Influence of Disorder on the Fidelity Susceptibility in the BCS-BEC Crossover

Transcription

1 Influence of Disorder on the Fidelity Susceptibility in the BCS-BEC Crossover 6th APCWQIS, December 2012 Bilal Tanatar December 6, 2012

2 Prologue 1 Introduction Prologue Cooling Techniques 2 BCS-BEC Crossover Motivation The Theory of Crossover 3 Disorder Anderson Localization (AL) Experiments 4 The Dirty Crossover Continuum Model (3D) Fidelity Susceptibility (FS) 5 Conclusion Epilogue Acknowledgement

3 Prologue 1 Introduction Prologue Cooling Techniques 2 BCS-BEC Crossover Motivation The Theory of Crossover 3 Disorder Anderson Localization (AL) Experiments 4 The Dirty Crossover Continuum Model (3D) Fidelity Susceptibility (FS) 5 Conclusion Epilogue Acknowledgement

4 Prologue Observation of superconductivity in Theoretical Prediction of Bose-Einstein Condensate in Theory of Superconductivity (BCS) in Experimental Observation of Bose-Einstein Condensate in Image Source: 1.

5 Cooling Techniques Doppler Cooling Six lasers applied opposite to each other from each direction. Light is red detuned to activate Doppler effect. Temperature attain 1mK. Sisyphus Cooling Opposite polarization of the laser beams create a potential crest and valley. Takes down temperature to 1µK. Image Source: 2. aspects.html 3.

6 Cooling Techniques Doppler Cooling Six lasers applied opposite to each other from each direction. Light is red detuned to activate Doppler effect. Temperature attain 1mK. Sisyphus Cooling Opposite polarization of the laser beams create a potential crest and valley. Takes down temperature to 1µK. Image Source: 2. aspects.html 3.

7 Cooling Techniques Evaporative Cooling Sympathetic Cooling Cut the higher edge of the magnetic trap with rf spectra. Atoms with higher energy will leave the pot leaving cooler atoms inside. Takes down temperature to nk level. Evaporative cooling does not work well for fermions. Mix evaporative cooled bosons to fermions to cool the laser cooled fermions sympathetically. Image Source: 4. aspects.html

8 Cooling Techniques Evaporative Cooling Sympathetic Cooling Cut the higher edge of the magnetic trap with rf spectra. Atoms with higher energy will leave the pot leaving cooler atoms inside. Takes down temperature to nk level. Evaporative cooling does not work well for fermions. Mix evaporative cooled bosons to fermions to cool the laser cooled fermions sympathetically. Image Source: 4. aspects.html

9 Cooling Techniques Feshbach Resonance a = a 0 [ 1 B B B 0 ] Image Source: 5. group/experimental setup/bec I/background.html

10 1 Introduction Prologue Cooling Techniques 2 BCS-BEC Crossover Motivation The Theory of Crossover 3 Disorder Anderson Localization (AL) Experiments 4 The Dirty Crossover Continuum Model (3D) Fidelity Susceptibility (FS) 5 Conclusion Epilogue Acknowledgement

11 Motivation BCS-BEC Crossover

12 Motivation Leggett in 1980 showed that starting from a BCS ground state for a Fermi gas with attractive interaction one can reach composite boson limit 1 at T = 0 for sufficiently strong attraction. Nozieres and Schmitt-Rink in 1985 extended this to finite temperature 2. Evolution from weak coupling to strong coupling is smooth. So it is a crossover, not a phase transition. 1. A. J. Leggett, J. Phys. 41, C7-19 (1980). 2. P. Nozieres, S. Schmitt-Rink, J. Low. Temp. Phys. 59, 195 (1985). Image Source: 6. archive/2006 spring/images/fermisea.jpg

13 Motivation Phase diagram BCS-BEC Crossover Image Source:

14 Motivation Image Source: 7. Modern Quantum Mechanics, J. J. Sakurai

15 Motivation Discovery of High Temperature Superconductors. High transition temperature and pseudogap. The order of k F ξ pair is similar in the crossover and in HTSC. Image Source: 8. P.hD. thesis of Cindy Regal

16 Motivation Beauty of BCS Wave function BCS ground state wave function can be extended to BEC limit under some constraint. Ψ = k [ uk + v k c ] k c k 0 Set g k = v k /u k, ( Ψ = k u k ) [ exp k g(k)c k c k ] 0 Define a new operator: b = k g(k)c k c k [b, b ] = k g(k) 2 (1 n k n k ) c number. Provided n kσ << 1 [b, b ] = c number. Ψ = exp (b ) 0 represents a Bosonic coherent state.

