π phases in balanced fermionic superfluids on spin-dependent optical lattices.

Abstract

We study a balanced two-component system of ultracold fermions in one dimension with attractive interactions and subject to a spin-dependent optical lattice potential of opposite sign for the two components. We find states with different types of modulated pairing order parameters which are conceptually similar to π phases discussed for superconductor-ferromagnet heterostructures. Increasing the lattice depth induces sharp transitions between states of different parity. While the origin of the order parameter oscillations is similar to the Fulde-Ferrel-Larkin-Ovchinnikov phase for paired states with spin imbalance, the current system is intrinsically stable to phase separation. We discuss experimental requirements for creating and probing these novel phases.

Full text

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π-phases in balanced fermionic superfluids in spin-dependent optical lattices
I. Zapata,1B. Wunsch,2N. T. Zinner,2and E. Demler2
1Departamento de F´ ısica de Materiales, Universidad Complutense de Madrid, E-28040 Madrid, Spain
2Department of Physics, Harvard University, 17 Oxford Street, Cambridge, MA 02138, USA
(Dated: September 29, 2010)
We study a balanced two-component system of ultracold fermions in one dimension with attractive
interactions and subject to a spin-dependent optical lattice potential of opposite sign for the two
components. We find states with different types of modulated pairing order parameters which
are conceptually similar to π-phases discussed for superconductor-ferromagnet heterostructures.
Increasing the lattice depth induces sharp transitions between states of different parity. While the
origin of the order paramter oscillations is similar to the FFLO phase for paired states with spin
imbalance, the current system is intrinsically stable to phase separation. We discuss experimental
requirements for creating and probing these novel phases.
PACS numbers: 67.85.-d,03.75.Ss,71.10.Pm,74.45.+c
One of the most intriguing examples of the interplay
of superconductivity and magnetism is the Fulde-Ferrel-
Larkin-Ovchinnikov (FFLO) phase, where Zeeman split-
ting of the Fermi surfaces is expected to lead to spatial
oscillations of the pairing amplitude. It is difficult to
obtain such phases in superconductors, since the orbital
effect of the magnetic field is typically much larger than
the spin Zeeman splitting. Several proposals have been
made, however they remain controversial [1]. For exam-
ple, FFLO phase has been discussed in the context of
heavy fermion CeCoIn5 superconductors [2, 3], but al-
ternative interpretation in terms of competing magnetic
order has also been given [4].
biguous demonstration of FFLO-like physics has been
achieved in heterostructures of ferromagnetic and super-
conducting (F/SC) layers [5], where proximity coupling
through ferromagnetic layers results in superconducting
π-junctions (see [1] for a review). We note that π-phases
arising from a different mechanism than FFLO have also
been discussed for high-Tccuprates [1, 6].
Recently, in cold atoms, there has been a large body
of work, both experimental and theoretical, aimed at
achieving FFLO states.The biggest difficulty is that
FFLO phases are fragile and extremely susceptible to
phase separation and the experimental situation remains
unclear [7–12]. In this paper we propose a novel system
of ultracold fermions in an optical lattice [13, 14] which
can be used to observe FFLO type states with oscillat-
ing pairing amplitude. The system which we discuss is
somewhat similar to F/SC heterostructures and should
be stable against phase separation. Our proposal relies
on the ability to create spin dependent optical lattices
and we find that beyond a certain critical strength of
such optical potential, the superconducting pairing am-
plitude becomes a sign changing function (we will refer
to such states as as π-phases, see Fig. 2).
The gap profile in the ground-state depends on the
wavelength of the lattice, λ, the strength of the poten-
tial, V0, and the interaction strength. In Fig. 1 we present
So far the only unam-
0.20.40.60.81.0
20
30
40
50
BCS
FF
3π
2π
π
∆max / EF
0.34
kF λ
V0 / EF
0.06
0.12
0.17
0.23
0.29
BCS-FF
FIG. 1: Phase diagram showing the emergence of π-phases
for a spin-dependent lattice potential of wavelength λ and
strength V0 for interaction strength g1DkF/EF
and zero temperature. A gradient of colors gives ∆max :=
max|∆˜ m|, the largest Fourier component amplitude of the
gap. The black lines indicate transitions from gap profiles
with zero (BCS), two (π), four (2π) and six (3π) zero-crossings
per unit cell. The dashed FF line and the BCS-FF arrow are
from a Fulde-Ferrell calculation in the homogeneuos system
and is explained in the text.
= −2.04
the (V0,λ) phase diagram showing the transitions from
constant gap to the π-phases with several zero-crossings
per unit cell in the pairing amplitude. A color gradient
gives the largest Fourier component amplitude of the gap
(see below) and the black lines indicate the transitions.
We clearly see π-phases occurring in a broad range of
λ restricted from below only by the coherence length as
we will discuss. The emergence of oscillations in the gap
gives clear signatures in the Fourier transform. We will
demonstrate how the rapid-ramp techniques can be used
to observe these states in time-of-flight measurements.
We also suggest ways to make spin dependent large wave-
arXiv:0910.1803v3 [cond-mat.quant-gas] 28 Sep 2010

