Supplementary Material for Anomalous conductances in an ultracold quantum wire

M. Kanasz-Nagy1, L. Glazman2, T. Esslinger3, and E. A. Demler1

1Department of Physics, Harvard University, Cambridge, MA 02138, U.S.A.2Department of Physics, Yale University, New Haven, CT 06520, U.S.A. and

3Department of Physics, ETH Zurich, 8093 Zurich, Switzerland

2

I. HAMILTONIAN

We determine the superconducting profile inside the elliptic constriction within the local density approximation(LDA). Assuming harmonic confinement in the x and z transverse directions, the kinetic and interacting part of thelocal Hamiltonian are given by

Hkin =

∫d3r

∑σ=↑,↓

φ†σ(r)

(−~2∆

2m+m

2

(ω2xx

2 + ω2zz

2)− µσ − Vg

)φσ(r), (S1)

Hint = g

∫d3r φ†↑(r)φ†↓(r)φ↓(r)φ↑(r), (S2)

where ωx and ωz denote the local values of the trapping frequencies, whereas Vg stands for that of the gate potential.In equilibrium, the chemical potentials µσ are constant across the constriction, and they are set by the leads. Thebare coupling constant g is related to the scattering lengths, as we discuss in Sec. III A. We decompose the fermionicfields φσ(r) in terms of the eigenmodes of the transverse confining potential

φσ(r) ≡ 1√Ly

∑n=(nx,nz)

∑q

eiqy ϕnx(x)ϕnz (z) an,σ,q,

where n = (nx, nz) denotes the transverse harmonic oscillator quantum numbers, and Ly is the length of the system.ϕnx(z) stands for the harmonic oscillator wave functions of oscillator length lx(z). We rewrite the Hamiltonian in termsof these transverse modes as

Hkin =∑σ=↑,↓

∑n,q

ξn,σ,q a†n,σ,q an,σ,q

Hint =g

Ly

∑n1n2n3n4

∑k,k′,q

〈〈n1n2||n3n4〉〉 a†n1,↑,k+qa†n2,↓,k′−qan3,↓,k′an4,↑,k,

with the single particle energies ξn,σ,q given below Eq. (1) in the main text. The matrix element of the Dirac deltainteraction potential with transverse oscillator modes is given by

〈〈n1n2||n3n4〉〉 ≡ 〈ϕnx1(x1), ϕnx2(x2)| δ(x1 − x2) |ϕnx3(x2), ϕnx4(x1)〉 (S3)

· 〈ϕnz1(z1), ϕnz2(z2)| δ(z1 − z2) |ϕnz3(z2), ϕnz4(z1)〉 .

II. SUPERCONDUCTING FREE ENERGY

We calculate the superconducting free energy inside the constriction within Bardeen–Cooper–Schrieffer (BCS)theory. Our analysis is an extension of the approach of Ref. 1 that considered BCS mean-field theory in quasi-two-dimensional geometry. As a first step, we transform the Hamiltonian into the basis of relative and center of masscoordinates (X,Z) = ((x1 + x2)/2, (z1 + z2)/2) and (x, z) = (x1 − x2, z1 − z2). The associated harmonic oscillatormodes can be written as

|Nx〉 ≡ 21/4 ϕNx

(√2X), |νx〉 ≡

1

21/4ϕνx

(x/√

2)≡ ϕνx(x),

and similarly for the z axis. Within this basis, the interaction matrix element Eq. (S3) is given by

〈〈n1,n2||n3,n4〉〉 =∑

N1,N2,ν1,ν2

〈n1,n2|N1,ν1〉 〈N2,ν2|n3,n4〉 ϕν1(0)ϕν2(0) δN1,N2 , (S4)

where we introduced the notations |N〉 ≡ |Nx, Nz〉, |ν〉 ≡ |νx, νz〉, and ϕν(0) ≡ ϕνx(0) ϕνz (0). Since the interactiononly depends on the relative coordinates of the colliding atoms, the center of mass quantum number is conserved.Using the orthogonality property of single particle eigenstates, one finds that only those matrix elements 〈n1,n2|N,ν〉are non-zero, for which n1 + n2 = N + ν.

We define the dimensionless matrix elements V n1n2

N ≡√lx lz ϕν(0) 〈Nν|n1n2〉, where the oscillator mode of trans-

verse motion is given by ν ≡ n1 + n2 −N. We rewrite the interaction vertex Eq. (S4) as

〈〈n1n2||n3n4〉〉 =1

lx lz

∑N

(V †N)n1n2V n3n4

N . (S5)

3

Using these matrix elements, we express the interaction Hamiltonian in the form

Hint =1

g

∑N,q

∆†N,q∆N,q,

introducing the operators ∆N,q ≡ g∑

n3n4V n3n4

N

∑k an3,↓,q−kan4,↑,k, and the bare coupling g = g/(Lylxlz), which

has dimension energy. We introduce separate order parameters for every one of the transverse center of mass channelsN , but assume that only q = 0 acquires non-zero expectation value, ∆N ≡ 〈∆N,q=0〉. This is analogous to theassumption of the zero momentum pairing for homogeneous superconductors. We decouple the interaction withinBCS theory, and arrive at the mean-field Hamiltonian

HMF =∑

n1,n2,q

(a†n1,↑,q, an1,↓,−q

)(ξ↑,q ∆∆† −ξ↓,q

)(an2,↑,qa†n2,↓,−q

)+∑n,q

ξn1,↓,q −∑N

|∆N|2

g, (S6)

with the kinetic and pairing terms given by(ξσ,q

)n1n2

= δn1n2ξσ,n1,q and ∆n1n2 =∑

N ∆N(V †N)n1n2 , respectively.

