Dynamical phases transitions in periodically driven Bardeen-Cooper-Schrieffer systems H. P. Ojeda Collado1Gonzalo Usaj2 3C. A. Balseiro2 3Dami an H. Zanette2 4and Jos e Lorenzana1

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Dynamical phases transitions in periodically driven Bardeen-Cooper-Schrieffer systems

H. P. Ojeda Collado,1, ∗Gonzalo Usaj,2, 3 C. A. Balseiro,2, 3 Dami´

an H. Zanette,2, 4 and Jos´

e Lorenzana1, †

1ISC-CNR and Department of Physics, Sapienza University of Rome, Piazzale Aldo Moro 2, I-00185, Rome, Italy

2Centro At´

omico Bariloche and Instituto Balseiro, Comisi´

on Nacional de Energ´

ıa

At´

omica (CNEA)–Universidad Nacional de Cuyo (UNCUYO), 8400 Bariloche, Argentina

3Instituto de Nanociencia y Nanotecnolog´

ıa (INN), Consejo Nacional de

Investigaciones Cient´

ıﬁcas y T´

ecnicas (CONICET)–CNEA, 8400 Bariloche, Argentina

4Consejo Nacional de Investigaciones Cient´

ıﬁcas y T´

ecnicas (CONICET), Argentina

(Dated: October 31, 2022)

We present a systematic study of the dynamical phase diagram of a periodically driven BCS system as a

function of drive strength and frequency. Three different driving mechanism are considered and compared:

oscillating density of states, oscillating pairing interaction and oscillating external paring ﬁeld. We identify

the locus in parameter space of parametric resonances and four dynamical phases: Rabi-Higgs, gapless,

synchronized Higgs and time-crystal. We demonstrate that the main features of the phase diagram are quite

robust to different driving protocols and discuss the order of the transitions. By mapping the BCS problem to a

collection of nonlinear and interacting classical oscillators, we shed light on the origin of time-crsytalline phases

and parametric resonances appearing for subgap excitations.

I. INTRODUCTION

The manipulation of many-body systems by periodic

drives, usually referred to as “Floquet engineering”, has

become a powerful tool to control properties of materials [1–

4]. Floquet engineering has beneﬁt from the tremendous

advances in laser technologies and the advent of highly

controllable systems [5] like ultracold atomic gases in optical

traps [6–8] or cavities [9–11], ion chains [12] and nuclear

spins. Some experimental demonstrations include, quantum

control of magnetism [13], topology [14], electron-phonon

interactions [15] and broken symmetry phases such as

superconductivity [16,17].

Superconductors are also one of the most popular platforms

for quantum technologies. Quantum devices require the

manipulation of the out-of-equilibrium system for as long as

possible without loosing coherence because of coupling to the

environment. Thus, in the last years a large effort has been

done to improve the materials and devices to increase the

energy relaxation time and the coherent dynamics [18–20].

Superconducting and ultracold atomic superﬂuid

condensates present a unique opportunity to study many-body

Floquet effects [21–29]. Because of a gap in the excitation

spectrum, these systems tend to have long relaxation times.

In addition, in weak-coupling, the dynamics can be described

by a mean-ﬁeld like Hamiltonian with effective all-to-all

interactions. Both effects contribute to provide a large window

of time where energy relaxation processes are suppressed

and the out-of-equilibrium dynamics can be studied.

Furthermore, all-to-all interacting systems have aroused

interest in the context of mean-ﬁeld time-crystals [27,30–33].

Notwithstanding all this growing interest, Floquet

engineering in superconducting or superﬂuid condensates has

not been addressed until recently [21,23,24,27,34–40].

∗hector.pablo.ojedacollado@roma1.infn.it

†jose.lorenzana@cnr.it

For periodically driven BCS systems, Rabi-Higgs

oscillations [21], parametric resonances and Floquet

time-crystal phases [24,27] have been demonstrated by

considering a periodic time-dependent pairing interaction

λ(t).

