PHYSICAL REVIEW B 99, 214505 (2019)

Atomtronics with a spin: Statistics of spin transport and nonequilibriumorthogonality catastrophe in cold quantum gases

Jhih-Shih You,1,2 Richard Schmidt,3,4 Dmitri A. Ivanov,5 Michael Knap,4,6 and Eugene Demler1

1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA2Institute for Theoretical Solid State Physics, IFW Dresden, Helmholtzstr. 20, 01069 Dresden, Germany

3Max Planck Institute of Quantum Optics, Hans-Kopfermann-Str. 1, 85748 Garching, Germany4Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 München, Germany

5Institute for Theoretical Physics, ETH Zürich, 8093 Zürich, Switzerland6Department of Physics and Institute for Advanced Study, Technical University of Munich, 85748 Garching, Germany

(Received 13 December 2018; revised manuscript received 23 May 2019; published 13 June 2019)

We propose to investigate the full counting statistics of nonequilibrium spin transport with an ultracold atomicquantum gas. The setup makes use of the spin control available in atomic systems to generate spin transportinduced by an impurity atom immersed in a spin-imbalanced two-component Fermi gas. In contrast to solid-staterealizations, in ultracold atoms spin relaxation and the decoherence from external sources is largely suppressed.As a consequence, once the spin current is turned off by manipulating the internal spin degrees of freedom of theFermi system, the nonequilibrium spin population remains constant. Thus one can directly count the number ofspins in each reservoir to investigate the full counting statistics of spin flips, which is notoriously challenging insolid-state devices. Moreover, using Ramsey interferometry, the dynamical impurity response can be measured.Since the impurity interacts with a many-body environment that is out of equilibrium, our setup provides away to realize the nonequilibrium orthogonality catastrophe. Here, even for spin reservoirs initially preparedin a zero-temperature state, the Ramsey response exhibits an exponential decay, which is in contrast to theconventional power-law decay of Anderson’s orthogonality catastrophe. By mapping our system to a multistepFermi sea, we are able to derive analytical expressions for the impurity response at late times. This allows us toreveal an intimate connection of the decay rate of the Ramsey contrast and the full counting statistics of spin flips.

DOI: 10.1103/PhysRevB.99.214505

I. INTRODUCTION

Some of the most interesting applications of condensedmatter theory are concerned with transport [1–3]. Most studiesof transport focus on averaged quantities such as currentsof charge, concentrations, or heat. However, transport ex-periments contain more information than just those averagequantities. Indeed, one of the important ideas that emergedin the studies of transport in condensed matter physics isthat fluctuations contain more information than accessiblefrom sole measurements of averaged quantities. In particular,the study of quantum noise that arises from fluctuations thatpersist even at zero temperature became of great practicalrelevance since it presents the ultimate limit to noise inelectronic and spintronic devices. From a more fundamentalperspective, the analysis of noise in transport [4,5] made thedemonstration of charge fractionalization in quantum Hallsystems possible [6,7], and provided a new means to sep-arate ballistic and diffusive quasiparticle transport in low-dimensional materials [8].

Likewise, achieving a high level of control over transportrequires a study beyond average quantities [9]. In particular,gaining control on the level of single electrons and spinsnecessitates the understanding of the intrinsic quantum noisein such systems [10–12]. A theoretical tool for this pur-pose is the full counting statistics (FCS) that contains the

information about all moments of the desired observable [13].In solid-state experiments, the control of the quantum noiseis, however, challenging since it is difficult to change systemparameters [5,14–22].

In recent years, ultracold atoms have emerged as a toolboxto study the transport of in- and out-of-equilibrium systems ina controlled setting [23–27], where a high degree of isolationfrom the environment is realized and single-atom resolutionis achievable. This allowed one to study analogs of elec-tronic transport, where neutral atoms correspond to chargecarriers in solid-state systems, giving name to the field ofatomtronics [28–31]. First examples range from the expansionof fermions in optical lattices [32] to the conductivity of aFermi gas [33], localization induced by disorder in Hubbardmodels [34–36], the realization of the analogs of diodes [37],transistors [38], and squids [39], as well as the study of atomtransport through point contacts [40–42], anomalous transportin quantum Hall systems [43–45], and topological chargepumping in bosonic quantum gases [46,47].

Inspired by the recent ultracold atom experiments onquantum impurities [48–56], in this work we propose a newtype of transport experiment that allows one to realize spin-atomtronic systems, analogs of spintronics in ultracold atomicsystems. Our setup provides a new platform for studying thefull counting statistics of transport, and allows one to revealits remarkable relation to the nonequilibrium orthogonality

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YOU, SCHMIDT, IVANOV, KNAP, AND DEMLER PHYSICAL REVIEW B 99, 214505 (2019)

Spin rotationbetween Fermi seas

impurity

FIG. 1. Schematic representation of our setup. An impurity atomis coupled to a spin-imbalanced two-component Fermi gas with mis-matched chemical potentials μ↑ �= μ↓. Top row: the impurity atomin the internal state |u〉 resonantly interacts with the first component|↑〉, but not with the second one |↓〉. Bottom row: applying a spinrotation mixes the two spin states and introduces impurity-inducedspin-flips between the two fermionic components.

catastrophe. In contrast to solid-state systems, our proposedultracold atom setup does not suffer from limited coherencetimes resulting from phonon relaxation and electron interac-tions and has the advantage that dynamics takes place on amuch longer time scale due to the diluteness of the atomicquantum gas.

Specifically, our proposed setup consists of a single quan-tum impurity that is coupled to two reservoirs of fermions,see Fig. 1. These two imbalanced Fermi reservoirs can beexperimentally realized by preparing fermionic atoms in twodifferent hyperfine states. We show that, despite atom colli-sions being originally spin conserving, by creating a superpo-sition of the two hyperfine states, spin changing collisions canbe engineered [57]. Combined with controllably switchingthe interactions between the impurity and the Fermi seas, anonequilibrium spin-flip dynamics between the reservoirs isinduced that can be directly measured using, e.g., absorptionimaging. Moreover, the full counting statistics of the scatteredfermionic particles is accessible, which is characterized by theprobability distribution PN (t ) of finding N scattered particlesat time t . With cold quantum gases this can be achievedusing time-of-flight measurements [58] or quantum gas mi-croscopy [59–66], both techniques that are not available insolid-state systems.

In addition, decoherence dynamics of the system canbe studied by applying a Ramsey sequence on the impu-rity [53,67–70]. We find that the Ramsey response of theimpurity, S(t ), is governed by a nonequilibrium orthogonalitycatastrophe (NOC). Quite counterintuitively the NOC featuresan exponential decay in S(t ) even at zero temperature. Thisis in contrast to the conventional orthogonality catastrophewhere an exponential decay is a signature of thermal deco-herence [67] (for a review see Ref. [69]). Remarkably, inthe long-time limit we find, up to logarithmic corrections, asimple relation between the decay of the Ramsey signal S(t )

and the FCS of spin flips at zero temperature

|S(t )| ∼√

PN=0(t ). (1)

This equation highlights the intimate relation betweenRamsey interferometry and the counting statistics of spin flips.

This work is organized as follows. In Sec. II we introducethe model. In Sec. III we discuss spin transport and fullcounting statistics for various parameter regimes. In Sec. IVwe present the results for the impurity decoherence dynamics,which can be measured by Ramsey interferometry and discussthe NOC. The full-time Ramsey response is evaluated numer-ically, but also long-time analytical expressions are provided.We present an analysis for both zero and finite temperatureand establish the relation between S(t ) and PN=0(t ). In Sec. Vwe summarize our results and discuss future prospects.

II. MODEL

We consider a single immobile impurity immersed in anoninteracting two-component Fermi gas. Experimentally thefermionic atoms of mass m are initially prepared in two(hyperfine) spin states denoted by (↑,↓). Furthermore, weassume that the impurity has two internal states |u〉 and |d〉.For simplicity we assume that interactions occur only betweenthe impurity in the state |u〉 and fermions in the |↑〉 state; ouranalysis can, however, be easily generalized. The Hamiltonianis given by

H =∑kσ

(εk − μσ )c†kσ ckσ + |u〉 〈u| ⊗ 1

V∑kq

Vqc†k+q↑ck↑,

(2)

where V is the system volume and c†kσ and ckσ denote the

fermion creation and annihilation operators, respectively. Thedispersion relation of the fermions is εk = k2/2m and theiroccupation number Nσ in the two spin states σ = (↑,↓)can be tuned individually by the chemical potentials μσ .Unless indicated otherwise we work in units where h = 1.In the following we consider contact interactions between theimpurity and the Fermi gas so that the momentum dependenceof the potential Vq = V0 can be neglected. Consequently, theresulting s-wave scattering phase shift δk at scattering momen-tum k = |k| is fully parametrized by the scattering length a.While our analytical results hold for general δk , in the specificexample of contact interactions considered in this work, thephase shift δk is then given by

δk = − tan−1(ak). (3)

The Hamiltonian (2) conserves spin and hence does notsuffice to study spin transport. In order to introduce therequired spin-changing interactions we make use of coherentspin-control available in atomic systems. To this end we startfrom the state |FS↑〉 ⊗ |FS↓〉, where |FSσ 〉 represent filledFermi seas (at zero temperature). Then a spin rotation isapplied that rotates the spin state of fermions on the Blochsphere at an arbitrary polarization angle θ leading to atoms ina superposition state described by

dk1 = cos(θ/2)ck↑ − sin(θ/2)ck↓,

dk2 = sin(θ/2)ck↑ + cos(θ/2)ck↓ (4)

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50

t EF100

E(k)/ EF1

0.5 1 01.5 2

0.5

n 1(t,k)

1

(a)

50

t EF100

E(k)/ EF1

0.5 1 01.5 2

0.5

n 2(t,k)

1

0

0.2

0.4

0.6

0.8

1(b)

FIG. 2. Nonequilibrium momentum population. Occupations (a) n1(t, k) and (b) n2(t, k) in the two different fermionic spin states at zerotemperature. Initially, only the state |1〉 is occupied up to the Fermi energy and no atom are in the second state |2〉. We have chosen thedimensionless interaction parameter kF1a = −6.