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2
length lattice potentials in the high-field regime as is
needed to access π-phases.
The quasi-1D system we study is described by the ef-
fective Hamiltonian [15]
H − µ↓N↓− µ↑N↑=
?
+ g1D
σ=↑↓
?
dx؆
σ(x)[−?2
2m
∂2
∂x2+ Vσ(x) − µσ]Ψσ(x)
?
dx؆
↑(x)Ψ†
↓(x)Ψ↓(x)Ψ↑(x), (1)
where g1Dis the effective 1D coupling constant. We use
g1DkF/EF= −2.04 as in [15] corresponding to an inter-
action to kinetic energy ratio of −mg1D/?2= 1.6. This
is neither weak coupling nor the unitary limit.
We consider a balanced system but introduce chem-
ical potentials µσ since the optical lattice potential is
spin-dependent. For the main part of this work we use
V↑(x) = V0cos(2πx/λ) and V↓(x) = −V↑(x) in which
case µ↑= µ↓. In the non-interacting system, the spin-
dependent lattice spatially displaces the degenerate so-
lution of the two components as V0 is increased. In a
simple-minded picture, pairing of these states will gener-
ate spatial variation in the order parameter which is the
origin of π-phases.
We solve the Hamiltonian of Eq. (1) in mean-
field theory by using the inhomogeneous Bogoliubov-
deGennes (BdG) ansatz for the field operator Ψσ(x,t) =
?
ωkσ > 0 with k the quasiparticle index composed of a
quasimomentum and the band index.
equations are
?
Hσ= −?2
2m
k[ukσ(x)e−iωkσtckσ+ σ¯ vk¯ σ(x)eiωk¯ σtc†
and c†denote the quasiparticles and the sum runs over
k¯ σ], where the c
The mean-field
Hσ
¯∆(x) −H−σ
∆(x)
??ukσ(x)
vkσ(x)
?
= ωkσ
?ukσ(x)
vkσ(x)
?
,
∂2
∂x2+ Vσ(x) − µσ+ g1Dn−σ(x), (2)
where
−g1D?Ψ↓(x)Ψ↑(x)?.
self-consistently for densities and gap through
nσ(x)=
?Ψ†
These equations can be solved
σ(x)Ψσ(x)?
and ∆(x)=
n↑(x) =
?
?
ωk↑
f(ωk↑)|uk↑(x)|2=
∞
?
?
˜ m=−∞
n↑ ˜ mei2π ˜ mx/λ
n↓(x) =
ωk↑
f(−ωk↑)|vk↑(x)|2=
?
∞
?
∞
˜ m=−∞
n↓ ˜ mei2π ˜ mx/λ
∆(x) = g1D
ωk↑
f(ωk↑)uk↑(x)¯ vk↑(x)
=
˜ m=−∞
∆˜ mei2π ˜ mx/λ, (3)
where f(ωkσ) = 1/(1 + exp(?ωkσ/kBT)) is the Fermi-
Dirac distribution. In Eq. (3) we use the periodicity of
the optical lattice to do a Fourier decomposition. Notice
that these equations only contain u,v for σ =↑ and the
sums over ωk↑are unrestricted [15]. We have explicitly
checked the convergence of our numerical solutions by
extending the cut-off on the basis size.
The mean-field BdG ansatz does not take into account
soft collective modes of the order parameter which, in
principle, lead to the power law decay of the supercon-
ducting correlations. However, the BdG approach de-
scribes the ground state energy in our parameter regime
