Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Exact Self-Consistent Condensates in (Imbalanced) Superfluid Fermi Gases Giacomo Marmorini Department of Physics, Keio University, Hiyoshi campus
YITP, September 7th, 2010
Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Work in progress with...
Muneto Nitta (Keio U.) Shunji Tsuchiya (Tokyo U. of Science) Ryosuke Yoshii (Keio U.)
Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Outline
1
Introduction
2
Exact self-consistent condensates
3
Ring geometry
4
Population imbalance
5
Conclusion and discussion
Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Keywords
1d geometry Integrability Solitons and BCS-BEC crossover
Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Peculiarities of 1d Cold atoms: If the inter-array potential is strong enough we have a series of genuinely 1d systems with inter-atomic potential V1d (x) = g1d δ(x)
g1d =
2~2 a 1 , ma⊥ 1 − Ca/a⊥
a⊥ =
p ~/mω⊥
Confinement induced resonance Strong coupling by tuning a and ω⊥ Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Integrability: an example Lieb-Liniger Hamiltonian ~2 X d 2m dxi i X + g1d δ(xi − xj )
H =−
A Tonks-Girardeau gas of 87 Rb has been realized and illustrated by Paredes et al.
i
Bethe Ansatz integrable. In the strong coupling regime g1d → ∞ (Tonks-Girardeau), the asymptotic correlations can be found analytically (bosonization, Fisher-Hartwig, replica trick) Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Solitons and BCS-BEC crossover b) Direction of sweep BEC
z y
x Dipole beam
Amount of excitation x [a.u.]
a)
8 7 6 5 4 3 2
0.0 0.2 0.4 0.6 0.8
Sweep velocity [mm/s]
c)
d)
e)
f)
g)
h)
i)
j)
k)
Observation of dark soliton (left) [Burger et al. ’99] and grey soliton (right) [Engels et al. ’07] Question: do they survive BCS-BEC crossover? Dark soliton: yes [Antezza et al. ’07] Grey soliton: ? Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
NJL in 1+1d [Nambu-Jona-Lasinio ’61] i g 2 h ¯ 2 ¯ 5ψ 2 L = ψ¯ i ∂/ ψ + ψψ + ψiγ 2 Upon introducing an auxiliary complex field ∆(x) = σ(x) − iπ(x) and integrating out fermions we arrive at the gap equation ∆(x) = ∗ δ 1 1 5 5 − 2ig ln det i ∂/ − 1 − γ ∆(x) − 1 + γ ∆ (x) δ∆(x)∗ 2 2 2
Very difficult functional equation if ∆(x) is not constant! Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
NJL in 1+1d (cont’d) A possible approach i) Define the Hamiltonian d −i dx ∆(x) H= d ∆∗ (x) i dx ii) Solve the eigenvalue problem Hψ = E ψ subject to the consistency condition X ¯ − ihψiγ ¯ 5 ψi = 2 un vn∗ (1 − 2fn ) = −∆/g 2 hψψi n
Bogoliubov-de Gennes approach to superconductivity in the Andreev approximation Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Andreev approximation
∆0 EF (equiv. kF−1 ξ0 ) justifies Andreev (quasicalssical) approximation (−∇2 /2m − EF )u(x)e ikF x ' −vF ∇u(x) Dirac-type dispersion relation for fermions around the Fermi point Typical in solid state physics: ∆0 /EF ∼ 10−3 ÷ 10−4 Well suited for cold Fermi gases: ∆0 /EF ∼ 10−1
Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
From BdG to Eilenberger Bogoliubov-de Gennes equation Hψ = E ψ d ∆(x) −i dx H= d ∆∗ (x) i dx Resolvent G (x, y ; E ) (Green’s function) (H − E )G (x, y ; E ) = δ(x − y ) Diagonal resolvent (Gor’kov Green’s function) R(x; E ) = hx|
1 1 |xi = lim+ [G (x + , x; E ) + G (x, x + ; E )] H −E →0 2 Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
From BdG to Eilenberger (cont’d) The diagonal resolvent is directly related to the density of states ρ(E ) =
1 Im TrD,x [R(x; E + i)] π
Properties: R = R† trD (R(x; E )σ3 ) = 0 det R(x; E ) = − 14 For any static condensate ∆(x) the Eilenberger eq. holds: ∂ E −∆(x) R(x; E ) σ3 = i , R(x; E ) σ3 ∆∗ (x) −E ∂x Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Consistent Ansatz and NLSE Hints from gap equation i) In terms of gran canonical potential Z 1 ∞ ln det [. . .] = − dE ρ(E ) ln 1 + e −β(E −µ) β −∞ ∆(x) ∝ δ ln det[. . .]/δ∆(x)∗ ⇒ R11 (x, E ) ∼ |∆(x)|2 ii) ∆(x) = −ig 2 trD,E γ 0 1 + γ 5 R(x; E ) ⇒ R12 (x, E ) ∼ ∆(x) Ba¸sar-Dunne’s Ansatz a(E ) + |∆(x)|2 b(E )∆(x) − i∆0 (x) R(x; E ) = N (E ) b(E )∆∗ (x) + i∆0 ∗ (x) a(E ) + |∆(x)|2 The simplest Ansatz that allows an inhomogeneous condensate and is consistent with all the properties of R(x, E )! Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Consistent Ansatz and NLSE (cont’d)
Diagonal components of Eilenberger eq. are automatically satisfied, while off-diagonal components yield ∆00 − 2|∆|2 ∆ + i (b − 2E ) ∆0 − 2 (a − Eb) ∆ = 0 The (very complicated) problem of finding a self-consistent condensate ∆(x) is reduced to solving a nonlinear Schr¨odinger eq.!
Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
General solution of the NLSE Twisted kink crystal [Ba¸sar-Dunne] σ(A x + iK0 − iθ/2) σ(A x + iK0 )σ(iθ/2) × exp [iA x (−i ζ(iθ/2) + i ns(iθ/2)) + i θη3 /2]
∆(x) = − A
Parameters: A sets the overall scale, ν ∈ [0, 1] Elliptic parameter, θ ∈ [0, 4K0 ]. Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
General solution of the NLSE (cont’d) Properties Complex (in general) Quasi-periodic, L = 2K/A ∆(x + L) = e 2iϕ ∆(x) ηθ ϕ = K −i ζ(iθ/2) + i ns(iθ/2) − 2K All previously known solutions can be recovered as various limits
Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Various limits ν → 0 Plane wave ∆(x) = m sech2 (θ/4)e −2im tanh
2
(θ/4)x
which reduces to constant (BCS) ∆(x) = m for θ = 0 ν → 1 Single complex kink [Shei ’76] (“infinite period”) ∆(x) = m
cosh (m sin (θ/2) x − iθ/2) iθ/2 e cosh (m sin (θ/2) x)
(cf. BEC grey soliton) which reduces to hyperbolic-tangent kink (first in polyacetylene [Takayama-Lin-Liu-Maki]) for θ = π (cf. BEC dark soliton) Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Various limits (cont’d) θ = 2K0 Real case [Brazovskii et al.; Machida et al.] ∆(x) = m sn(x; ν) 1.0
0.5
-20
10
-10
20
-0.5
-1.0
namely an array of hyperbolic-tangent kinks, which assumes a sinusoidal form for small ν (FFLO form)
Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Fermionic spectrum E
E
m
m
Θ 2K’
Θ Π
4K’
-m
2Π
-m
BdG energy spectrum versus θ for 0 < ν < 1 and ν = 1.
Besides the continuum states, there is a midgap band that merges the continuum when the soliton degenerates to a constant (θ = 0, 4K0 ) The midgap band shrinks to a single bound state for ν → 1 (single kink) Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Fermion number fractionalization
Single kink (ν = 1): the bound state carries fractional fermion number (well known in relativistic QFTs [Niemi-Semenoff]; proposals of detection in cold atoms confined into optical lattices [Dunne et al.]) θ 1 Nf = − tan−1 (cot ) π 2 Kink crystal: the midgap band is partially filled; the filling fraction is given by ξ = θ/4K0
Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Ring geometry: quantization condition
Radius of the ring R = n0 L/2π = n0 K/πA ∼ 1/m. The quantization condition for ∆(x) implies ∆(x + n0 L) = e 2inπ ∆(x)
⇒ ϕ=
n π n0
n, n0 ∈ Z
n0 number of periods in the ring n total “vorticity” (phase rotation) Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Periodicity For a given ν, the period of ∆(x) depnds only on n/n0 and is determined numerically 4.0
3.5
3.0
0.3
0.4
0.5
0.6
0.7
0.8
Symmetry about n/n0 = 1/2: time-reversal symmetry n0 = 2n, θ = 2K0 : Real case Angular-FFLO Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Superfluid current 1 ∗ (∆ (x)∆0 (x) − ∆(x)∆0∗ (x)) 2i 1 0 3 0 = iA ns(iθ/2) ℘(Ax + iK ) − ℘(iθ/2) − ℘ (iθ/2) 2
J(x, θ) =
JHxL 0.30 0.25 0.20 0.15 0.10 0.05 x -1.0
-0.5
0.5
1.0
Inhomogeneous current (comoving frame) Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Superfluid current (cont’d) Current peak as a function of θ 1 0 3 0 J(0, θ) = iA ns(iθ/2) ℘(iK ) − ℘(iθ/2) − ℘ (iθ/2) 2 JH0,ΘL 0.3 0.2 0.1 Θ 2
4
6
8
10
-0.1 -0.2 -0.3
Josephson-like behavior Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Lessons from solid state physics Bloch theorem (relativistic) ψ± (x + L) = e ±ikL e iϕγ5 ψ± (x) Brillouin zones Band structure: an energy gap forms at the edge of a Brillouin zone 3
2
1