(for an illustration see Fig. 1). In the absence of impurities inthe |u〉 state this process is fully coherent. It initializes the state|ψF 〉 ≡ |FS1〉 ⊗ |FS2〉 with |FSα〉 =∏|k|<kFα

d†kα |0〉 and α =

(1, 2), where the Fermi momenta kF1 = kF↑ and kF2 = kF↓

are invariant under the spin rotation [similarly, N1(t = 0) =N↑ and N2(t = 0) = N↓].

Expressing the fermionic operators in Eq. (2) in terms ofdk1 and dk2 yields

H =H0︷ ︸︸ ︷∑

kα

εkd†kα dkα + 1

V∑kq

Vq

(d†

k+q1

d†k+q2

)T(cos2

(θ2

)cos(

θ2

)sin(

θ2

)cos(

θ2

)sin(

θ2

)sin2

(θ2

) )(dk1

dk2

)︸ ︷︷ ︸

H1

⊗ |u〉 〈u| −∑

α

μα d†kα dkα. (5)

Here, the second term generates spin-flip processes betweenthe states 1 to 2 of the atoms in the Fermi seas when scatteringwith the impurity and thus Eq. (5) allows one to realize theanalog of a quantum spin pump. In Appendix A we providea solution to the single-particle problem corresponding tothe Hamiltonian (5) where the spin-dependent interaction iscontrolled by the polarization angle θ and interaction strengthV0. Both are fully tunable in real time in ultracold atomic sys-tems. In the following we study the dynamical and statisticalproperties of this Hamiltonian.

III. SPIN TRANSPORT

In our setup the Fermi seas |FS1〉 and |FS2〉 representtwo spin reservoirs 1 and 2. We choose EF2 = 0 so that thesystem is initially far from the state of equal spin population.Switching the impurity state from |d〉 to |u〉 leads to spin flipsthat result in a spin current from reservoir 1 to 2.

A. Spin current

First we study the discharging dynamics of the two-component Fermi gas. In our setup the spin transport rate (wedenote it as ‘spin current’) between the reservoirs |FS1〉 and|FS2〉 is controlled by the rotation angle θ . There are twoprocesses contributing to the dynamics: First, a fermion inreservoir 1 can scatter with the impurity leading to a changein its momentum state, while it remains in the same spin state.This is a spin-conserving process. By contrast, in the secondtype of process the impurity can additionally flip its spin in the

scattering event, leading to a transfer of spins from reservoir1 to 2.

In the time evolution, the spin current generated by thespin flips is accompanied by a buildup of a nontrivial mo-mentum distribution in both spin components n1,2(k, t ) =〈ψF |eiHt n1,2(k)e−iHt |ψF 〉. We consider an ultracold, diluteFermi gas and short-range interactions. Hence only s-wavestates contribute to the dynamics and we will only considerthese modes in the following. The two main processes con-tributing to the dynamics are reflected in the s-wave con-tributions n1,2(t, k) shown in Fig. 2 (k refers to the s-waveradial momentum). First, in |FS1〉 the sudden switch on of theimpurity leads to the generation of particle-hole fluctuationswithin the Fermi sea that are the origin of the Andersonorthogonality catastrophe [71]. This dynamics that originatesfrom the momentum-changing collisions of the fermions withthe impurity is well studied [67,69]. There is, however, alsothe second process corresponding to the spin flips between thestates 1 to 2, and, since we have chosen the second Fermi sea|FS2〉 to be initially empty, one can attribute all atoms foundin the state 2 to such spin-flip processes.

The spin-flip probability �(E ) inherits its energy depen-dence from the phase shift δ(E ) ≡ δk=√

2mE that increasesmonotonically with energy E = k2/2m. It is determined byrecognizing that scattering occurs according to |↑〉 ⊗ |u〉 →e2iδ(E ) |↑〉 ⊗ |u〉, and |↓〉 ⊗ |u〉 → |↓〉 ⊗ |u〉. From this rela-tion it follows (see Appendix B)

�(E ) = sin2 θ sin2 δ(E ). (6)

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YOU, SCHMIDT, IVANOV, KNAP, AND DEMLER PHYSICAL REVIEW B 99, 214505 (2019)

Since the phase shift δ(E ) increases monotonously in mag-nitude with energy, the spin-flip probability is largest forfermions close to the Fermi surface. Hence we find the largestbuildup of occupations in the reservoir 2 close to the Fermienergy EF1 of the first Fermi sea.

Experimentally the momentum occupation n2(t, k) can bemeasured by transferring the impurity back to its noninter-acting state |d〉 at time t and simultaneously rotating theFermi seas back to their ↑,↓ states. Following the separationof the spin states ↑ and ↓ by a Stern-Gerlach procedure,the momentum distribution is obtained from a time-of-flightmeasurement. Since the dynamics has been initialized withan empty reservoir 2, all observed atoms in the atomic↓-state can be attributed to the spin-flip dynamics. This allowsone to achieve measurements with a high signal-to-noiseratio.

We find that the current flow is not only unidirectionalfrom reservoir 1 to 2 at early times, but remains so alsoat long times. This effect can be understood in a picturewhere the Fermi sea is decomposed into wave packets thatare localized both in energy and space [69]. When thesewave packets are scattered off the impurity they move bal-listically outwards and can not rescatter. If their spin hasbeen flipped in the scattering process they are thus forcedto remain in the final spin state. Note that in the scatteringprocess the wave packet becomes a superposition of spin-flipped and spin-conserved components. In real space thiseffect will be visible as an ever growing cloud of atoms withspin-flipped components moving outwards from the impuritycenter.

Summing over the occupation numbers Nσ (t ) =∑k nσ (k, t ) we numerically find that after a short initial time a

steady current N2(t ) = Jt is established [72]. Here the currentis defined as J = dN (t )

dt with N (t ) = N2(t ) − N2(0). Thecurrent can also be determined analytically by integrating thespin-flip probability �(E ) in Eq. (6) over the occupation of thereservoir 1. With the phase shift δ(E ) = − tan−1(a

√2mE ),

we arrive at

J =∫ EF1

0

dE

2π�(E ) = sin2 θ

2ma2EF1 − ln(1 + 2ma2EF1)

4πma2.

(7)

Figure 3 demonstrates that the data, obtained by the functionaldeterminant approach (FDA), see Appendix A for details, isfully described by the analytical expression. This figure alsoillustrates how the spin current J can be controlled in variousways. For instance, changing the dimensionless scatteringlength kF a, the largest current is achieved at resonance wherea diverges and the scattering rate is thus maximal. The sym-metry between positive and negative values of kF a, directlyapparent from the analytical result Eq. (7) [cf. also Fig. 3(a)],indicates that the bound state, existing for a > 0 is not relevantfor the spin transport dynamics at long times. Moreover, asshown in Fig. 3(b), the spin current J can be adjusted bythe polarization angle θ , which determines the ratio of theoff-diagonal to diagonal matrix elements in Eq. (5). As canbe seen from Fig. 3(b) J increases monotonically with θ andreaches its maximum at θ = π/2.

0

0.05

0.1

0.15

0.2 kF1 a = 12

kF1 a = -1.5

(b)

-1 -0.5 0 0.5 10

0.05

0.1

0.15

0.2 = /4 = /2 = 3 /4

(a)

0

AnalyticalNumerics

FIG. 3. Nonequilibrium spin current J . (a) The current J isshown for EF2/EF1 = 0 as function of the inverse scattering length1/kF1a for θ = π/4, π/2, 3π/4. (b) J as function of the spin rotationθ for fixed interaction kF1a = −1.5 and 12. The current J is sym-metric with respect to θ and π − θ . Both cases are evaluated at zerotemperature T = 0. The numerically evaluated current J , symbols,agrees well with the analytical expression (7), solid lines.

B. Full counting statistics of spin current

In solid-state systems it is notoriously difficult to micro-scopically observe spin transport dynamics on the level ofa few spins. By contrast, with cold atoms one can directlycount the number of transferred spins by absorption imaging.Moreover, spin counting can be achieved in real time bydestructively measuring the particle number at arbitrary timesbecause of the characteristically slow dynamics of cold atomicsystem [54]. This brings about the possibility to study time-resolved shot-to-shot fluctuations.

