A direct derivation of the dark soliton excitation energy
aa r X i v : . [ n li n . PS ] D ec A direct derivation of the dark soliton excitation energy
Li-Chen Zhao,
1, 2
Yan-Hong Qin,
1, 2
Wenlong Wang, ∗ and Zhan-Ying Yang
1, 2 School of Physics, Northwest University, Xi’an 710127, China Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi’an 710127, China College of Physics, Sichuan University, Chengdu 610065, China (Dated: December 11, 2019)Dark solitons are common topological excitations in a wide array of nonlinear waves. The darksoliton excitation energy, crucial for exploring dark soliton dynamics, is necessarily calculated in arenormalized form due to its existence on a finite background. Despite its tremendous importanceand success, the renormalized energy form was firstly only suggested with no detailed derivation,and was then “derived” in the grand canonical ensemble. In this work, we revisit this fundamentalproblem and provide an alternative and intuitive derivation of the energy form from the fundamentalfield energy by utilizing a limiting procedure that conserves number of particles. Our derivationyields the same result, putting therefore the dark soliton energy form on a solid basis.
I. INTRODUCTION
Dark solitons are fundamental topological excitationsof numerous nonlinear waves [1], and have received muchattention in the past several decades. A dark soliton is aspatially localized density “dip” on top of a finite back-ground, accompanied with a phase step through the dip[2]; the phase step is π if the dark soliton is stationary.Dark solitons exist in numerous physical systems, suchas liquids [3], thin magnetic films [4], optical media [5–7],and Bose-Einstein condensates [8, 9] among others. Onetheme of research in this broad field is to study the effec-tive nonlinear dynamics of the dark solitons, highlightingthe particle aspect of these solitary waves. In order todiscuss the effective mass and related kinetic dynamicsof dark solitons [10–19], it is essential to characterize theexcitation energy of a dark soliton. Importantly, thisenergy also serves as a starting point for investigatingdynamics of dark soliton filaments and surfaces in higherdimensions [20, 21].It is not straightforward to extract the dark solitonexcitation energy as one needs to properly subtract thecontribution of the finite background. This is in contrastto a bright soliton, where there is a density “hump” on azero background. The renormalized dark soliton energywas firstly suggested in 1994 [10], and is proven to bevery effective, yielding excellent agreement between the-ory and experiment [22–24]. The dark soliton energy wasalso “derived” from the “grand canonical energy” or freeenergy [12, 14]. This formalism works indirectly with aconstant chemical potential rather than a constant num-ber of particles. However, these pioneering argumentsand calculations are not entirely rigorous or clear.The main purpose of this work is to provide an alter-native and importantly intuitive derivation of the darksoliton excitation energy with details. We focus on a fi-nite domain and calculate the excitation energy using the ∗ Electronic address: [email protected] fundamental field energy definition by keeping the num-ber of particles conserved. By taking the infinite limit ofthe domain, our result remarkably recovers to the well-known expression of the dark soliton excitation energy.Our derivation is therefore conceptually clear and natu-ral compared with the free energy calculation, since theHamiltonian of interest (see the next section) does not ac-tually admit particle number variation. Our derivation,due to the finite domain and the limiting procedure, istechnically more complicated but nevertheless straight-forward and readily tractable.This paper is organized as follows. In Sec. II, we intro-duce the model and the method. Next, we present ouranalytical results and discussions in Sec. III. Finally, ourconclusions and open problems for future considerationare given in Sec. IV.