-1.0
0.5
-0.5
1.0
-1
-2
-3
Energy vs. Bloch momentum (illustrative) Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Lessons from solid state physics Bloch theorem (relativistic) ψ± (x + L) = e ±ikL e iϕγ5 ψ± (x) Brillouin zones Band structure: an energy gap forms at the edge of a Brillouin zone 3
2
1
-1.0
0.5
-0.5
1.0
-1
-2
-3
1st Brillouin zone Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Lessons from solid state physics (cont’d) The “true” period L0 is defined by ∆(x + L0 ) = ∆(x) and the corresponding fundamental Brillouin zone is −π/L0 < k ≤ π/L0 (“folded”) Example: if n, n0 are coprime, then L0 = n0 L Even for fixed length of the ring 2πR, one can accomodate different pairs (n, n0 ) Possibility of Hofstadter’s butterfly
Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Hofstadter’s butterfly Electrons in 2d periodic potential and transverse magnetic field If the magnetic flux through the unit cell α is a rational number (in units of hc/e), namely α = p/q, then an analog of Bloch theorem holds and the energy spectrum plotted against α gives rise to a fractal, the Hofstadter’s butterfly [Hofstadter ’76]
Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
(Pseudo-)spin imbalance Typically two hyperfine states of, e.g., 6 Li, whose imbalance is controlled by radiofrequency pulses Imbalanced BdG equation −ivF↑ ∂x ∆(x) ψ = E ψ, ∆∗ (x) ivF↓ ∂x The modified Eilenberger equation reads " ! # −1 vF↑ 0 E −∆(x) ∂x R(x; E ) σ3 = i , R(x; E ) σ3 −1 ∆∗ (x) −E 0 vF↓
Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Population imbalance (cont’d) Upon rescaling of the parameters (only!) c 2 = vF↑ vF↓ , ˜ = c −1 ∆, ∆
˜a = c −2 a, b˜ = c −2 b, 1 −1 −1 E˜ = (vF↑ + vF↓ )E 2
we get the same nonlinear Schr¨ odinger eq. ˜ 00 − 2∆| ˜ ∆| ˜ 2 + i(b˜ − 2E˜ )∆ ˜ 0 − 2(˜a − E˜ b) ˜ ∆ ˜ =0 ∆ ˜ The solutions for the condensate ∆(x) will have the same analytic form (“shape”) as in the balanced case!
Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Angular FFLO state
Numerical calculations indeed show that the A-FFLO not only exists, but is stable in a certain region of the phase space [imbalance-temperature] (a toroidal trap is assumed)
(a) P=0
-0.4
0
0.4
(b) P=0.21
0.8
1.2
1.6
(d) P=0.44
-1.5
-1
-0.5
0
0.5
1
-0.4
0
0.4
0.8
1.2
(e) P=0.49
1.5
-1.5
-1
-0.5
0
0.5
1
(c) P=0.39
1.6 -0.4
0
0.4
0.8
1.2
1.6
(f) P=0.69
1.5 -1.5
-1
-0.5
0
0.5
1
1.5
[Yanase ’09]
Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Summary
Exact method to find self-consistent superfluid condensates in 1d Most general solution in linear and circular geometry and its physicsl implications (supercurrent, bound states, fermion number fractionalization, etc.) Extension to population imbalance
Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
Problems and directions Finite temperature Stability of the complex kink crystal - phase diagram Exact spinor solution of BdG eq. in the imbalanced case (preliminary results for perturbatively small imbalance are already available) How to create the complex kink crystal in the lab Different pairing Interfering 1d arrays Multi-gap Co-dimension bigger than one Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
Introduction Exact self-consistent condensates Ring geometry Population imbalance Conclusion and discussion
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Giacomo Marmorini
Exact Self-Consistent Condensates in Superfluid Fermi Gases
