We investigate the thermodynamic properties of a Bose-Einstein condensate
with negative scattering length confined in a toroidal trapping potential. By
numerically solving the coupled Gross-Pitaevskii and Bogoliubov-de Gennes
equations, we study the phase transition from the uniform state to the
symmetry-breaking state characterized by a bright-soliton condensate and a
localized thermal cloud. In the localized regime three states with a finite
condensate fraction are present: the thermodynamically stable localized state,
a metastable localized state and also a metastable uniform state. Remarkably,
the presence of the stable localized state strongly increases the critical
temperature of Bose-Einstein condensation.Comment: 4 pages, 4 figures, to be published in Physical Review A as a Rapid
  Communication. Related papers can be found at
  http://www.padova.infm.it/salasnich/tdqg.htm

A. L. Fetter

A. Parola

L. Reatto

L. Salasnich

M. Mitchell

English

arXiv

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Thermodynamics of solitonic matter waves in a toroidal trap

We investigate the thermodynamic properties of a Bose-Einstein condensate with negative scattering length confined in a toroidal trapping potential. By numerically solving the coupled Gross-Pitaevskii and Bogoliubov–de Gennes equations, we study the phase transition from the uniform state to the symmetry-breaking state characterized by a bright-soliton condensate and a localized thermal cloud. In the localized regime, three states with a finite condensate fraction are present: the thermodynamically stable localized state, a metastable localized state, and also a metastable uniform state. Remarkably, the presence of the stable localized state strongly increases the critical temperature of Bose-Einstein condensation

SALASNICH, LUCA

PAROLA A

REATTO L.

Archivio istituzionale della ricerca - Università di Padova

We investigate the thermodn. properties of a Bose-Einstein condensate with neg. scattering length confined in a toroidal trapping potential.  By numerically solving the coupled Gross-Pitaevskii and Bogoliubov-de Gennes equations, we study the phase transition from the uniform state to the symmetry-breaking state characterized by a bright-soliton condensate and a localized thermal cloud.  In the localized regime, three states with a finite condensate fraction are present: the thermodynamically stable localized state, a metastable localized state, and also a metastable uniform state.  Remarkably, the presence of the stable localized state strongly increases the crit. temp. of Bose-Einstein condensation. [on SciFinder (R)

