[PPT] - Gas in the Ring Geometry Eugene Zaremba Queens University, PowerPoint Presentation

SLIDE 1

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

SLIDE 2

Experiments on Persistent Currents

S. Moulder et al., Phys. Rev. A 86, 013629 (2012)
A. Ramanathan et al., Phys. Rev. Lett. 106, 130401 (2011)

SLIDE 3

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

Bloch argued that, if E0(L) exhibits local minima at ,

persistent currents are stable

E0(L) = L2 2MT R2 + e0(L)

where e0(L) is even and periodic: Ln = nN~

*F. Bloch, Phys. Rev. A7, 2187 (1973)

L = ν~

Yrast Spectrum

e0(−L) = e0(L), e0(L + N~) = e0(L)

SLIDE 4

Yrast spectrum of the Lieb-Liniger model and connection to the soliton solutions of the GP equation

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 many-body wavefunctions can be obtained using the Bethe

ansatz (Lieb and Liniger, 1963)

ˆ H = − ~2 2MR2 ∂2 ∂θ2 + 1 2U X

ij

δ(θi − θj)

the excitations corresponding to the yrast spectrum can be identified

as solitons (Ishikawa and Takayama, 1980)

the Hamiltonian for 1D bosons interacting via a delta function

potential is given by

SLIDE 5

Mean-field Analysis for the Single-component Case

the Gross-Pitaevskii energy functional for bosons on a ring is
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

this can be achieved by minimizing the functional

where and are Lagrange multipliers Ω

¯ E[ψ] = Z 2π dθ

dψ

dθ

2

+ πγ Z 2π dθ|ψ(θ)|4

¯ L = 1 i Z 2π dθ ψ∗ dψ dθ = l ¯ F[ψ] = ¯ E[ψ] − Ω¯ L − µ Z 2π dθ|ψ(θ)|2

this leads to the GP equation

−ψ00(θ) + iΩψ0(θ) + 2πγρ(θ)ψ(θ) = µψ(θ)

µ

SLIDE 6

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 − Ωt)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)

3 2 1 1 2 3 ¯ E0(l) − γ/2 l

SLIDE 7

Extension of Bloch’s Argument to the Two-species System

we consider an ideal 1D ring geometry with NA particles of mass

MA and NB particles of mass MB; N = NA+NB, MT = NAMA+NBMB

the many-body wave function can be written as

ΨLα(θ1, ..., θN) = exp(iNlΘcm)χLα(θ1, ..., θN)

Θcm = 1 MT

N

X

i=i

Miθi, L = lN~

where

χLα(θ1,…,θN) is a function of coordinate differences θi – θj and is

therefore a zero angular momentum wave function

SLIDE 8

Extension of Bloch’s Argument, cont’d

if MA/MB = p/q, a rational number, eα(L) is a periodic function with

period where

χLα(θ1,…,θN) satisfies the SchrÖdinger equation

HχLα = eα(L)χLα

with the boundary conditions

˜ N = pNA + qNB

for MA = MB = M, p = q = 1 and the periodicity is N as for the

single-species case

χLα(· · · , θi + 2π, · · · ) = exp ✓ −i2π νMi MT ◆ χLα(· · · , θi, · · · ) ˜ N~ eα(L) = Eα(L) − L2 2MT R2

SLIDE 9

Connection with Landau’s Criterion

where we have defined the angular velocity

MA = MB = M; Bloch’s argument allows for persistent currents at

Ln = nN~

E0(Ln + ∆L) = 1 2MT R2Ω2

n + Ωn∆L + E0(∆L)

Ωn = Ln MT R2

assuming E0(ΔL) to correspond to a single quasiparticle excitation

with energy ε(m) and angular momentum , we have

E0(Ln + ∆L) = E0(Ln) + ε(m) + m~Ωn

Bloch’s criterion for persistent currents, E0(Ln + ΔL) > E0(Ln), then

implies

Ωn < ✓ε(m) ~|m| ◆

min

this is the Landau criterion for a ring

∆L = m~

; for

L = Ln + ∆L

SLIDE 10

Bogoliubov Excitations in a Ring

the Landau criterion is satisfied for most choices of the parameters,

implying the stability of superfluid flow at Ln

the two-species system has Bogoliubov excitations with energies

E2

± = 1

2

E2

A + E2 B

± 1

2 q (E2

A + E2 B)2 + 4 (4✏A✏Bg2 AB − E2 AE2 B)

