the second Chern number changes from unity to zero as the Yang monopole moves out of the sphere[27]
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[1] Lee T D, Yang C N 1956 Phys. Rev. 104 254
Google Scholar
[2] Lee T D, Yang C N 1956 Phys. Rev. 105 1119
Google Scholar
[3] Lee T D, Huang K, Yang C N 1957 Phys. Rev. 106 1135
Google Scholar
[4] Anderson M H, Ensher J R, Matthews M R, Wieman C E, Cornell E A 1995 Science 269 198
Google Scholar
[5] Davis K B, Mewes M O, Andrews M R, van Druten N J, Durfee D S, Kurn D M, Ketterle W 1995 Phys. Rev. Lett. 75 3969
Google Scholar
[6] Huang K, Yang C N 1957 Phys. Rev. 105 767
Google Scholar
[7] Cornell E A, Wieman C E 2001 Rev. Mod. Phys. 74 875
Google Scholar
[8] Zhai H 2021 Ultracold Atomic Physics (Cambridge: Cambridge Universit Press)
[9] Chou T T, Yang C N, Yu L H 1996 Phys. Rev. A 53 4257
Google Scholar
[10] Chou T T, Yang C N, Yu L H 1997 Phys. Rev. A 55 1179
Google Scholar
[11] Altmeyer A, Riedl S, Kohstall C, Wright M J, Geursen R, Bartenstein M, Chin C, Denschlag J H, Grimm R 2007 Phys. Rev. Lett. 98 040401
Google Scholar
[12] Shin Y I, Schirotzek A, Schunck C H, Ketterle W 2008 Phys. Rev. Lett. 101 070404
Google Scholar
[13] Papp S B, Pino J M, Wild R J, Ronen S, Wieman C E, Jin D S, Cornell E A 2008 Phys. Rev. Lett. 101 135301
Google Scholar
[14] Navon N, Nascimbene S, Chevy F, Salomon C 2010 Science 328 729
Google Scholar
[15] Navon N, Piatecki S, Günter K, Rem B, Nguyen T C, Chevy F, Krauth W, Salomon C 2011 Phys. Rev. Lett. 107 135301
Google Scholar
[16] Skov T G, Skou M G, Jorgensen N B, Arlt J J 2021 Phys. Rev. Lett. 126 230404
Google Scholar
[17] Bethe H A 1931 Z. Physik 71 205
Google Scholar
[18] Yang C N 1967 Phys. Rev. Lett. 19 1312
Google Scholar
[19] Yang C N, Yang C P 1969 J. Math. Phys. 10 1115
Google Scholar
[20] Lieb E, Liniger W 1963 Phys. Rev. 130 1605
Google Scholar
[21] van Amerongen A H, van Es J J P, Wicke P, Kheruntsyan K V, van Druten N J 2008 Phys. Rev. Lett. 100 090402
Google Scholar
[22] Yang C N, You Y Z 2011 Chin. Phys. Lett. 28 020503
Google Scholar
[23] Pagano G, Mancini M, Cappellini G, Lombardi P, Schafer F, Hu H, Liu X J, Catani J, Sias C, Inguscio M, Fallani L 2014 Nat. Phys. 10 198
Google Scholar
[24] Zhai H 2015 Rep. Prog. Phys. 78 026001
Google Scholar
[25] Yang C N, Mills R L 1954 Phys. Rev. 96 191
Google Scholar
[26] Yang C N 1978 J. Math. Phys. 19 320
Google Scholar
[27] Sugawa S, Salces-Carcoba F, Perry A R, Yue Y, Spielman I B 2018 Science 360 1429
Google Scholar
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