II. MODEL AND METHOD
We now focus on a specific system of Bose-Einsteincondensate for ease of discussion, although our followingresults are also valid for many other systems describedby the same equation, particularly for optics. Here, wework in the quasi-1D regime, as we are interested in darksoliton states [8, 9]. In the framework of the mean-fieldtheory, the dynamics of a repulsive cigar-shaped conden-sate at sufficiently low temperatures can be described bythe following dimensionless Gross-Pitaevskii (GP) equa-tion: iψ t = − ψ xx + | ψ | ψ, (1)where ψ ( x, t ) is the macroscopic wavefunction. It is well-known that the system conserves among others the fieldenergy H = R ∞−∞ [ ψ ∗ ( − ∂ x ) ψ + | ψ | ] dx , and the num-ber of particles N = R ∞−∞ | ψ | dx . The two energy termsare the kinetic energy and the interaction energy, respec-tively.The integrable GP equation has the following travel-ling dark soliton solution: ψ d = { p µ − v tanh[ p µ − v ( x − vt )] + iv } e − iµt , (2)where v is the dark soliton velocity, and µ is the chem-ical potential. It is immediately clear that the field en-ergy H is infinite, and we need to properly subtract thebackground energy to extract the energy of the spatiallylocalized dark soliton. The correct renormalized energyform of the dark soliton was suggested firstly by Y.S.Kivshar et al. in 1994 [10] as: E s = Z ∞−∞ (cid:20) | ∂ x ψ d | + 12 ( | ψ d | − µ ) (cid:21) dx, (3)= 43 ( µ − v ) / . (4)This soliton energy form has been successfully appliedto numerous studies of dark soliton dynamics in exter-nal potentials, yielding good agreement with experiments[11–16, 22–24]; see also the pertinent two-component gen-eralization to the dark-bright soliton [25–30]. The energyis also a key element for more exotic topics such as thenegative mass of the dark soliton [17–19]. However, thiscrucial renormalized energy was suggested in a somewhatvague way, namely, it was stated that “the soliton partof the total Hamiltonian (the system energy) may be de-fined as Eq. (18)” [10].Before introducing our method, it is instructive to ex-amine a wrong but instructive derivation of the dark-soliton energy. At the same chemical potential µ , theground state is uniform ψ g = √ µ exp( − iµt ). Consider-ing simply the field energy difference, we find: E s (wrong) = H [ ψ d ] − H [ ψ g ] , (5)= Z ∞−∞ (cid:20) | ∂ x ψ d | + 12 | ψ d | (cid:21) dx − Z ∞−∞ (cid:20) | ∂ x ψ g | + 12 | ψ g | (cid:21) dx, = −
23 ( µ + 2 v ) p µ − v . (6)The result is strikingly wrong, i.e., the excitation energyis negative, but the dark soliton is an excited state im-printed on the ground state. It is not hard to see thatthis negative energy stems from the “lose of matter” atthe density dip. The gain in the kinetic energy is not suf-ficient to compensate the missing interaction energy. Ouridea is exactly to correct this particle number differenceas follows:1. For a chosen interval around the dark soliton(Eq. 2) center [ − L, L ], calculate the dark solitonfield energy H L [ ψ d ] and number of atoms N L [ ψ d ].Note that the integral is restricted within the in-terval, and we assume that L is much larger thanthe healing length ξ . 2. Calculate the ground state (under periodic bound-ary conditions) in the same interval such that ithas the same number of atoms as the dark soli-ton, i.e., N L = N L [ ψ d ] = N L [ ψ gL ]. The chemicalpotential and the field are denoted as µ g ( L ) and ψ gL = p µ g ( L ) e − iµ g ( L ) t , respectively.3. Evaluate the dark soliton energy, again for the finiteinterval, as E s ( L ) = H L [ ψ d ] − H L [ ψ gL ].4. The dark soliton energy is finally extracted as E s =lim L →∞ E s ( L ).In the next section, we present the detailed results. Wewill see that the final solution is remarkably identical tothat of Eq. 4. III. RESULT
