Time-dependent method for many-body problems and its application to nuclear resonant systems
TTime-dependent method for many-body problems and itsapplication to nuclear resonant systems ∗ Tomohiro Oishi , † and Lorenzo Fortunato Department of Physics and Astronomy “Galileo Galilei”, University of Padova,and I.N.F.N. Sezione di Padova, Italy † E-mail: [email protected] decay process of the schematic one-dimensional three-body systemis considered. A time-dependent approach is used in combination with aone-dimensional three-body model, which is composed of a heavier corenucleus and two nucleons, with the aim of describing its evolution in two-nucleon emission. The process is calculated from the initial state, in whichthe three ingredient particles are confined. In this process, two differenttypes of emission can be found: the earlier process includes the emissionof spatially correlated two-nucleon pair, like a dinucleon, whereas, at asubsequent time, all the particles are separated from each other. The time-dependent method can be a suitable option to investigate the meta-stableand/or open-quantum systems, where the complicated many-body dynam-ics should necessarily be taken into account.PACS numbers: 03.65.Xp, 21.10.Tg, 23.50.+z, 24.10.Cn
1. Introduction
Quantum resonance or meta-stability is a basic concept to understandseveral dynamical processes in atomic nuclei. Those include, e.g. two-protonor two-neutron emission [1, 2, 3], tetra neutron [4, 5], and alpha-clusteringresonant states ( c.f.
Hoyle state of C) [6, 7, 8, 9]. By investigating theseprocesses, we expect to obtain fundamental information on nuclear interac-tion, multi-spin dynamics, and/or quantum tunneling effect in systems withmany degrees of freedom.On the theoretical side, however, the description of these meta-stablesystems has been a long-standing problem. The usual quantum mechanics ∗ Presented at the XXXV Mazurian Lakes Conference on Physics, Piaski, Poland,September 3-9, 2017 (1) a r X i v : . [ nu c l - t h ] A p r Oishi printed on November 8, 2018 x -X x -x m m m Fig. 1. Three-body system in one dimension. X = ( x + x ) / for bound states should be extended to deal with the meta-stability and themulti-particle degrees of freedom on equal footing [1, 8, 10]. For this pur-pose, we have developed a time-dependent three-body model for theoreticaland computational approach [11, 12, 13, 14]. This method can provide anintuitive way to discuss even the broad-resonance system, whose lifetime isconsiderably short, and thus the multi-particle dynamics should be takeninto account.In this work, we perform a toy-model calculation to investigate thebroad-resonance state. We utilize the time-dependent method to describethe scattering emission from the three-body localized state. In contrast tothe radioactive processes, it is not guaranteed that this emission process canbe attributed to a single quasi-stationary state, but we have to take intoaccount the contribution from all the possible components.In the next section, we employ one-dimensional three-body model as ourtesting field for time-dependent calculation. Section 3 is devoted to presentour results and discussions. Finally, we summarize this article in section 4.
2. Model and Formalism
In this work, we give an example of the time-dependent (TD) calculation,implemented into a three-body system in one dimension (1D) [15]. The totalHamiltonian is H tot = (cid:88) i =1 p i m i + V ( | x − x | ) + V ( | x − x | ) + V ( | x − x | ) . (1)We employ the masses of particles defined as m = m = 939 MeV /c and m = 16 ·
939 MeV /c . Namely, we assume a heavy core nucleus and twonucleons moving on the one-dimensional x -axis (see Fig.1), mimicking the O nucleus but without pretending a realistic description. For the nucleon-nucleon subsystem, we employ a square-well attractive potential. That is, V ( x ) = (cid:26) − .
84 MeV ( | x | ≤ . , | x | > . . (2) ishi printed on November 8, 2018 -10-5 0 5 -15 -10 -5 0 5 10 15 20 25 P o t e n t i a l s ( M e V ) x ij (fm)V =V (t>0)V =V (t=0)V (t>0) = V (t=0) Fig. 2. Two-body potentials as functions of the relative distances, x ij ≡ ( x i − x j ). For the core-nucleon channel, on the other hand, V ( x ) = V ( x ) = V r exp (cid:32) − x d r (cid:33) + V a exp (cid:32) − x d a (cid:33) , (3)where d r = 5 .
04 fm, d a = 3 .
15 fm, V r = 24 MeV and V a = −
32 MeV. Thesepotentials are shown in Fig.2. The bump in the core-nucleon potential canbe associated with the centrifugal barrier in realistic nuclei. Note that,in this work, we focus on the broad-resonance state. For this purpose,the two-body potentials are fixed shallower than the usual potentials inthe three-dimensional calculations. Also, instead of the Woods-Saxon type,we employ the Gaussian potential, which enables us to utilize the analyticformula to obtain the matrix elements with the harmonic oscillator (HO)basis employed in the next subsection.