While the current J gives the averaged particle numberN2(t ) transferred per time between the Fermi seas, in eachindividual experimental measurement the observed numberN2 will fluctuate. The corresponding probability PN2 to mea-sure a certain transferred particle number N2 in an individualexperimental realization, also called the full counting statistics(FCS) of N2, is given as the Fourier transformation of thecharacteristic function

χ (λ, t ) ≡ 〈eiλN 〉(t ) =∑

N

PN (t )eiλN (8)

with respect to the counting parameter λ.The characteristic function χ (λ, t ) contains all information

about the distribution of counted particles. In particular arbi-trary moments of the distribution PN (t ) can be computed bydifferentiation 〈Nn〉t = dn

d (iλ)n χ (λ, t )|λ=0. Since N is a bilinear,one can compute χ (λ, t ) exactly using the functional determi-nant approach (FDA), from which PN (t ) then follows from aFourier transform.

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0 50 100 150 2000

5

10

15

20

25

30

35

40

0

0.2

0.4

0.6

0.8P

20 25 30 35N

0

0.1

0.2

0.3

0.4

P t E =10

t E =200

t E F1

N2

(a) (b)

(c) (d)

0 50 100 150 2000

2

4

6

8

10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

P N

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

P N

0 1 2 3 4 5 6

0 2 4 60

0.5

1

PN

0 2 4 6 8 10N2

0

0.2

0.4

PN

t E =10

t E =200

FDA (numerics)

FDA (numerics)

Analytical

kFa = -0.5

kFa = -6

FIG. 4. Full counting statistics. Probability PN2 to measure N2

atoms for rotation angle θ = π/2 and EF2/EF1 = 0 at zero tem-perature for kF1a = −0.5 [(a, b)], and kF1a = −6 [(c, d)]. The leftpanels show PN2 (t ) as a function of the time tEF1 and transferred spinnumber N2 as obtained from the numerically exact FDA calculation.The right panels show PN2 (t ) at fixed times tEF1 = 10 and tEF1 =100. The numerical results are shown as blue bars and the analyticalprediction from Eq. (9) is shown as red circles. In the top panel of(d) we do not show analytical data as at such short times and stronginteractions Eq. (9) becomes invalid.

In Figs. 4(a), 4(c) we show the time evolution of PN2

for intermediate and strong interactions at zero temperatureand polarization angle θ = π/2 as function of time t andN2. After sufficiently long times, the distribution is peakedaround its mean value, and tracking the linear evolution ofthe mean with time makes the development of the steadyspin current J evident. However, what is the distribution ofmeasured N2 away from its mean? This question is studied inFigs. 4(b), 4(d) where the distribution PN2 (t ) obtained fromthe FDA is shown at fixed times tEF1 = 10 and 100 as bluebars.

The observed distributions can again be studied in a wavepacket picture. Over time wave packets reach the impurityand either remain in the original spin reservoir (only pickingup a scattering phase shift) or they undergo a spin-changingcollision. For N incoming particles within a time span t thereare N trials to flip the spin. This line of argument leads us toLevitov’s formula that describes fermions transmitted througha multichannel barrier at zero temperature [3,13],

ln χ (λ, t ) = t∫ EF1

0

dE

2π hln[1 + �(E )(eiλ − 1)]. (9)

The data obtained from this expression, which is valid in thelong-time limit, is shown as red circles in Figs. 4(b), 4(d).The excellent agreement with the exact numerical result un-derlines the accuracy of the intuitive picture of wave packetsof fermions scattering of the impurity and thereby flipping

their spin with a finite probability. One can understand theFCS derived from Eq. (9) in various regimes analytically.For very weak coupling |kF a| � 1, where δk = −ka, Eq. (9)reduces to the characteristic function of a Poisson distribution.At unitarity (where a tends to infinity), δk = π/2, �(E ) isindependent of energy, and Eq. (9) becomes the characteristicfunction of a binomial distribution. Finally, in the regimein between, Eq. (9) represents a superposition of binomialdistributions; see Appendix C. We note that a finite numberof impurities leads to deviations from the FCS studied in thissection, as discussed in Appendix G.

IV. NONEQUILIBRIUM ORTHOGONALITYCATASTROPHE

So far we have discussed how to employ the fermionicmedium as a probe to study transport. However, our systemalso allows us to use the impurity as a probe of the many-bodydynamics to study the nonequilibrium orthogonality catastro-phe (NOC). In the conventional orthogonality catastrophe, asintroduced by Anderson [71] and then extended to dynamicsby Nozieres et al. [73], one considers a single-componentFermi gas in its ground state into which a scattering potentialis suddenly introduced. This results in a quantum quenchdynamics exhibiting a characteristic power-law decay of theimpurity Green’s function [73–76]. Extending this scenario,where the Fermi sea is initially in an equilibrium state, thenonequilibrium orthogonality catastrophe refers to the situa-tion where the system is initially in a nonequilibrium state.This scenario is realized in our setup since the system, despitebeing in a pure state, is initially not in its energetic groundstate of the noninteracting Hamiltonian H0 due to the largespin imbalance between the two reservoirs.

Previously it has been shown that quite generally thesudden introduction of a scattering potential into a systemexhibiting Fermi baths with multiple Fermi edges (in ourcase two), leads to a dynamical response of the system thatfeatures modified power laws accompanied by exponentialdampening [77–82]. Here we bring together the results ofthese previous works as well as the study of subleading excita-tion branches and bottom-of-the-band dynamics introduced inRefs. [67,69], and show how the dynamics can be observed inultracold atom experiments. Combining both analysis allowsus to analytically uncover a nontrivial connection between thedecay of the Ramsey contrast and the tail of the FCS of spintransport. However, before we turn to the analytical analysisof the NOC, we consider the exact numerical solution of theproblem and outline how it can be probed in experiments.

A. Ramsey spectroscopy

One of the key signatures of the NOC is contained in theimpurity Green’s function that can be probed directly inRamsey spectroscopy. In contrast to the previousworks [54,67,69], here the Ramsey sequence is performedon both the spin degree of freedom of the impurity as wellas the bath atoms: the experimental sequence starts with theimpurity atom prepared in a hyperfine state |d〉, for whichthe interaction between the impurity and the Fermi gas,|FS↑〉 ⊗ |FS↓〉, is absent. As a next step, the Hamiltonian (5),

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YOU, SCHMIDT, IVANOV, KNAP, AND DEMLER PHYSICAL REVIEW B 99, 214505 (2019)

describing spin-flip interactions between the two componentsof the Fermi gas, is realized by performing a spin rotationon the internal spin degree of freedom of the fermionicatoms with polarization angle θ . This leads to the state ofthe Fermi bath |ψF 〉 = |FS1〉 ⊗ |FS2〉 (cf. Fig. 1). Note,in this section we allow for a finite Fermi energy EF2. Inorder to probe the many-body dynamics of the Fermi gassubject to the Hamiltonian (5), next, a Ramsey sequence isperformed on the impurity spin degree of freedom [67]. Tothis end, using a π/2 radio-frequency pulse, the impurityis prepared in a superposition of its hyperfine states |u〉and |d〉, so that the initial state of the system is given by| (0)〉 = 1√

2(|u〉 + |d〉) ⊗ |ψF 〉. Following this preparation,

the system is evolved for a time t . During this time, thesystem evolves in a many-body superposition state, sincethe Hamiltonian dynamics of the Fermi gas is different forthe impurity states |d〉 and |u〉. Accordingly, the state of thesystem at time t is given by

| (t )〉 = 1√2|u〉 ⊗ e−iH0t |ψF 〉 + 1√

2|d〉 ⊗ e−iH1t |ψF 〉.

(10)

Finally, at time t , the Ramsey sequence is completed bya measurement of σx of the impurity spin. This yields theRamsey signal [54,67,69,83]

〈σx〉 = Re〈ψF |eiH0t e−iH1t |ψF 〉 = ReS(t ). (11)

By choosing the phase of the closing π/2 pulse acting onthe impurity spin, it is, moreover, possible to measure thecomplex signal S(t ) [67], thus providing access to the fulltime-dependent response of the impurity spin [67,83].

As described in Appendix A, the overlap S(t ) can be ob-tained numerically exactly using the FDA. The FDA allows usto map the calculation of many-body wave function overlapsonto the evaluation of determinants in single-particle Hilbertspace. For S(t ) one obtains

S(t ) = 〈ψF | eiH0t e−iH1t |ψF 〉 = det[1 + n(R − 1)]. (12)

Additional to 1 = diag(1, 1), Eq. (12) contains two noncom-muting block matrices: the two-component distribution matrixn = diag(n1, n2) that is diagonal in the (1,2) basis [ni =1/(eβ(h0,i−μi ) + 1)] and the matrix R = diag(eih0,↑t e−ih1,↑t , 1),which acts diagonally in the (↑,↓) basis. Here h0,↑,

h1,↑, and h0,i are the single-particle representations ofthe many-body Hamiltonian H0,↑ =∑k εkc†

k↑ck↑, H1,↑ =H0,↑ + V0

V∑

kq c†k↑cq↑, and H0,i=1/2 =∑k εkd†

kidki, respec-tively.

