Maximum Energy Growth Rate in Dilute Quantum Gases
MMaximum Energy Growth Rate in Dilute Quantum Gases
Ran Qi, ∗ Zheyu Shi, and Hui Zhai † Department of Physics, Renmin University of China, Beijing, 100872, P. R. China Key State Laboratory of Precision Spectroscopy,East China Normal University, Shanghai 200062, China Institute for Advanced Study, Tsinghua University, Beijing 100084, China (Dated: February 16, 2021)In this letter we study how fast the energy density of a quantum gas can increase in time, when theinter-atomic interaction characterized by the s -wave scattering length a s is increased from zero witharbitrary time dependence. We show that, at short time, the energy density can at most increase as √ t , which can be achieved when the time dependence of a s is also proportional to √ t , and especially,a universal maximum energy growth rate can be reached when a s varies as 2 (cid:112) (cid:126) t/ ( πm ). If a s variesfaster or slower than √ t , it is respectively proximate to the quench process and the adiabatic process,and both result in a slower energy growth rate. These results are obtained by analyzing the shorttime dynamics of the short-range behavior of the many-body wave function characterized by thecontact, and are also confirmed by numerical solving an example of interacting bosons with time-dependent Bogoliubov theory. These results can also be verified experimentally in ultracold atomicgases. The ability of tuning interactions between particles isa major advantage of ultracold atomic systems [1, 2].Especially, by ultilizing magnetic and optical tools, theinteraction strength between atoms, usually character-ized by the s -wave scattering length a s , can be tunedin a time scale much faster than the many-body relax-ation time. This has led to a number of interesting ultra-cold atomic experiments reported in recent years, suchas universal quench dynamics observed by quenching in-teraction to unitarity [3, 4], and coherent excitation ofthe Higgs model in superfluid Fermi gases and the Bo-goliubov quasi-particles in Bose condensate by periodi-cally modulating interactions [5–8]. These experimentalprogresses are also accompanied by lots of theoretical in-terests on studying non-equilibrium dynamics driven bytime-dependent interactions [9–30].Motivated by these progresses, here we address a fun-damental issue whether there is a universal upper limitfor the energy increasing rate. To be concrete, supposethat we start with a non-interacting quantum gas with a s = 0 and then change a s in time, and suppose that a s ( t )can be controlled in any function form, the question iswhether there is an upper bound for the rate of how fastthe total energy can increase as a function of time. In thisletter we show that there does exist such a universal ratelimit, as far as the initial growth rate is concerned. Thisresult is quite counter-intuitive, because normally theinteraction energy increases as the interaction strengthincreases. Thus, intuitively, one would think that afaster increasing of interaction strength should result in afaster increasing of interaction energy, and consequently,a faster increasing of the total energy. Since we considerthat a s can be increased as fast as one wants, it seems toindicate that there should not be such a bound.However, our results show that this intuition is not cor-rect. Before presenting rigorous mathematical statement, we first emphasize that our result is closely tied to a keyquantity of ultracold atomic gases called the contact [31–39]. It is now well known that, for quantum gases withzero-range interactions, one can define contact C throughthe short-range behavior of the many-body wave functionwhen any two atoms are brought close to each other, orequivalently, through the high-momentum tail of the mo-mentum distribution. It has been shown that the totalenergy of a quantum gas is directly related to the contact[31–39].To gain an intuitive understanding of our results, let usfirst consider two limits. On one limit, the fastest changeof the interaction strength is the quench process, duringwhich a s instantaneously jumps from zero to any non-zero value. However, it can be shown that the contactdoes not change and retains zero right after the quench[32], and therefore, the total energy also does not changeafter the quench. This means that the fastest change ofinteraction actually does not result in a fast change ofthe total energy, and in contrast, the interaction energydoes not change at all. On the opposite limit, we canconsider an adiabatic varying of the interaction strength,during which the interaction energy does vary in timebut it varies adiabatically with sufficiently slow rate. Thephysical pictures in these two limits motivate us to expecta universal maximum growth rate driven by intermediatespeed of varying the interaction strength. General Expression for the Contact Growth.
