Persistent Currents in a Two-component Bose Gas in the Ring Geometry � Eugene Zaremba Queen’s University, Kingston, Ontario, Canada Financial support from NSERC Work done in collaboration with Konstantin Anoshkin and Zhigang Wu; also Smyrnakis, Magiropoulos, Efremidis and Kavoulakis
Experiments on Persistent Currents � A. Ramanathan et al ., Phys. Rev. Lett. 106 , 130401 (2011) S. Moulder et al ., Phys. Rev. A 86 , 013629 (2012)
Bloch’s Criterion for Persistent Currents* � • for a single-species system in the one-dimensional ring geometry, Bloch showed that the ground state energy takes the form L 2 Yrast Spectrum E 0 ( L ) = 2 M T R 2 + e 0 ( L ) L = ν ~ where e 0 ( L ) is even and periodic: e 0 ( − L ) = e 0 ( L ) , e 0 ( L + N ~ ) = e 0 ( L ) • Bloch argued that, if E 0 ( L ) exhibits local minima at , L n = nN ~ persistent currents are stable *F. Bloch, Phys. Rev. A 7 , 2187 (1973)
Yrast spectrum of the Lieb-Liniger model and connection to the soliton solutions of the GP equation � • the Hamiltonian for 1D bosons interacting via a delta function potential is given by ~ 2 ∂ 2 ∂θ 2 + 1 ˆ X H = − δ ( θ i − θ j ) 2 U 2 MR 2 ij • the many-body wavefunctions can be obtained using the Bethe ansatz ( Lieb and Liniger, 1963 ) • particle momentum shows no BEC in the thermodynamic limit • the yrast spectrum corresponds to Lieb’s type II excitations ( Lieb , 1963 ); it can be determined explicitly using the Lieb-Liniger solution ( Kaminishi et al., 2011 ) • the excitations corresponding to the yrast spectrum can be identified as solitons ( Ishikawa and Takayama, 1980 )
Mean-field Analysis for the Single-component Case � • the Gross-Pitaevskii energy functional for bosons on a ring is Z 2 π Z 2 π 2 � � d ψ ¯ � � d θ | ψ ( θ ) | 4 E [ ψ ] = + πγ d θ � � d θ � � 0 0 • the yrast spectrum is obtained by minimizing the GP energy with respect to ψ subject to the constraint that the average angular momentum has the value Z 2 π d θ ψ ∗ d ψ L = 1 ¯ d θ = l i 0 • this can be achieved by minimizing the functional Z 2 π F [ ψ ] = ¯ ¯ E [ ψ ] − Ω ¯ d θ | ψ ( θ ) | 2 L − µ 0 where and are Lagrange multipliers Ω µ • this leads to the GP equation − ψ 00 ( θ ) + i Ω ψ 0 ( θ ) + 2 πγρ ( θ ) ψ ( θ ) = µ ψ ( θ )
Mean-field Solutions and Yrast Spectrum � • the mean-field solutions for a general value of l are solitons • this stationary state solution represents a travelling soliton as viewed in a rotating frame; in the lab frame, the soliton is the time- dependent state ψ ( θ , t ) = ψ ( θ − Ω t ) e − iµt • the energy of the mean-field soliton agrees with the exact many- body energy if the interactions are not too strong ( Kanamoto et al. , 2010 ) E 0 ( l ) − γ / 2 ¯ � 3 � 2 � 1 0 1 2 3 l
Extension of Bloch’s Argument to the Two-species System � • we consider an ideal 1D ring geometry with N A particles of mass M A and N B particles of mass M B ; N = N A +N B , M T = N A M A +N B M B • the many-body wave function can be written as Ψ L α ( θ 1 , ..., θ N ) = exp( iNl Θ cm ) χ L α ( θ 1 , ..., θ N ) where N 1 X Θ cm = M i θ i , L = lN ~ M T i = i • χ L α ( θ 1 ,…, θ N ) is a function of coordinate differences θ i – θ j and is therefore a zero angular momentum wave function
Extension of Bloch’s Argument, cont’d � • χ L α ( θ 1 ,…, θ N ) satisfies the Schr Ö dinger equation L 2 H χ L α = e α ( L ) χ L α e α ( L ) = E α ( L ) − 2 M T R 2 with the boundary conditions ✓ ◆ − i 2 π ν M i χ L α ( · · · , θ i + 2 π , · · · ) = exp χ L α ( · · · , θ i , · · · ) M T • if M A / M B = p / q , a rational number, e α ( L ) is a periodic function with period where ˜ N ~ ˜ N = pN A + qN B • for M A = M B = M , p = q = 1 and the periodicity is N as for the single-species case
Connection with Landau’s Criterion � • M A = M B = M ; Bloch’s argument allows for persistent currents at ; for L = L n + ∆ L L n = nN ~ E 0 ( L n + ∆ L ) = 1 2 M T R 2 Ω 2 n + Ω n ∆ L + E 0 ( ∆ L ) where we have defined the angular velocity L n Ω n = M T R 2 • assuming E 0 ( Δ L ) to correspond to a single quasiparticle excitation with energy ε ( m ) and angular momentum , we have ∆ L = m ~ E 0 ( L n + ∆ L ) = E 0 ( L n ) + ε ( m ) + m ~ Ω n • Bloch’s criterion for persistent currents, E 0 ( L n + Δ L ) > E 0 ( L n ), then implies ✓ ε ( m ) ◆ Ω n < ~ | m | min • this is the Landau criterion for a ring