We diagonalize HMF using the Bogoliubov transformation(γ†↑,qγ↓,q

)=

(uq −v∗qvq u∗q

) (a†↑,qa↓,−q

).

where the unitarity of the transformation ensures that operators γσ,n,q obey fermionic anti-commutation relations.After the Bogoliubov transformation,(

uq −v∗qvq u∗q

)†(ξ↑,q ∆∆† −ξ↓,q

)(uq −v∗qvq u∗q

)=

(Eq 00 −Eq

)(S7)

the Hamiltonian becomes diagonal, HMF = EMF +∑

n,q,σ En,q γ†σ,n,qγσ,n,q, with the ground state energy

EMF =∑n,q

(ξ↓,n,q − En,q)−∑N

|∆N|2

g. (S8)

In order to determine the superconducting order parameters ∆N across the constriction, we need to find the minimumof the BCS free energy FMF = EMF − kBT

∑n,σ,q log

(1 + e−Enσq/kBT

)with respect to ∆N, at a finite temperature

T .One of the challenges in analyzing equation (S8) is that taken directly with the coupling constant g, it is UV

divergent. However this divergence should disappear when one uses the Lippmann–Schwinger relation between themicroscopic interaction and the scattering length a. This is similar to the three dimensional case, where the freeenergy can be regularized after expressing the bare coupling in terms of the scattering length, a [2]. Details of suchregularization procedure for quasi-one dimensional gases are presented in Sec. III. One of the aspects of this analysisis the effect of the confinement potential on the T -matrix describing two particle scattering processes. In particular, itdescribes the well known phenomenon that tighter confinement gives rise to stronger pairing [1]. Thus the constrictioncan be superconducting even at temperatures where the leads are in the normal phase. This is the key ingredient ofour analysis.

III. CONFINEMENT INDUCED RENORMALIZATION OF INTERACTIONS IN THETWO-PARTICLE PROBLEM

As has been pointed out by Olshanii [3], and Petrov and Shlyapnikov [4, 5], confining cold atoms into low dimensionalstructures leads to strong renormalization of their interactions, which can even lead to the so-called confinementinduced resonances [6–11]. The effective interaction of atoms within the confined system are characterized by the T -matrix describing two particle scattering processes. In this section, we use the vacuum T -matrix as a tool to determinethe bare coupling g in terms of physically measurable quantities, the scattering length a and the vacuum two-particlebound state energy EB of the confined system. As we show in Sec. III A, this makes Eq. (S8) divergence-free. TheT -matrix can be determined using the Bethe–Salpeter equation [12, 13], which takes on a particularly simple form inthe vacuum, where only ladder diagrams contribute to the T -matrix,

T~n,~n′(ω,q) = g 〈〈~n||~n′〉〉+∑~n′′

g 〈〈~n||~n′′〉〉 Π(0)~n′′ (ω,q) T~n′′,~n′(ω,q), (S9)

4

as depicted in Fig. S1. The solid lines stand for vacuum propagators, whereas the dashed lines denote the interactionvertex g 〈〈~n||~n′〉〉, with ~n = (n1,n2) denoting the channel indices of the particle pairs. The polarization operator,corresponding to the pairs of propagators, is defined as [13]

Π(0)(n1,n2)(ω,q) =

∫dk

2π

1

~ω + i0+ −(

~2k2

m + ~2q2

4m + ~(n1 + n2)ω) , (S10)

with ω = (ωx, ωz).

FIG. S1. Bethe–Salpeter equations in the confined system. Solid lines denote vacuum propagators, whereas dashed lines standfor the interaction vertex g 〈〈n1n2||n3n4〉〉.

In the two particle problem, parabolic confinement potentials have the special feature of complete separation ofcenter of mass and relative motion. In the basis of center of mass and relative coordinates, the Bethe–Salpeter equationtakes a simpler form

T(0)Nν;N′ν′(ω, q) =

∑n1,n2,n′1,n

′2

〈N,ν|n1,n2〉T (0)n1,n2;n′1,n

′2(ω, q) 〈n′1,n′2|N′,ν′〉 .

Note that in the last equation, the vacuum polarization operator Eq. (S10) depends on the center of mass coordinatesonly via an energy shift,

Π(0)n1,n2

(ω, q) = Π(0)N+ν(ω, q) = Π(0)

ν (ω −N · ω, q),

with n1 + n2 = N + ν. We solve the Bethe–Salpeter equation using the following ansatz [13]

T(0)Nν;N′ν′(ω, q) = δN,N′ ϕν(0)ϕν′(0) T (ω −N · ω, q), (S11)

that allows us to express the T -matrix in terms of the bare coupling in Eq. (S9)

1

g=

1

T (ω, q)+∑ν

|ϕν(0)|2 Π(0)ν (ω, q). (S12)

A. Regularization of the bare coupling

Eq. (S12) is UV-divergent as expressed in terms of the bare coupling. To deal with this singularity, we express g interms of the scattering length a, which is related to the T -matrix of a three dimensional gas as [13]

4π~2a

m= T 3D(ω → 0−,q = 0). (S13)

In the absence of confining potentials, the vacuum Bethe–Salpeter equations take on the simple form

T 3D(ω,q) = g + gΠ3D(ω,q)T 3D(ω,q), (S14)

where the polarization operator is given by

Π3D(ω,q) =

∫d3k

(2π)3

1

~ω + i0+ − ~2(

k2

m + q2

4m

) .