Here weconsider a uniform Bose gas or spin-1/2 Fermi gas start-ing from any non-interacting state | Ψ (cid:105) at t = 0, and thenthe s -wave scattering length a s ( t ) can vary with arbitrarytime dependence. Below we use n and n σ to denote thedensities of bosons and fermions with spin- σ ( σ = ↑ , ↓ ),respectively, and ˆ ψ and ˆ ψ σ to denote boson operator andfermion operator with spin- σ , respectively. One of themain results of this work states as follows: a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b In the short-time limit, the dynamics of the contact C ( t ) is given by C ( t ) = g (0) | η ( t ) | . (1)Here g ( r ) is defined as (cid:104) Ψ | ˆ ψ † ( r ) ˆ ψ † ( − r ) ˆ ψ ( − r ) ˆ ψ ( r ) | Ψ (cid:105) for bosons and (cid:104) Ψ | ˆ ψ †↑ ( r ) ˆ ψ †↓ ( − r ) ˆ ψ ↓ ( − r ) ˆ ψ ↑ ( r ) | Ψ (cid:105) forfermions, and g (0) means g ( r ) evaluated at r = 0. Es-pecially, if | Ψ (cid:105) is the non-interacting ground state, then g ( r ) = n or n ↑ n ↓ for the Bose or the spin-1 / η ( t ) obeys thefollowing integral equation (cid:20) ˆ L + 14 πa s ( t ) (cid:21) η ( t ) = − , (2)where ˆ L is a linear operator acting on η ( t ) asˆ Lη ( t ) = (cid:16) m (cid:126) (cid:17) π / √ i × lim (cid:15) → + (cid:20)(cid:90) t − (cid:15) η ( τ )( t − τ ) dτ − η ( t ) √ (cid:15) (cid:21) . (3)This result is motivated by solving the two-body prob-lem, which satisfies the following Schr¨odinger equation inthe relative coordinate r frame as i (cid:126) ∂ψ∂t = − (cid:126) ∇ ψm + 4 π (cid:126) a s ( t ) m δ ( r ) ∂∂r rψ. (4)Starting from an initial state ψ ( r ) = 1 / √ V ( V is the totalvolume of the system), the time evolution of the wavefunction always obeys the following asymptotic form inthe short-range r → ψ ( r , t ) = η ( t )4 π √ V (cid:20) r − a s ( t ) (cid:21) + O ( r ) , (5)and it can be shown that η ( t ) satisfies Eq. 28 [40]. Gen-eralizing this result from the two-body problem to themany-body problem utilizes the short-time expansionand is quite straightforward, which yields Eq. 1 [40].Here we note that for the two-body problem, η ( t ) sat-isfies Eq. 28 for all time scales, but for the many-bodyproblem, the result is only valid for the short-time scale.Here short-time is defined as the time scale much shorterthan the typical many-body time scale t n = (cid:126) /E n , where E n = (cid:126) k / (2 m ) and k n = (6 π n ) / (with n replaced by n σ for fermions). In other word, in such short time scale,the short-range behavior of many-body wave function isstill dominated by the two-body physics. Contact Growth Rate.
Here, without loss of generality,we consider that a s ( t ) grows from zero to a positive valuein a power-law function as a s ( t ) = √ βl (cid:18) tt (cid:19) α , (6) FIG. 1: (a1-a2) The time-dependence of the scattering length a s ( t ) (in unit of l ) with different power-law functions of Eq. 6.(a1) α = , , , and β is fixed at β = 1. (a2) α is fixed at α = and β = , . , .