AIR Universita degli studi di Milano

SALASNICH L

PAROLA, ALBERTO

Archivio istituzionale della ricerca - Università dell'Insubria

Thermodynamics of solitonic matter waves in a toroidal trapL. Salasnich,1 A. Parola,2 and L. Reatto31CNR-INFM and CNISM, Unità di Milano Università, Via Celoria 16, 20133 Milano, Italy2Dipartimento di Fisica e Matematica, Università dell’Insubria, Via Valleggio 11, 22100 Como, Italy3Dipartimento di Fisica and CNISM, Università di Milano, Via Celoria 16, 20133 Milano, ItalysReceived 13 April 2006; published 19 September 2006dWe investigate the thermodynamic properties of a Bose-Einstein condensate with negative scattering lengthconfined in a toroidal trapping potential. By numerically solving the coupled Gross-Pitaevskii andBogoliubov–de Gennes equations, we study the phase transition from the uniform state to the symmetry-breaking state characterized by a bright-soliton condensate and a localized thermal cloud. In the localizedregime, three states with a finite condensate fraction are present: the thermodynamically stable localized state,a metastable localized state, and also a metastable uniform state. Remarkably, the presence of the stablelocalized state strongly increases the critical temperature of Bose-Einstein condensation.DOI: 10.1103/PhysRevA.74.031603 PACS numberssd: 03.75.KkIn recent experiments, a repulsive Bose-Einstein conden-sate sBECd has been produced and studied in a quasi-one-dimensional s1Dd ring f1,2g. These experiments and alsoprevious experimental investigations with cold atoms in tor-odial traps f3g are important steps to achieve ultrahigh-precision sensors with atom interferometry ssee, for instance,Ref. f4gd. The case of an attractive BEC in a ring has not yetbeen experimentally investigated but appears very interest-ing: a quantum phase transition from an azimuthally uniformcondensate to a bright soliton has been predicted f5,6g.Dynamically stable multipeak bright solitons may alsoappear f7g.In this paper, we investigate the effect of temperature onthe uniform-to-localized phase transition by solving theGross-Pitaevskii equation sGPEd of the macroscopic wavefunction of the Bose condensate and the Bogoliubov–deGennes equations for the quantum depletion and the thermalcloud. We show that, for a fixed number N of atoms, theuniform solution always exists but it is thermodynamicallystable only up to a critical interaction strength that dependson the temperature. Above this critical interaction strength,both the condensate and the thermal cloud become localized.In addition, we determine the finite-temperature phase dia-gram of the system for a sample of alkali-metal atoms in aring. Our results are relevant not only for the physics of coldatoms but also for the nonlinear science of solitons. In fact, amixture of condensed and noncondensed attractive particlesf8g correspond to incoherent solitons of light in Kerr mediaf9g.We consider a Bose gas with negative scattering lengthsas,0d confined in a toroidal potential and model the trans-verse confinement with a harmonic potential of frequencyv'. The two characteristic lengths of the toroidal trap are theazimuthal radius R and the transverse harmonic lengtha'= f" / smv'dg1/2. To avoid the confinement-induced reso-nance at uas u .a' f10g, we impose that uas u≪a' f11g. Atlow temperature T and with a finite number N of atoms, inthe Bose gas there is the thermal cloud but also the BEC.Under the condition R≫a', the azimuthal wave functionfszd of the BEC in the ring satisfies the 1D nonpolynomialSchrödinger equation sNPSEd f7,12g,F− "22m]z2 + m„n0szd…Gfszd = m¯fszd , s1dwhere z=Ru is the azimuthal coordinate with u the azimuthalangle. This equation has been deduced from the 3D GPE byusing a Gaussian transverse wave function with a density-dependent width and neglecting curvature effects f7,12g. Thenonpolynomial term m(n0szd) is a function of the BEC den-sity n0szd=N0 ufszdu2 and it is given by m=]E /]n0, whereE= "v'n0s1+2asn0d1/2 and as,0 is the 3D s-wave scatter-ing length f7g. The wave function fszd is normalized to 1and satisfies the condition of periodicity fsz+Ld=fszd,where L=2pR. The chemical potential m¯ is fixed by thenormalization condition eufszdu2dz=1. Quantum and thermaldepletions of the BEC are obtained by solving theBogoliubov–de Gennes sBdGd equations f13g for thequasiparticle amplitudes uaszd and vaszd, given byLSuaszdvaszdD = eaSuaszdvaszdD , s2dwhere ea are the energies of quasiparticle excitations and theoperator L isL =1−"22m]z2− m¯ +]sn0md]n0N0f2]m]n0− N0sf*d2]m]n0"22m]z2 + m¯ −]sn0md]n02 .s3dThe local density of the quantum depletion readsnoutszd=oa uvaszdu2, while