“Ground states” of periodically driven systems
Le M. Cristian1

Akashi Ryosuke2

Shinji Tsuneyuki2,3

1 Max Planck Institute for Structure and Dynamics of Matter
2 Department of Physics, University of Tokyo
3 Institute for Solid State Physics, University of TokyoNQS 2022
Nov 11, 2022

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Contents

1

Floquet physics

2

Redefining the Floquet states

3

Floquet ground state

4

Conclusion

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Floquet physics

Contents

1

Floquet physics
Overview of Floquet physics
Roadblock2

Redefining the Floquet states

3

Floquet ground state

4

Conclusion

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet physics

Overview of Floquet physics

Amazing what we can achieve nowadays

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet physics

Overview of Floquet physics

Amazing what we can achieve nowadays
Simulate materials just from composition: DFT, CC-SD

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet physics

Overview of Floquet physics

Amazing what we can achieve nowadays
Simulate materials just from composition: DFT, CC-SD
Precise control of atoms: optical tweezers, STM

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet physics

Overview of Floquet physics

Amazing what we can achieve nowadays
Simulate materials just from composition: DFT, CC-SD
Precise control of atoms: optical tweezers, STM
Exotic states of matter: superconductors, topological material

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet physics

Overview of Floquet physics

Amazing what we can achieve nowadays
Simulate materials just from composition: DFT, CC-SD
Precise control of atoms: optical tweezers, STM
Exotic states of matter: superconductors, topological material
Explore ultra-fast dynamics: fs-lasers, TD-DFT

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet physics

Overview of Floquet physics

Amazing what we can achieve nowadays
Simulate materials just from composition: DFT, CC-SD
Precise control of atoms: optical tweezers, STM
Exotic states of matter: superconductors, topological material
Explore ultra-fast dynamics: fs-lasers, TD-DFT

cristian.le@mpsd.mpg.de-Is this what we really want?

“Ground states” of periodically driven systems

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Floquet physics

Overview of Floquet physics

Holy grail of computational physics
Amazing what we can achieve nowadays
Simulate materials just from composition: DFT, CC-SD
Precise control of atoms: optical tweezers, STM
Exotic states of matter: superconductors, topological material
Explore ultra-fast dynamics: fs-lasers, TD-DFT-Is this what we really want?

Unsolved systems: Material + Laser
Long time dynamics
Many-body interaction
Steady state with environment

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet physics

Overview of Floquet physics

Holy grail of Floquet physics
Amazing what we can achieve nowadays
Simulate materials just from composition: DFT, CC-SD
Precise control of atoms: optical tweezers, STM
Exotic states of matter: superconductors, topological material
Explore ultra-fast dynamics: fs-lasers, TD-DFT-Is this what we really want?

Unsolved systems: Material + Laser
Time-periodic systems
Many-body interaction
Steady state with environment
From first principles

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet physics

Overview of Floquet physics

Why Floquet physics?

Dynamics in Quantum Mechanics
Static systems:

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet physics

Overview of Floquet physics

Why Floquet physics?

Dynamics in Quantum Mechanics
Static systems: Just solve Schrodinger equation

cristian.le@mpsd.mpg.deĤ |Ψn i = En |Ψn i“Ground states” of periodically driven systems(e−iEn t |Ψn i)

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Floquet physics

Overview of Floquet physics

Why Floquet physics?

Dynamics in Quantum Mechanics
Static systems: Schrodinger equation(Not for lasers)

cristian.le@mpsd.mpg.deĤ |Ψn i = En |Ψn i“Ground states” of periodically driven systems

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Floquet physics

Overview of Floquet physics

Why Floquet physics?

Dynamics in Quantum Mechanics
Static systems: Schrodinger equation(Not for lasers)Ĥ |Ψn i = En |Ψn i

General time-dependent:

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet physics

Overview of Floquet physics

Why Floquet physics?

Dynamics in Quantum Mechanics
Static systems: Schrodinger equation(Not for lasers)Ĥ |Ψn i = En |Ψn i

General time-dependent: TD-Schrodinger equation|Ψ(t + δt )i = |Ψ(t)i − iδt Ĥ(t) |Ψ(t)icristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet physics

Overview of Floquet physics

Why Floquet physics?

Dynamics in Quantum Mechanics
Static systems: Schrodinger equation(Not for lasers)Ĥ |Ψn i = En |Ψn i

General time-dependent: TD-Schrodinger equation(Slow and inefficient - only laser pulses)|Ψ(t + δt )i = |Ψ(t)i − iδt Ĥ(t) |Ψ(t)icristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet physics

Overview of Floquet physics

Why Floquet physics?

