An impurity in a Fermi sea on a narrow Feshbach resonance: A variational study of the polaronic and dimeronic branches

Transcription

1 An impurity in a Fermi sea on a narrow Feshbach resonance: A variational study of the polaronic and dimeronic branches Christian Trefzger Laboratoire Kastler Brossel ENS Paris

2 Introduction: The system Homogeneous (3D) Fermi gas of particles of same spin state m, k F Distinguishable impurity (boson/fermion) M,K s-wave interactions are described by two parameters: Scattering length a Feshbach length R

3 The Narrow Feshbach resonance x a = a bg B B B V(x) = interatomic potential E mol x = interparticle distance a = scattering length B B B = magnetic field a bg The resonance width : B =) R / / B a, R (i) Theoretically: Competition of (ii) Experimentally: 4 K 6 Li narrow Feshbach resonances R > nm R VdW 2nm

4 Ground state The ground state has two quasi-particle branches: POLARONIC a) c DIMERONIC /k F a C. Lobo et. al., PRL 97, 243 (26) F. Chevy, PRA 74, (26) N. Prokof ev, B. Svistunov, PRB 77, 248 (28) M. Punk et. al., PRA 8, 5365 (29) C. Mora, F. Chevy, PRA 8, 3367 (29) R. Combescot et. al., EPL 88, 67 (29)

5 Outlook. Two-channel model Hamiltonian 2. Variational ansätze, integral equations 3. Properties of the two branches at ħk= a. Polaron-to-dimeron crossing point b. Non-trivial weakly interacting limit 4. Polaronic quasi-particle residue 5. Polaronic pair correlation function 6. A moving polaron

6 Ĥ = X k Hamiltonian of the system + p V X h " k û kûk + E k ˆd k ˆd k + k,k Two-channel model " i k +M/m + E mol ˆb kˆb k (k rel )(ˆb k+k û k ˆdk +h.c.) V(x) = interatomic potential E mol x = interparticle distance E mol 2 = µ 2 ~ 2 a + Kinetic energies in the open channel " k = ~2 k 2 2m Z E k = ~2 k 2 2M d 3 k (2 ) 3 2 (k) 2µ ~ 2 k 2

7 Polaron: Integral equations pol (~K)i = ˆd K + X q qˆb K+qûq + X k,q kq ˆd K+q kû kûq FSi K q q k K + q K + q k Minimise: h pol (~K) Ĥ pol(~k)i, with respect to, q, k,q E pol (~K) =E K + Thermodynamic limit: Z d 3 q (2 ) 3 D q [ E pol (~K), ~K]

8 Dimeron: Integral equations dim (~K)i = ˆb K + X k k ˆd K kû k + X k,q kqˆb K+q kû kûq+ X k,k,q k kq ˆd K+q k k û k û kûq FS : N i K k q k q k k K k K + q k K + q k k Minimise: h dim (~K) Ĥ dim(~k)i, with respect to, k, kq, k kq Z d 3 k d 3 q Thermodynamic limit: (2 ) 6 M[ E dim(~k), K; k, q; k, q ] k q =

9 A discrete state coupled to a continuum E Lower border of the continuum K K Λ Λ Λ Λ Λ...

10 A discrete state coupled to a continuum E... K Λ Λ Lower border of the continuum K Λ Λ Λ pol (~K)i = ˆd K + X q qˆb K+qûq + X k,q kq ˆd K+q kû kûq FSi K q q k K + q K + q k

11 A discrete state coupled to a continuum E K Λ... Λ Lower border of the continuum K Λ Λ Λ dim (~K)i = ˆb K + X k k ˆd K kû k + X k,q kqˆb K+q kû kûq+ X k,k,q k kq ˆd K+q k k û k û kûq FS : N i K k q k q k k K k K + q k K + q k k

12 A discrete state coupled to a continuum E K Λ Λ... Lower border of the continuum K Λ Λ Λ () Discrete state EXPELLED from the continuum, remains a discrete state. (2) Discrete state DILUTED in the continuum, becomes a resonance: E = < E + i= E Of physical interest if = E < E