In order to solve the eigen-states of H b , first we employ the mass-scaled Jacobi coordinates (MSJC) [16, 17]. Using the common-relative mass, µ ≡ (cid:113)(cid:81) i =1 m i / (cid:80) i =1 m i , those are defined as, ξ = (cid:114) µ µ ( x − x ) , ξ = (cid:114) µ µ (cid:18) x − x + x (cid:19) , (4)and ξ ≡ ( m x + m x + m x ) / (cid:80) i =1 m i , which is the center-of-mass coor-dinate. Partial relative masses are defined as µ k ≡ m k +1 (cid:80) ki =1 m i / (cid:80) k +1 j =1 m j , Oishi printed on November 8, 2018 for k = 1 and 2. With MSJC, the total Hamiltonian reads H tot = T CM + π µ + π µ + V + V + V ,T CM = π m + m + m ) , (5)where { π i } are the conjugate momenta to { ξ i } . In the following, we neglectthe center-of-mass motion, T CM . We diagonalize the remaining Hamilto-nian, H b = H tot − T CM , by calculating its matrix elements, (cid:104) Ψ cd | H b | Ψ ab (cid:105) ,within the harmonic oscillator (HO) basis:Ψ ab ( ξ , ξ ) = ψ a ( ξ ) ψ b ( ξ ) , (6)where a and b are non-negative integers. Notice that ψ n is the HO wavefunction corresponding either to the relative motion of particles 1 and 2,or the motion of particle 3 with respect to the center-of-mass between 1and 2, with HO-energy, ( n + 1 / hω . Our model space is truncated as a, b ≤
15 with ¯ hω = 0 . | S = 0 (cid:105) = ( |↑↓(cid:105) − |↓↑(cid:105) ) / √
2. Thus, the spatial part should besymmetric against the exchange between particles 1 and 2. It means thatonly { ψ a ( ξ ) } with even a can be included in our basis.Within the chosen MSJC scheme, the matrix elements of V are diag-onal, whereas V and V yield non-diagonal components. For computa-tion of these non-diagonal elements, we utilized a kinetic rotation tech-nique, whose details can be found in Ref. [15]. Then, all the eigen-states, H b | Φ M (cid:105) = E M | Φ M (cid:105) , can be solved by diagonalization: | Φ M (cid:105) = (cid:80) ab c M,ab | Ψ ab (cid:105) . We employ the confining potential method for time-evolution. Thismethod has provided a good approximation for quantum meta-stable phe-nomena especially in nuclear physics [11, 12, 13]. For the confining potential, V ( c )13 = V ( c )23 at t = 0 fm /c , we fix the wall potential from | x i − x j | ≥ . V between the light two particles is unchanged.See Fig.2 for visual plots of these potentials. Our initial state, | Υ( t = 0) (cid:105) , issolved by diagonalizing the confining Hamiltonian including V ( c )13 and V ( c )23 .It is also worthwhile to note that the initial state can be expanded on theeigen-states of the true Hamiltonian: | Υ( t = 0) (cid:105) = (cid:88) M d M (0) | Φ M (cid:105) . (7) ishi printed on November 8, 2018 ct = 0 (fm)-20 -10 0 10 20x - x (fm)-20-10 0 10 20 x - X ( f m ) ct = 200 (fm)-20 -10 0 10 20x - x (fm)-20-10 0 10 20 ct = 400 (fm)-20 -10 0 10 20x - x (fm)-20-10 0 10 20 x - X ( f m ) ct = 600 (fm)-20 -10 0 10 20x - x (fm)-20-10 0 10 20 Fig. 3. Density distribution, ρ ( t ) = | Υ( t ) | , for ct = 0 , ,
400 and 600 fm. Theseare plotted as functions of x − x and x − X , where X is the center-of-massbetween the 1st and 2nd particles. For this initial state, after the subtraction of the center-of-mass motion, theexpectation value of the relative Hamiltonian is given as (cid:104)
Υ(0) | H b | Υ(0) (cid:105) =0 .