5

At negative frequencies, the denominator of this expression is negative, we can thus make use of the integral identityA−1 = −

∫∞0dτ eτA, and rewrite it as

Π3D(ω,q) = −∫ ∞

0

dτeτ(~ω+i0+−~2q2/4m)

∫d3k

(2π)3e−

τm~2k2

= −∫ ∞

0

dτeτ(~ω+i0+−~2q2/4m)( m

4π~2τ

)3/2

.

Taking the ω → 0− limit, and using Eq. (S13) allows one to express the coupling constant g in terms of the scatteringlength

1

g=

m

4π~2a−∫

d3k

(2π)3

k

~2k2 + i0+=

m

4π~2a−∫ ∞

0

dτ( m

4π~2τ

)3/2

. (S15)

This expression is equivalent to the first line of Eq. (5) in the main text.

B. Confinement effects

The T -matrix Eq. (S12) of the confined system can thus be written in terms of the scattering length instead ofthe bare coupling. Although Eqs. (S12) and (S15) are both divergent, their singular parts exactly cancel each other.The T -matrix thus becomes regular, as we show in the remaining part of this section, through a somewhat technicalderivation. First, we split the summation in Eq. (S12) over transverse harmonic oscillator modes into two parts. Letνx and νz be two arbitrary transverse mode indices, such that the motion of the particle pair in the correspondingchannels are evanescent, ~ω + ~2q2/4m < ~νx, ~νz. For νx ≤ νx and νz ≤ νz, we represent the polarization operatorsusing the following analytical form

Π(0)ν (ω, q) =

∫dk

2π

1

~ω + i0+ −(

~2k2

m + ~2q2

4m + ~ν · ω) = −i

√√√√ m/(4~2)

~ω + i0+ −(

~2q2

4m + ~ν · ω) . (S16)

In case of all other channels, the denominator of the momentum integral in Eq. (S16) is negative, and we can rewritethe polarization operator as an exponential integral

Π(0)ν (ω, q) =

∫ ∞0

dτ eτ(~ω+i0+− ~2q2

4m −~ν·ω) ∫

dk

2πe−τ

~2k2

m =

√m

4π~2

∫ ∞0

dτ√τeτ(~ω+i0+− ~2q2

4m −~ν·ω). (S17)

This exponential form allows one to sum up the harmonic oscillator indices to infinite order, using the followingidentity,

∞∑νx=0

|ϕνx(0)|2 e−τ~νxωx =1√

4πlx

√e~ωxτ

sinh(~ωxτ). (S18)

This expression follows from the expansion of the density matrix of a single particle in a harmonic oscillator potential,at temperature 1/τ [12, 14].

By combining Eqs. (S15, S16, S17, S18), we can express the vacuum T -matrix in the analytic form

1

T (ω, q)=

m

4π~2

(1

a+

1√4π lxlz

W

(~ω + i0+ − ~2q2

4m

~√ωxωz

)), (S19)

where the function W is regular, and it is defined as

W(α) =

∫ ∞0

dx√x

(eαx

[√e(η+1/η)x

sinh(ηx) sinh(x/η)− 4πlxlz

νx∑νx=0

νz∑νz=0

|ϕνx(0)|2 |ϕνz (0)|2 e−x(νxη+νz/η)

]− 1

x

)

+ i√π

νx∑νx=0

νz∑νz=0

4πlxlz |ϕνx(0)|2 |ϕνz (0)|2 1√α+ i0+ − (νxη + νz/η)

, (S20)

with the parameter η =√ωx/ωz characterizing the ellipticity of the confining potential. W incorporates the effects of

confinement on the interaction [7], and renormalizes the T -matrix through Eq. (S19). In case of circularly symmetricwaveguides, ωx = ωz, Eqs. (S19, S20) agree with earlier results of Refs. 3 and 15.

6

Two-particle bound states are defined as the poles of the vacuum T -matrix in Eq. (S19). Their energy EB is relatedto the scattering length as

−√

4πlxlza

=W(

EB~√ωxωz

). (S21)

Although in three dimensional systems, bound states appear only on the BEC side of the Feshbach resonance [16],quasi-one dimensional systems always exhibit a bound state, irrespective of the sign of the scattering length [3–5].

IV. MANY-BODY PROBLEM: REGULARIZATION OF THE BCS ENERGY

In this section, we show that the BCS mean-field energy of the elliptical constriction Eq. (S8) is free of divergences,when expressed in terms of the scattering length a or the bound state energy EB . As a first step, we identify thesingular part of the sum in Eq. (S8) by expanding the Bogoliubov energies in terms of the superconducting orderparameter, and show that this UV divergence is canceled by Eq. (S15). In three dimensional gases, the singularity ofthe BCS mean-field energy is associated with the second order term in the expansion of the Bogoliubov energies, andall higher order terms give a finite contribution [16, 17]. This is also the case in the quasi-two dimensional confinedsystem, as we will show by expanding En,q in terms of the order paramters ∆N within perturbation theory [18].

At large momenta and channel indices, the pairing gap becomes negligible as compared to the kinetic energy,

ξn,q ∑

n2

˜|∆n,n2|, the anomalous terms of the Hamiltonian Eq. (S6) can thus be treated as perturbation, HMF (q) =

H0(q) + δH, with

H0 =

(ξ(q) 0

0 −ξ(q)

),

δH =

(0 ∆

Ơ

0

).

