JĄDRA ATOMOWE. Badania naukowe: struktura jądra atomowego. Profesor Stefan Ćwiok Jądra stabilne. Jądra znane. Jądra ciężkie i superciężkie

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1 ur. 1933r. Przeryty Bór k. Tarnowa. Studia: Akademia Górniczo-Hutnicza w Krakowie, Wydział Fizyki Uniwersytetu Moskiewskiego im.łomonosowa. Doktorat: Wydział Fizyki Uniwersytetu Warszawskiego (1969) Adiunkt: Wydział Fizyki Uniwersytetu Warszawskiego ( ) Od 1973: Instytut Fizyki Politechniki Warszawskiej 1998: tytuł Profesora Fizyki Wieloletni z-ca Dyrektora Instytutu Fizyki PW i Kierownik Zakładu Fizyki Jądrowej Badania naukowe: struktura jądra atomowego JĄDRA ATOMOWE Jądra stabilne? Jądra ciężkie i superciężkie Profesor Stefan Ćwiok Jądra znane protony Neutron Gwiazdy star neutronowe neutrony

2 ur. 1933r. Przeryty Bór k. Tarnowa. Studia: Akademia Górniczo-Hutnicza w Krakowie, Wydział Fizyki Uniwersytetu Moskiewskiego im.łomonosowa. Doktorat: Wydział Fizyki Uniwersytetu Warszawskiego (1969) Adiunkt: Wydział Fizyki Uniwersytetu Warszawskiego ( ) Od 1973: Instytut Fizyki Politechniki Warszawskiej 1998: tytuł Profesora Fizyki Wieloletni z-ca Dyrektora Instytutu Fizyki PW i Kierownik Zakładu Fizyki Jądrowej Badania naukowe: struktura jądra atomowego Profesor Stefan Ćwiok Osiągnięcia: -hipoteza stabilności bardzo ciężkich jąder o N=162 potwierdzona przez doświadczenia w GSI Darmstadt i Berkeley (USA). -wyjaśnienie zagadki istnienia dwóch kanałów rozszczepienia w izotopach fermu. -hipoteza istnienia (metastabilnych) stanów hiperzdeformowanych w aktynowcach - potwierdzona eksperymentalnie. -systematyczne obliczenia dla nieparzystych jąder superciężkich (do dziś interpretacja wielu eksperymentów bazuje na tych wynikach). -hipoteza istnienia liczby magicznej Z=126, a nie Z=114 jak dotychczas sądzono. -hipoteza współistnienia kształtów w jądrach superciężkich (Nature, 433(2005)705)

3 O pewnych własnościach kwantowych gazów atomowych Piotr Magierski (WF PW) Współpracownicy: Aurel Bulgac, Joaquin E. Drut (University of Washington, Seattle)

4 Outline Particle scattering at low energies. BCS-BEC BEC crossover. What is the unitary regime? How one can manipulate the two-body interaction in experiments with atomic gases? Theoretical approach: path integral description of strongly interacting Fermi gases. Equation of state for the Fermi gas in the unitary regime. Critical temperature. Conclusions.

5 Scattering at low energies (s-wave scattering) If ik r e ikr ψ () r = e + f ; f - r 1 f = 1 1 ik + r0 k a 2 2 2π λ = >> R k R - radius of the interaction potential scattering amplitude, a - scattering length, r - effective range k 0 then the interaction is determined by the scattering length alone. 0

6 Scattering at low energies: attractive interaction 1) 2) 3) a < 0 there is no bound state a = ± V(r) ψ () r What is the energy of the dilute Fermi gas? kf kf εf = ; n = ( k 2 Fr 0 << 1) 2m 3π a > 0 a bound state exists Fermi gas? E( k a ) =? F - particle density Perturbation series: E 10 6 = 1+ ( ka F ) 1 ( ka F )( 11 2ln2 )... EFG 9π π 3 EFG = εfn - Energy of the noninteracting Fermi gas 5 PAIRING NOT INCLUDED YET!