well, which is determined by correlations on the scale of
the BCS correlation length, ξ = ?vF/∆ [15, 16]. Thus
we expect it to also correctly capture the competition
between 0- and π-phases.
In Fig. 2 we show (n↑(x)−n↓(x))/kFand ∆(x)/EFas
functions of 2x/λ with kFλ = 30 for amplitudes V0/EF=
0.39, 0.40, 0.77 and 0.78. Here we notice a sudden change
in the gap profile from an even to an odd function (around
x = 0) at the definite value V0/EF = 0.39, and again at
V0/EF = 0.78. We will give arguments as to why this
occurs in the following sections. The signature of the
new phases are even more clear in Fig. 3 which shows
the largest components of |∆˜ m|. Here we see a very clear
jump between even and odd Fourier components as V0
is increased. This transition constitutes the main result
of our paper and below we propose a way to observe
the π-phases which is clearly distinguishable from other
oscillatory behaviors in the gap.
The spin-dependent lattice we use here is invariant un-
der spatial reflection. In addition, for the balanced sys-
tem we study, the symmetry of the lattice potential im-
plies that the densities are even and interchanged every
half wavelength (λ/2). We can use this observation to
restrict the functional form of the gap. In the absence
of any currents, the gap obeys ∆(x + λ/2) = ±∆(x).
Combined with the full periodicity ∆(x+λ) = ∆(x), we
see that either only even or only odd Fourier components
survive which facilitates its unmistakable detection.
Our spin-dependent lattice potential effectively acts as
a spatially varying magnetic field. In contrast, in homo-
geneous systems the wavevector of a Fulde-Ferrell (FF)
state increases monotonically with increasing chemical
potential difference. Using the results presented in [15],
we determine which chemical potential difference V0(at
fixed density) is needed so that the order parameter of
the FF state has the wavevector 2π/λ corresponding to
the lattice used in our calculation. The result is pre-
sented as a dashed line in Fig. 1. This line indicates
where FF solutions exists, however only for V0 ≥ 0.27
(the BCS-FF arrow in Fig. 1) is it a global energy min-
imum. The qualitative agreement clearly demonstrates
the connection to FFLO physics. The difference for our
system is that the wavelength of variations in the gap
does not vary continuously but changes at discrete val-
ues of V0 since the oscillations must be commensurate
with the lattice potential. FFLO states are similar to