the thermal local density is nTszd=oafuuaszdu2+ uvaszdu2gn¯a, where n¯a= fexpsea /kBTd−1g−1 arethe Bose numbers of occupation for the noninteracting qua-siparticles, kB is the Boltzmann constant, and T is theabsolute temperature of the thermal cloud. The number N˜ ofnoncondensed atoms is thus given by N˜ =Nout+NT, whereNout=enoutszddz and NT=enTszddz. In the weak-couplinglimit, where asn0≪1, the energy density E becomesE= "v'n0+ "v'asn02, the NPSE s1d reduces to the familiarPHYSICAL REVIEW A 74, 031603sRd s2006dRAPID COMMUNICATIONS1050-2947/2006/74s3d/031603s4d ©2006 The American Physical Society031603-11D GPE f7g, and the BdG equations s2d and s3d reduce to theBdG equations of the strictly 1D problem. Note that Eqs.s1d–s3d could be improved by taking into account Popov andBeliaev corrections, which are usually not negligible onlywhen the condensed fraction is very small f14g.In our approach, by fixing the total number N=N0+N˜ ofatoms and the temperature T, the equilibrium configurationsare found from the Helmholtz free energy f13g,F = E0 + E˜ + m¯sN − N0d − TS , s4dwhereE0 = N0E F− "22mf*szd]z2fszd + E„n0szd…Gdz s5dis the BEC energy. The out-of-condensate energy E˜ and en-tropy S are instead given byE˜ = oaeaFn¯a −E uvaszdu2dzG , s6dS = − kBoafn¯alogsn¯ad − s1 + n¯adlogs1 + n¯adg . s7dThe solution of the 1D GPE with periodic boundaryconditions is known analytically f5g: it is uniform for0,gN0,p2a' /L, with g=2 uas u /a' the interatomicstrength, and becomes localized for gN0.p2a' /L. By usingthe NPSE, we find that the transition strength approaches the1D GPE one for large L f7g. The main difference between 1DGPE and NPSE is that the NPSE correctly gives the collapseof the localized solution for gN0.4/3, while the 1D GPEdoes not predict collapse f7g. It is important to observe thatthe uniform-to-localized transition depends on N0 and onlyimplicitly on N=N0+Nout+NT. We shall show that, for afixed number N of atoms, the uniform solution exists for anyvalue of the interatomic strength g but it is thermodynami-cally stable only below a critical strength gc. Above gc, thestable state is a localized solution that minimizes the freeenergy.The BdG equations s2d and s3d are easily solved if fszd isuniform: fszd=1/ÎL. In this case, the quasiparticle ampli-tudes ukszd= u¯keikz and vkszd=v¯ke−ikz are plane waves and thewave vector k is quantized: k= s2p /Ldj, with j an integernumber. When fszd is not uniform, one must numericallysolve Eqs. s1d–s3d. We use a finite-difference space discreti-zation and diagonalize the matrix associated to the operatorL of Eq. s3d. In our calculations, we use various matrix di-mensions up to 400034000. The numerical procedure is dis-cussed in Ref. f15g. In our approach, the transverse excita-tions do not contribute to the thermodynamics: thisassumption is reasonable only if the system is quasi-1D.In Fig. 1, we plot the results of our numerical calculationsof the free energy F as a function of the scaled interatomicstrength g=2 uas u /a' for increasing values of the tempera-ture T. We choose N=104 atoms and a toroidal geometrywith L /a'=100. While the uniform solution sdashed linedexists for any strength g, the localized solution ssolid linedexists only above a critical strength ge. As expected, atT.0 we find gc.p2a' /L=0.0987. For a given g sg.ged,there are two localized states: one is thermodynamicallystable slower solid lined and the other is metastable suppersolid lined. The metastability is guaranteed by the Bogoliu-bov eigenvalues being real. At very low temperature, seepanel sad of Fig. 1, for g.ge the solid line of the metastablelocalized solution is very close to the dashed line of themetastable uniform solution. At higher temperature, see pan-els sbd and scd of Fig. 1, the free energy of the metastablelocalized solution, but also the free energy of the stable oneat small g.ge, is higher than the free energy of the uniformsolution. The strength gc at which the dashed line crosses thelower solid line determines the point where the first-ordertransition takes place. At T.0, we find that gc.ge. Thestrength gc grows with the temperature T, but in general gcdoes not coincide with the strength ge at which the localizedsolutions appear. Thus, at finite temperature, for 0,g,gethere is only the stable uniform state; for ge,g,gc, thereare two metastable localized states and the stable uniformstate; for g.gc, the stable state is one of the two localizedstates while the other localized state and the uniform state aremetastable.A coherent bright soliton and a thermal cloud coexist inthe stable state, as in the metastable localized state. This isclearly