The time evolution of S(t ) at zero temperature is shown inFig. 5 for EF1 �= EF2. We find that S(t ) develops oscillationsand an exponential damping at long times that persists even atzero temperature. In the conventional orthogonality catastro-phe, an exponential decay of S(t ) is observed only for finitetemperature T > 0. There it indicates thermal decoherencedue to the thermal occupation of single-particle states givenby the Fermi distribution nk. Thus one might be tempted toassume that the exponential decay observed in the NOC mightbe related to the development of a quasithermal state of theFermi bath, which in turn induces quasithermal decoherence.

0 50 100 150 200 t EF1

-1

-0.5

0

0.5

1

ReS

(t),

ImS

(t)

ReS(t),FDAImS(t),FDAanalytical

(b)

0 20 40 60 80 100 t EF1

0.2

0.4

0.60.8

|S(t

)|

EF2

/EF1

=0.75

EF2

/EF1

=0.4

EF2

/EF1

=0

(a)

analyticalFDA

FIG. 5. Ramsey signal of the impurity. (a) Ramsey contrast |S(t )|for θ = 3π/4, scattering length kF1a = 1.5 and temperature T = 0.Red, blue, green symbols correspond to the numerical FDA resultfor EF2/EF1 = 0.75, EF2/EF1 = 0.4, and EF2/EF1 = 0, respectively,while the solid lines show the analytical prediction obtained fromEq. (15), with coefficients C obtained from fits to the data. (b) Realand imaginary part of the Ramsey signal computed numerically usingthe FDA (symbols) for θ = 3π/4, kF1a = 1.5 and EF2/EF1 = 0.75.The solid lines are obtained from the asymptotic form, Eq. (24),using the coefficients C as fit parameters.

However, as we have seen in the previous discussion thatnσ (t, k) does not reach a thermal state; see, e.g., Fig. 2.Therefore, the exponential decay of S(t ) must have a differentorigin and we will discuss below by analytical means.

B. Analytical approach to the asymptotic behaviorof S(t ) at zero temperature

Building on the insight from the numerically exact solutionusing the FDA, one may use the theory of Toeplitz determi-nants to derive analytical expressions that describe the exactdynamics with high accuracy also at intermediate times. Tofind such a description we first map the problem of an impurityinteracting with two Fermi seas to the case of an impurityinteracting with a single-component Fermi sea. To this endwe express both n and R in the (↑,↓) basis using a unitarytransformation (|1〉 , |2〉)T = U (|↑〉 , |↓〉)T . A straightforwardcalculation (see Appendix D) shows that Eq. (12) can beexpressed as

S(t ) = det[1 + (eih0,↑t e−ih1,↑t − 1)n(E )], (13)

where the associated single-particle occupation operator n(E )corresponds to the momentum distribution

n(E ) = (1 − p)nF (E − EF2) + pnF (E − EF1). (14)

This distribution is shown in Fig. 6. It exhibits two Fermisurfaces at energies EF1 and EF2 and the polarization p =cos2(θ/2) determines the occupation of the middle plateau

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FIG. 6. Effective two-step distribution function. The expecta-tion value of the effective single-particle occupation operator n(E )is given by a two-step function n(E ) = (1 − p)nF (E − EF2) +pnF (E − EF1) with polarization p = cos2(θ/2).

in n(E ). Using this transformation we have thus mapped thedynamics of the two-component Fermi gas onto the dynamicsof a one-component Fermi gas featuring two Fermi edgesfor which long-time solutions have been discussed in theliterature [81,82].

In fact, Eq. (13) already allows one to qualitatively un-derstand the source of the observed exponential decoherencepersistent in the NOC at T = 0. Indeed comparing Eq. (13) tothe functional determinant formula Eq. (A1) in Appendix Areveals that the dynamics is effectively governed by a many-body density matrix that describes a single-component Fermigas not in a pure but in a mixed state. It is the classical natureof this state that provides the resource of exponential decoher-ence of the observed dynamics. We now turn to support thisargument by a quantitative derivation.

Following Refs. [67,69], we decompose S(t ) in terms ofbranches of different excitations of the Fermi system. Theseso-called excitation branches are

(i) particle-hole excitations near the two Fermi surfaces[denoted as (FS1) and (FS2)];(ii) excitations from the bottom-of-the-band (FB);(iii) for a > 0, excitations involving the bound state (BS).We focus first on the attractive interaction regime, where

the scattering length a, as determined by the low-energyexpansion of the phase shift δk = −ka, is negative, a < 0.Using the formulation in terms of a single Fermi sea, theasymptotic behavior of S(t ) can be organized as

S(t ) =∑

n1+n2+n3=0

Cn1,n2,n3e−iκ0t

· S(FS1)n1

(t )S(FS2)n2

(t )S(FB)n3

(t ). (15)

Here the subscript ni=1,2 refers to the number of particlesadded to or removed from the first and second Fermi edge,respectively, while n3 < 0 refers to the number of particlesremoved from the bottom of the Fermi sea. Particle numberconservation imposes the constraint n1 + n2 + n3 = 0.

While the coefficients Cn1,n2,n3 depend on the microscopicdetails, the other contributions in Eq. (15) can be cast in ana-lytical form. The complex-valued constant κ0 is, for instance,given by [81,82] (see also Appendix E)

κ0 = E0 − iγ = i∫ ∞

0

dE

2πln[1 + n(E )(e2iδ(E ) − 1)]

= −∫ EF2

0

dE

πδ(E ) −

∫ EF1

EF2

dE

πδeff(E ). (16)

Here the first term of the last expression is obtained from theintegration from 0 . . . EF2 where n(E ) = 1. The second termoriginates from the remaining integration region EF2 . . . EF1

where n(E ) = p < 1. It involves the effective phase shiftdefined by

δeff(E ) = − i

2ln[1 + p(e2iδ(E ) − 1)], (17)

and represents a generalization of Fumi’s theorem (whichexpresses the ground-state energy as a sum over scatteringphase shifts [84,85]) of the conventional OC to the case ofspin-flip interactions.

The analytical calculation of the time-dependent factorsSi(t ) in Eq. (15) is challenging in a naive bosonization ap-proach. Instead, the use of Szego formula [86–89] to secondorder allows one to approach the problem. Indeed, Gutmanand coworkers showed that the contributions involving exclu-sively particle-hole fluctuations close to the two Fermi edgesare given by [81,82]

S(FS1)0 (t ) ∝ t−( δ1

π)2

(18)

S(FS2)0 (t ) ∝ t−( δ2

π)2. (19)

These expressions represent the Fermi edge singularities andexhibit a nontrivial power-law behavior with exponents deter-mined by (see Appendix E)

δ1 = δeff(EF1 − 0+) (20)

δ2 = δ(EF2 + 0+) − δeff(EF2 − 0+). (21)

Generalizing this analysis to the case where n particlesare added or removed from the Fermi edges at EF1 and EF2

allows one to describe analytically not only the long- but alsothe intermediate-time dynamics with high accuracy [81,82].In Appendix E we provide a detailed derivation that leads tothe expressions

S(FS1)n (t ) ∝ e−inEF1t

(1

t

)( δ1π

−n)2

,

S(FS2)n (t ) ∝ e−inEF2t

(1

t

)( δ2π

−n)2

(22)

that are valid in the zero-temperature limit. Note that herewe include the phase factors that depend on the Fermi en-ergies into the definitions of S(FS1,2)

n (t ), which is a differentconvention compared to Ref. [69]. To reflect this choice weintroduced the subindex n = 0 in κ0 given by Eq. (16).

A further contribution, which has so far not been studiedin the context of NOC dynamics, originates from processeswhere particles are excited from the bottom of the bandto the two edges of the Fermi sea, leaving holes behind.The corresponding contribution can be found from few-bodytheory and reads [67,69]

S(FB)n ∝

[∫ ∞

0

dE√E

sin2 δ(E )eiEt

]−n

, (23)

with n � 0.We now turn to the interaction regime for a > 0, where a

weakly bound state of energy Eb < 0 exists. Here, the overlap

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YOU, SCHMIDT, IVANOV, KNAP, AND DEMLER PHYSICAL REVIEW B 99, 214505 (2019)

S(t ) can be expressed as

S(t ) =∑

n1+n2+n3+n4=0

Cn1,n2,n3,n4 e−iκ0t

· S(FS1)n1

(t )S(FS2)n2

(t )S(FB)n3

(t )S(BS)n4

(t ). (24)

The index n4 takes on values 0 or 1, depending on whetherthe bound state is occupied or empty; i.e., S(BS)

1 = e−iEbt orS(BS)

0 = 1, respectively. In Fig. 5(b) we compare the analyticalexpression to the numerical results. Here the coefficients Cserve as fit parameters and we keep only leading contributionswith

∑i |ni| � 2. We find that the asymptotic form reproduces

the exact numerical results with remarkable precision downto small evolution times. Here the superposition of oscillat-ing factors from bottom-of-the-band contributions [given byEq. (23)], bound-state (proportional to eiEbt ) and Fermi surfacecontributions [proportional to ∼e−inEFit , c.f. Eq. (22)] resultsin the oscillations visible in Fig. 5.