7 a < 0 Fermi gas How pairing emerges? Cooper s argument (1956) Gap 2 Cooper pair = kf π 1 1 ε exp, iff 1 and F BCS k a - size of the Cooper pair 2 F << << η = 2m 2k k k e Fa F F 2 EHF+BCS 10 5 BCS 10 = 40 + kfa + = + kfa + 4 EFG 9π 8 εf 9π e π 1 ( )... 1 ( )... exp kfa Hartree-Fock term BCS term

8 Superconductivity and superfluidity in Fermi systems 20 orders of magnitude over a century of (low temperature) physics Dilute atomic Fermi gases T c ev Liquid 3 He T c 10-7 ev Metals, composite materials T c ev Nuclei, neutron stars T c ev QCD color superconductivity T c ev units (1 ev 10 4 K)

9 What is the unitary regime? A gas of interacting fermions is in the unitary regime if the average separation between particles is large compared to their size (range of interaction), but small compared to their scattering length. n r 3 n a 3 1 n - particle density 0 1 a - scattering length r 0 - effective range ie.. r 0, 0 The only scale: E a FG ± N = 3 ε 5 ( ) UNIVERSALITY: ( ) = ξ T ε E T E FG QUESTIONS: What is the shape of? What is the critical temperature for the superfluid-to-normal transition?... F F NONPERTURBATIVE REGIME System is dilute but strongly interacting! σ = ξ ( ) T ε F 4π a 2

10 Expected phases of a two species dilute Fermi system BCS-BEC BEC crossover EASY! weak interaction BCS Superfluid T Strong interaction UNITARY REGIME? EASY! weak interactions Molecular BEC and Atomic+Molecular Superfluids 1/a a<0 no 2-body bound state a>0 shallow 2-body bound state Bose molecule

11 A little bit of history Bertsch Many-Body X challenge, Seattle, 1999 What are the ground state properties of the many-body system composed of spin ½ fermions interacting via a zero-range, range, infinite scattering-length contact interaction. Why? Besides pure theoretical curiosity, this problem is relevant t to neutron stars! In 1999 it was not yet clear, either theoretically or experimentally, whether such fermion matter is stable or not! A number of people argued that under such conditions fermionic matter is unstable. - systems of bosons are unstable (Efimov effect) - systems of three or more fermion species are unstable (Efimov effect) Baker (winner of the MBX challenge) concluded that the system is stable. See also Heiselberg (entry to the same competition) Carlson et al (2003) Fixed-Node Green Function Monte Carlo and Astrakharchik et al. (2004) FN-DMC provided the best theoretical estimates for the ground state energy of such systems: ξ ( T = 0) 0.44 Thomas Duke group (2002) demonstrated experimentally that such systems are (meta)stable.

12 In dilute atomic systems experimenters can control nowadays almost anything: The number of atoms in the trap: typicallyt about atoms divided among the lowest two hyperfine states. The density of atoms Mixtures of various atoms The temperature of the atomic cloud The strength of this interaction is fully tunable! Who does experiments? Jin s group at Boulder Grimm s group in Innsbruck Thomas group at Duke Ketterle s group at MIT Salomon s group in Paris Hulet s group at Rice Physics Today, v54, 20 (2001)

13 One fermionic atom in magnetic field Fm F F= I+ J ; J= L+ S Nuclear spin Electronic spin Two hypefine states are populated in the trap Collision of two atoms: At low energies (low density of atoms) only L=0 (s-wave) scattering is effective. Due to the high diluteness atoms in the same hyperfine state do not interact with one another. Atoms in different hyperfine states experience interactions only in s-wave. s

14 Feshbach resonance 2 2 p H = + ( V + V ) + V ( r) P + V ( r) P + V 2µ i= 1 hf Z d i i hf ahf Z V = I J, V = ( γ ) 2 ejz γniz B Tiesinga, Verhaar, Stoof, Phys. Rev. A47, 4114 (1993) Channel coupling E resonance: a ± Regal and Jin, PRL 90, (2003) Interatomic distance