596 and 4. (b1-b2) The short timebehavior of the contact C (in unit of g (0) l ) with a s ( t ) plottedin (a1) and (a2), respectively. (c1-c2) The time-dependenceof the energy density change δ E (in unit of g (0) l (cid:126) /m ) with a s ( t ) plotted in (a1) and (a2), respectively. where l is an arbitrary length unit and t is the timeunits, and l and t are both related to the same energyunit as (cid:126) /t = (cid:126) / (2 ml ). α, β are two constants de-scribing the power and the coefficient, respectively anda factor √ L has an important property thatˆ Lt α = (cid:16) m (cid:126) (cid:17) B ( α ) t α − , (7)where B ( α ) is a constant given by B ( α ) = i / Γ( α +1) / (4 π Γ( α + 1 / η ( t ) is apower-law function in t , when ˆ L acts on η ( t ), it lowersthe power of η ( t ) by 1 /
2. This property plays a crucialrole in the following analysis because it means whether α in Eq. 6 is greater or smaller than 1 / α > /
2. In this case, the 1 / (4 πa s ) term domi-nates Eq. 28, and thus, to the leading order of t , η ( t ) and C ( t ) are given by η ( t ) = − πa s ( t ); C ( t ) = 16 π a ( t ) g (0) . (8)This is consistent with the adiabatic regime where thephysical quantities only depend on the instantaneousscattering length at time t .Case II: α < /
2. In this case, the ˆ L term dominatesEq. 28, and thus, to the leading order of t , η ( t ) and C ( t )are given by η ( t ) = − (cid:18) (cid:126) m (cid:19) B (1 / √ t ; C ( t ) = (cid:126) m g (0) B (1 / t, (9)where B (1 /
2) = − / (8 √ iπ ). Surprisingly, in this casethis result shows that the growth of contact at the short-time is independent of parameters l , t , α and β in Eq.6. That is to say, it is independent of how fast a s variesin time. Even if l or β is infinitely large, or α is infinites-imally small, and then a s ( t ) initially grows infinitely fast,the contact always grows linearly in time with a constantrate. This means that as long as α < /
2, the short-rangephysics at the short time is the same as a quench pro-cess where the scattering length instantaneously jumpsto unitarity.Case III: α = 1 /
2. In this case, Eq. 6 becomes a s ( t ) = β (cid:114) (cid:126) tm , (10)By dimension analysis, it is easy to see that l and t cancel each other and only the coefficient β enters theexpression. In this case, both ˆ L term and the 1 / (4 πa s )term are equally important. Also to the leading order of t , we obtain η ( t ) = − A ( β ) √ t ; C ( t ) = | A ( β ) | g (0) t, (11)where A ( β ) is also a constant given by A ( β ) = (cid:18) (cid:126) m (cid:19) B (cid:0) (cid:1) + πβ . (12)As one can see from here, this is a critical case. In theCase III, by taking β → ∞ , Eq. 11 recovers Eq. 9, consis-tent with the quench limit, and by taking β →
0, Eq. 11recovers Eq. 8, consistent with the adiabatic limit.Here an important point is that | A ( β ) | is not a mono-tonic function in β . For a given initial state, g (0) isfixed, and we can then define the initial growth rate forcontact as v C = lim t → d C ( t ) /dt . As one can see fromEq. 8, v C = 0 for case I. And for both the Case II andthe Case III, v C is a constant, given by | A ( β ) | g (0)for the Case III and | A ( β = ∞ ) | g (0) for the CaseII. It turns out that | A ( β ) | reaches its maximum at β c1 = 2 (cid:112) /π ≈ . v max C = ( (cid:126) /m )128 πg (0). FIG. 2: Initial growth rate for contact (a) and for energy(b) as a function of β for a s ( t ) = β (cid:112) (cid:126) t/m . Arrows mark β c1 and β c2 where the maximum contact growth rate and themaximum energy growth rate are reached. υ C and υ E areplotted in units of g (0) (cid:126) /m and g (0) (cid:112) (cid:126) /m respectively. Energy Growth Rate.
The total energy density of auniform zero-range interacting quantum gas can be mea-sured through its momentum distribution n k . For exam-ple, for spin-1 / E = (cid:90) d k (2 π ) (cid:15) k (cid:18) n k − C k (cid:19) + C πma s , (13)where n k = n k ↑ + n k ↓ , (cid:15) k = (cid:126) k / (2 m ), and the contact C is related to n k σ through C ≡ lim k →∞ k n k σ [31]. Thesame expression, replacing all C by C /
2, also holds forthe spinless Bose gas as long as the three-body contactcan be ignored [38].On the other hand, there is a direct relation betweenthe time evolution of the energy and the contact. Forspin-1 / ddt E ( t ) = (cid:126) C ( t )4 πma ( t ) da s dt . (14)For spinless bosons, an extra 1 / α > /
2. With the help of Eq. 8, one canobtain that δ E ( t ) = 4 π (cid:126) a s ( t ) m g (0) . (15)where δ E ( t ) = E ( t ) − E ( t = 0). This result again showsthat the physics in this regime is consistent with adia-batic regime where the energy is determined by the in-stantaneous scattering length. Since α > /
2, the energyincreases slower than √ t at the short time.Case II: α < /
2. In this regime, Eq. 9 gives rise to δ E ( t ) = 16 √ αβ (1 − α ) (cid:126) m g (0) l (cid:18) tt (cid:19) − α . (16)Since α < /
2, the energy also increases slower than √ t at the short time. When taking the α → FIG. 3: Dynamics of the total energy density of Bose gas for a s ( t ) with different power-law functions of Eq. 6. (a) β = 1and α = , , . (b) α = and β = 4 , . , . δ E isplotted in unit of n l (cid:126) /m and we have set t = t n and thus l = 1 /k n in the numerical calculation. β → ∞ or l → ∞ limit, δ E ( t ) →
0, and it is consistentwith the fact there is no energy change for the quenchprocess as discussed above.Case III: α = 1 /
2. In this regime, Eq. 11 yields δ E ( t ) = (cid:114) (cid:126) m | A ( β ) | πβ g (0) √ t. (17)It is in this case that the energy growth at the shorttime is the fastest. Now we can define an energygrowth rate v E = lim t → d E ( t ) /d √ t . For case Iand II, this rate is zero. In case II, v E is givenby (cid:112) (cid:126) /m | A ( β ) | g (0) / (4 πβ ), which reaches its max-imum at β c2 = 2 / √ π ≈ .