C. Role of finite temperature

In the previous discussion we have found that a key sig-nature of the nonequilibrium orthogonality catastrophe is theexponential decay of |S(t )| ∼ e−γ t that is present even at zerotemperature. We now focus on the temperature dependence ofthe decay rate γ . Using the Szego theorem for the asymptoticproperties of Toeplitz determinants, we find

γ = −Re∫ ∞

0

dE

2πln[1 + n(E )(e2iδ(E ) − 1)]. (25)

In this expression, which follows from Eq. (16) (for detailssee Appendix E), we take into account the energy-dependentphase shift δ(E ) and the temperature-dependent distributionfunction n(E ) given by Eq. (14).

In Fig. 7 this analytical result is compared to the decayrate obtained from fitting |S(t )| ∼ e−γ t to the exact FDAresults at long times. We find excellent agreement betweenthe numerical FDA data and the analytical expression bothwhen studying the θ and 1/kF1a dependence of γ for the twotemperatures T/TF = 0 and T/TF = 0.1.

Using the relation Re ln[1 + p(e2iδ(ε) − 1)] = Re ln[1 +(1 − p)(e2iδ(ε) − 1)] one finds from the analytical predictionEq. (25) that the decay rate is symmetric with respect top = 1/2 at zero temperature, as shown by the compari-son of p = cos2(π/4) and p = cos2(3π/4) in Fig. 7(a). Atfinite temperatures this symmetry is absent and, as shown inFig. 7(b), we find that p > 1/2 exhibits a larger decay ratethan p < 1/2. The reason for the different decay rates lies inthe fact that it is spin-conserving collisions within a reservoir,as determined by the diagonal element of the scattering matrixin Eq. (5), which give rise to additional thermal decoherence;and since we have chosen the reservoir 1 to have a largeroccupation, polarizations p = cos2(θ/2) < 1/2 will give alarger decoherence rate compared to p > 1/2.

D. Relation between Ramsey interferometryand the FCS of spin flips

It turns out that the decay rate of the Ramsey signal hasa remarkable relation to the FCS of spin flips. Specifically,we find that the decay rate γ in Eq. (25) and the FCS at

(a)

(b)

FIG. 7. Asymptotic decay rate γ of the Ramsey signal. Thedecay rate γ is shown as a function of scattering length kF1a for threedifferent values of the polarization angle θ = {π/4, π/2, 3π/4},an initially empty second reservoir EF2/EF1 = 0, and temperatures(a) T/TF1 = 0 and (b) T/TF1 = 0.1. The solid lines are obtainedfrom Eq. (25).

zero temperature and EF2 = 0, as described by the time-dependent generating function χ in Eq. (9), are related by (seeAppendix F):

|S(t )| → e12 ln χ (eiλ→0). (26)

From this equation directly follows the relation Eq. (1),|S(t )| ∼ √

PN=0(t ), which holds up to logarithmic corrections.This relation implies that the Ramsey decoherence is given

by the square root of the probability of having no spin flipsin the time interval 0 . . . t , which fits the notion of PN=0(t ) asan idle-time probability, similar to the emptiness probabilitydiscussed in other contexts [90,91]. Thus the Ramsey signalis related directly to the FCS and thus the intrinsic quantumnoise in the number of observed spin flips. Therefore, thedecay of the Ramsey signal can serve as an indirect probe ofthe tail of the FCS at low particle number.

The relation Eq. (1) can be understood as follows: theRamsey contrast |S(t )| is determined by the overlap of many-body states. When the spin of one of the fermions is flipped,a state of the Fermi system results that is orthogonal to theinitial state, leading to a vanishing Ramsey contrast. There-fore, finding a finite Ramsey contrast requires configurationsthat have no fermion spin flipped. The probability of such aconfiguration is PN=0. The Ramsey contrast |S(t )| measures,however, an amplitude [cf. Eq. (12)] so that |S(t )| is propor-tional to

√PN=0.

V. CONCLUSION AND DISCUSSION

In this work we proposed an ultracold atom experimentwhere impurities are coupled to a spin-imbalanced two-

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ATOMTRONICS WITH A SPIN: STATISTICS OF SPIN … PHYSICAL REVIEW B 99, 214505 (2019)

component Fermi gas. The setup allows one to study fun-damental relations between quantum fluctuations in transportand dephasing dynamics. Specifically, we showed that apply-ing rf pulses to the Fermi system provides a means to realizeinitial nonequilibrium spin populations that are required tostudy spin transport. Based on a functional determinant ap-proach we explored the full counting statistics of the spin flipsthat accompany the spin current generated in our setup.

Furthermore, we showed that the dynamics of the many-body wave function can be explored using Ramsey interfer-ometry. This opens the path toward the study of the nonequi-librium orthogonality catastrophe (NOC) with ultracold quan-tum gases. The NOC is characterized by a decay of theRamsey signal, which is exponential although the systemis initially in a pure quantum state, and one thus mighthave naively expected a power-law decay as obtained forthe Fermi edge singularity. By mapping the problem ontoa multi-Fermi-edge scenario in energy space, we obtainedanalytic predictions for the long-time impurity response and,in particular, for its exponential decay rate. This allowed usto uncover a relation between the FCS of spin flips and therate at which the Ramsey contrast of the impurity decays.In this work we considered local quench-type dynamics, inwhich the impurity strength is changed only once. In orderto explore a broader class of nonequilibrium phenomena,one may include multiple quenches of the scattering phaseshift. Mathematically handling such multiple discontinuitieswill require a further generalization of the theory of Toeplitzdeterminants with Fisher-Hartwig singularities [82]. In thisrespect ultracold atom experiments might provide a quantumtool to explore mathematical problems for which solutionshave yet to be found.

Moreover, in the present work we did not attempt toexplore ways to explicitly control the FCS of spin flips.In this regard it will be interesting to study whether it ispossible to suppress fluctuations imprinted in the FCS bycontrolling and manipulating the impurity potential similarlyto the realization of a source of pure single-particle spintransmission [10,13,92]. Finally, it has recently been shownthat von Neumann and the Renyi entanglement entropies canbe expressed in terms of even order cumulants [93,94]. Thefact that the full counting statistics contains the informationabout moments of arbitrary order thus suggests that our pro-posed scheme might enable one to further explore the relationbetween entanglement dynamics and full counting statistics incold atom experiments.

ACKNOWLEDGMENTS

We thank Rudolf Grimm and Dimitri Abanin for fruit-ful discussions. We acknowledge support from Harvard-MIT CUA, NSF Grant No. DMR-1308435, AFOSR-MURI:Quantum Phases of Matter, AFOSR-MURI: Photonic Quan-tum Matter, award FA95501610323. J.-S.Y. was supportedby the Ministry of Science and Technology, Taiwan (GrantNo. MOST 104-2917-I-564-054). M.K. acknowledges sup-port from the Technical University of Munich - Institutefor Advanced Study, funded by the German Excellence Ini-tiative and the European Union FP7 under Grant Agree-ment No. 291763, DFG Grant No. KN1254/1-1, and DFG

TRR80 (Project F8). M.K. and R.S. acknowledge supportfrom the Deutsche Forschungsgemeinschaft (DFG, GermanResearch Foundation) under Germany’s Excellence Strategy–EXC-2111-390814868.

APPENDIX A: FUNCTIONAL DETERMINANT APPROACHAND SOLUTION OF THE SINGLE-PARTICLE PROBLEM

For any bilinear many-body operator Xα =∑i j 〈i| xα | j〉 c†

i c j , we can make use of the identity

〈eX1 · · · eXN 〉 = Tr[ρeX1 · · · eXN ] = det(1 − n + nex1 · · · exN ),

(A1)

with ρ the density matrix and n denotes the correspondingsingle-particle occupation operator. Hence, S(t ) can be ex-pressed as

S(t ) = 〈eiH0t e−iH1t 〉 = det[1 − n + neih0t e−ih1t ], (A2)

where h0 and h1 are the single-particle Hamiltonians in theabsence and presence of impurity, respectively. To evaluatethe functional determinant numerically, we work in the basisof single-particle eigenstates of h0 and h1.

To this end, we solve the single-particle problem in thepresence of localized impurity. The Schrödinger equation forthe two-component host fermions is given by(

−∇2

2m + V (r) cos2(

θ2

)V (r) cos

(θ2

)sin(

θ2

)V (r) cos

(θ2

)sin( θ

2 ) −∇2

2m + V (r) sin2(

θ2

))(

ψ1(r)

ψ2(r)

)

= E

(ψ1(r)

ψ2(r)

), (A3)

where V (r) is the short-range potential. For our numericswe consider a finite system confined in a sphere of radiusR chosen large enough so that finite-size corrections arenegligible. For short-range interactions only the s-wave com-ponents of the scattering wave functions experience a phaseshift. Defining the radial wave function φn(r) via ψn(r) =φn(r)/(

√4πr) with nodal quantum number n, Eq. (A3) is

expressed as a radial one-dimensional Schrödinger equation.The interaction between the impurity and itinerant fermions isfully characterized by the scattering length a with the s-wavescattering phase shift given by δk = − tan−1 ka.