13 Outlook. Two-channel model Hamiltonian 2. Variational ansätze, integral equations 3. Properties of the two branches at ħk= a. Polaron-to-dimeron crossing point b. Non-trivial weakly interacting limit 4. Polaronic quasi-particle residue 5. Polaronic pair correlation function 6. A moving polaron

14 Properties of the two branches at ħk= polaron dim pol. is GS < m* > dim Im(!Edim) < m* < pol dim. is GS m* > pol m*dim < Im(! E m* < dim pol. is GS m* pol 6 kfr* Im(! E dim ) > 8 < dim. is GS m* < pol 4 m* dim > dim ) pol. is GS 2 M/m =.55 m* M/m = dim. is GS m* > pol M/m = /(kfa) dimeron m*pol < /(kfa) /(kfa) kfr* m* dim > < 8

15 (a) Polaron-to-dimeron crossing point [E-E FS (N)]/2E F [E-E FS (N)]/2E F [E-E FS (N)]/2E F M/m =.55 M/m = M/m = a) c =.75 a) c = 2.6 (a) (a2) (a3) a) dimeron a) c =.84 polaron a) c =.36 (b) (b2) (b3) a) /(k F a) c =.7 a) c = -4.3 a) c = a) c =.25 a) c = -.58 (c) (c2) (c3) a) k F R * = k F R * = k F R * =

16 (a) Polaron-to-dimeron crossing point a) c M/m=.55 M/m= M/m= (a) k F R * R In vacuum, for ANY value of, one has a two-body bound state iff a>

17 Intuitive picture Question: When do we have a two-body bound state? V(x) = interatomic potential E mol x = interparticle distance () Two particles in vacuum: (i) Naïve answer: Ẽ mol < a> E mol < (ii) Lamb shift due to vacuum fluctuations in the open channel: Z d ee 3 k (k) 2 2 mol = E mol (2 ) 3 ) e 2 µ E ~ 2 k 2 mol = 2 ~ 2 a 2µ

18 Intuitive picture Question: When do we have a two-body bound state? (2) One impurity in a Fermi sea: TWO EFFECTS (i) Effect on the Lamb shift Ẽ mol = E mol k>k F d 3 k χ(k)λ 2 (2π) 3 2 k 2 (ii) Change of the dissociation threshold, Ẽ mol <E F = k F a > 2 π 2µ E F M m + M k FR

19 Test of intuitive picture a) c M/m=.55 M/m= M/m= (a) k F R * k F a = 2 π M m + M k FR

20 Quantitative analytical result described later a) c M/m=.55 M/m= M/m= (a) k F R * Same slope as intuitive picture, but exact calculation of the INTERCEPT of the asymptote

21 (b) Non-trivial weakly interacting limit Standard weakly interacting limit: a,r fixed! Loose information on the narrowness of the resonance f k = a + ik + k2 R Non-trivial weakly interacting limit: a, a and R proportional

22 (b) Non-trivial weakly interacting limit a!, ar = const. E () pol + E() pol k F a = E () dim + E() dim k F a ( ) k F a c + 2 ( +r π r = R + ) r +r k FR [ (+r ) 5/2 r ( ) ] /2 r ln +r + ( ( ) /2 r +r ) /2 r +r

23 (b) Non-trivial weakly interacting limit a) c C.T. and Yvan Castin, Phys. Rev. A 85, 5362 (22) M/m=.55 M/m= M/m= (a) k F R * M/m =,R =:analytics numerics 3% R. Grimm, Nature 485, (22)

24 Outlook. Two-channel model Hamiltonian 2. Variational ansätze, integral equations 3. Properties of the two branches at ħk= a. Polaron-to-dimeron crossing point b. Non-trivial weakly interacting limit 4. Polaronic quasi-particle residue 5. Polaronic pair correlation function 6. A moving polaron