Page 3

3
?1.0
?0.50.51.0
?0.4
?0.2
0.2
0.4
V0?0.39 EF
?1.0
?0.50.5 1.0
?0.4
?0.2
0.2
0.4
V0?0.40 EF
?1.0
?0.50.51.0
?0.4
?0.2
0.2
0.4
V0?0.77 EF
?1.0
?0.50.5 1.0
?0.4
?0.2
0.2
0.4
V0?0.78 EF
FIG. 2: The polarization (n↑(x) − n↓(x))/kF in dot-dashed blue and pairing ∆(x)/EF in thick red as functions of 2x/λ, for
the case of 0-phase (left), π-phase (two figures in the middle), and 2π-phase (right) at zero temperature. The dashed and
dotted black lines in the left plot show the spin-dependent lattice potential. Here kFλ = 30,T = 0,g1DkF/EF = −2.04, and
V↑(x) = −V↓(x) = V0cos(2πx/λ).
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ?????
? ??1?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
0.40.6
V0?EF
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
0.00.2
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ???? ?????
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
0.8
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
1.0
0.1
0.2
0.3
??m?? ?EF
? ??4?
? ??3?
? ??2?
? ??0?
FIG. 3: Plot of the absolute value of the Fourier components
of ∆(x) in Eq. (3) as function of the lattice potential strength
V0 (parameters as in Fig. 2). Only even components are non-
zero for V0/EF ≤ 0.38, whereas only odd ones are non-zero
for 0.39 ≤ V0/EF ≤ 0.77, and so forth. This is the tell-tale
sign of the transition from the 0- to the π-phase.
π-phases since modulation of the pairing field generates
a lower energy solution. However, the present proposal
differs from FFLO since we do not have a global spin
imbalance and the system is intrinsically stable to phase
separation.
The transition between the different π-phases can be
explained by an energy balance argument.
driven by the competition between interaction (pairing
and Hartree terms) and potential energy in the lattice.
First consider the situation where the gap and densities
are almost constant and even functions, ∆e(x) = ∆0
and n↑(x) = n↓(x) = n0/2 (V0/EF ≤ 0.38 in Fig 2),
and contrast this with the situation where the gap is
an odd function. Let us for simplicity assume the same
magnitude of the gap and introduce a corresponding os-
cillation in the densities, thus ∆o(x) = ∆0sin(2πx/λ)
and no
↑/↓(x) = n0/2 ∓ (δn/2)cos(2πx/λ). In the long-
wavelength limit we can neglect the kinetic energy and
the energy densities of the even and odd state can be
They are
written
ve/o:=
?
g1Dne/o
dx
??
↓(x)ne/o
σ
Vσ(x)ne/o
σ (x)+
↑(x) + |∆e/o(x)|2/g1D
From our ansatz we get
0/g1Dand vo= g1Dn2
V0δn/2 − (g1D/2)(δn/2)2. If we determine the density
variation in the odd state by requiring minimal energy,
we find vo = g1Dn2
which it follows that the constant even solution is lower in
energy until V0= ∆0. Taking ∆0/EF ∼ 0.3 from Fig. 2
gives V0/EF ∼ 0.3. Numerically we find V0/EF ∼ 0.39,
about 30% higher. At small λ we expect large deviations
from this estimate. This is caused by the neglected ki-
netic term that grows with decreasing λ and pushes the
jump to larger V0. Non-commensurate solutions should
be thermodynamically unstable by the same argument as
they do not benefit from lowering of the potential energy.
Initially we did search for non-commensurate solutions
but as expected we found only commensurate ones.
The transitions we find are very sharp as illustrated in
Figs. 2 and 3. The 0- and π-phase have different parities
and we therefore have a crossing of ground-states as we
tune V0. We test the stability of our predictions by us-
ing a potential that has V↑(x) = −3V↓(x)/2. As Fig. 4
shows, a sharp transition occurs also in this case. Even
though this potential breaks λ/2 symmetry, there is still
conservation of parity, thus ∆− ˜ m= ±∆˜ m, and a sharp
transition still occurs. However, now the order parameter
constains both even and odd Fourier components.
The results presented in Figs. 3 and 4 have kFλ = 30.
For smaller λ ∼ ξ, the lattice drives the system into the
normal state before showing any noticeable oscillations
of the gap. For the interaction strength g1DkF/EF =
−2.04, we estimate that kFλ ? 12 is necessary to support
observable π-phases. In the phase diagram in Fig. 1 the
suppression of the gap at the transitions at small λ is
clearly seen. Fig. 1 also demonstrates that more zeros of
the gap per unit cell could be accessible in experiments.
The presented results are for the zero temperature case.
?
/L,(4)
where L is the system size.
ve= g1Dn2
0/4+∆2
0/4+∆2
0/(2g1D)−
0/4 + ∆2
0/(2g1D) + V2
0/(2g1D) from

