Two-dimensional polaron spectroscopy of Fermi superﬂuids Ivan Amelio1 1Institute of Quantum Electronics ETH Zurich CH-8093 Zurich Switzerland

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Two-dimensional polaron spectroscopy of Fermi superﬂuids

Ivan Amelio1

1Institute of Quantum Electronics ETH Zurich, CH-8093 Zurich, Switzerland

(Dated: October 5, 2022)

Multidimensional spectroscopy is becoming an increasingly popular tool and there is an ongoing

eﬀort to access electronic transitions and many-body dynamics in correlated materials. We apply

the protocol recently proposed by Wang [1] to extract two-dimensional polaron spectra in a Fermi

superﬂuid with an impurity. The bath is descibed by a BCS ansatz and it assumed that the impurity

can scatter at most one quasiparticle pair. The spectral response contains a symmetric contribution,

which carries the same information as Ramsey spectra, and an asymmetric one. While a priori it may

seem promising to probe the quasiparticle gap from the asymmetric contribution, we show explicitly

that this is not the case and, in the absence of incoherent processes, multidimensional spectroscopy

does not bring much additional information. Our calculation is suitable for 3D ultracold gases, but

we discuss implications for exciton-polarons in 2D materials.

Introduction. Multidimensional spectroscopy [2–4] is

an experimental tool that allows to study extremely fast

processes with high spectral resolution. It has been

tremendously successful in investigating the mechanisms

underlying photosynthesis [5] and it is an essential tool

to understand the incoherent and coherent energy trans-

fer in molecular aggregates and the dynamics electronic

transitions. Currently, there is growing interest in us-

ing this approach to explore the many-body properties

of correlated materials [6, 7], including cuprates. Onging

attempts to extend the technique to the terahertz range

are also worth to be mentioned [8].

Recently, Wang proposed [1, 9] an extension of multi-

dimensional spectroscopy suitable for cold atoms experi-

ments, consisting in immersing an impurity in a Fermi sea

and performing a sequence of four Rabi pulses. Here, we

will refer to this approach as to two-dimensional polaron

spectroscopy (2DPS), since the dressing of the impurity

by the excitations of the bath to form polaronic states

plays a central role for the impurity dynamics.

In this Letter we are interested in applying the 2DPS

protocol to an impurity immersed in a three-dimensional

Fermi superﬂuid, where spin up fermions pair with spin

down fermions [10]. Polaron formation along the BEC-

BCS crossover is in itself a topic of intense theoretical

research [11?–15]. The main challenge is to provide an

accurate description both in the BCS and BEC regime,

to recover the Fermi and Bose polaron cases respectively.

Using a generalized Chevy ansatz [16] on top of a

BCS variational state [17], we compute the Ramsey one-

dimensional spectra as well as the 2DPS ones. We show

that, in the absence of incoherent energy transfer, the

2DPS does not bring much additional information. In

particular, there is no direct signature of the quasipar-

ticle gap in the asymmetric contribution to the 2DPS.

Our results may help to elucidate experimental data in

future studies of multidimensional spectroscopy in super-

conductors.

Polarons in Fermi superﬂuids. We consider a zero tem-

perature gas of spin one-half fermionic atoms described

by the annihilation operators ckσ, where σ=↑,↓, and by

the dispersion ξk=k2

2m−µ, where mis the mass and µthe

chemical potential. A single impurity of mass Mis also

present and it has two internal states, splitted by a large

energy ω0and described by dkσ. Each fermion interacts

with a spin-ﬂipped atom via a contact attractive inter-

action with coupling strength g. This is related to the

scattering length ain the usual way 1

g=m

4πa −1

VPΛ

k2,

where Vis the volume of the system and Λa cutoﬀ.

We assume that the impurity interacts only with the ↑

fermions and only when in its ↑state, with strength g↑

and scattering length a↑. This is reasonable experimen-

tally, since Feschbach resonances involve a speciﬁc spin

conﬁguration.

The overall Hamiltonian reads

H=X

kσ

ξkc†

kσckσ+g

kpq

c†

k+q↑c†

p−q↓cp↓ck↑+

kσω0δσ↑+k2

2Md†

kσdkσ+g↑

kpq

c†

k+q↑d†

p−q↑dp↑ck↑

(1)

and it is actually convenient to split it as H=H↑+H↓

depending on the internal state of the impurity.

Pairing in the ground-state of the bath can be qualita-

tively captured by the variational BCS ansatz along all

the BEC-BCS crossover [17]. A convenient approach is

to introduce the fermionic quasiparticle operators

γk↑=ukck↑+vkc†

−k↓, γk↓=ukck↓−vkc†

−k↑.(2)

We deﬁne |BCSias the vacuum of the quasi-particles

γkσ|BCSi= 0.Writing uk= cos θk, vk= sin θk, the vari-

ational parameter θkminimizes the energy for tan 2θk=

−∆

ξk,where ∆ = g↑

VPkhck↑ck↓iBCS =−g↑

VPkukvk

is the pairing order parameter. In the following we

use EF=kF

2m=3π2

mnas unit, which ﬁxes kFand

the density of each spin n=1

VPkhc†

kσckσiBCS, where

hc†

kσckσiBCS =v2

arXiv:2210.01174v1 [cond-mat.quant-gas] 3 Oct 2022

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Two-dimensionalpolaronspectroscopyofFermisuperuidsIvanAmelio11InstituteofQuantumElectronicsETHZurich,CH-8093Zurich,Switzerland(Dated:October5,2022)Multidimensionalspectroscopyisbecominganincreasinglypopulartoolandthereisanongoingeorttoaccesselectronictransitionsandmany-bodydynamicsincorrelatedmate...

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Two-dimensional polaron spectroscopy of Fermi superﬂuids Ivan Amelio1 1Institute of Quantum Electronics ETH Zurich CH-8093 Zurich Switzerland

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