When the host fermions do not interact with the impurity,the eigenfunctions are given by

φ1,n(r) =√

2

Rsin(knr) ⊗ |1〉, φ2,n(r) =

√2

Rsin(knr) ⊗ |2〉,

(A4)

with the boundary condition knR = nπ.

In presence of the scattering potential, Eq. (A3) has so-lutions with energies En = k′

n2/2m that are determined by

k′nR + δk′

n= nπ and eigenstates

φn(r) = An

√2

Rsin(k′

nr + δkn

)⊗[

cos

(θ

2

)|1〉 + sin

(θ

2

)|2〉], (A5)

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YOU, SCHMIDT, IVANOV, KNAP, AND DEMLER PHYSICAL REVIEW B 99, 214505 (2019)

0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

1

E(k)/ EF1

n(k

)

n1(k, t E

F1=0)

n1(k, t E

F1=100)

n2(k, t E

F1=100)

(a)

kF1

a= − 0.5

0 2 4 60

0.2

0.4

0.6

0.8

1

N2

PN

Binomial

t EF1

=10

(b)

0 2 4 60

0.2

0.4

0.6

0.8

1

N2

PN

Binomial

t EF1

=100

(c)

0.6 0.7 0.8 0.9 1 1.10

0.2

0.4

0.6

0.8

1

E(k)/ EF1

n(k

)

n1(k, t E

F1=0)

n1(k, t E

F1=100)

n2(k, t E

F1=100)

(d)

kF1

a= − 0.5

0 2 4 60

0.2

0.4

0.6

0.8

1

N2

PN

Binomial

t EF1

=10

(e)

0 2 4 60

0.2

0.4

0.6

0.8

1

N2

PN

Binomial

t EF1

=100

(f)

FIG. 8. Nonequilibrium momentum population and FCS obtained in a scenario where in the initial state fermions occupy only a smallenergy interval. In the top panel fermions occupy a low-energy interval while in the bottom panel higher energies are occupied. The secondcomponent |2〉 is initially empty while the first component |1〉 has a finite occupation. The interaction strength is characterized by kF1a = −0.5.(a), (d) Energy-resolved occupation by the first and second spin component. The FCS of the number of spin flips PN2 (t ) is shown in (b) and (e)for tEF1 = 10, and at tEF1 = 100 in (c) and (f). The numerical FDA results (blue bars) are compared to a binomial distribution (red crosses).

where An = 1/

√1 + sin 2δk′

n2k′

nR . There exists also a second setof solutions that is given by the noninteracting solutionsdetermined by E0(n) = (kn)2/(2m) and

φ0,n(r) =√

2

Rsin(knr) ⊗

[sin

(θ

2

)|1〉 − cos

(θ

2

)|2〉].

(A6)

Finally, for a > 0 a bound state exists with energy Eb =−1/(2ma2) and eigenfunction

φb(r) = Abe−r/a ⊗[

cos

(θ

2

)|1〉 + sin

(θ

2

)|2〉]. (A7)

Here Ab =√

2a up to corrections that vanish as R → ∞.

APPENDIX B: SPIN-FLIP PROBABILITY �(E )

Here we derive an analytical expression for the spin-flipprobability �(E ) given by Eq. (6) in the main text. Scatteringoccurs only between fermions in their | ↑〉 spin state and theimpurity in the |u〉 state:

|↑〉 ⊗ |u〉 �→ ei2δ(E )|↑〉 ⊗ |u〉|↑〉 ⊗ |d〉 �→ |↑〉 ⊗ |d〉|↓〉 ⊗ |u〉 �→ |↓〉 ⊗ |u〉|↓〉 ⊗ |d〉 �→ |↓〉 ⊗ |d〉, (B1)

where δ(E ) is the energy-dependent phase shift.Initially we apply a spin rotation such that each fermion is

prepared in a superposition state

|1〉 = cos(θ/2)|↑〉 − sin(θ/2)|↓〉, (B2)

|2〉 = sin(θ/2)|↑〉 + cos(θ/2)|↓〉. (B3)

When the impurity is switched into the interacting state |u〉,the bath fermions, now prepared in states |1〉 to |2〉, undergospin-flip interactions. Using Eq. (B1) this scattering process isdescribed by

|1〉 �→ cos(θ/2)ei2δ(E )|↑〉 − sin(θ/2)|↓〉, (B4)

|2〉 �→ sin(θ/2)ei2δ(E )|↑〉 + cos(θ/2)|↓〉. (B5)

When rewriting this process in the basis of |1〉, |2〉|1〉 �→ [ei2δ(E ) cos2(θ/2) + sin2(θ/2)]|1〉

+ (ei2δ(E ) − 1) sin(θ/2) cos(θ/2)|2〉, (B6)

|2〉 �→ (ei2δ(E ) − 1) sin(θ/2) cos(θ/2)|1〉+ [ei2δ(E ) sin2(θ/2) + cos2(θ/2)]|2〉, (B7)

one can directly read of the spin flip probability

�(E ) = |(ei2δ(E ) − 1) sin(θ/2) cos(θ/2)|2 = sin2 θ sin2 δ(E ).

(B8)

APPENDIX C: NONEQUILIBRIUM MOMENTUMPOPULATION AND FCS IN A GIVEN ENERGY INTERVAL

Equation (9) shows that the FCS of the total number ofspin flips is determined as a sum involving the scatteringprobability for each momentum mode of the fermions. Hence,according to this expression, the FCS of spin flips in eachindividual momentum mode gives rise to a binomial dis-tribution. In this Appendix, we show that this argument isindeed confirmed by exact numerical simulation using theFDA.

To this end, we prepare an initial state where the secondFermi sea of component |2〉 is empty and where the momen-tum distribution of the fermions in the state |1〉 has only a

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ATOMTRONICS WITH A SPIN: STATISTICS OF SPIN … PHYSICAL REVIEW B 99, 214505 (2019)

small interval of energy levels that are occupied. The spin-fliprate, as given by Eq. (6), depends on the scattering phase shiftδ(E ) that increases monotonously with E (considering a < 0).Consequently, at a fixed interaction strength kF1a, fermionsin lower-energy modes should experience a smaller spin-fliprate.

This is confirmed by the numerical simulation shown inFig. 8 for moderate interaction strength kF1a = −0.5. In thetop panel we show the time evolution of the FCS for an initialstate occupation confined to a low-energy interval, while forthe bottom panel higher-energy modes are occupied initially.Confirming our expectation from the analytical result Eq. (9),in both cases the FCS of total spin flips [Figs. 8(b), 8(c)and 8(e), 8(f), respectively] obeys a binomial distribution.Furthermore, for higher energies the spin-flip probability isindeed enhanced. In Fig. 9 we repeat the simulation for ainteraction strength kF1a = −12 further corroborating ourfindings.

Note that in the momentum resolved distributions shownin the left panels of the figures a broadening of the ini-tially sharp distribution function can be seen. This broad-ening is due to the sudden quench of interactions, whichprojects the initially occupied states into the eigenstates ofthe interacting Hamiltonian. The overlaps to these states

are nonzero also for states outside of the initial energywindow, which represents the scattering of the fermionsto different momentum state upon collisions with the im-purity and that leads to the broadening of the momentumdistribution.

APPENDIX D: MAPPING ONTO ASINGLE-COMPONENT FERMI GAS

The time-dependent response S(t ) is obtained from thedeterminant det(1 + n(R − 1)), where the two-componentoccupation matrix n = diag(n1, n2) is diagonal in the ro-tated atomic (1,2) basis. The matrix representing the dynam-ics, R = diag(eih0,↑t/he−ih1,↑t/h, 1) = diag(e2iδθ (t ), 1), is on theother hand diagonal in the nonrotated basis (↑,↓). In theseexpressions n1/n2 are the number operators, and δ is thephase shift operator that applies the scattering phase shift toscattering wave packets.

To compute S(t ), we first write both n and R in the basis|↑〉 and |↓〉 using the unitary transformation (|1〉, |2〉)T =U (|↑〉, |↓〉)T , with

U =(

cos(

θ2

) − sin(

θ2

)sin(

θ2

)cos(

θ2

))

, (D1)

to express n as n = U †diag(n1, n2)U . We obtain

S(t ) = det

[(1 0

0 1

)+ U †

(n1 0

0 n2

)U

(e2iδθ (t ) − 1 0

0 0

)](D2)

= det

[(1 0

0 1

)+(

n1 cos2(

θ2

)+ n2 sin2(

θ2

) (n2−n1 )2 sin(θ )

(n2−n1 )2 sin(θ ) n2 cos2

(θ2

)+ n1 sin2(

θ2

))(

e2iδθ (t ) − 1 00 0

)](D3)

= det[1 + (e2iδθ (t ) − 1)n(E )] (D4)

where n(E ) = n1 cos2(θ/2) + n2 sin2(θ/2) represents a one-component distribution exhibiting two Fermi surfaces. It isdetermined by

n(E ) = (1 − p)nF (E − EF2) + pnF (E − EF1), (D5)

where we assumed EF2 < EF1 and defined the polarizationp = cos2(θ/2).