Bogoliubov Excitations in a Ring � • the two-species system has Bogoliubov excitations with energies ± = 1 ± 1 q B ) 2 + 4 (4 ✏ A ✏ B g 2 E 2 � E 2 A + E 2 � ( E 2 A + E 2 AB − E 2 A E 2 B ) B 2 2 where ✏ s = ~ 2 m 2 p ✏ 2 E s = s + 2 ✏ s g ss , 2 M s R 2 g ss = U ss 0 √ N s N s 0 2 π • the Landau criterion is satisfied for most choices of the parameters, implying the stability of superfluid flow at L n • however, if M A = M B and U AA U BB = U 2 AB , the E_ mode is particle- like and supercurrents are not stable at L n
Mean-field Analysis � • the stability of persistent currents can be analyzed using mean-field theory; this was first done by Smyrnakis et al .* • for the special case M A = M B and U AA = U BB = U AB = U , the so- called symmetric model, the Gross-Pitaevskii energy functional is Z 2 π Z 2 π 2 ! 2 � � � � d ψ A d ψ B x A | ψ A | 2 + x B | ψ B | 2 � 2 ¯ � � � � � E [ ψ A , ψ B ] = d θ + x B + πγ d θ x A � � � � d θ d θ � � � � 0 0 with γ = NMR 2 U/ π ~ 2 x A = N A /N, x B = N B /N, • the objective is to minimize the GP energy with respect to ψ A and ψ B subject to the constraint that the average total angular momentum has the value L = lN ~ • this can be achieved by minimizing the functional Z 2 π F [ ψ A , ψ B ] = ¯ ¯ E [ ψ A , ψ B ] − Ω ¯ X d θ | ψ s ( θ ) | 2 L − x s µ s 0 s where and are Lagrange multipliers Ω µ s *J. Smyrnakis et al ., Phys. Rev. Lett 103 , 100404 (2009)
Minimizing the GP Energy � • the condensate wave functions are expanded as X X ψ A ( θ ) = c m φ m ( θ ) , ψ B ( θ ) = d m φ m ( θ ) m m where φ m ( θ ) = e im θ √ 2 π • the expansion coefficients must satisfy the normalization constraints | c m | 2 = 1 , | d m | 2 = 1 X X m m and the angular momentum constraint m | c m | 2 + x B X X m | d m | 2 l = x A m m
Two-component Analysis � • the simplest variational ansatz is ψ A = c 0 φ 0 + c 1 φ 1 , ψ B = d 0 φ 0 + d 1 φ 1 • minimizing the energy with respect to c 0 , c 1 , d 0 and d 1 , one finds ¯ E 0 ( l ) = l + γ / 2 E 0 ( l ) • this result is exact if 0 ≤ l ≤ x B or x A ≤ l ≤ 1 0 x B x A 1 2 l • as predicted by the Landau criterion, superfluid flow is unstable at L n • however, there is a possibility that persistent currents might be stable for l in the range x B < l < x A • to examine this possibility, an improved variational ansatz is required
Persistent Currents at l = x A + n � E 0 ( l ) E 0 ( l ) = l 2 + ¯ ¯ e 0 ( l ) 0 x B x A 1 2 l • the stability of persistent currents at l = x A + n is determined by the slope d ¯ � E 0 ( l ) � l =( x A + n − 1) − = 2 n − 1 − λ � dl � • for n = 1 , the critical value of the interaction parameter is 3 γ cr = 2(4 x A − 3) • this gives the correct value of γ cr = 3/2 for x A = 1 ; however, the above expression predicts that persistent currents are not possible for n > 1 (Smyrnakis et al ., 2009)
Analytic Soliton Solutions * � • the coupled GP equation for equal interactions strengths − ψ 00 s ( θ ) + i Ω ψ 0 s ( θ ) + 2 πγρ ( θ ) ψ s ( θ ) = µ s ψ s ( θ ) • here, the angular velocity, Ω , is a Lagrange multiplier introduced to ensure the angular momentum takes a specific value l • modulus-phase representation Z 2 π p ρ s ( θ ) e i φ s ( θ ) , ψ s ( θ ) = d θ ρ s ( θ ) = 1 0 • boundary conditions ρ s ( θ + 2 π ) − ρ s ( θ ) = 0 φ s ( θ + 2 π ) − φ s ( θ ) = 2 π J s , J s = 0 , ± 1 , ± 2 , · · · • J s is the soliton winding number *Z. Wu and E. Zaremba, Phys. Rev. A 88 , 063640 (2013)
Density Ansatz � • the ansatz ( Porubov and Parker, 1994; Smyrnakis et al., 2012 ) ρ B = 1 − r + r ρ A 2 π reduces the coupled system to two uncoupled equations for the densities s − W 2 1 s − 1 s ) 2 − 2 πγ s ρ 3 µ s ρ 2 2 ρ s ρ 00 4( ρ 0 s + ˜ s = 0 4 with different interaction strengths γ A = ( x A + rx B ) γ γ B = ( r − 1 x A + x B ) γ • the density equation for the single-component system was solved by others; it has analytic solutions in terms of Jacobi elliptic functions u = jK ( m ) 1 + η s dn 2 ( u | m ) ⇥ ⇤ ρ s ( θ ) = N ( η s ) ( θ − θ 0 ) π where j is the soliton train index and η s is a number which depends on γ s (hence x A , γ and r ); m is the elliptic parameter defining the complete elliptic integral K ( m )
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