128 with v max E = 4(2 + √ √ πg (0) (cid:112) (cid:126) /m ≈ . g (0) (cid:112) (cid:126) /m . Note that thisvalue of v max E applies for the spin-1 / a s ( t ) and byconsidering positive a s ( t ), it can be extended to otherfunction forms, such as including the logarithmic func-tion corrections, and including the situations where a s varies to negative values. The results discussed above aresummarized in Fig. 1 and Fig. 2. Fig. 1(a1) and (a2)show different power-law function of a s ( t ) given by Eq. 6,either with different power α , or with different coefficient β and fixed α = 1 /
2. Fig. 1(b1) and (b2) show the corre-sponding contact growth, and Fig. 1(c1) and (c2) showthe corresponding energy growth, using spinless bosonsas an example. It clearly shows that a faster increasing of a s does not necessarily lead to a faster increasing of thecontact and the energy density. One can see that for dif-ferent powers, α = 1 / α fixedat 1 / β = β c1 yields the fastest contact growth and β = β c2 yields the fastest energy growth, as also shownin Fig. 2. Example.
The analysis above is based on the short time expansion. Here, we consider a concrete example ofspinless bosons, which can be described by the followingtime-dependent Hamiltonianˆ H ( t ) = (cid:88) k (cid:15) k ˆ b † k ˆ b k + U ( t )2 V (cid:88) k , k (cid:48) , q ˆ b † k ˆ b † q − k ˆ b q − k (cid:48) ˆ b k (cid:48) , (18)where ˆ b k are boson creation operators with momentum k . U ( t ) is related to a s ( t ) through the renormalizationrelation 1 U ( t ) = m π (cid:126) a s ( t ) − V (cid:88) k (cid:15) k . (19)We solve this Hamiltonian by adopting the Bogoliubov-type variational ansatz as | Ψ( t ) (cid:105) = N ( t ) exp g ( t )ˆ b † + (cid:88) k (cid:54) =0 g k ( t )ˆ b † k ˆ b †− k | (cid:105) , (20)where N ( t ) is a normalization factor, | (cid:105) is vacuum ofparticles, and g and g k are all variational parameters.This approach is not restricted to the short time andhas been successfully used in the previous studies of de-generate Bose gas quenched to unitarity [10, 13, 26].The evolution of variational parameters g ( t ) and g k ( t )can be obtained from the Euler-Lagrange equation forthe Lagrangian L = [ (cid:104) Ψ( t ) | ˙Ψ( t ) (cid:105) − (cid:104) ˙Ψ( t ) | Ψ( t ) (cid:105) ] −(cid:104) Ψ( t ) | ˆ H ( t ) | Ψ( t ) (cid:105) , which yields a set of differential equa-tions for g and g k . Since we start with a non-interactingBose condensate, we take g = 1 and g k = 0 at t = 0 asthe initial conditions for these differential equations. Wecan obtain the variational wave function by solving theseequations, and subsequently, we can determine the totalenergy density with Eq. 13. The results for the totalenergy density are shown in Fig. 3 for different powersand different coefficients. One can see that the short timebehaviors agree very well with that given in Fig. 1(c1)and (c2). Summary.