illustrated in Fig. 2, where we plot the condensatefraction N0 /N as a function of g for different temperatures.At low temperature, see panel sad of Fig. 2, the uniform statesdashed lined is a quasipure BEC for 0,g,ge. For g.ge,the condensate fraction of the uniform state quickly de-creases. At low temperature, the uniform-to-localized phasetransition does not produce a relevant quantum depletionNout /N at the transition point. This result is obtained with a0 0.1 0.2 0.3 0.4 0.5 0.6 0.7N γ  −0.09−0.08−0.070 0.1 0.2 0.3 0.4 0.5 0.6 0.7−0.04−0.03−0.02F/N  0 0.1 0.2 0.3 0.4 0.5 0.6 0.7−0.025−0.015−0.005(c)(b)(a)kBT=2kBT=4kBT=1FIG. 1. Free energy F as a function of the scaled interatomicstrength g=2 uas u /a' for an attractive Bose gas of N=104 atoms ina toroidal trap with L /a'=100. Dashed line: uniform solution.Solid line: localized solutions. The energies F and kBT are in unitsof "v'.SALASNICH, PAROLA, AND REATTO PHYSICAL REVIEW A 74, 031603sRd s2006dRAPID COMMUNICATIONS031603-2fixed number of atoms sN=104d. We have verified that byfixing N0 instead of N sas done in Ref. f5gd, the condensatefraction N0 /N goes to zero for gN0.p2a' /L.At higher temperature, see panels sbd and scd of Fig. 2,BEC and noncondensate components coexist also in the uni-form state. Figure 2 shows that the condensate fraction of thelocalized states is higher than the condensate fraction of theuniform state. Remarkably, at the transition strength gc thereis a sizeable jump in the condensate fraction. Moreover, forthe stable localized state, the ratio N0 /N initially grows byincreasing the interatomic strength. As discussed previously,the BEC soliton of the localized state will collapse atgN0.4/3, but the true collapse is avoided by populating theout-of-condensate cloud with N˜ =N−N0 atoms for Ng.4/3.We have verified that in the stable localized state, the numberN0 of condensed atoms is always very close to the collapsevalue 4/ s3gd and, for a fixed g, the number N0 has a verysmall temperature dependence. It follows that one can esti-mate the condensed fraction of the stable localized state asN0 /N.4/ s3Ngd. This estimation is not so rough: choosing,for instance, Ng=2, one finds N0 /N.2/3, in good agree-ment with the result shown in the upper solid line of Fig. 2ssee where N0 /N=0.66d.In the previous calculations, transverse modes have beenneglected and the thermally excited quasiparticles are al-lowed to populate only the lowest transverse state. This as-sumption can be justified by a simple argument: In theweakly localized regime slow Ngd, the spectrum of elemen-tary excitations is well approximated by the free Bose gas,where the longitudinal modes have energies Os"v'a'2 /L2dmuch lower than the Os"v'd transverse modes. In this limit,we have explicitly verified that, with the chosen parameterssL /a'=100d, for kBT&5"v' the number of quasiparticlesin the excited transverse modes is indeed negligible. In thestrongly localized region, both longitudinal and transverselow-energy excitations are known in the large-L limit f6,17g,where a finite Os"v'd gap separates the two branches.Moreover, our calculations show that for Ng,4/3, thermaldepletion does not affect deeply the thermodynamics of themodel up to the transition temperature.For completeness, we have repeated the calculations byusing the 1D GPE. Also in this case we have found twolocalized solutions, but the stable one always maintains acondensate fraction close to 1, due to the fact that the 1DGPE does not predict the collapse. Thus the transverse struc-ture of the soliton, taken into account by the NPSE, stronglymodifies the thermodynamics of the attractive Bose gas.In Fig. 3, we plot the density profile n0szd of the conden-sate cloud ssolid lined and of the density profile nTszdof the thermal cloud sdot-dashed lined for the stablelocalized state and two values of g. We have verifiedthat the density profile noutszd of the quantum depletionreduces by increasing the temperature T, and thatthe quantum depletion Nout /N becomes relevant onlywhen kBT is of the order of the lowest Bogoliubovenergy level. This energy can be easily estimated askBT / s"v'd= f"2 / s2mdgs2p /Ld2 / s"v'd=2p2a'2 /L2.0.002.In Fig. 4, we plot the phase diagram of the atomic cloudof bosons in the plane sNg ,Td for L /a'=100 and N=104. Inthe figure, we insert a dashed line at kBT / s"v'd=6.01,which is the transition temperature TBEC of the Bose-Einsteincondensation above which the condensate fraction of the uni-form state is zero. The transition temperature TBEC reducesby increasing L /a'. In fact, it is easy to show f16g thatkBTBEC / s"v'd scales as Na'2 /L2. Figure 4 displays as a solidline the curve of the uniform-to-localized transition. Remark-0 0.4 0.8 1.2 1.6 2 2.4N γ 00.250.50.7510 0.4 0.8 1.2 1.6 2 2.400.250.50.751N0/N0 0.4 0.8 1.2 1.6 2 2.400.250.50.751(c)(b)(a)k BT=1k BT=2k BT=4FIG. 2. Condensate fraction N0 /N as a function of thescaled interatomic strength g=2 uas u /a' for N=104 atoms withL /a'=100. Dashed line: uniform solution. Solid line: localized so-lutions. The vertical dot-dashed line indicates the strength Ngc atwhich the uniform-to-localized phase transition takes place.