15 Theoretical approach: Fermions on 3D lattice Coordinate space L limit for the spatial correlations in the system kcut π = ; x x - Spin up fermion: - Spin down fermion: External conditions: T µ - temperature - chemical potential n(k) Momentum space 2π/L k cut =π/ x k π x k y π x k x 2π/L 2 2 π εf,, kt << 2 m ( x ) π δε > 2 ml 2π δ p > L 2

16 Hamiltonian 2 ˆ ˆ ˆ 3 3 H = T + V = d r ψˆs ( r) ψˆs( r) g d r nˆ ( r) nˆ ( r) s= 2m N d r n ( r) n ( r) ; n ( r) ( r) ( r) ˆ 3 ( ) = ˆ + ˆ ˆs = ψˆs ψˆ s 1 g mk cut m = + 4π a 2π Grand-canonical ensemble: { ρ } Running coupling constant g defined by lattice 1 g = m 2π 2 x ˆ 1 { ˆ ˆ ˆ } 1 E( T ) = H = Tr H ρ ( H, N, T ) = En e Z ( T ) Z - UNITARY LIMIT 1 kt ( E µ N ) 1 ( E µ N ) ( H µ N ) 1 ˆ ˆ n n kt kt Z ( T ) = Tr ( Hˆ, Nˆ, T ) = e ; ρ( Hˆ, Nˆ, T )= e Eigenenergies of the Hamiltonian are unknown! n n n n

17 Single-particle Path integral approach: particle quantum mechanics: r e r = D[ r( t)] e ihˆ ( t t ) 1 1 ( ) 0 t0 ( ( ), ( )) t ( ( ), ( ) mr () t ) = = 2 i L r t r t dt is[ r ( t)] t0 L r(), t r() t V ( r); e e Quantum statistical mechanics: t i L r t r t dt is 1 r 1 is 2 e e { } Z( β) = Tr exp( β ( Hˆ µ Nˆ) = nexp( β ( Hˆ µ Nˆ) n β = 1 ; imaginary time: τ = it kt n many body states [ ] + σ β = ln{det[1 Uˆ ({ })]} Z( ) D σ( r, τ) e S[ σ( r, τ)] = ln{det[1 + Uˆ ({ σ})]} - action

18 1 kt τ β 0 Uˆ({ σ}) = T exp{ dτ[ hˆ({ σ }) µ ]}; hˆ({ σ}) one-body operator U({ σ}) = ψ Uˆ ({ σ}) ψ ; ψ - single-particle wave function kl k l l S [ σ ] σ τ = ˆ D[ ( r, )] e ET ( ) H = EU [ ({ σ })] ZT ( ) EU [ ({ σ })]- Quantum Monte-Carlo: Sigma space sampling energy associated with a given sigma field σ N σ 1 ET ( ) = EU({ k}) N σ k= 1 ( σ ) E( T) - stochastic variable E( T) = E( T) P( σ) e S[ σ ] N σ 2 E( T) E( T) number of uncorrelated samples N σ

19 Quantum Monte-Carlo: parallel computing τ β 0 Uˆ({ σ}) = T exp{ dτ[ hˆ({ σ }) µ ]}; hˆ({ σ}) one-body operator U({ σ}) = ψ Uˆ ({ σ}) ψ ; ψ - single-particle wave function kl k l l For each sigma n single particle states have to be evolved.... ψ n ψ 3 ψ 2 ψ 1 U ˆ({ σ }) ˆ({ σ}) ˆ({ }) U U σ... U ˆ({ σ }) U({ σ}) = ψ Uˆ ({ σ}) ψ kl k l