Since δH contains only anomalous terms, odd orders in perturbation theory automatically become zero. The lowest

order term in the expansion En,q = E(0)n,q + δE

(2)n,q + δE

(4)n,q +O

(‖∆‖6

)is simply given by the normal state spectrum,

E(0)n,q = 〈n ↑ |H0|n ↑〉 = ξn,q, that is exactly canceled in Eq. (S8). The second order term has, however, a singular

contribution to mean-field energy,

δE(2)n,q =

∑m

|〈m ↓ |δH|n ↑〉|2

〈n ↑ |H0|n ↑〉 − 〈m ↑ |H0|m ↑〉=∑m

∣∣∣∑N ∆N(V †N)m,n∣∣∣2

ξn,q + ξm,q. (S22)

As we show in the second part of this section, this singularity is canceled by the regularization of the coupling constantg. Finally, the contribution of the fourth order term

δE(4)n,q =

∑m1,m2 6=n,m3

〈n ↑ |δH|m1 ↓〉〈m1 ↑ |δH|m2 ↑〉〈m2 ↑ |δH|m3 ↓〉〈m3 ↑ |δH|n ↑〉(ξn + ξm1

)(ξm1+ ξm2

)(ξm2+ ξn)

−

(∑m

|〈m ↓ |δH|n ↑〉|2

ξn,q + ξm,q

)2

(S23)is regularized, similarly to all higher order terms. The somewhat technical derivation of this result is given in Ref. 19.

We remove the singularity of Eq. (S22), by expressing the bare coupling g in terms of the T -matrix of the confinedsystem in Eq. (S12). In the center of mass channel N, the energy ω of incoming particles is simply shifted by N · ω(see Eq. (S11)),

1

g=

1

T (ω −N · ω, q)+∑ν

|ϕν(0)|2 Π(0)ν (ω −N · ω, q)

=1

T (ω −N · ω, q)+∑ν

|ϕν(0)|2 Π(0)N+ν(ω, q)

By inserting a full basis 1 =∑

n1,n2|n1,n2〉〈n1,n2|, we can rewrite g as

1

g=

1

T (ω −N · ω, q)+

∑ν,n1,n2

(〈N,ν|n1,n2〉ϕν(0)) (ϕν(0)〈n1,n2|N,ν〉) Π(0)N+ν(ω, q).

7

We make use of the definition of the matrices V n1n2

N above Eq. (S5) and that of the polarization operator in Eq. (S16),to rewrite the coupling as

1

g=

1

T (ω −N · ω, q)+

1

lxLylz

∑q

∑n1,n2

(V †N)n1,n2 V n1n2

N

~ω + i0+ −(

~2q2

m + ~(n1 + n2) · ω) . (S24)

Using this relation, we rewrite the second term in the mean-field energy EMF in Eq. (S8) as

∑N

|∆N|2

g=∑N

|∆N|2

T (ω −N · ω, q)+

1

lxLylz

∑n1,n2,q

∑N

∣∣∣∆N(V †N)n1n2

∣∣∣2~ω + i0+ −

(~2q2

m + ~(n1 + n2) · ω) . (S25)

This expression has the same asymptotic behavior in the (q,n1,n2) → ∞ limit as the second order expansion of∑n,q En,q in Eq. (S22). Therefore, it exactly cancels the singular part of mean-field energy, and we can combine

Eqs. (S19 S8, S25), to rewrite EMF in the regularized form

EMF

Ly=

∫dq

2π

∑n

(ξn,q − En,q)−∑n1,n2

∑N

∣∣∣∆N(V †N)n1n2

∣∣∣2ω + i0+ −

(~2q2

m + ~(n1 + n2) · ω) (S26)

−∑N

|∆N|2

4π√ωxωz

(1

a+

1√4π lxlz

W(ω + i0+ −N · ω√ωxωz

)),

with the function W defined in Eq. (S20). Note that in the last equation, the parameter ω is completely arbitrary,since the ω dependence of the two terms cancel.

Although Eq. (S26) allows for pairing within all transverse channels N, only ∆N=0 takes on non-zero value for theparameters of the experiment [20]. This allows for a further simplification of the mean-field energy. By choosing ω atthe energy of the two-particle bound state in Eq. (S26), the second term vanishes due to Eq. (S21), and the mean-fieldenergy becomes

EMF

Ly=

∫dq

2π

∑n

(ξn,q − En,q)−∑n1,n2

|∆0|2 |V n1n20 |2

EB −(

~2q2

m + ~(n1 + n2) · ω) .

This expression is equivalent to Eq. (6) in the main text.Furthermore, Eq. (S24) allows one to derive the dependence of the bound state energy on the bare coupling, as

shown in Eq. (5). We choose the incoming energy at EB , so that the first term in Eq. (S24) is zero, and we find

1

g=

1

lxlz

1

Ly

∑q

∑n1,n2

|V n1n20 |2

EB −(

~2q2

m + ~(n1 + n2) · ω)

in the N = 0 channel.