Despite this progress, several questions remain open.

Different dynamical phases have been identiﬁed [21,23,

24,36] and a partial dynamical phase diagram has been

presented for the driven BCS system in Ref. [27]. On

the other hand, the order of the transition has not been

discussed. Also, so far studies have concentrated on a

driving mechanism in which the interaction parameter λis

time-dependent (λ-driving). However, it is also possible

to envisage that the density-of-states (DOS) could be time

dependent (DOS-driving).

Here we present a systematic study of the dynamical phase

diagram of a driven BCS system including driving frequencies

such that 

hω lies below and above the gap and a large

range of drive amplitudes and both λ−and DOS-driving

mechanisms. The DOS-driving protocol is relevant for

ultracold atoms setups as well as in condensed-matter systems

where the electrons can couple to an electromagnetic ﬁeld

in the THz regime. In addition, a systematic comparison of

driving mechanisms allow to separate universal features from

mechanism-dependent details.

We show that the phase diagram is, in general, surprisingly

rich with at least four dynamical phases (Rabi-Higgs, gapless,

synchronized-Higgs and time-crystal) ubiquitously appearing

for both driving protocols. Dynamical phase transitions

(DPTs) are analyzed in detail and we demonstrate the

existence of ﬁrst and second-order like phase transitions. We

analyze the parametric resonances discovered before [27] and

discuss their origin in the context of the mapping to a classical

dynamical system. In order to clarify the essential ingredients

leading to parametric resonances, we compare the phase

diagram for λ−and DOS-driving with the one corresponding

to an external pairing ﬁeld (third driving mechanism). Also,

to highlight the relevance of the many-body interactions in

the emergence of parametric resonances, we compare these

arXiv:2210.15693v1 [cond-mat.supr-con] 27 Oct 2022

phases diagrams to the one obtained in the case that the

self-consistency of the BCS order parameter is neglected.

The paper is organized as follows: Section II introduces the

model and the methods used. Sec. III presents the dynamical

phase diagrams. Section IV discusses the dynamics in each

phase. Section Vanalyzes the order of the transitions. In

Sec. VI we present the mapping to a classical system of

non-linear oscillators. Finally, in Sec. VII we present our

conclusions.

II. PERIODICALLY DRIVEN BCS MODEL

A. The pseudospin model

We consider the following time-dependent BCS

Hamiltonian written in terms of Anderson pseudospins [41],

HBCS =−2∑

ξk(t)ˆ

k−λ(t)∑

k,k′

S+

kˆ

S−

k′.(1)

Here, ξk=εk−µmeasures the energy of the fermions (εk)

from the Fermi level µand λis the pairing interaction. Either

ξk(t)or λ(t)is taken as time-dependent. In the ﬁrst case, for

a uniform rescaling of the fermionic band, we can consider the

DOS itself νto be time-dependent (DOS-driving) while the

second case deﬁnes λ-driving. More details of the protocols

will be given in the next subsection.

The 1

2-pseudospin operators are given in terms of fermionic

operators as,

k=1

2ˆc†

k↑ˆc†

−k↓+ˆc−k↓ˆck↑,

k=1

2iˆc†

k↑ˆc†

−k↓−ˆc−k↓ˆck↑,(2)

k=1

21−ˆc†

k↑ˆck↑−ˆc†

−k↓ˆc−k↓,

and ˆc†

kσ(ˆckσ) is the usual creation (annihilation) operator for

fermions with momentum kand spin σ. The operator ˆ

S±

k≡

k±iˆ

kcreates or annihilates a Cooper pair (k,−k).

Due to the all-to-all interaction, assumed in the second

term of Eq. (1), one can use a time-dependent mean-ﬁeld

treatment [42–54] which yields the exact dynamics in the

thermodynamic limit. The BCS mean-ﬁeld Hamiltonian can

be written as,

HMF =−∑

Sk⋅bk.(3)

where, bk(t)=(2∆ (t),0,2ξk)is the mean-ﬁeld acting

on the the 1

2-pseudospin operator ˆ

Sk=(ˆ

k,ˆ

k). The

pseudomagnetic ﬁeld bkhas to be obtained in a self-consistent

manner during the dynamics.