First, we set the dark soliton center to 0 for our inte-grals without loss of generality. The number of atoms forthe dark soliton state is: N L [ ψ d ] = Z L − L | ψ d | dx, = 2 µL − p µ − v tanh[ p µ − v L ] . (7)The particle number for the ground state is N L [ ψ gL ] =2 Lµ g ( L ). Requiring the ground state and the dark soli-ton state have the same number of atoms, the chemicalpotential of the ground state is calculated as: µ g ( L ) = µ − p µ − v tanh[ p µ − v L ] L . (8)Note that lim L →∞ µ g ( L ) = µ as expected. But clearly wecannot simply set it to µ for the field directly, otherwisewe would get the wrong expression again. Physically, thesubtle difference means that the dark soliton excitationshould lead to a small increase of the background densityif the particle number is conserved. But the infinite sizeof the background hides this tiny variation.Having the two fields in the finite interval in place, weare now ready to evaluate the tedious but straightforwarddark soliton energy in the finite interval as: E s ( L ) = H L [ ψ d ] − H L [ ψ gL ] , (9)= Z L − L (cid:20) | ∂ x ψ d | + 12 | ψ d | (cid:21) dx − Z L − L (cid:20) | ∂ x ψ gL | + 12 | ψ gL | (cid:21) dx, = 16 ( µ − v ) / sech [ L ′ ](3 sinh[ L ′ ] + sinh[3 L ′ ])+ 13 ( µ − v ) / sech [ L ′ ] tanh[ L ′ ](2 + cosh[2 L ′ ]) − L ( µ − v ) sech [ L ′ ] tanh[ L ′ ] sinh[2 L ′ ] , (10)where L ′ = p µ − v L . Finally, taking the L → ∞ limit, E s = lim L →∞ E s ( L ) = 43 ( µ − v ) / , = Z ∞−∞ (cid:20) | ∂ x ψ d | + 12 ( | ψ d | − µ ) (cid:21) dx. (11)It is remarkable that these expressions are exactly whatY.S. Kivshar et al. insightfully suggested. Here, we showthat it is possible to derive these results analytically fromthe field energies by carefully keeping track of the densityvariation caused by the dark soliton excitation on top ofthe ground state.Finally, we summarize here for completeness the“derivation” of the soliton energy in the grand canoni-cal ensemble. In this setting, the dark soliton energy isdefined from the difference of the “grand canonical en-ergy” or free energy Ω = H − µN [12, 14]. Using thethermodynamic dark soliton state and the ground stateof the same chemical potential µ , the dark soliton energycan be rather straightforwardly calculated as: E s = Z ∞−∞ (cid:20) | ∂ x ψ d | + 12 | ψ d | − µ | ψ d | (cid:21) dx − Z ∞−∞ (cid:20) | ∂ x ψ g | + 12 | ψ g | − µ | ψ g | (cid:21) dx, (12)= 43 ( µ − v ) / . (13)While this approach is very mathematically efficient, theinterpretation of this free energy difference as the darksoliton energy is a bit confusing. The system is actuallynot a grand canonical system, and the number of atomsis strictly conserved. This is also true experimentally,e.g., during the phase imprinting process or other exci-tation processes [8, 9]. The best way to understand thisis perhaps to reasonably assume that these two ensem-bles here are equivalent. Our approach therefore in this sense is much more direct and intuitive. It is interesting,however, that all of these approaches yield the same darksoliton excitation energy E s = ( µ − v ) / . IV. CONCLUSION
In this work, we revisited the fundamental problem ofthe dark soliton excitation energy and provided an alter-native and intuitive approach to derive this energy. Weuse a limiting process to keep track of the tiny densityvariations of the background due to the dark soliton ex-citation, ensuring that the number of atoms is conserved.Our derivation uses only the elementary definition of thedark soliton excitation energy, field energy, and numberof atoms, putting the well-known dark soliton excitationenergy on a firm basis.One possible future direction is to use this approachto investigate more exotic states such the vortex [31–33].This is conceptually straightforward but appears to beanalytically very challenging. But some approximationsperhaps can be introduced. Research efforts along thisline is currently in progress and will be presented in fu-ture publications.