−4 −3 −2 −1 0z01000200030004000n0(z), nT(z) thermalcondensate−4 −3 −2 −1010002000300040005000n0(z), nT(z) thermalcondensateNγ=2Nγ=343210 4321FIG. 3. Azimuthal density profiles of the thermodynamicallystable state for N=104 atoms with L /a'=100 and kBT / s"v'd=2.Length z is in units of a'= f" / smv'dg1/2.THERMODYNAMICS OF SOLITONIC MATTER WAVES IN¼ PHYSICAL REVIEW A 74, 031603sRd s2006dRAPID COMMUNICATIONS031603-3ably, the BEC transition temperature of the localized solutionis much larger than the uniform one. Figure 4 showsthat with Ng.0.68, there is a first-order transition fromthe localized BEC phase to the uniform thermal phasewithout BEC. Instead, for Ng,0.68 the no-BEC phase isobtained starting from the uniform phase by increasing thetemperature.In conclusion, we have shown that an attractive Bose con-densate in a quasi-one-dimensional toroidal trap displays atfinite temperature new and interesting features, such asthe coexistence of a coherent self-bound bright soliton with athermal cloud, the avoidance of true collapse via populationof the out-of-condensate sthermald component, and theenhancement of the Bose-Einstein transition temperaturefor the localized solution. We stress that our predictions canbe verified by using a magnetic ring with geometric param-eters not too far from those used in Ref. f1g for trapping arepulsive condensate of 87Rb atoms. For instance, a sampleof 7Li atoms, which have a negative scattering lengthsas=−1.4 nmd, can be trapped in a ring with transverse widtha'.6 mm, which gives "v' /kB="2 / s2a'2 d.2 nK. Inthis way, having N=104 atoms, the interaction strength isNg=N2 uas u /a'.5. Then, by taking the azimuthal radiusR=L / s2pd.100 mm, one can test the predictions of Fig. 4varying the temperature or tuning the scattering length asaround a Feshbach resonance.f1g S. Gupta, K. W. Murch, K. L. Moore, T. P. Purdy, and D. M.Stamper-Kurn, Phys. Rev. Lett. 95, 143201 s2005d.f2g A. S. Arnold, C. S. Garvie, and E. Riis, e-print cond-mat/0506142.f3g J. A. Sauer, M. D. Barrett, and M. S. Chapman, Phys. Rev.Lett. 87, 270401 s2001d; A. Kasper et al., J. Opt. B: QuantumSemiclassical Opt. 5, S143 s2003d; A. Hopkins, B. Lev, andH. Mabuchi, Phys. Rev. A 70, 053616 s2004d.f4g M. A. Kasevich, Science 298, 1363 s2002d.f5g G. M. Kavoulakis, Phys. Rev. A 67, 011601sRd s2003d; R.Kanamoto, H. Saito, and M. Ueda, ibid. 67, 013608 s2003d.f6g R. Kanamoto, H. Saito, and M. Ueda, Phys. Rev. A 68,043619 s2003d; G. M. Kavoulakis, ibid. 69, 023613 s2004d.f7g A. Parola, L. Salasnich, R. Rota, and L. Reatto, Phys. Rev. A72, 063612 s2005d.f8g H. Buljan, M. Segev, and A. Vardi, Phys. Rev. Lett. 95,180401 s2005d; B. Pozzi, L. Salasnich, A. Parola, and L.Reatto, J. Low Temp. Phys. 119, 57 s2000d.f9g M. Mitchell and M. Segev, Nature sLondond 387, 880 s1997d;M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoul-ides, Phys. Rev. Lett. 79, 4990 s1997d.f10g M. Olshanii, Phys. Rev. Lett. 81, 938 s1998d.f11g L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A 70,013606 s2004d; 72, 025602 s2005d.f12g L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A 65,043614 s2002d.f13g A. L. Fetter, in Bose Einstein Condensation in Atomic Gases,Proceedings of the Enrico Fermi School, Course CXL, editedby M. Inguscio, S. Stringari, and C. W. Wieman sSIF, Bologna,1999d.f14g A. Griffin, Phys. Rev. B 53, 9341 s1996d; H. Shi and A. Grif-fin, Phys. Rep. 304, 1 s1998d.f15g L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A 60, 4171s1999d.f16g V. I. Yukalov, Phys. Rev. A 72, 033608 s2005d.f17g L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A 66,043603 s2002d.0 0.2 0.4 0.6 0.8 1 1.2Nγ02468101214k BTUNIFORMLOCALIZEDNO BECFIG. 4. sColor onlined Phase diagram in the plane temperature Tvs interatomic strength g for N=104 atoms in a toroidal trap, withL=100. The energy kBT is in units of "v' and the length in units ofa'.SALASNICH, PAROLA, AND REATTO PHYSICAL REVIEW A 74, 031603sRd s2006dRAPID COMMUNICATIONS031603-4

Thermodynamics of Solitonic Matter Waves in a Toroidal Trap

Thermodynamics of Solitonic Matter Waves in a Toroidal Trap

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