17 The Theory of Crossover The appropriate Hamiltonian: { } H eff = dr Ψ (rα)h e (r)ψ(rα) + V (r)ψ (rα)ψ(rα) + dr α { } (r)ψ (r )Ψ (r ) + (r)ψ(r )Ψ(r ) Perform unitary transformation 3 : Ψ(r ) = ( ) γ n u n (r) γ n v n (r)) n Ψ(r ) = ( ) γ n u n (r) + γ n v n (r)) n 3. P. D. de GennesSuperconductivity of Metals and Alloys, Addition-Wesley Publishing Company, Inc

18 The Theory of Crossover Bogoliubov degennes Equation ɛu(r) = [H e + V (r)] u(r) + (r)v(r) ɛv(r) = [H e + V (r)] v(r) + (r)u(r) The order parameter is (r) = g n v n (r)u n (r)(1 2f n ). Gap & Density Equation (r) = g u n (r)vn (r) ɛ n>0 n(r) = 2 v n (r) 2 ɛ n>0

19 The Theory of Crossover Bogoliubov degennes Equation ɛu(r) = [H e + V (r)] u(r) + (r)v(r) ɛv(r) = [H e + V (r)] v(r) + (r)u(r) The order parameter is (r) = g n v n (r)u n (r)(1 2f n ). Gap & Density Equation (r) = g u n (r)vn (r) ɛ n>0 n(r) = 2 v n (r) 2 ɛ n>0

20 The Theory of Crossover Consider no external potential fermion-fermion interaction is mediated via short range contact potential. Gap & Density Equation m 4πa = k n = k [ 1 1 ] 2E k 2ɛ k [ 1 ξ ] k. E k

21 The Theory of Crossover Consider no external potential fermion-fermion interaction is mediated via short range contact potential. Gap & Density Equation m 4πa = k n = k [ 1 1 ] 2E k 2ɛ k [ 1 ξ ] k. E k

22 1 Introduction Prologue Cooling Techniques 2 BCS-BEC Crossover Motivation The Theory of Crossover 3 Disorder Anderson Localization (AL) Experiments 4 The Dirty Crossover Continuum Model (3D) Fidelity Susceptibility (FS) 5 Conclusion Epilogue Acknowledgement

23 Anderson Localization (AL) Definition Beyond a critical amount of impurity, motion of the electron can come to a complete halt. The electron becomes trapped and the conductivity vanishes. Direct observation of AL for electron is very difficult. Number of phenomena can mask single particle quantum effects genuinely induced by disorder. Most of the evidences are indirect and stem from conductivity measurement. Cold atoms are good candidate for observation of AL: Genuine quantum particles described as matter waves. Single atom matter waves can be directly visualized by different imaging techniques in Bose-Einstein Condensate.

24 Anderson Localization (AL) Definition Beyond a critical amount of impurity, motion of the electron can come to a complete halt. The electron becomes trapped and the conductivity vanishes. Direct observation of AL for electron is very difficult. Number of phenomena can mask single particle quantum effects genuinely induced by disorder. Most of the evidences are indirect and stem from conductivity measurement. Cold atoms are good candidate for observation of AL: Genuine quantum particles described as matter waves. Single atom matter waves can be directly visualized by different imaging techniques in Bose-Einstein Condensate.

25 Anderson Localization (AL) Definition Beyond a critical amount of impurity, motion of the electron can come to a complete halt. The electron becomes trapped and the conductivity vanishes. Direct observation of AL for electron is very difficult. Number of phenomena can mask single particle quantum effects genuinely induced by disorder. Most of the evidences are indirect and stem from conductivity measurement. Cold atoms are good candidate for observation of AL: Genuine quantum particles described as matter waves. Single atom matter waves can be directly visualized by different imaging techniques in Bose-Einstein Condensate.

26 Experiments Optical Disorder Bichromatic Lattice, 39 K Optical Speckle, 87 Rb Nature 453, 891 (2008). Nature 453, 895 (2008).

27 Experiments In One Dimension Optical Speckle, 87 Rb Bichromatic Lattice, 39 K Nature 453, 891 (2008). Nature 453, 895 (2008).

28 Experiments In Three Dimension Optical Speckle, 87 Rb Optical Speckle, 40 K arxiv: [cond-mat.other]. Science 334, 66 (2011).

29 1 Introduction Prologue Cooling Techniques 2 BCS-BEC Crossover Motivation The Theory of Crossover 3 Disorder Anderson Localization (AL) Experiments 4 The Dirty Crossover Continuum Model (3D) Fidelity Susceptibility (FS) 5 Conclusion Epilogue Acknowledgement

30 Continuum Model (3D) What to Expect? Disorder should not affect the BCS superfluid. Disorder should seriously affect the molecular BEC. What happens in the crossover?