where

gss = Uss0√NsNs0 2π

Es = p ✏2

s + 2✏sgss,

✏s = ~2m2 2MsR2

however, if MA = MB and UAAUBB = U2

AB , the E_ mode is particle-

like and supercurrents are not stable at Ln

SLIDE 11

Mean-field Analysis

*J. Smyrnakis et al., Phys. Rev. Lett 103, 100404 (2009)

with

for the special case MA = MB and UAA = UBB = UAB = U, the so-

called symmetric model, the Gross-Pitaevskii energy functional is

the stability of persistent currents can be analyzed using mean-field

theory; this was first done by Smyrnakis et al.*

¯ E[ψA, ψB] = Z 2π dθ xA

dψA

dθ

2

+ xB

dψB

dθ

2!

+πγ Z 2π dθ

xA|ψA|2 + xB|ψB|22

xA = NA/N, xB = NB/N, γ = NMR2U/π~2

the objective is to minimize the GP energy with respect to ψA and

ψB subject to the constraint that the average total angular momentum has the value

L = lN~

this can be achieved by minimizing the functional

¯ F[ψA, ψB] = ¯ E[ψA, ψB] − Ω¯ L − X

s

xsµs Z 2π dθ|ψs(θ)|2

where and are Lagrange multipliers Ω

µs

SLIDE 12

Minimizing the GP Energy

the condensate wave functions are expanded as
the expansion coefficients must satisfy the normalization

constraints

ψA(θ) = X

m

cmφm(θ), ψB(θ) = X

m

dmφm(θ) φm(θ) = eimθ √ 2π

where

X

m

|cm|2 = 1, X

m

|dm|2 = 1

and the angular momentum constraint

l = xA X

m

m|cm|2 + xB X

m

m|dm|2

SLIDE 13

Two-component Analysis

the simplest variational ansatz is

ψA = c0φ0 + c1φ1, ψB = d0φ0 + d1φ1

minimizing the energy with respect to c0, c1, d0 and d1, one finds

¯ E0(l) = l + γ/2

this result is exact if

0 ≤ l ≤ xB

r

xA ≤ l ≤ 1

xA

l

1 2 xB

E0 ) l (

as predicted by the Landau criterion, superfluid flow is unstable at Ln
however, there is a possibility that persistent currents might be stable

for l in the range xB < l < xA

to examine this possibility, an improved variational ansatz is required

SLIDE 14

Persistent Currents at l = xA + n

the stability of persistent currents at l = xA + n is determined by the

slope

xA

l

1 2 xB

E0 ) l (

d ¯ E0(l) dl

l=(xA+n−1)− = 2n − 1 − λ
for n = 1, the critical value of the interaction parameter is

γcr = 3 2(4xA − 3)

this gives the correct value of γcr = 3/2 for xA = 1; however, the

above expression predicts that persistent currents are not possible for n > 1 (Smyrnakis et al., 2009)

¯ E0(l) = l2 + ¯ e0(l)

SLIDE 15

Analytic Soliton Solutions*

the coupled GP equation for equal interactions strengths
Js is the soliton winding number
here, the angular velocity, Ω, is a Lagrange multiplier

introduced to ensure the angular momentum takes a specific value l

modulus-phase representation

−ψ00

s (θ) + iΩψ0 s(θ) + 2πγρ(θ)ψs(θ) = µsψs(θ)

boundary conditions

ρs(θ + 2π) − ρs(θ) = 0 φs(θ + 2π) − φs(θ) = 2πJs, Js = 0, ±1, ±2, · · ·

ψs(θ) = p ρs(θ)eiφs(θ), Z 2π dθ ρs(θ) = 1

*Z. Wu and E. Zaremba, Phys. Rev. A 88, 063640 (2013)

SLIDE 16

Density Ansatz

the ansatz (Porubov and Parker, 1994; Smyrnakis et al., 2012)
the density equation for the single-component system was

solved by others; it has analytic solutions in terms of Jacobi elliptic functions reduces the coupled system to two uncoupled equations for the with different interaction strengths

1 2ρsρ00

s − 1

4(ρ0

s)2 − 2πγsρ3 s + ˜

µsρ2

s − W 2 s

4 = 0 ρs(θ) = N(ηs) ⇥ 1 + ηsdn2 (u|m) ⇤

u = jK(m) π (θ − θ0)