In summary, we have studied the energygrowth rate of degenerate quantum gas driven by in-creasing the s -wave scattering length a s from zero, byboth analyzing the short time behavior on general sit-uations and numerically solving a concrete example ofinteracting bosons. Two main results are summarized asfollows: (i) For energy increasing as t α at the short time, α cannot be smaller than 1 / α = 1 / a s ( t ) varies as ∝ √ t . (ii) For energy increasingas √ t at the short time, the fastest energy increasing isachieved when a s ( t ) = 2 (cid:112) (cid:126) t/ ( πm ), with a maximum en-ergy growth given by 4(2 + √ (cid:112) π (cid:126) t/mg (0) for thespin-1 / a s varies as √ t , the entire many-body Schr¨odinger equation is invari-ant under a space-time scaling transformation t → λ t and r → λr . Similar examples of such scale invari-ant many-body dynamics have been studied in [41–43].Hence, this result ties together the fastest energy growthwith the scaling symmetry, and this is reminiscent of anequilibrium analogy, where the interaction effect is thestrongest at unitarity where the system is also scale in-variant. Acknowledgment.
We thank Peng Zhang for helpfuldiscussions. The project was supported by NSFC un-der Grant No. 12022405 (RQ), No. 11774426 (RQ) andNo. 11734010 (HZ and RQ), Beijing Outstanding YoungScholar Program (HZ), the National Key R and D Pro-gram of China under Grant No. 2018YFA0306501(RQ),the Research Funds of Renmin University of China un-der Grant No. 19XNLG12 (RQ), the Beijing NaturalScience Foundation under Grant No. Z180013 (RQ),and Program of Shanghai Sailing Program Grant No.20YF1411600 (ZYS). ∗ Electronic address: [email protected] † Electronic address: [email protected][1] T. K¨ohler, K. G´oral, and P. S. Julienne, Rev. Mod. Phys. , 1311 (2006).[2] C. Chin, R. Grimm, P. S. Julienne, and E. Tiesinga, Rev.Mod. Phys. , 1225 (2010).[3] P. Makotyn, C. E. Klauss, D. L. Goldberger, E. A. Cor-nell, and D. S. Jin, Nat. Phys. , 116 (2014).[4] C. Eigen, J. A. Glidden, R. Lopes, E. A. Cornell, R. P.Smith, and Z. Hadzibabic, Nature (London) , 221(2018).[5] A. Behrle, T. Harrison, J. Kombe, K. Gao, M. Link, J.-S. Bernier, C. Kollath and M. K¨ohl, Nat. Phys. , 781(2018)[6] L. W. Clark, A. Gaj, L. Feng and C. Chin, Nature ,356 (2017).[7] L. Feng, J. Hu, L. W. Clark, and C. Chin, Science ,521 (2019).[8] J. Hu, L. Feng, Z. Zhang, and C. Chin, Nat. Phys. ,785 (2019).[9] X. Yin and L. Radzihovsky, Phys. Rev. A , 063611(2013).[10] A. G. Sykes, J. P. Corson, J. P. D’Incao, A. P. Koller, C.H. Greene, A. M. Rey, K. R. Hazzard, and J. L. Bohn,Phys. Rev. A , 021601 (2014).[11] A. Ran¸con and K. Levin, Phys. Rev. A , 021602 (2014).[12] B. Kain and H. Y. Ling, Phys. Rev. A , 063626 (2014).[13] J. P. Corson and J. L. Bohn, Phys. Rev. A , 013616 (2015)[14] F. Ancilotto, M. Rossi, L. Salasnich, and F. Toigo, Few-Body Syst. , 801 (2015).[15] X. Yin and L. Radzihovsky, Phys. Rev. A , 033653(2016).[16] A. Eckardt, Rev. Mod. Phys. , 011004 (2017).[17] V. E. Colussi, J. P. Corson, and J. P. D’Incao, Phys. Rev.Lett. , 100401 (2018).[18] V. E. Colussi, S. Musolino, and S. J. J. M. F. Kokkel-mans, Phys. Rev. A , 051601 (2018).