31 Continuum Model (3D) The Scheme Image Source: 9. L. Han & C. A. R. Sa de Melo, New J. Phys. 13, (2011).

32 Continuum Model (3D) Theory-I The Hamiltonian H = Ψ σ [( 2 2m µ) + V (r) ] Ψ σ gψ Ψ Ψ Ψ The random potential originating from the scattering of fermions against impurity atoms: V (r) = j g d δ(r R j ). Corresponding correlation function: V ( q)v (q) = βδ iωm,0κ, where q = (q, iω m ) and κ is the disorder strength.

33 Continuum Model (3D) Theory-II Modified gap and density equation 3 : m = [ 1 4πa k n = k 2E k 1 2ɛ k ] [ 1 ξ k E k ] Ω B µ. Disorder induced thermodynamic potential: 1 Ω B = lim β 0 2β q ln M κ 2 q,ω W m=0 M 1 W 4. G. Orso, Phys. Rev. Lett. 99, (2007).

34 Continuum Model (3D) Theory-II Modified gap and density equation 3 : m = [ 1 4πa k n = k 2E k 1 2ɛ k ] [ 1 ξ k E k ] Ω B µ. Disorder induced thermodynamic potential: 1 Ω B = lim β 0 2β q ln M κ 2 q,ω W m=0 M 1 W 4. G. Orso, Phys. Rev. Lett. 99, (2007).

35 Continuum Model (3D) Theory-II Modified gap and density equation 3 : m = [ 1 4πa k n = k 2E k 1 2ɛ k ] [ 1 ξ k E k ] Ω B µ. Disorder induced thermodynamic potential: 1 Ω B = lim β 0 2β q ln M κ 2 q,ω W m=0 M 1 W 4. G. Orso, Phys. Rev. Lett. 99, (2007).

36 Continuum Model (3D) Theory-II Modified gap and density equation 3 : m = [ 1 4πa k n = k 2E k 1 2ɛ k ] [ 1 ξ k E k ] Ω B µ. Disorder induced thermodynamic potential: 1 Ω B = lim β 0 2β q ln M κ 2 q,ω W m=0 M 1 W 4. G. Orso, Phys. Rev. Lett. 99, (2007).

37 Continuum Model (3D) Analytical Extensions 5 : 1 = 2 [ 2 ] 1/3I1 (x 0 ) k F a π 3I 2 (x 0 ) [ 2 ] 2/3. = ɛ F 3I 2 (x 0 ) Disorder induced Density Equation 6 : ( 2 ) 2/3 η = + ɛ F 3I 1 (x 0 ) π 2 I 3(x 0 ), (η = 0) = η ɛ F π 2 I 3(x 0 ), η = κm 2 /k F 5. M. Marini, F. Pistolesi, and G. C. Strinati, Eur. Phys. J. B 1, 151, (1998). 6. Personal communication with G. Orso is acknowledged.

38 Continuum Model (3D) Analytical Extensions 5 : 1 = 2 [ 2 ] 1/3I1 (x 0 ) k F a π 3I 2 (x 0 ) [ 2 ] 2/3. = ɛ F 3I 2 (x 0 ) Disorder induced Density Equation 6 : ( 2 ) 2/3 η = + ɛ F 3I 1 (x 0 ) π 2 I 3(x 0 ), (η = 0) = η ɛ F π 2 I 3(x 0 ), η = κm 2 /k F 5. M. Marini, F. Pistolesi, and G. C. Strinati, Eur. Phys. J. B 1, 151, (1998). 6. Personal communication with G. Orso is acknowledged.

39 Continuum Model (3D) & µ What they say? Order parameter ( ) remains unaffected by weak disorder in the BCS limit but follows the approximation to the hard core bosons in BEC limit i.e 0 η/k F a. Chemical potential (µ) remains unaffected. A. Khan, IJMPCS 11, 120 (2012).

40 Continuum Model (3D) Condensate Fraction n c = k [ (η) ] 2 2E k (η) Nonmonotonicity of condensate fraction. BCS side follows mean field approximation i.e n c. BEC side follows correction due to disorder i.e n c n c0 η. Enhancement of condensate fraction around the crossover. A. Khan, S. Basu, S. W. Kim, J. Phys. B 45, (2012).

41 Continuum Model (3D) v c & v s What they say? Critical velocity (v c ) is nonmonotonic and the maxima is pinned in the vicinity of a. Sound velocity (v s ) get depleted in the BEC side might be due to additional random scattering rendered by impurity. A. Khan, S. Basu, S. W. Kim, J. Phys. B 45, (2012).