Without loss of generality, we consider that the equilibrium

superconducting order parameter ∆0is real. We will assume

this remains valid over time and show below that this is indeed

the case because of the electron-hole symmetry.

The real part of the instantaneous BCS order parameter is

given by

∆(t)=λ(t)∑

k,(4)

where Sx

k, without hat, denotes the expectation value of the

operator ˆ

kin the time-dependent BCS state. Hereon, we

will use this notation for all pseudospins components.

In practice, since the pseudomagnetic ﬁeld depends on k

only through ξk, rather than solving the equations for each

kwe solved the equations for a generic DOS converting the

sums into integrals over the fermionic energy ξ,

∆(t)=λ(t)∫dξν(ξ)Sx(ξ),(5)

with Sx(ξk)≡Sx

kand similar for the other components—we

shall use Sx

kand Sx(ξk)interchangeably, keeping in mind

that in actual computations the ξ-dependent form was used.

At equilibrium, in the absence of periodic perturbations,

the 1

2-pseudospins align in the direction of their local ﬁelds

k=(2∆0,0,2ξk)in order to minimize the system’s

energy [described by Eq. (3)]. This corresponds to the

zero-temperature paired ground state in which the pseudospin

texture (the expectation value of pseudospin operators as a

function of momentum k) is given by

Sx,0

k=∆0

2ξ2

k+∆2

, Sy,0

k=0, Sz,0

k=ξk

2ξ2

k+∆2

.(6)

Such pseudospin texture is used as initial condition and once

the pairing interaction or the DOS is modulated in time,

the expectation values of the pseudospins evolve obeying a

Bloch-like equation of motion

dSk

dt =−bk(t)×Sk,(7)

where we set 

h≡1.

We assume that the time dependent solutions do not

spontaneously break particle-hole symmetry. From the

equations of motion one can check that if by(ξ)=∆′′ =0

then since bx(ξ)=bx(−ξ)=2∆ and bz(ξ)=bz(−ξ)the

self-consistent solution preserves the following symmetries,

Sx(ξ)=Sx(−ξ),

Sy(ξ)=−Sy(−ξ),(8)

Sz(ξ)=−Sz(−ξ).

Indeed, the imaginary part of the order parameter is given by,

∆′′(t)=λ(t)∫dξ ν(ξ)Sy(ξ),(9)

which vanishes if Eq. (8) holds [and ν(ξ)=ν(−ξ)as

assumed]. Now, by considering ∆′′ at a time t+dt,

∆′′(t+dt)−∆′′(t)=dtλ(t)

×∫dξ ν(ξ)[bx(ξ)Sz(ξ)−bz(ξ)Sx(ξ)]

=0.(10)

This shows that the ∆′′(t)=0is preserved at all times. Thus,

our initial assumption and also Eqs. (8) are self-consistently

satisﬁed at all times.

B. Numerical implementation

In our computations, we consider typically N=104

pseudospins associated to equally spaced discrete energy

states ξkwithin an energy range of W=40∆0around µ

with an energy constant density of states ν. The Ncoupled

differential equations arising from Eq. (7) are solved using a

standard Runge-Kutta 4th-order method with a small enough

dt ensuring the convergence of dynamics. Some selected

points in the phase diagram were also checked using an

adaptive step-size Runge-Kutta method (Fehlberg method).

C. Driving protocols

In the following, we consider two different driving

protocols. In the λ-driving case, the pairing interaction is

taken periodic in time, as

λ(t)=λ0[1+αsin(ωdt)],(11)

while ξkdoes not depend on time. Here, λ0is the equilibrium

coupling constant, αis the driving strength, and ωdis the drive

frequency.