Dynamics in Quantum Mechanics
Static systems: Schrodinger equation(Not for lasers)Ĥ |Ψn i = En |Ψn i

General time-dependent: TD-Schrodinger equation(Slow and inefficient - only laser pulses)|Ψ(t + δt )i = |Ψ(t)i − iδt Ĥ(t) |Ψ(t)i

Time-periodic:

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet physics

Overview of Floquet physics

Why Floquet physics?

Dynamics in Quantum Mechanics

1

Static systems: Schrodinger equation(Not for lasers)Ĥ |Ψn i = En |Ψn i

General time-dependent: TD-Schrodinger equation(Slow and inefficient - only laser pulses)|Ψ(t + δt )i = |Ψ(t)i − iδt Ĥ(t) |Ψ(t)i

Time-periodic: Floquet solution1[Ĥ(t) − i∂t ] |Φn (t)i = n |Φn (t)iShirley, J. H. (1965)
cristian.le@mpsd.mpg.de

Physical Review, 138, B979; Sambe, H. (1973)“Ground states” of periodically driven systems

Physical Review A, 7, 2203
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Floquet physics

Overview of Floquet physics

Why Floquet physics?

Dynamics in Quantum Mechanics

1

Static systems: Schrodinger equation(Not for lasers)Ĥ |Ψn i = En |Ψn i

General time-dependent: TD-Schrodinger equation(Slow and inefficient - only laser pulses)|Ψ(t + δt )i = |Ψ(t)i − iδt Ĥ(t) |Ψ(t)i

Time-periodic: Floquet solution1(Stable/slowly varying laser fields)[Ĥ(t) − i∂t ] |Φn (t)i = n |Φn (t)iShirley, J. H. (1965)
cristian.le@mpsd.mpg.de

Physical Review, 138, B979; Sambe, H. (1973)“Ground states” of periodically driven systems

Physical Review A, 7, 2203
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Floquet physics

Overview of Floquet physics

It works!

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet physics

Overview of Floquet physics

It works!

Key features:
Energy Brillouin zone
Stable eigenstates
Adiabatic or Thermodynamic

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet physics

Overview of Floquet physics

It works!
Key features:
Energy Brillouin zone
Stable eigenstates
Adiabatic or Thermodynamic

Figure: Floquet state in Bi2 Se3 22

Mahmood, F. et al. (2016)
cristian.le@mpsd.mpg.deNature Physics, 12, 306.

“Ground states” of periodically driven systems

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Floquet physics

Overview of Floquet physics

It works!
Key features:
Energy Brillouin zone
Stable eigenstates
Adiabatic or Thermodynamic

Promising applications:
Floquet engineering
Topological material
Quantum computing

2

Mahmood, F. et al. (2016)
cristian.le@mpsd.mpg.de

Figure: Floquet state in Bi2 Se3 2Nature Physics, 12, 306.

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Floquet physics

RoadblockWhat’s the issue?

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet physics

Roadblock

Floquet + ab-initio = incompatible

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet physics

Roadblock

Floquet + ab-initio = incompatible

First-principles methods:
Derived from ground-state→ Excited state, thermal, time-dependent…
No assumption or prior experimentSelf-consistent interaction

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet physics

Roadblock

Floquet + ab-initio = incompatible

Floquet systems:
First-principles methods:
Derived from ground-state→ Excited state, thermal, time-dependent…

Models fitted to experiment or excited statecalculations

No assumption or prior experimentUnclear what’s the target state

Self-consistent interaction

3
4Energy Brillouin zone → No lowest “energy”3

No adiabatic continuation4

Maitra, N. T. and Burke, K. (2007) Chemical Physics Letters, 441, 167
Hone, D. W. et al. (1997) Physical Review A, 56, 4045 .
cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems.
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Floquet physics

Roadblock

No Floquet ground state → No Floquet DFT

Quasi-energy 

8

6

4

2

0

0

10

20
30
Driving strength v1

40

50

Figure: Typical Floquet picutre55Hone, D. W. et al. (1997)
cristian.le@mpsd.mpg.de

Physical Review A, 56, 4045.

“Ground states” of periodically driven systems

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Floquet physics

Roadblock

No Floquet ground state → No Floquet DFT

Figure: Floquet DFT breaking down55Maitra, N. T. and Burke, K. (2002)
cristian.le@mpsd.mpg.de

Chemical Physics Letters, 359, 237“Ground states” of periodically driven systems.
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Floquet physics

Roadblock

Floquet + ab-initio = incompatible

Floquet systems:
First-principles methods:

Energy Brillouin zone → No lowest “energy”3

Derived from ground-state→ Excited state, thermal, time-dependent…

Models fitted to experiment or excited statecalculations

No assumption or prior experimentUnclear what’s the target state

Self-consistent interaction

No adiabatic continuation4

Due to quasi-energy formalism

3
4

Maitra, N. T. and Burke, K. (2007) Chemical Physics Letters, 441, 167
Hone, D. W. et al. (1997) Physical Review A, 56, 4045 .
cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems.
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Floquet physics

Roadblock

Can we overcome this?