V. TRANSPORT THROUGH THE CHANNEL

We determine the conductance and spin conductance of the quasi-one dimensional constriction within Landauertheory, by calculating its reflection and Andreev reflection coefficients. These can be obtained from the asymptoticbehavior of scattering eigenstates of the constriction. To determine these states, one has to solve the spatially non-uniform Bogoliubov–de Gennes (BdG) equations, since the confinement frequencies ωx(y) and ωz(y), as well as thepotential V (y) and the superconducting order parameters ∆N(y) all change along the constriction. We do this bydividing the constriction into short intervals (yi−1, yi), in which the system’s parameters can be approximated asconstants, as shown in Fig. S2. We require furthermore that the wave functions of the eigenstates change smoothlyat the interfaces between the intervals. In each interval, the scattering eigenstates of energy ε above the Fermi energyare described by the local BdG equation(

H0,i ∆i

∆†i −H∗0,i

)(ui(x, z)vi(x, z)

)eiqy = ε

(ui(x, z)vi(x, z)

)eiqy, (S27)

8

FIG. S2. Conductance calculation. (a) We divide the constriction into finite intervals, in which the potential V (y) and thesuperconducting order parameters ∆N(y) are approximated as constants. (b) We derive the transfer matrix Mi by requiringthe wave function to be continuous and differentiable at each interface yi.

where the kinetic and anomalous terms in the Hamiltonian are given by

H0,i(x, z) = q2/2m− εF,i +∑n

n · ωi |ni(x, z)〉〈ni(x, z)| ,

∆i(x, z) =∑n1n2

∆i,n1n2 |ni,1(x, z)〉〈ni,2(x, z)| .

Here, the value of the local Fermi energy is given by εF,i = µ−Vi− (ωx,i+ωz,i)/2, and the states |ni(x, z)〉 denote thetransverse wave functions of the harmonic oscillator. We use the index λ to label the modes corresponding to energyε,

|Ψi(x, z)〉 ≡(ui(x, z)vi(x, z)

)=∑λ

(aiλe

iqiλy + biλe−iqiλy

)∑n

(uiλnviλn

)|ni(x, z)〉, (S28)

with aiλ and biλ denoting the amplitudes of left and right moving modes. Most of these modes are evanescent, and theqiλ wave vector is imaginary. We solve Eq. (S27), in order to determine the wave vectors. This leads to an eigenvalueequation for qiλ [21],

q2iλ

2m

(uiλviλ

)=

(ε+ εF,i −Ωi −∆i

∆†i −ε+ εF,i −Ωi

)(uiλviλ

), (S29)

where Ωi,n1n2≡ δn1n2

n1 · ωi denotes the harmonic oscillator energies, whereas (∆i)n1n2≡ ∆i,n1n2

stands forthe matrix of superconducting order parameters. We also introduced the vectorial notations (uiλ)n ≡ uiλn and(viλ)n ≡ viλn for the Bogoliubov modes of the eigenstates. The transfer matrix at point yi describes the relationbetween the amplitudes on the two sides of the interface(

~ai~bi

)= Mi

(~ai−1

~bi−1

).

Here, we introduced the vectorial notation (~ai)λ = aiλ and (~bi)λ = biλ for the amplitudes of the BdG states. Mi canbe determined by requiring the continuity and differentiability of the wave function at yi,

|Ψi−1(yi)〉 = |Ψi(yi)〉,∂y|Ψi−1(yi)〉 = ∂y|Ψi(yi)〉.

To determine the transfer matrix, we take the matrix elements of these equations with the transverse oscillator modes〈ni(x, z)|, and make use of the expansion Eq. (S28) of the wave function, to get∑

λ

(ai−1λe

iqi−1λyi + bi−1λe−iqi−1λyi

)∑n′

⟨ni|n′i−1

⟩(ui−1λn′

vi−1λn′

)=∑λ

(aiλe

iqiλyi + biλe−iqiλyi

)(uiλnviλn

),

∑λ

qi−1λ

(ai−1λe

iqi−1λyi − bi−1λe−iqi−1λyi

)∑n′

⟨ni|n′i−1

⟩(ui−1λn′

vi−1λn′

)=∑λ

qiλ(aiλe

iqiλyi − biλe−iqiλyi)(uiλn

viλn

).

9

To simplify our notations, we rewrite these equations in a matrix form. We introduce the matrix (Wi)nn′ ≡⟨ni|n′i−1

⟩that describes the unitary transformation from the harmonic oscillator basis in the interval (yi−1, yi) to that of(yi, yi+1). A short calculation shows that the transfer matrix is given by,

Mi =1

2

(e−i~qi yi −e−i~qi yiei~qiyi ei~qiyi

)U−1i

(Wi 00 Wi

)Ui−1 0

0 (Ui~qi)−1

(Wi 00 Wi

)Ui−1~qi−1

( ei~qi yi−1 e−i~qiyi−1

−ei~qi yi−1 e−i~qiyi−1

)(S30)

where ~qi denotes the matrix that has the momenta qiλ on its diagonal, and the matrix Ui contains the Bogoliubovcoefficients

Ui =

((ui,λ=1

vi,λ=1

),

(ui,λ=2

vi,λ=2

),

(ui,λ=3

vi,λ=3

), . . .

).

One of the challenges in working with transfer matrices is that they are numerically unstable due to evanescentmodes of the constriction. Although the transfer matrices of small intervals are well defined, that of the entireconstriction, M =

∏i Mi contains elements that blow up or vanish exponentially with the system length Ly. We

circumvent this difficulty by representing the scattering problem in terms of scattering matrices in each interval, whichconnect incoming states to outgoing ones, (

~bi−1

~ai

)= Si

(~ai−1

~bi

).

The transfer matrix and the scattering matrix are trivially related to each other [22]. However, the latter is freefrom the divergences associated with evanescent modes. We follow Ref. 23 to obtain the scattering matrix S of theconstriction from those of individual intervals.