APPENDIX E: FERMI SURFACE DYNAMICS FROMTOEPLITZ MATRICES

In this Appendix we study the Fermi surface contributionsto the time-dependent overlap function

S(t ) = 〈ψF |eiH0t/he−iH1t/h|ψF 〉 = det[1 − n + neih0t/he−ih1t/h]

(E1)

using the theory of Toeplitz matrices. Here h1 and h0 arethe single-particle representations of the many-body Hamil-tonian describing the interaction of an impurity with a single-component Fermi gas. In Eq. (E1) we have used the mappingonto a single-component Fermi gas so that n is the occupa-tion operator given by Eq. (D5). By inspection of Eq. (A1)it is evident that in this representation the system can be

understood to be described by a mixed density matrix. Incontrast, without the mapping the ket |ψF 〉 on the left-handside of Eq. (E1) represents the pure initial state of the systemgiven by |ψF 〉 = |FS1〉 ⊗ |FS2〉.

In the following we work in a basis of wave packetslocalized in time and energy [88]. In this basis the time evo-lution operator eih0t/he−ih1t/h acts approximately diagonallyin energy. Following Refs. [69,81], time may be discretizedaccording to t = Nt where we introduce the time intervalt = hπ/� and a high-energy cutoff �. The overlap S(t ) canthen be rewritten in terms of an N × N Toeplitz matrix σ :

S(t ) = det

⎛⎜⎜⎜⎜⎜⎜⎜⎝

σ0 σ−1 σ−2 · · · σ−N+1

σ1 σ0 σ−1. . .

...

σ2 σ1. . . · · · σ−2

.... . .

. . . σ0 σ−1

σN−1 · · · σ2 σ1 σ0

⎞⎟⎟⎟⎟⎟⎟⎟⎠

. (E2)

The matrix elements σk (k is here a time index) follow fromFourier transformation

σk =∫ 2�

0

dEeiEkt /h

2�hσ (E ), (E3)

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YOU, SCHMIDT, IVANOV, KNAP, AND DEMLER PHYSICAL REVIEW B 99, 214505 (2019)

0.06 0.08 0.10

0.2

0.4

0.6

0.8

1

E(k)/ EF1

n(k

)

n1(k,tE

F=0)

n1(k,tE

F=100)

n2(k,tE

F=100)

(a)

kF1

a=−12

0 2 4 60

0.2

0.4

0.6

0.8

1

N2

PN

2

Binomial

t EF1

=10

(b)

0 2 4 60

0.2

0.4

0.6

0.8

1

N2

PN

2

Binomial

t EF1

=100

(c)

0.7 0.8 0.9 1 1.10

0.2

0.4

0.6

0.8

1

E(k)/ EF

n(k

)

n1(k,tE

F=0)

n1(k,tE

F=100)

n2(k,tE

F=100)

kF a = −12

(d)

0 2 4 60

0.2

0.4

0.6

0.8

1

N2

PN

2

Binomial

t EF1

=10

(e)

0 2 4 60

0.2

0.4

0.6

0.8

1

N2

PN

2

Binomial

t EF1

=100

(f)

FIG. 9. Nonequilibrium momentum population and FCS starting from initial occupations in a small energy interval as in Fig. 8, here forinteractions characterized by kF1a = −12.

of the kernel

σ (E ) = eiEδ/�(1 − n(E ) + n(E )ei2δ ), (E4)

which is diagonal in energy. The high-energy regularizationof the integral in Eq. (E3) follows from the definition ofthe time-interval t = hπ/� so that energies are restrictedto the interval E ∈ (0, 2�]. Furthermore, following Gutmanet al. [81], we have imposed a phase factor eiEδ/� in thekernel σ in Eq. (E4). Introducing the angular variable θ ≡Et/h defined on a unit circle θ ∈ (0, 2π ] to express σk =∫ 2π

0dθeiθk

2πσ (θ ) the phase factor ensures periodicity of the ker-

nel σ (θ ) on (0, 2π ] in Eq. (E4). In the end of the calculationwe will take the limit � → ∞ so that the phase factor willdisappear.

The kernel now obeys periodic boundary conditionslimE→0 σ (E ) = limE→2� σ (E ) [81]. This allows us to applythe Szego theorem [95] to find the asymptotic properties ofthe Toeplitz matrix SN defined by Eq. (E2) in the limit of largeN . Since N = t/t , considering large N corresponds to thelimit of long times t . For large N the Szego theorem statesthat

ln det σ ∼ N[ln σ (θ )]0 +∞∑

k=1

k[ln σ (θ )]k[ln σ (θ )]−k. (E5)

The Szego theorem demands ln σ (θ ) to be be a smooth func-tion with Fourier harmonics [ln σ (θ )]k = ∫ 2π

0dθ2π

ln σ (θ )e−ikθ .In our case the smoothness of σ (E ) is, however, not guar-anteed and we rely on the Fisher-Hartwig (FH) conjec-ture that extends the applicability of Eq. (E5) [81,82,89].In fact in Ref. [81,82] it was shown that also for Fermidistributions with multiple steps the naive formula follow-ing from the strong Szego theorem still leads to correctresults.

Expressing the real-time overlap function as

det σ ∝ e−iκt+c(t ), (E6)

the first term of Eq. (E5) yields a term with a linear depen-dence on time,

−itκ ={

N∫ 2π

0

dθ

2πln σ (θ )

}={

N∫ 2�

0

t dE

2π hln σ (E )

}

→{

t∫ ∞

0

dE

2π hln(1 − n(E ) + n(E )ei2δ )

}, (E7)

where in the last line we have taken the limit � → ∞ so thatσ (E ) → 1 − n(E ) + n(E )ei2δ . Accordingly the exponentialdecay rate γ , defined by |S(t )| ∼ e−γ t , is given by

γ = −Re

{∫ ∞

0

dE

2π hln(1 − n(E ) + n(E )ei2δ )

}.

(E8)

Remarkably, Gutman and coworkers [81] showed that alsofor subleading contribution c(t ) defined in Eq. (E6) an analyt-ical expression can be found. It is determined by the secondterm

∑∞k=1 k[ln σ (θ )]k[ln σ (θ )]−k in Eq. (E5) and as shown in

Ref. [81] it leads to a nontrivial power-law behavior in time.Following Ref. [81] we consider zero temperature T = 0 andthe double step distribution function (EF2 < EF1),

n(E ) = (1 − p)θ (EF2 − E ) + pθ (EF1 − E ). (E9)

The regularized kernel in Eq. (E4) takes the form [81]

σ (E ) = eiEδ/� ×⎧⎨⎩

e2iδ , 0 < E < EF2

1 + p(ei2δ − 1), EF2 < E < EF1

1 , EF1 < E .

(E10)

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ATOMTRONICS WITH A SPIN: STATISTICS OF SPIN … PHYSICAL REVIEW B 99, 214505 (2019)

Thus 12i ln σ (E ) can be expressed as:

1

2iln σ (E ) = Eδ

2�+⎧⎨⎩

δ , 0 < E < EF2

δeff(E ) ≡ − i2 ln[1 + (e2iδ − 1)p], EF2 < E < EF1

0 , EF1 < E .

(E11)

The Fourier harmonics [ln σ (E )]k �=0 =∫ 2�

0t dE2π h ln σ (E )e−ikt E/h required for the evaluation of

Eq. (E5) are given by

[ln σ (θ )]k �=0 = − 1

πk[δ1e−ikEF1t + δ2e−ikEF2t ] (E12)

where δ1 = δeff(EF1 − 0+) and δ2 = δ − δeff(EF2 − 0+) takeinto account the phase jumps at Fermi energies EF1 and EF2.From Eq. (E12) the second term of Eq. (E5) follows:

∞∑k=1

k[ln σ (θ )]k[ln σ (θ )]−k

= −∞∑

k=1

1

π2k

[δ2

1 + δ22 + 2δ1δ2 cos(kt (EF1 − EF2))

]

∼ −∫ t

t

dτ1

π2τ

[δ2

1 + δ22 + 2δ1δ2 cos(τ (EF1 − EF2))

].

(E13)

When EF1 = EF2, Eq. (E13) gives − δ2

π2 ln t�π

, which recoverscorrectly the power-law decay of S(t ) characteristic for theAnderson OC that considers a single-component Fermi seawith a single Fermi edge. For EF1 �= EF2 in the long-time limitdefined by t |EF1 − EF2| � 1, Eq. (E13) leads to

∞∑k=1

k[ln σ (θ )]k[ln σ (θ )]−k

∼ −(

δ21

π2+ δ2

2

π2

)ln

t�

π

− 2δ1δ2

[ln

π (EF1 − EF2)

�+γ − π2(EF1 − EF2)2

4�2

],

(E14)

where we applied the limit t → 0 (i.e., � → ∞) and per-formed the cosine integral Ci(x) = ∫∞

x du cos(u)u ≈ ln x + γ −

x2

4 with γ the Euler-Mascheroni constant. Note that the lastterm in Eq. (E14), which is proportional to δ1δ2, is timeindependent.