[19] M. Van Regemortel, H. Kurkjian, M. Wouters, and I.Carusotto, Phys. Rev. A , 053612 (2018).[20] J. P. D’Incao, J. Wang, and V. E. Colussi, Phys. Rev.Lett. , 023401 (2018).[21] H. Fu, L. Feng, B. M. Anderson, L. W. Clark, J. Hu, J.W. Andrade, C. Chin, and K. Levin, Phys. Rev. Lett. , 243001 (2018).[22] T. Chen and B. Yan, Phys. Rev. A , 063615 (2018).[23] S. Musolino, V. E. Colussi, and S. J. J. M. F. Kokkel-mans, Phys. Rev. A , 013612 (2019).[24] A. Mu˜noz de las Heras, M. M. Parish, F. M. Marchetti,Phys. Rev. A , 023623 (2019).[25] Z. Wu and H. Zhai, Phys. Rev. A , 063624 (2019).[26] C. Gao, M. Sun, P. Zhang, and H. Zhai, Phys. Rev. Lett. , 040403 (2020).[27] M. Sun, P. Zhang, and H. Zhai, Phys. Rev. Lett. ,110404 (2020).[28] Y.-Y. Chen, P. Zhang, W. Zheng, Z. Wu, and H. Zhai,Phys. Rev. A , 011301(R) (2020).[29] Y. Cheng and Z. Y. Shi, arXiv:2004.12754[30] C. Lv, R. Zhang, Q. Zhou, Phys. Rev. Lett. , 253002(2020)[31] S. Tan, Ann. Phys. (N.Y.) , 2952 (2008).[32] S. Tan, Ann. Phys. (N.Y.) , 2971 (2008).[33] S. Tan, Ann. Phys. (N.Y.) , 2987 (2008).[34] M. Punk and W. Zwerger, Phys. Rev. Lett. , 170404(2007)[35] G. Baym, C. J. Pethick, Z. Yu and M. W. Zwierlein,Phys. Rev. Lett. , 190407 (2007)[36] E. Braaten and L. Platter, Phys. Rev. Lett. , 205301(2008).[37] S. Zhang and A. J. Leggett. Phys. Rev. A , 023601(2009)[38] E. Braaten, D. Kang, and L. Platter, Phys. Rev. Lett. , 153005 (2011).[39] Z. Yu, J. H. Thywissen, and S. Zhang, Phys. Rev. Lett. , 135304 (2015).[40] See the supplementary material for the detailed proof.[41] S. Deng, Z.-Y. Shi, P. Diao, Q. Yu, H. Zhai, R. Qi, andH. Wu, Science , 371 (2016).[42] Z.-Y. Shi, R. Qi, H. Zhai, and Z. Yu, Phys. Rev. A ,050702(R) (2017).[43] S. Deng, P. Diao, F. Li, Q. Yu, S. Yu, and H. Wu, Phys.Rev. Lett. , 125301 (2018). Supplementary material:Maximum Energy Growth Rate in Dilute Quantum Gases
In this supplementary material, we provide the details of proof for Eq. (1)-(3) in the main text.
SOLUTION OF TWO-BODY PROBLEM
In this section, we solve the time-dependent two-body problem and establish Eq. (2) and (3). We consider thefollowing time-dependent Schr¨odinger equation in the relative coordinate r frame i (cid:126) ∂∂t ψ ( r , t ) = − (cid:126) m ∇ ψ ( r , t ) + 4 π (cid:126) a s ( t ) m δ ( r ) ∂∂r rψ ( r , t ) , (21)which is Eq. (4) in the main text and we choose the initial state ψ ( r ) = 1 / √ V . We first define an auxiliary function η ( t ) as η ( t ) = −√ V πa s ( t ) lim r → ∂∂r [ rψ ( r , t )] . (22)Then Eq. (21) can be rewritten as i (cid:126) ∂∂t ψ ( r , t ) = − (cid:126) m ∇ ψ ( r , t ) − √ V (cid:126) m δ ( r ) η ( t ) . (23)Now Eq. (23) can be solved with the standard Green’s function approach and the solution is given as ψ ( r , t ) = ψ ( r , t ) + i √ V (cid:126) m (cid:90) t G rel0 ( r , t − τ ) η ( τ ) τ (24)where G rel0 ( r , t − τ ) is the noninteracting Green’s function in the relative coordinate frame given by G rel0 ( r , t − τ ) = (cid:20) mi π (cid:126) ( t − τ ) (cid:21) / exp (cid:20) i mr (cid:126) ( t − τ ) (cid:21) , (25)and ψ ( r , t ) satisfies the non-interacting