In the normal leads, the wave function of scattering states can be decomposed as

|ψ(r)〉 =∑n

(eiq

pny

(apn0

)+ e−iq

pny

(bpn0

))|n(x, z)〉√

vpn+

(eiq

hny

(0ahn

)+ e−iq

hny

(0

bhn

))|n(x, z)〉√

vhn, (S31)

with the amplitudes ap/hn and b

p/hn corresponding to right and left moving states in transverse channel n. The

momenta qp/hn are those of particle (p) and hole states (h) at energy ε, and the associated group velocities are given

by vp/hn = ~ qp/hn /m. With the above normalization, all modes correspond to unit flux, and the scattering matrix of

the constriction is unitary in this basis due to charge conservation [22].The scattering matrix can be written in the usual representation in terms of reflection (r) and transmission matrices

(t) as bpLbhLapRahR

= S

apLahLbpRbhR

=

rpp rph t′pp t′phrhp rhh t′hp t′hhtpp tph r′pp r′phthp thh r′hp r′hh

apLahLbpRbhR

,

where the indices L and R denote states in the left and right leads, respectively. Although the elements of thescattering matrix depend on the energy of incoming particles ε, we omitted the energy arguments for brevity. Thezero bias charge and spin conductance of the constriction in Eq. (7) can be obtained from the reflection rpp(ε) andAndreev reflection coefficients rph(ε) using the Landauer formula [22].

A. Coefficients, Wi

In order to calculate the transfer matrix Mi in Eq. (S30), we need to determine the unitary operator Wnn′ =〈ni|n′i−1〉 = 〈nxi|n′x i−1〉 〈nzi|n′z i−1〉, describing the change of harmonic oscillator basis between intervals i and i− 1.

The harmonic oscillator eigenstates along x are given by |nxi(x)〉 = ϕnx (x/lxi) /√lxi, with the Hermite function

ϕn(x) =1√

2n n!√πe−x

2/2Hn(x),

10

and a similar expression holds in case of the z direction. We make use of the following identity of the Hermitepolynomials [24]

Hn(γx) =

bn/2c∑l=0

γn−2l(γ2 − 1)l(n

2l

)(2l)!

l!Hn−2l(x),

to show that the matrix elements are given by

〈nxi|n′x i−1〉 =√γxi αxi

bn/2c∑l=0

bn′/2c∑l′=0

δn−2l,n′−2l′ αn−2lxi βn

′−2l′

xi (α2xi − 1)l(β2

xi − 1)l′(n

2l

)(n′

2l′

)√(n− 2l)! (n′ − 2l′)!

22l+2l′ n!n′!,

with αxi ≡√

2/(1 + γ2xi), βxi ≡ αxiγxi, and γxi ≡ lx i−1/lxi =

√ωxi/ωx i−1. A similar equation holds for 〈nzi|n′z i−1〉,

from which Wnn′ can be determined.

VI. DYNAMICS OF SUPERCONDUCTING ORDER PARAMETER BEYOND MEAN-FIELD

In the main text we used a simple mean-field model of the superconductivity in the constriction. We now analyzefluctuations of the superconducting order parameter which arise due to a finite size of the superconducting islandand its coupling to the reservoirs of normal fermions. To understand the role of phase fluctuations on the transportprocesses, we estimate the fluctuations of the superconducting phase ϕ during the timescale τ of Andreev reflections.We use an effective impedance model to describe dynamics of the phase [25–27]. This model adopts a simple Gaussiandescription of phase fluctuations and requires specifying a frequency dependent impedance of the environment, whichwill be presented in the framework of a resistor-capacitor model. From the Gaussian nature of phase fluctuations wehave

〈ei(ϕ(τ)−ϕ(0))〉 = e−12 〈(ϕ(τ)−ϕ(0))2〉. (S32)

For numerical estimates, it will be more convenient to work with the phase variation averaged over time τ

Var2ϕτ ≡1

τ

∫ τ

0

dt 〈(ϕ(t)− ϕ(0))2〉. (S33)

One can use Var2ϕτ to obtain phase fluctuations 〈(ϕ(τ)− ϕ(0))2〉 = ∂τ (τ Var2ϕτ ). We express fluctuations in terms

of the spectral power density

Sϕϕ(ν) =

∫ ∞0

dt e2πi νt〈ϕ(t)ϕ(0)〉,

at frequency ν. Making use of the symmetry Sϕϕ(ν) = Sϕϕ(−ν), the fluctuations can be written in the form

Var2ϕτ = 4

∫ ∞0

dν Sϕϕ(ν) ζτ (ν), (S34)

with the kernel ζτ (ν) = 1− sin(2πντ)2πντ , that vanishes as ζτ (ν) ' 2π2

3 (ντ)2 at small frequencies ν . 1/(2τ), thus providing

a low frequency cutoff. At higher frequencies, ζτ (ν) ' 1.

FIG. S3. Resistance capacitor model of phase fluctuations in the superconducting constriction.

11

To analyze the dynamics of phase fluctuations in a superconducting constriction, we use an effective resistor-capacitor model, shown in Fig. S3 [25–27]. The superconducting region is coupled to the leads through effectiveresistors R = 1/(2GA), where GA is the Andreev conductance of the constriction. The thermal Johnson–Nyquistnoise associated with the leads is indicated by the µL/R voltage sources. The quantum capacitance CQ of theconstriction arises from the energy cost of adding δN atoms to the superconducting region, leading to a shift of itschemical potential

µsc = µ(0)sc +

δN

CQ,

and the constant µ(0)sc denotes the time averaged equilibrium value of the chemical potential of the superconducting

region. The physical motivation for Fig. S3 comes from the observation that it reproduces the semiclassical equationsof motion for the current across the constriction. The current through the left and right superconducting interfaceare given by

IL =µsc − µL

R,

IR =µR − µsc

R,

and charge conservation in the superconducting region gives

∂t δN = IL − IR.