Combining all results for the case of two Fermi stepsassuming EF2 < EF1 we obtain the long-time behavior of thecontribution from particle-hole excitations at the two Fermiedges as

S(FS)(t ) ∝ t−(

δ21

π2 + δ22

π2

)e−iκ0t . (E15)

Thus we identify S(FS1)0 (t ) ∼ t− δ2

1π2 at Fermi edge EF1 and

S(FS2)0 (t ) = t− δ2

2π2 at the Fermi edge EF2 where

δ1 = δeff(EF1 − 0+) (E16)

δ2 = δk (EF2 + 0+) − δeff(EF2 − 0+). (E17)

Inspired by previous studies of Fermi surface contributionswith n �= 0 for the case of an impurity interacting with asingle-component Fermi gas in its ground state [69] we maynow straightforwardly conjecture the generalization to ourcase of a spin-flip Hamiltonian (5) and arrive at

S(FS1)n (t ) ∝ e−inEF1t

(1

t

)( δ1π

−n)2

,

S(FS2)n (t ) ∝ e−inEF2t

(1

t

)( δ2π

−n)2

. (E18)

Finally we note, that in Eqs. (E7) and (E8) one mayreintroduce the energy-dependent phase shift δ(E ) on a phe-nomenological basis and also apply those results to the case offinite temperature. Indeed we find that these expressions yieldexcellent agreement with exact numerical results for a largerange of temperatures (see Fig. 7). In fact Eqs. (E7) and (E8)represent a direct generalization of previous findings [69],which were restricted to the case of an impurity interactingwith a Fermi gas with a single Fermi-step distribution n(E ),to the case of nonequilibrium fermions with a multistepdistribution that fulfills n(E ) = 1 for E = 0 and n(E ) = 0 forE → ∞.

APPENDIX F: RELATION OF RAMSEYDECOHERENCE AND FCS

The exponential decay rate of the Ramsey signal at longtimes at T = 0 and EF2 = 0 is determined by

A = −γ = ln |S(t )| = Re ln S(t )

= t∫ EF1

0

dE

2πRe ln[1 + p(e2iδ(E ) − 1)] (F1)

with polarization p = cos2 θ/2. Using Re ln z = ln |z| onefinds

A = t∫ EF1

0

dE

2πln |1 + p(e2iδ(E ) − 1)|

= t∫ EF1

0

dE

2π

1

2ln[1 − 2p(1 − p)(1 − cos 2δ(E ))]. (F2)

Now consider the quantity

B = 1

2ln χ (eiλ → 0), (F3)

where χ is given by Eq. (9), so that

ln χ (eiλ → 0) = t∫ EF1

0

dE

2π hln[1 − �(E )]. (F4)

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YOU, SCHMIDT, IVANOV, KNAP, AND DEMLER PHYSICAL REVIEW B 99, 214505 (2019)

Using �(E ) = sin2 θ sin2 δ(E ), sin2 θ = 4p(1 − p) andsin2 δ(E ) = (1 − cos 2δ)/2 one finds that indeed

B = t∫ EF1

0

dE

2π

1

2ln[1 − 2p(1 − p)(1 − cos 2δ(E ))], (F5)

which equals Eq. (F2) and hence we have shown

|S(t )| →√

χ (eiλ → 0). (F6)

This prescription projects out the contribution N2 = 0 inEq. (8), so that we can indeed conclude that up to logarithmiccorrections,

|S(t )| =√

PN=0(t ) =[∫

dλχ (λ, t )

]1/2

. (F7)

APPENDIX G: FCS FOR A FINITE NUMBEROF IMPURITIES

Experiments that use impurities as probes, are natu-rally subject to relatively small signal-to-noise ratios due tothe small numbers of impurities. By using the many-bodymedium itself as a probe, our experimental scheme circum-vents this challenge. In particular, the measured signal canbecome large at late times, because the impurity can flip anarbitrary number of spins in the background gas. The fact thatmany spin flips occur has also a consequence for theoreticalapproaches to the impurity-induced spin-transport problem:Since the number of spin-flipped atoms easily exceeds one,simple variational wave functions based on few-fermion exci-tations [96–103] are bound to fail.

In typical experimental setups the impurity number will befinite, which raises the question of what the influence of afinite density of impurities is on the observed dynamics. In thisregard the typical interparticle distance d ∼ n−1/3

I betweenimpurities of a density nI becomes a relevant length scale. As avery conservative estimate, the dynamics will be governed bythe physics of independent scattering centers as long as timestvF < d (with vF the Fermi velocity) are considered. Onlywhen tvF > d fermions will be able to scatter from multipleimpurities leading to correlated scattering events that are,for instance, the basis for bath-mediated, Ruderman-Kittel-Kasuya-Yosida (RKKY)-type, impurity-impurity interactions.

Here, we focus on the regime of a low-impurity densitywhere induced interactions can be neglected. In this casescattering events are independent and each impurity (repre-senting an independent stochastic variable) is characterized bya FCS with generating function χ (λ). The probability PNTot

2to

measure a total number NTot2 of spin-flipped atoms in a sample

of NI impurities (localized in a central region of a Fermi gasof constant density) is then derived from the characteristicfunction

χTot(λ, t ) = [χ (λ, t )]NI . (G1)

The evaluation of the Fourier transform of this expressionyields the desired probability

PNTot2

(t ) =∫

dλ

[∑N2

PN2 (t )eiλN2

]NI

e−iλNTot2 . (G2)

This equation renders the constraint NTot2 =∑i N2(i), where

N2(i) is the number of spin flips produced by the ith impurity,particularly transparent. As we have seen, the distribution PN2

is well described by a sum over binomials, c.f. Eq. (9), sothat PN2 has well-defined moments. Thus, by virtue of thecentral limit theorem, the distribution of total observed spinflips, PNTot

2, approaches a normal distribution for a sufficiently

large number of impurities NI .This can be seen explicitly as follows: let us assume that the

impurities represent independent and identically distributedrandom variables N (1), . . . , N (NI ), each with mean value 〈N〉and variance σ 2

N . Consequently∑NI

x=1 N (x) has mean valeNI · 〈N〉 and variance NI · σ 2

N . As dictated by the central limittheorem, the probability PNTot

2will tend towards a normal

distribution as the number of independent random variablesincreases. To make this statement more precise we followstandard textbooks [104], and define the sum of rescaledvariables

ZNI =NI∑

x=1

1√NI

Yx, (G3)

where the variables Yx = N (x)−〈N〉σN

have zero mean and unitvariance. The characteristic function of ZNI is

χZNI(λ) = χ∑NI

x=11√NI

Yx(λ)

= χY1

(λ√NI

)χY2

(λ√NI

)· · · χYNI

(λ√NI

)

=[χY1

(λ√NI

)]NI

, (G4)

where we made use of the fact that χ Yx√NI

(λ) = χYx( λ√

NI). By

expanding the characteristic function χY1( λ√

NI),

χY1

(λ√NI

)=∑N1

PN1 ei λ√

NI

N1−〈N〉σN

= 1 + i2λ2

2NI+ O

((λ√NI

)3)

(G5)

the characteristic function χZNI(λ) can be written as

χZNI(λ) �

(1 + i2λ2

2NI

)NI

→ e− λ2

2 , (G6)

where we have used ex = limn→0(1 + x/n)n. This last ex-pression shows that, even when the probability distributionof a single impurity, obtained from 〈eiλN 〉, is not Gaussian,the distribution of the

∑NIx=1 N (x) indeed becomes a normal

distribution as NI → ∞, in accordance with the central limittheorem.

In Fig. 10 we show the spin-flip dynamics at strong inter-actions kF1a = −6, for up to NI = 16 impurities immersed ina Fermi gas. At long times and for such strong interactionsa normal distribution is quickly approached. In this figure weassume that the spatial interimpurity separation is chosen suchthat up to the maximal times shown, tEF1 = 60, scattering

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ATOMTRONICS WITH A SPIN: STATISTICS OF SPIN … PHYSICAL REVIEW B 99, 214505 (2019)

0 1 2 3 4 50.00.10.20.30.40.50.60.7

0 2 4 6 8 100.0

0.1

0.2

0.3

0.4

10 15 20 25 300.0

0.1

0.2

6 8 10 120.0

0.1

0.2

0.3

0.4

0.5

25 30 35 40 450.00

0.05

0.10

0.15

0.20

0.25

120 130 140 1500.00

0.02

0.04

0.06

0.08

0.10

0.12

(a)

(b)

FDANormal

FIG. 10. Influence of multiple impurities. FCS PNTot2

of total number of spin flips NTot2 for strong interactions kF1a = −6 at times (a) tEF1 =

10 and (b) tEF1 = 60 for NI = 1, 4, 16 impurities immersed in the Fermi gas (left to right). The blue squares represent the exact result fromFDA. For the first figure in (a) the normalized Gaussians is not shown. It does not fit the data since NTot

2 is bound by zero from below.

events can be treated as independent. As discussed above,beyond this time scale, multi-impurity collisions will affect

the normal distribution at late times in a nontrivial way, whichwould be intriguing to measure experimentally.

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