Schr¨odinger equation i (cid:126) ∂∂t ψ ( r , t ) = − (cid:126) m ∇ ψ ( r , t ) , ψ ( r , t = 0) = ψ ( r ) (26)where ψ ( r ) is the initial wave function of ψ ( r , t ).For ψ ( r ) = 1 / √ V , we have ψ ( r , t ) = 1 / √ V and, based on Eq. (24), we obtain the following asymptotic expansionat r → ψ ( r , t ) = 1 √ V η ( t )4 πr + (cid:16) m (cid:126) (cid:17) π / √ i lim r → + (cid:90) t η ( τ ) exp (cid:104) i mr (cid:126) ( t − τ ) (cid:105) ( t − τ ) dτ − (cid:90) t −∞ η ( t ) exp (cid:104) i mr (cid:126) ( t − τ ) (cid:105) ( t − τ ) dτ + O( r ) , = 1 √ V (cid:26) η ( t )4 πr + (cid:16) m (cid:126) (cid:17) π / √ i lim (cid:15) → + (cid:20)(cid:90) t − (cid:15) η ( τ )( t − τ ) dτ − (cid:90) t − (cid:15) −∞ η ( t )( t − τ ) dτ (cid:21) + O( r ) (cid:27) , = 1 √ V (cid:26) η ( t )4 πr + (cid:16) m (cid:126) (cid:17) π / √ i lim (cid:15) → + (cid:20)(cid:90) t − (cid:15) η ( τ )( t − τ ) dτ − η ( t ) √ (cid:15) (cid:21) + O( r ) (cid:27) , = 1 √ V (cid:20) η ( t )4 πr + ˆ Lη ( t ) + O( r ) (cid:21) , (27)where ˆ L is defined in Eq. (3) in the main text. Then substituting Eq. (27) into the r.h.s. of Eq. (22), we immediatelyobtain (cid:20) ˆ L + 14 πa s ( t ) (cid:21) η ( t ) = − , (28)which is exactly the Eq. (2) in the main text. GENERALIZATION TO MANY-BODY WAVE FUNCTION
In this section, we generalize the two-body solution to many-body wave functions and establish Eq. (1) in themain text. To simplify notations, we set (cid:126) = m = 1 in this section. Since we are only interested in the short timeevolution of the short distance behavior of the many-body wave function, it is convenient to divide the entire 3 N dimensional space into two regions: D (cid:15) and I (cid:15) . D (cid:15) denotes the region in which the distance between any two bosonsis larger than (cid:15) , and I (cid:15) is the complementary space to D (cid:15) . In the short time limit t (cid:28) t n , it is possible to choose anintermediate length scale (cid:15) such that √ t (cid:28) (cid:15) (cid:28) /k n ( t n and k n was already defined in the main text). Such dividingof configuration space will be very useful in the following proof of this section.We consider the following many-body Schr¨oinger equation for N-interacting bosons with a time-dependent s-wavescattering length a s ( t ) i ∂∂t ψ ( r , r , · · · r N ; t ) = − N (cid:88) i =1 ∇ i ψ ( r , r , · · · r N ; t ) + 4 πa s ( t ) (cid:88) i 0, and we obtain the following asymptotic behavioras r ij ≡ | r ij | → ψ int ( r i , r j ; R ij ; t ) = Φ( R ij , R ij ; t )4 πr ij + i Z (cid:90) t dτ Φ t − τ (cid:16) R ij ; R ij ; τ (cid:17) [4 πi ( t − τ )] / + O ( r ij ) , (40)where we have definedZ (cid:90) t dτ Φ t − τ ( R ; Z ; τ )[4 πi ( t − τ )] / = 1(4 πi ) / lim (cid:15) → + (cid:20)(cid:90) t − (cid:15) dτ Φ t − τ ( R ; Z ; τ )( t − τ ) / − R ; Z ; t ) √ (cid:15) (cid:21) . (41)Finally, combining Eq. (31), (35) and (40) we obtain an closed integral equation for Φ( R ; Z ; t ) − Φ( R ; Z ; t )4 πa ( t ) = ψ ( R , R , Z ; t ) + iZ (cid:90) t dτ Φ t − τ ( R ; Z ; τ )[4 πi ( t − τ )] / + 2 N − (cid:88) i =1 ψ int ( z i , R ; ¯ Z i ; t ) + N − (cid:88) i