The spectral density of the Johnson–Nyquist noise of the chemical potential µsc at frequency ν is given by [28]

Sµµ(ν) = 4hν ReZ(ν)(ηqm(ν) + ηth(ν)

), (S35)

where Z denotes the complex impedance Z(ν) = R/(1+iν/νL) of the circuit between the constriction and the ground,with a relaxation frequency νL = (4π RCQ)−1. Note that Eq. (S35) incorporates Johnson noise from both reservoirs,that are assumed to be independent. The frequency dependent factors

ηqm(ν) = 1/2, ηth(ν) =1/2

ehν/kBT − 1(S36)

denote the noise contribution due to quantum and thermal fluctuations, respectively, as given by the fluctuation-dissipation theorem [28]. At small frequencies, ν νth ≡ kBT/h, thermal fluctuations dominate, ηth(ν) ∼ kBT

2hν ,leading to the Johnson–Nyquist noise formula of classical systems [29]. In the opposite limit, at high frequencies,thermal fluctuations vanish and the noise is dominated by quantum fluctuations.

Fluctuations of the phase ϕ of the superconducting region are related to chemical potential fluctuations in the leadsby the Josephson relation [17]

~ ∂tϕ = 2µsc, (S37)

which determines the spectral density

Sϕϕ(ν) =

(2

hν

)2

Sµµ(ν). (S38)

Sϕϕ can be determined by combining Eqs. (S35) and (S38),

Sϕϕ(ν) =8

hν

(ηqm(ν) + ηth(ν)

) R

1 +(ννL

)2 .

Although the spectral power density is singular at small frequencies, its divergence is canceled in Eq. (S34) by thelow frequency cut-off of the kernel ζτ (ν) ∼ (ντ)2 at frequencies ν . τ−1.

To estimate the growth of phase fluctuations with the time τ , we first consider the system at zero temperature,where only quantum fluctuations contribute to Eq. (S34),

Var2ϕqm

τ =16R

h

∫ ∞0

dν

ν

ζτ (ν)

1 +(ννL

)2 . (S39)

12

At long times, τ ν−1L , the integral in the last equation is dominated by the frequency range (2τ)−1 < ν < νL, where

Var2ϕqm

τν−1L' 16R

h

∫ νL

1τ

dν

ν∼ R

hlog(τ νL), (S40)

which indicates that phase correlations 〈ei ϕ(τ) e−i ϕ(0)〉 decay algebraically in time. In the opposite, short time limit,we can make use of the quadratic low frequency dependence of ζτ (ν) below Eq. (S34). The integral is dominated bythe same frequency range,

Var2ϕqm

τν−1L' 32π2

3

R

h(τνL)2

∫ 1τ

νL

dν

ν∼ R

h(τ νL)2 log

(1

τνL

). (S41)

At finite temperatures, the low frequency fluctuations are dominated by thermal effects. Making use of the approx-imation ηth(ν . νth) ∼ νth/ν, phase fluctuations in Eq. (S34) can be approximated as

Var2ϕth

τ '16R

hνth

∫ νth

0

dν

ν2

ζτ (ν)

1 +(ννL

)2 .

A simple calculation shows that thermal phase fluctuations grow linearly at long times

Var2ϕth

τν−1L ,ν−1

th∼ R

h(τ νth), (S42)

which indicates phase diffusion. This is in accordance with our expectations that a mesoscopic superconductingregion should have exponentially decaying temporal correlations with the timescale set by the temperature. The lastequation also shows that thermal fluctuations dominate quantum effects, Eq. (S40) in the long time limit. At shorttimes, on the other hand, thermal fluctuations are quadratic in τ

Var2ϕth

τν−1L ,ν−1

th∼ R

h(τ2 νthνL), (S43)

they are thus of similar order as quantum fluctuations, Eq. (S41).It is instructive to estimate the strength of the fluctuations over the time scale τ it takes to accommodate a Cooper

pair in the process of Andreev reflection. We will use estimates of the relevant time and frequency scales τ , νL andνth from the experimental parameters of Ref. 20. The width of the superconducting-normal interface is of the orderof the superconducting correlation length ξ, thus the time scale of Andreev processes is given by

τ =ξ

vs,

where vs is the superflow velocity of the condensate. vs can be obtained by considering the supercurrent through theconstriction

Is =∆µ

R= n0vs, (S44)

where ∆µ = (µL − µR)/2 denotes the chemical potential bias. µL and µR are the chemical potentials of the left andright leads respectively.

We determine the resonance frequency νL of the resistor-capacitor model by estimating the quantum capacitanceCQ of the constriction. CQ is related to the energy cost of adding extra atoms to the superconductor, and it is givenby

1

CQ=∂2Esc∂ N2

,

where E denotes the energy of the superconducting region. To estimate the quantum capacitance, we assume thatthis region of length L0 is uniform, with a one-dimensional density n0 = N/L0. Since the chemical potential is givenby

µ =∂Esc∂N

=1

L0

∂Esc∂n0

,

13

the quantum capacitance can be estimated as

1

CQ=

∂µ

∂N=

1

L0

1∂n0

∂µ

.