20 More details of the calculations: Lattice sizes used from 8 3 x 257 (high Ts) to 8 3 x (low Ts),, <N>=50, and 6 3 x 257 (high Ts) to 6 3 x (low Ts),, <N>=30. Effective use of FFT(W) makes all imaginary time propagators diagonal (either in real space or momentum space) and there is no need to store large e matrices. Update field configurations using the Metropolis importance sampling algorithm. Change randomly at a fraction of all space and time sites the signs s the auxiliary fields σ(x, (x,τ)) so as to maintain a running average of the acceptance rate between 0.4 and 0.6. Thermalize for 50, ,000 MC steps or/and use as a start-up field configuration a σ(x, (x,τ)-field configuration from a different T At low temperatures use Singular Value Decomposition of the evolution operator U({σ}) to stabilize the numerics. Use 200,000-2,000,000 σ(x, (x,τ)- field configurations for calculations MC correlation time time steps at T T c

21 a = ± Superfluid to Normal Fermi Liquid Transition Normal Fermi Gas (with vertical offset, solid line) ξ ( T= 0) 0.41(2) Bogoliubov-Anderson phonons and quasiparticle contribution (dashed line ) Bogoliubov-Anderson phonons contribution only (dotted( line) People never consider this??? Quasi-particle contribution only (dotted line) π T E ( T) = ε N exp T quasi-particles F ε F 7/3 2 π = ε F exp e 2 kfa 4 3 3π T Ephonons( T) = εfn, /2 ξs 5 16ξs εf A. Bulgac, J.E. Drut, P. Magierski, cond-mat/

22 Phase transition Ideal Fermi gas entropy S E µ 3 T E = µ N - PV + TS = εf ( n) N e = ε( n) nv 5 ε F ( n) N kf kf n = =, ε ( ) = 2 F n V 3π 2m 5 en ( ) µ 2 = 3 T S N = Nσ, P = e( n) n T ε F ( n) 3

23 Low temperature behaviour of a Fermi gas in the unitary regime 3 T µ ( T) ET ( ) = ε Nξ and ξ 0.41(2) for T < T εf εf 5 F s C de( T) T 2 T T µ ( T) = = ε ξ ξ F εfξs dn εf 5 εf εf 5/2 T T ξ = ξs + ςs, ςs 11(1) εf εf 3 T ET ( ) = εfn ξs + ςs 5 εf n Lattice results disfavor either n 3 or n 2 and suggest n=2.5(0.25) This is the same behavior as for a gas of noninteracting (!) bosons below the condensation temperature.

24 Conclusions Fully non-perturbative calculations for a spin ½ many fermion system in the unitary regime at finite temperatures are feasible and apparently the system undergoes a phase transition in the bulk at a T c = 0.23 (2) ε F (Exp: T c = 0.27(2) ε F, J. Kinast et al. Science,, 307, 1296 (2005): Based on theoretical assumptions). Chemical potential is constant up to the critical temperature note similarity with Bose systems! Below the transition temperature, both phonons and fermionic quasiparticles contribute almost equaly to the specific heat. In more than one way the system is at crossover between a Bose and Fermi systems. There are reasons to believe that below the critical temperature this system is a new type of fermionic superfluid, with unusual properties.

25 Quest for unitaryu point critical temperature Analytics Numerics 0.55 Experiment + assumptionns 0.50 This work T c /E F M. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo, and R. Walser, PRL 87, (2001) P. Nozieres, S. Schmitt-Rink, J. Low. Temp. Phys 59, 195 (1985) J. Kinhast, A. Turlapov, J.E. Thomas, Q. Chen, J. Stajic, K. Levin, Science 307, 1296 (2005) A. Ours Bulgac, J. E. Drut, P. Magierski, cond-mat/ X.-J. Liu, H. Hu, cond-mat/ T.Lee, D. Schafer, nucl-th/ E. Burovski, N. Prokofev, B. Svistunov, M. Troyer Private communication M. Wingate, cond-mat/ Boris Svistunov s talk (updated), Seattle 2005

26 Evidence for fermionic superfluidity: vortices! 6 system of fermionic Li atoms Feshbach resonance: B=834G BEC side: a>0 UNITARY REGIME BCS side: a<0 M.W. Zwierlein et al., Nature, 435, 1047 (2005)