25 Polaronic quasi-particle residue POLARONIC DIMERONIC The quasi-particle residue Z quantifies the quasi-particle nature of the ground state of the system Z = 2 = K 2 Z MEANING Free Impurity No Quasi-Particle

26 Polaron: Test the ansatz Test the ansatz with quantum Monte Carlo results for equal masses and wide resonance: Test: Energy ansatz Monte Carlo.8 m = M,R = Test: State DiagMC Chevy s ansatz N * =, bold MIT experiment E pol () - Z p /k F a N. Prokof ev, B. Svistunov, PRB 77, 248 (28) /k F a J. Vlietinck, J. Ryckebusch, K. Van Houcke, Phys. Rev. B 87, 533 (23) The variational ansatz seems to be good to describe both the ground energy and the ground state of the system

27 Polaronic quasi-particle residue Z = 2 = 2 K Z Z Z M/m =.55 M/m = M/m = (a) (a2) a) analytics (b) numerics (b2) a) (a3) (b3) (c3) (c) (c2) a) k F R * = k F R * = k F R * = C.T. and Yvan Castin, Europhys. Lett., 36 (22)

28 Polaronic quasi-particle residue Non-trivial weakly attractive limit a!, ar = const. Z exact a! = + " 2 M k F a c M + m + c 2 # 2 kf a + O(k F a) 3 Compare with an exactly solvable model: The infinite-mass impurity c 2 ln(m/m) M/m!+ The divergence is a signature of the Anderson ortogonality catastrophe

29 Outlook. Two-channel model Hamiltonian 2. Variational ansätze, integral equations 3. Properties of the two branches at ħk= a. Polaron-to-dimeron crossing point b. Non-trivial weakly interacting limit 4. Polaronic quasi-particle residue 5. Polaronic pair correlation function 6. A moving polaron

30 Pair correlation function Due to the symmetry of interactions, the density that surrounds the impurity is a spherically symmetric function of the radial coordinate x x = x u x d G(x u x d )= h ˆ u (x u ) ˆ d (x d) ˆd(x d ) ˆu(x u )i d ˆu(x u ) ˆd(x d ) Fermion field operator Impurity field operator G(x) is a measure of the spatial extension of the polaron in a Fermi gas, G(x) presents Friedel-like oscillations and a multiscale structure

31 (k F x) 2 [G(x)-] hxi = Pair correlation function (a) At large distance Z x k F x G(x) x!+ d 3 xx[g(x) ] x!+ A 4 Z + A 4 + B 4 cos(2k F x) (k F x) 4 x 3 (k F x) 4 x!+ A 4 ln(x)

32 Pair correlation function A 4 & B 4 A 4 & B 4 A 4 & B (a) (a2) M/m =.55 M/m = M/m = a) 2 - (b) A 4 B 4 (b2) a) (a3) (b3) (c3) 2 - (c) (c2) a) C.T. and Yvan Castin, Europhys. Lett., 36 (22) k F R * = k F R * = k F R * =

33 Multiscale structure (k F x) 4 [G(x)-] (i) (ii) (iii) k F x ln x intermediate x 4 Ci(x), Si(x) x 4 (b)

34 Outlook. Two-channel model Hamiltonian 2. Variational ansätze, integral equations 3. Properties of the two branches at ħk= a. Polaron-to-dimeron crossing point b. Non-trivial weakly interacting limit 4. Polaronic quasi-particle residue 5. Polaronic pair correlation function 6. A moving polaron

35 A moving polaron Homogeneous (3D) Fermi gas of same spin state m, k F Distinguishable impurity (boson/fermion) M,K No interaction between the Fermions Attraction g! R

36 What do we know (R * )? (K = ) ) Perturbative regime: The ground state is the Fermi polaron N. Prokof ev, B. Svistunov, PRB 77, 248 (28) R.F. Bishop, Nucl. Phys B 7, 573 (97) 2) Any regime: Variational ansatz F. Chevy, PRA 74, (26)