Acknowledgements
This work is supported by National Natural ScienceFoundation of China (Contact No. 11775176), Basic Re-search Program of Natural Science of Shaanxi Province(Grant No. 2018KJXX-094), The Key Innovative Re-search Team of Quantum Many-Body Theory and Quan-tum Control in Shaanxi Province (Grant No. 2017KCT-12), and the Major Basic Research Program of NaturalScience of Shaanxi Province (Grant No. 2017ZDJC-32).W.W. acknowledges support from the Fundamental Re-search Funds for the Central Universities, China. [1] V.E. Zakharov, A.B. Shabat, Interac-tion between solitons in a stable medium,Sov. Phys. JETP 37, 823 (1973).[2] Y.S. Kivshar, B. Luther-Davies, Dark optical solitons:physics and applications, Phys. Rep. 298, 81 (1998).[3] B. Denardo, S. Wright, W. Putterman, A. Larraza, Ob-servation of a kink soliton on the surface of a liquid,Phys. Rev. Lett. 64, 1518 (1990).[4] M. Chen, M.A. Tsankov, J.M. Nash, and C.E. Patton,Microwave magnetic-envelope dark solitons in yttriumiron garnet thin films, Phys. Rev. Lett. 70, 1707 (1993).[5] P. Emplit, J.P. Hamaide, F. Reynaud, C.Froehly, A. Barthelemy, Picosecond steps anddark pulses through nonlinear single mode fibers,Opt. Commun. 62, 374 (1987).[6] D. Kr¨okel, N.J. Halas, G. Giuliani, and D.Grischkowsky, Dark-pulse propagation in optical fibers, Phys. Rev. Lett. 60, 29 (1988).[7] A.M. Weiner, J.P. Heritage, R.J. Hawkins, R.N.Thurston, E.M. Kirschner, D.E. Leaird, andW.J. Tomlinson, Experimental observation ofthe fundamental dark soliton in optical fibers,Phys. Rev. Lett. 61, 2445 (1988).[8] S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sen-gstock, A. Sanpera, G.V. Shlyapnikov, and M. Lewen-stein, Dark solitons in Bose-Einstein condensates,Phys. Rev. Lett. 83, 5198 (1999).[9] S. Stellmer, C. Becker, P. Soltan-Panahi, E.M.Richter, S. Drscher, M. Baumert, J. Kronjger,K. Bongs, and K. Sengstock, Collisions of darksolitons in elongated Bose-Einstein condensates,Phys. Rev. Lett. 101, 120406 (2008).[10] Y.S. Kivshar, X. Yang, Perturbation-induced dynamicsof dark solitons, Phys. Rev. E 49, 1657 (1994). [11] Y.S. Kivshar, W. Kr´olikowski, Lagrangian approach fordark solitons, Opt. Commun. 114, 353-362 (1995).[12] T. Busch and J.R. Anglin, Motion of darksolitons in trapped Bose-Einstein condensates,Phys. Rev. Lett. 84, 2298 (2000).[13] A. Muryshev, G.V. Shlyapnikov, W. Ertmer, K.Sengstock, and M. Lewenstein, Dynamics ofdark solitons in elongated Bose-Einstein condensates,Phys. Rev. Lett. 89, 110401 (2002).[14] V.V. Konotop and L.P. Pitaevskii, Landau dy-namics of a grey soliton in a trapped condensate,Phys. Rev. Lett. 93, 240403 (2004).[15] V.A. Brazhnyi, V.V. Konotop, and L.P. Pitaevskii,Dark solitons as quasiparticles in trapped condensates,Phys. Rev. A 73, 053601 (2006).[16] D.E. Pelinovsky and P.G. Kevrekidis,Dark solitons in external potentials,Z. angew. Math. Phys. 59, 559599 (2008).[17] M.L. Aycocka, H.M. Hurst, D.K. Emkin, D. Genk-ina, H. Lu, V.M. Galitski, and I.B. Spielman, Brow-nian motion of solitons in a Bose-Einstein condensate,PNAS , 2503-2508 (2017).[18] H.M. Hurst, D.K. Efimkin, I.B. Spielman, and V. Galit-ski, Kinetic theory of dark solitons with tunable friction,Phys. Rev. A ,053604(2017).[19] V.N. Serkin, Busch-Anglin effect for matter-waveand optical dark solitons in external potentials,Optik173