91 MeV. This is equivalent to the energy release (Q-value) carried out bythe emitted particles.
3. Result and Discussion
In the first panel of Fig.3, we plot the density distribution of the initialstate: ρ ( t = 0) = | Υ( t = 0; ξ , ξ ) | . As expected, the three ingredientparticles are spatially localized at t = 0. Oishi printed on November 8, 2018 s ( E ) E (MeV) P s u r v ( t ) ct (fm)TotalE < 1.8 MeV Fig. 4. (Left panel) Energy spectrum of the emission state, | Υ( t ) (cid:105) . E T = 1 . From Eq.(7), time-evolution via H b can be calculated as | Υ( t ) (cid:105) ≡ exp (cid:20) − it H b ¯ h (cid:21) | Υ( t = 0) (cid:105) = (cid:88) M d M ( t ) | Φ M (cid:105) , where d M ( t ) = e − itE M / ¯ h d M (0) . (8)The time-evolution of the density distribution is shown in Fig. 3. That is, ρ ( t ; ξ , ξ ) = | Υ( t ; ξ , ξ ) | . (9)Notice that the energy distribution is invariant during the time-evolution: s ( E M ) ≡ | d M (0) | = | d M ( t ) | . In Fig.4, we plot the energy distribution.From this result, we can find that the state of interest, | Υ( t ) (cid:105) , can be mostlyattributed to the low-lying components with continuum energies up to E ≤ . ct = 200 fm, the emission process proceeds mainly with x = x and x − X = ±
10 fm, where X indicates ( x + x ) /
2. Thisearlier process means that the two light particles, m and m , are spatiallycorrelated and emitted as a pair from the core. Namely, we observe adinucleon emission in 1D space [13].After ct ≥
400 fm as shown in Fig.3, on the other hand, the processshows a different pattern with | x − x | (cid:39)
15 fm and | x − X | (cid:39)
10 fm.In this process, the two light particles are not localized anymore, and threeparticles move away from each other. Thus, the total emission should bea superposition of the primary dinucleon emission and the secondary sep-arated emission. This superposition is quite in contrast to Ref.[13], whereonly the dinucleon emission is dominant with the pairing force. In such a ishi printed on November 8, 2018 way, our time-dependent method can provide a direct and intuitive solutionto describe this complex quantum dynamics. First we define the decay state, | Υ d ( t ) (cid:105) , such as | Υ d ( t ) (cid:105) ≡ | Υ( t ) (cid:105) − β ( t ) | Υ(0) (cid:105) = (cid:88) M y M ( t ) | Φ M (cid:105) , (10)where β ( t ) ≡ (cid:104) Υ(0) | Υ( t ) (cid:105) and y M ( t ) = d M ( t ) − β ( t ) d M (0). Notice that (cid:104) Υ(0) | Υ d ( t ) (cid:105) = 0. Also, the decay probability can be formulated as P decay ( t ) ≡(cid:104) Υ d ( t ) | Υ d ( t ) (cid:105) = 1 − P surv ( t ), where P surv ( t ) is the so-called survival proba-bility. That is, P surv ( t ) = | β ( t ) | = |(cid:104) Υ(0) | Υ( t ) (cid:105)| . (11)In the second panel of Fig.4, the survival probability is plotted in logarithmicscale: there is an oscillatory decay along time-evolution. Thus, this processis not alike the radioactive emission, since the exponential decay-rule ishardly observed.Indeed, the process can be interpreted as a superposition of the well-converged exponential decay and the fluctuation due to high-energy com-ponents. To confirm this, remembering that P surv ( t ) = 1 − P decay ( t ), wedecompose the decay probability into the low- and high-energy componentsby fixing the border of E T = 1 . P decay ( t ) = (cid:88) E M 500 fm. Inthis exponential decay, P surv ( ct ≥ 500 fm; E < E T ) (cid:39) exp( − t Γ / ¯ h ), wherethe decay-width is approximated as Γ (cid:39) . 41 MeV in our calculation. Noticethat this decay-width value is similar to the empirical values observed inseveral light one- and two-proton emitters [1, 2]. 4. Summary We have performed the time-dependent analysis of the emission pro-cess in the 1D three-body system. By monitoring the time-evolution fromthe initially confined state, we confirmed that two different types of emis-sion are taking place: the earlier dinucleon emission, and the secondary Oishi printed on November 8, 2018 separated emission. It is shown that, even for such a superposition of dif-ferent processes, our time-dependent calculation can be a suitable tool tounderstand its multi-particle dynamics with an intuitive procedure. By an-alyzing the survival probability, we have also found that this process canbe interpreted mainly as the exponential decay with E < . e.g. tetra neutron, whosemeasured decay-width is considerably wide and hardly allows us to inferan exponential-decay behavior [4]. Our extension of the time-dependentmethod applied to these realistic 3D nuclear systems is in progress now.This work is financially supported by the P.R.A.T. 2015 project IN:Theory in the University of Padova (Project Code: CPDA154713).REFERENCES [1] L. V. Grigorenko et al. , Phys. Lett. B 677, 30 (2009).[2] M. Pf¨utzner, M. Karny, L. V. Grigorenko, and K. Riisager, Rev. Mod. Phys.84, 567 (2012).[3] Y. Kondo et al. , Phys. Rev. Lett. 116, 102503 (2016).[4] K. Kisamori et al. , Phys. Rev. Lett. 116, 052501 (2016).[5] K. Fossez, J. Rotureau, N. Michel and P`loszajczak, Phys. Rev. Lett. 119,032501 (2017).[6] E. E. Salpeter, Astrophys. Journal Vol. 115, 326 (1952).[7] F. Hoyle, Astrophys. Journal, Suppl. Ser. Vol. 1, 121 (1954).[8] H. Suno, Y. Suzuki and P. Descouvemont, Phys. Rev. C 94, 054607 (2016).[9] R. Smith et al. , Phys. Rev. Lett. 119, 132502 (2017).[10] T. Myo et al.et al.