In DOS-driving, we consider a time-periodic DOS with

a time independent pairing interaction λ0. This can be

achieved with a periodic modulation of the Fermi velocity

which corresponds to a change in the band structure as

ξk(t)=ξ0

k[1+βsin (ωdt)],(12)

yielding a time-dependent DOS given by

ν(t)=ν0

1+βsin (ωdt),(13)

where ν0is the constant DOS at equilibrium. The equilibrium

Tcand order parameter depend on the product λν so an

adiabatic change in either parameter is equivalent. In contrast,

the two protocols are rather different when the system is out

of equilibrium and produce different dynamics. DOS-driving

implies that there is a momentum and time dependent

pseudomagnetic ﬁeld along z[through ξk(t)] which acquires

x-components once ∆becomes time dependent. On the

other hand, λ−drive means a time-dependent pseudomagnetic

ﬁeld only along the x−direction. Possible experimental

implementations of both protocols in ultracold atoms and

condensed matter systems have been discussed in detail in

Ref. [21].

III. DYNAMICAL PHASE DIAGRAMS

In this section, we present the dynamical phase diagram

with both driving methods and in a wide range of frequency

and driving strengths. Previous studies focused on λ-driving

and the subgap regime [27] or speciﬁc frequencies [21,23].

A ﬁrst screening of the phase diagram can be obtained [27]

using the time-averaged superconducting order parameter ¯

∆

Figure 1. (Color online) (a) Temporal average of superconducting

order parameter ¯

∆as a function of amplitude and frequency of

the drive, considering a λ−driving protocol. We have computed ¯

∆

using the time window t∆0∈[0,200].Dashed orange lines indicate

cuts to be show later (Figs. 9,10). The orange dots indicate the

parameters where the detailed dynamics is displayed (Fig. 2). They

correspond to the different dynamical phases found: Synchronized

Higgs (circle), time-crystal (triangle), Rabi-Higgs (square) and

gapless (rhombus). (b) Schematic representation of the phase

diagram showing the dominant phases in each region. Dashed curves

represent continuous phase transitions with fractal-like boundaries

while solid lines represent ﬁrst-order DPTs as shown below.

as dynamical order parameter. In Fig. 1we show a false color

map of ¯

∆as a function of the amplitude and frequency of the

drive for the λ−driving protocol. At ﬁrst look, there are two

main regions that can be easily distinguished: in the light blue

regions the average of the superconducting order parameter

is near the equilibrium value ( ¯

∆≈∆0) while regions with

zero order parameter average (ZOPA) appear in dark blue. We

identify four different dynamical phases within these regions,

which are schematized and labeled in panel (b). However,

these need a more reﬁned analysis to be distinguished, as

explained below.

Two dynamical phases appear for subgap excitations, and

two when the system is driven above the gap ωd>2∆0. This

rather strong distinction could be anticipated as in one case it

is not possible to directly excite quasiparticles in the system

(for subgap excitations in an off-resonant regime) while for

ωd>2∆0it is possible.

For ωd<2∆0, dark indentations or “Arnold tongues”

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DynamicalphasestransitionsinperiodicallydrivenBardeen-Cooper-SchrieffersystemsH.P.OjedaCollado,1,GonzaloUsaj,2,3C.A.Balseiro,2,3Dami´anH.Zanette,2,4andJos´eLorenzana1,1ISC-CNRandDepartmentofPhysics,SapienzaUniversityofRome,PiazzaleAldoMoro2,I-00185,Rome,Italy2CentroAt´omicoBarilocheandInstitutoBal...

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Dynamical phases transitions in periodically driven Bardeen-Cooper-Schrieffer systems H. P. Ojeda Collado1Gonzalo Usaj2 3C. A. Balseiro2 3Dami an H. Zanette2 4and Jos e Lorenzana1.pdf

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Dynamical phases transitions in periodically driven Bardeen-Cooper-Schrieffer systems H. P. Ojeda Collado1Gonzalo Usaj2 3C. A. Balseiro2 3Dami an H. Zanette2 4and Jos e Lorenzana1

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