Page 4

4
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ???
? ??1?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
0.60.8
V0?EF
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
0.00.20.4
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ???
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
1.0
0.1
0.2
0.3
??m?? ?EF
? ??4?
? ??3?
? ??2?
? ??0?
FIG. 4: Same as Fig.
V0cos(2πx/λ).
3 but with V↑(x) = −3V↓(x)/2 =
For the parameter regime in Fig. 1, we find that the phase
diagram is qualitatively the same up to T ∼ 0.1TF. We
have not performed a detailed analysis of the transition
to the normal state outside of this parameter regime or
for even higher temperatures. An interesting aspect of
this system is that the spin-dependent potential leads to
an appreciable triplet component in the pairing [17].
In order to detect the π-phase, the rapid ramp tech-
nique can be used to transfer opposite spin pairs into
molecules on the Bose-Einstein condensate (BEC) side
of the Feshbach resonance as shown in [18–20].
can calculate the number of molecules with center-of-
mass momentum q := 2π ˜ m/λ, with ˜ m an integer, by
projection of the state before the ramp into the bound
s-wave molecular state fs(r) ∼
where r is the relative distance and as the scattering
length on the BEC side [21, 22]. Assuming as,a⊥? ξ,
with a⊥ the transverse confinement length, we obtain
nq ∼ (8La2
density terms give only a featureless continuous contri-
bution to nq. For Vσ(x) = −V−σ(x), π-phases are linked
to either only odd or only even Fourier components of
the order parameter. After the rapid ramp process, the
BCS to π-phase transition is seen as the disappearance
of a single spot and the apperance of two new spots,
ruling out other types of oscillation in ∆. In contrast,
phase-separated densities located in the minima of the
potentials would require many Fourier components. Ob-
servation of sharp peaks in the distribution is therefore
a clear sign of π-phases.
For experimental realization we focus here on40K [24].
6Li is another possible candidate, although we note that
a spin-dependent lattice is harder to implement [25]. We
assume a 1D geometry of tubes that are optically trapped
with a superposed magnetic field to control the interac-
tion via the Feshbach resonance at B0= 202.1G. Using
N ∼ 100 per tube of length L ∼ 40µm, we have n ∼ 2.5
µm−1and kF = πn/2 ∼ 3.93 µm−1. For simplicity we
We
?2/asexp(−r/as)/r,
⊥/g2
1Das)κ[a⊥/as]2|∆˜ m|2[23].Additional
neglect the external confinement. With a⊥ = 60.3nm
and as= 132.3nm from [24], and using ∆1/EF = 0.15
and g1DkF/EF = −2.04, we find n2π/λ= 1.58. How-
ever, the number of tubes in [24] was 4900 so we expect
a signal of 7742 molecules, which should be detectable.
In the unpolarized system, the appearance of π-phases
in the order parameter requires a spin-dependent lattice
potential with a wavelength longer than the coherence
length; λ ? ξ. To fulfil both requirements multiple lasers
should be used [26–28]. To get spin-dependence there
are several proposals and we focus on the one of [28].
The splitting is controlled by the difference in laser inten-
sity and phase of left-circular and right-circular polarized
light. Furthermore the transition is between the2S1/2
and2P1/2,2P3/2lines with optical wavelength λopt∼ 770
nm. To change λ in the lattice one changes the angle be-
tween laser and tubes. The magnetic field is aligned par-
allel to the lasers. If θ is the angle between the tubes and
these lasers, the lattice wave-length is λ = λopt/2cos(θ).
kFλ = 30 translates to 7.6 µm and θ ∼ 87o. Since this
is almost perpendicular to the 1D tube, the heating will
also be reduced as most of the recoil is absorbed in the
confining potential.
In an actual experiment it has been suggested that 1D
tubes in an intermediate strength 2D optical lattice will
give the best conditions for observing the FFLO state in
1D [29], and we expect this to hold for our π-phases as
well. Our proposal differs from other studies on FFLO
states in cold atom since we use an unpolarized gas. The
Fermi surfaces are therefore identical for the two spins
and the non-trivial pairing properties of the system are
entirely due to the spin-dependent lattice potential.
We thank R. Sensarma, D. Pekker, L. Fritz, D. Weld,
M. A. Cazalilla and F. Sols for helpful discussions. The
authors acknowledge support from Real Colegio Com-
plutense en Harvard, MEC (Spain) grant FIS2007-65723,
the German Research Foundation grant WU 609/1-1,
the Villum Kann Rasmussen foundation, CUA, DARPA,
MURI, and NSF grant DMR-0705472.
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0
dx?∞
0
dyy exp(−√
x2+α2y2−y2/4)
x2+α2y2
√