Assuming for the moment that the constriction is in the normal phase, the density is n0 = 4/λF , with λF denotingthe Fermi wavelength. The quantum capacitance is thus given by

1

CQ∼ εF

2

λFL0

, (S45)

where εF is the Fermi energy. In case when the channel is superconducting, we cannot give such a simple analyticalformula for CQ. However, we checked numerically, that its value changes only slightly even in the presence of a pairinggap as large as ∆ ∼ εF . We therefore use the estimate in Eq. (S45) in the following.

We estimate the phase fluctuations in the strongly interacting regime, where the transport is dominated by Andreevprocesses. In this regime, the superconducting gap is of the order of the Fermi energy ∆ ∼ εF , as we discuss in themain text. We assume an Andreev conductance of GA = 4/h, the resistance in the model is thus R = h/8. We estimatethe correlation length as ξ ∼ ~vF

∆ = εF∆λFπ , with vF denoting the Fermi velocity. Furthermore, we assume a typical

chemical potential bias of ∆µ = 0.1 εF in Eq. (S44), leading to a low frequency cut-off of τ−1 = 0.45 εF /h in Eq. (S34).We use Eq. (S45) to estimate the resonance frequency of the resistor-capacitor circuit, νL = (4π RCQ)−1 = 0.03 εF /h.The temperature of the measurement Ref. 20 leads to the frequency νth = kBT/h = 0.15 εF /h. As these frequencyscales are much smaller than τ−1, phase fluctuations are dominated by quantum effects. Using these parameters, weestimate the phase fluctuations Eq. (S34) as

Var2ϕτ(2π)2

' 0.06.

The phase fluctuations are thus suppressed in this regime, justifying the use of the mean-field description presentedin the main text.

[1] A. M. Fischer and M. M. Parish, Phys. Rev. A 88, 023612 (2013).[2] A. J. Leggett and S. Zhang, The BEC-BCS Crossover and the Unitary Fermi-Gas (Springer, 2011).[3] M. Olshanii, Phys. Rev. Lett. 81, 938 (1998).[4] D. S. Petrov, M. Holzmann, and G. V. Shlyapnikov, Phys. Rev. Lett. 84, 2551 (2000).[5] D. S. Petrov and G. V. Shlyapnikov, Phys. Rev. A 64, 012706 (2001).[6] H. Moritz et al., Phys. Rev. Lett. 94, 210401 (2005).[7] E. Haller, M. Gustavsson, M. J. Mark, J. G. Danzl, R. Hart, G. Pupillo, and H.-C. Nagerl, Science 325, 1224 (2009).[8] E. Haller et al., Phys. Rev. Lett. 104, 153203 (2010).[9] B. Frohlich et al., Phys. Rev. Lett. 106, 105301 (2011).

[10] A. T. Sommer et al., Phys. Rev. Lett. 108, 045302 (2012).[11] S. Sala et al., Phys. Rev. Lett. 110, 203203 (2013).[12] M. Kanasz-Nagy, E. A. Demler, and G. Zarnd, Phys. Rev. A 91, 032704 (2015).[13] V. Pietila, D. Pekker, Y. Nishida, and E. Demler, Phys. Rev. A 85, 023621 (2012).[14] R. P. Feynman, Statistical Mechanics: A Set Of Lectures (Westview Press, 1998).[15] W. Fu, Z. Yu, and X. Cui, Phys. Rev. A 85, 012703 (2012).[16] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008).[17] M. Tinkham, Introduction to Superconductivity, (Dover Publications, 2004).[18] To simplify our discussion, we assume a spin symmetric chemical potentials µ↑ = µ↓. Systems with spin asymmetry can

be treated analogously.

[19] The slowest decaying term in terms of momentum q and channel index n in δE(4)n,q is given by the second term in Eq. (S23).

In order to investigate this term, we need to determine the asymptotic behavior of the matrix elements V n1n2N in the

limit n1,n2 → ∞. In a strongly confined system, pairing occurs only in the lowest few N channels, therefore we canassume that N is small. In this case, V n1n2

N and V n1n20 are asymptotically of the same order. Furthermore, using the

asymptotic formulas of harmonic oscillator wave functions, one can show that in the n1,n2 → ∞ limit V n1n2N only takes

on non-negligible values when n1 ∼ n2. In this case, V nn0 ∼ 1/

√nxnz. Using this asymptotic form, we gain an upper limit

on the asymptotic behavior of Eq. (S23), E(4)n,q ∼ n−2

x n−2z

(q2/m+ nxωx + nzωz

)−2. The sum

∑n,q E

(4)n,q thus gives a finite

contribution to Eq. (S8). Furthermore, all higher order terms are regularized as well, since they contain even higher powersof ξ−1

n,q and V n1n2N .

14

[20] S. Krinner et al., arXiv:1511.05961.[21] To make sure that the evanescent waves are indeed decaying in case of our right moving states, we take E → E + i0+.[22] Y. V. Nazarov and Y. M. Blanter, Quantum transport: Introduction to Nanoscience (Cambridge University Press, 2009).[23] P. W. Brouwer, M. Duckheim, A. Romito, and F. von Oppen, Phys. Rev. B 84, 144526 (2011).[24] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables

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1992).[27] Y. Makhlin, G. Schon and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001).[28] G. Schon and A. D. Zaikin, Phys. Rep. 198, 237 (1990).[29] H. Nyquist, Phys. Rev. 32, 110 (1998).