37 What do we know (R * )? (K = ) ) Perturbative regime: The ground state is the Fermi polaron N. Prokof ev, B. Svistunov, PRB 77, 248 (28) R.F. Bishop, Nucl. Phys B 7, 573 (97) 2) Any regime: Variational ansatz F. Chevy, PRA 74, (26) (K 6= ) The energy of the polaron becomes complex E(K) =< E(K)+i = E(K) 3) Perturbative regime: R.F. Bishop, Nucl. Phys B 7, 573 (97) M = m : = E(K) / K 4 for K<k F NOT REPRODUCED BY THE ANSATZ

38 A moving polaron (R * = ) E(K) = g! ~ 2 K 2 2M ( g)2 + g + f(k/k F )+O(g 3 ) E F f(apple) Sextuple integral (over the momenta of particle-hole pair) Presents singularities of the nth-order derivative apple = apple = apple = r M/m

39 A moving polaron (R * = ) E(K) = g! ~ 2 K 2 2M ( g)2 + g + f(k/k F )+O(g 3 ) E F f(apple) Sextuple integral (over the momenta of particle-hole pair) Presents singularities of the nth-order derivative apple = apple = apple = r M/m apple 3r 3r (r 2 ) <f(apple) = r + 2(r 2 ) 2 (apple 2 + 2r 9) (apple )3 apple (apple r)4 apple [apple(apple + 3)(r 2 2) + 6r 2 2] ln apple apple apple +4r r 2 ln apple r apple + apple! apple. apple

40 A moving polaron (R * = ) M/m =.55 M/m = M/m = (a) (b) (c) f( ).5 (a2) (b2) (c2) f ( ) K = k F (apple = ) Effective mass divergence is an artifact of the perturbative approach. We expect a behavior g 2 ln(g 2 )

41 A moving polaron (R * = ) 4 M/m =.55 M/m = M/m = (a) 4 (b) 4 (c) - f( ) (a2) (b2) (c2) - f ( ) 4 J = 9 /

42 Experiments? (R * ) f(apple) =f(apple) (c apple 2 + c 2 )k F R Measurable by RF-spectroscopy: Broadening and shift of the resonance Innsbruck: C. Kohstall et. al., Nature 485, 65 (22) 4 K/ 6 Li M/m = k F R = k F a =.46.2 (a) (b) - f ~ ( )/E F. k B T =.5 E F f ~ ( )/E F k B T =.5 E F.5.5 2

43 Conclusions s-wave narrow Feshbach resonance: a, R 2 M/m=.55 (a) a) c M/m= M/m= k F R * Ground state: polaron-to-dimeron crossing point Physical interpretation in terms of the Lamb shift

44 Conclusions Polaronic ansatz: Good for the energy and the state Z /k F a (c2) (k F x) 2 [G(x)-] (a) k F x Moving polaron: experimentaly observable E(K) = g! ~ 2 K 2 2M ( g)2 + g + f(k/k F )+O(g 3 ) E F

45 Yvan Castin Laboratoire Kastler Brossel Ecole normale supérieure 24 rue Lhomond, 7523 Paris, France Thank You!!

46 (b) Effective mass E(~K) = K! E() + ~2 K 2 2m + O(K4 ) POLARON a) c DIMERON /k F a K K m pol a M m dim a + m + M

47 (b) Effective mass m/m* m/m* m/m* m m pol a) m m dim Im(!E dim ) < M/m =.55 M/m = M/m = a a +.87 (a) (a2) polaron..2 dimeron Im(!E -.2 dim ) < (a3) (b3) (c3) Im(!E dim ) < (b) (b2) a) m m pol m m dim a a +.5. Im(!E dim ) < (c) (c2) a) m m pol m m dim k F R * = k F R * = k F R * = a.55 a +.3

48 (c) The closed-channel Closed-channel-molecule population: Ĥ = k E molˆb kˆb k +... Ĥ E mol = k N cc = k ˆb kˆb k ˆb kˆb k N cc = E E mol N cc = E (/a) 2µR 2 N cc M/m =.55 M/m = M/m = (a) (b) (c) dimeron 2 a) polaron a) a) k F R * =