ρB = 1 − r 2π + rρA

γA = (xA + rxB)γ γB = (r−1xA + xB)γ

where j is the soliton train index and ηs is a number which depends on γs (hence xA, γ and r); m is the elliptic parameter defining the complete elliptic integral K(m) densities

SLIDE 17

Phase Boundary Condition

with the density solutions in hand, the phase boundary

condition gives

this equation provides a relation between the elliptic index m

defining the density distributions and the parameter r appearing in the density ansatz

J ≡ JB − JA = 1 4π Z 2π dθ  WB ρB(θ) − WA ρA(θ)

the different branches

correspond to different soliton states in different ranges of angular momentum given by

xB = 0.04 xB = 0.2

γ = 23

kxB ≤ l ≤ (k + 1)xB

the winding numbers JA

and JB take specific values along each of the branches

SLIDE 18

Soliton States and Yrast Spectrum

xB = 0.04

γ = 23

xB = 0.2, 0.04, 0.008, 0

SLIDE 19

Persistent Currents at Higher Angular Momenta

xB = 0.04

the soliton solutions explain how

the limit is reached

xB → 0

persistent currents are in fact

possible at higher angular momenta for sufficiently small xB and sufficiently large γ

SLIDE 20

Asymmetric Interactions:

Can one obtain a criterion for determining whether a plane wave state is an yrast state and secondly, whether this state supports persistent currents?

there are no known analytic solutions to the coupled GP equations

for asymmetric interactions – the density ansatz used for the symmetric model does not work

the analysis of the symmetric model showed that certain plane

wave states are special in that they can be yrast states and can sustain persistent currents

we expect certain plane wave states to continue playing an

important role in the yrast spectrum of the asymmetric model γAA, γBB, γAB

SLIDE 21

Local Minima of the GP Energy Functional*

we suppose is a candidate plane-wave yrast state
for an arbitrary deviation , the

change in GP energy is

(φµ, φν) ψA = φµ + δψA, ψB = φν + δψB

the energy has a local minimum if is positive-definite; this

yields the following conditions:

2xAγAA + 1 − 4µ2 > 0 (2xAγAA + 1 − 4µ2)(2xBγBB + 1 − 4ν2) − 4xAxBγ2

AB > 0

if these conditions are not satisfied, persistent currents are not

possible

*Z. Wu et al., Phys. Rev. A 92, 033630 (2015)

δ ¯ E = ¯ E[ψA, ψB] ¯ E[φµ, φν] ' X

m>0

v†

mHmvm,

vm = (δcµ−m δc∗

µ+m δdν−m δd∗ ν+m)T

Hm

SLIDE 22

Effect of Interaction Asymmetry on the Stability of Persistent Currents

the inequalities allow one to determine the range of parameters

for which persistent currents at a particular plane-wave state are possible

(φ1, φ0)

50 100 150 0.2 0.4 0.6 0.8 1 γ xB

κB = −1/22 κB = −1/11 κB = −2/11

γ = γAB

κA = 0.1

example of
for κ = -1/11, ;

the xB-γ boundary is similar to that found in the symmetric model

γAA = (1 + κA)γ, γBB = (1 + κB)γ

γAAγBB − γ2

AB = 0

SLIDE 23

On the Cusp

for a plane-wave yrast state supporting

persistent currents at , the yrast spectrum looks locally like:

l0 = µxA + νxB

the slopes of the yrast spectrum can be obtained by minimizing

the GP energy functional subject to the constraint

¯ L = l0 + δl

Ω(l) = ∂ ¯ E0(l) ∂l

one obtains a quartic with roots

Ω1 < Ω2 < Ω3 < Ω4

the middle two give the slopes of the

yrast spectrum at the plane-wave state

if one of these roots goes to zero,

persistent currents are no longer possible; this happens precisely when the inequality is violated

SLIDE 24

Possible Criterion for Plane-wave Yrast States

a double root of the quartic signals when the plane-wave state

ceases to be an yrast state

whether this scenario is correct must

be checked by numerical calculations

SLIDE 25

Numerical Solutions; Preliminary Results

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Angular momentum

0.2 0.4 0.6 0.8 1 1.2

Energy, shifted Yrast spectrum

one can obtain solutions to the

coupled GP equations by imaginary time propagation

the results support the criterion for the

stability of persistent currents at plane-wave states

but there is evidence that the criterion

for a plane-wave state being an yrast state is not generally valid

there is no difficulty solving the GP