42 Fidelity Susceptibility (FS) Fidelity in Quantum Phase Transition Fidelity is the measure of closeness of two quantum states, F = Ψ Φ (for normalized states). Quantum Phase Transition is a sudden change in the ground state of a many body system when a controlling parameter λ of the Hamiltonian crosses critical value λ c. There should be an abrupt change in the fidelity F (λ + δλ, λ) = Ψ(λ + δλ) Ψ(λ) in the vicinity of λ c. A hasty drop of the ground-state fidelity at the critical point will then correspond to a divergence of the fidelity susceptibility, χ(λ) = 1 Ω Ψ(λ) λ Ψ(λ). λ

43 Fidelity Susceptibility (FS) Fidelity in BCS-BEC Crossover Start with BCS ground state wave function 7, Ψ(λ) = [ k uk (λ) + v k (λ)c ] k c k 0. The fidelity-susceptibility is: χ(λ) = ) 2 ( ) 2 ] + dvk dλ. [( dk duk (2π) 3 dλ The dependence of u k and v k on λ is determined by BCS gap and density equation: k = dk V (2π) 3 λ (k, k ) k 2E, n = dk 2v 2 k (2π) 3 k. The resulting fidelity susceptibility will be: dk 1 [ dµ χ(λ) = (2π) 3 4Ek 4 k dλ + ξ d ] k 2. k dλ 7. A. Khan and P. Pieri, Phys. Rev. A 80, (2009).

44 Fidelity Susceptibility (FS) Clean Fermi Gas Dirty Fermi Gas A. Khan, P. Pieri, Phys. Rev. A 80, (2009). A.Khan, S. Basu and BT, submitted in Phys. Lett. A. Notations χ is in dimensions of k 3 F λ = (k F a) 1. in both the plots.

45 Fidelity Susceptibility (FS) A.Khan, S. Basu and BT, submitted in Phys. Lett. A.

46 Fidelity Susceptibility (FS) Definition Skewness Statistical Analysis S = (x x )3 (x x ) 2 3/2. Kurtosis κ = (x x )4 (x x ) x = (k F a) 1. A.Khan, S. Basu and BT, submitted in Phys. Lett. A. Both S and κ monotonically moves towards zero. η c = obtained from the linear fit of the data.

47 Fidelity Susceptibility (FS) Density of States (DOS) N(ω) = k u 2 k δ(ω E k) + v 2 k δ(ω + E k) A.Khan, S. Basu and BT, submitted in Phys. Lett. A.

48 Fidelity Susceptibility (FS) Spectral Gap A.Khan, S. Basu and BT, submitted in Phys. Lett. A.

49 Fidelity Susceptibility (FS) Observations N(ω) is low in the BCS and BEC regions but the singular pile up is high at the unitarity. Distinct reduction of spectral gap at the unitarity. S and κ data predict for a phase transition at a moderate to high disorder value. From the behavior of DOS as well as FS, we consider the possible phases after transition might be Anderson glass for BCS superfluid, Fermi glass for unitary superfluid and Bose glass for BEC superfluid.

50 Epilogue Field of dirty crossover is introduced. Effect of weak disorder in three dimensional continuum model: Monotonic depletion of order parameter is observed. Nonmonotonic behavior of condensate fraction is discussed. Suppression of sound velocity is presented. Study of FS and DOS: The FS looses symmetric nature in presence of disorder, associated skewness and kurtosis approach zero for large disorder strength. Spectral gap is considerably reduced at unitarity where as BCS and BEC extremes remains unaffected.

51 Epilogue Field of dirty crossover is introduced. Effect of weak disorder in three dimensional continuum model: Monotonic depletion of order parameter is observed. Nonmonotonic behavior of condensate fraction is discussed. Suppression of sound velocity is presented. Study of FS and DOS: The FS looses symmetric nature in presence of disorder, associated skewness and kurtosis approach zero for large disorder strength. Spectral gap is considerably reduced at unitarity where as BCS and BEC extremes remains unaffected.

52 Epilogue Field of dirty crossover is introduced. Effect of weak disorder in three dimensional continuum model: Monotonic depletion of order parameter is observed. Nonmonotonic behavior of condensate fraction is discussed. Suppression of sound velocity is presented. Study of FS and DOS: The FS looses symmetric nature in presence of disorder, associated skewness and kurtosis approach zero for large disorder strength. Spectral gap is considerably reduced at unitarity where as BCS and BEC extremes remains unaffected.

53 Acknowledgement Thanks to my Collaborators Collaboration of Ayan Khan with Sang Wook Kim is acknowledged. Saurabh Basu, Indian Institute of Technology Guwahati, India. Ayan Khan, Bilkent University, Turkey. Thank You for Your Kind Attention