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Redefining the Floquet states

Contents

1

Floquet physics

2

Redefining the Floquet states
Energy and quasi-energy
Floquet average energy

3

Floquet ground state

4

Conclusion

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Redefining the Floquet states

Energy and quasi-energy

Why do we care about the energy?

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Redefining the Floquet states

Energy and quasi-energy

Why do we care about the energy?
E[Ψ] = hΨ|Ĥ|Ψi
Conserved quantum number
Observable → Can be measured
Does not change in timecristian.le@mpsd.mpg.de

dO
=
dt

“Ground states” of periodically driven systems* Ehrenfest+ theorem
DE∂ Ô(t)− i [Ô(t), Ĥ]
∂t
*
+
dE
∂ Ĥ
=
=0
dt
∂t

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Redefining the Floquet states

Energy and quasi-energy

Why do we care about the energy?
E[Ψ] = hΨ|Ĥ|Ψi
Conserved quantum number

dO
=
dt

Observable → Can be measured
Does not change in time

Observable of Hamiltonian
Determines dynamics/conserved observables* Ehrenfest+ theorem
DE∂ Ô(t)− i [Ô(t), Ĥ]
∂t
*
+
dE
∂ Ĥ
=
=0
dt
∂t
TD Schrodinger equation∂t |Ψ(t)i = −iĤ |Ψ(t)icristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Redefining the Floquet states

Energy and quasi-energy

Why do we care about the energy?
E[Ψ] = hΨ|Ĥ|Ψi

Determines dynamics/conserved observables* Ehrenfest+ theorem
DE∂ Ô(t)− i [Ô(t), Ĥ]
∂t
*
+
dE
∂ Ĥ
=
=0
dt
∂t

The observation does not (de-)excite the system

TD Schrodinger equation

Conserved quantum number

dO
=
dt

Observable → Can be measured
Does not change in time

Observable of Hamiltonian∂t |Ψ(t)i = −iĤ |Ψ(t)ihΨm |Ĥ|Ψn i = δmn En

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Redefining the Floquet states

Energy and quasi-energy

Why do we care about the energy?
E[Ψ] = hΨ|Ĥ|Ψi

Determines dynamics/conserved observables* Ehrenfest+ theorem
DE∂ Ô(t)− i [Ô(t), Ĥ]
∂t
*
+
dE
∂ Ĥ
=
=0
dt
∂t

The observation does not (de-)excite the system

TD Schrodinger equation

Conserved quantum number

dO
=
dt

Observable → Can be measured
Does not change in time

Observable of Hamiltonian

ThermodynamicallyEnergy flows: High → Low∂t |Ψ(t)i = −iĤ |Ψ(t)ihΨm |Ĥ|Ψn i = δmn En

0K system → Ground stateρ=cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

ThermalX equilibrium
e−βEn |Ψn ihΨn |n
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Redefining the Floquet states

Energy and quasi-energy

Quasi-energy ≡ Energy?

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Redefining the Floquet states

Energy and quasi-energy

Original Floquet formulation
Periodic HamiltonianĤ(t + T ) = Ĥ(t)First order periodic equation → Floquet

TD Schroedinger eq[Ĥ(t) − i∂t ] |Ψ(t)i = 0

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Redefining the Floquet states

Energy and quasi-energy

Original Floquet formulation
Periodic HamiltonianĤ(t + T ) = Ĥ(t)TD Schroedinger eq[Ĥ(t) − i∂t ] |Ψ(t)i = 0
Floquet decomposition−iĜtFirst order periodic equation → Floquet
Derived to satisfy Schrodinger equationÛ (t) = P̂ (t)eP̂ (t + T ) = P̂ (t)XĜ =n |Φn (0)ihΦn (0)|nP̂ (t) |Φ(0)i = |Φ(t)icristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Redefining the Floquet states

Energy and quasi-energy

Original Floquet formulation
Periodic HamiltonianĤ(t + T ) = Ĥ(t)First order periodic equation → Floquet

TD Schroedinger eq

Derived to satisfy Schrodinger equation[Ĥ(t) − i∂t ] |Ψ(t)i = 0
Floquet Schrodinger equation[Ĥ(t) − i∂t ] |Φn (t)i = n |Φn (t)icristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Redefining the Floquet states

Energy and quasi-energy

Original Floquet formulation
Periodic HamiltonianĤ(t + T ) = Ĥ(t)First order periodic equation → Floquet

TD Schroedinger eq

Derived to satisfy Schrodinger equation[Ĥ(t) − i∂t ] |Ψ(t)i = 0
Floquet Schrodinger equation

Ĥ F |Φn (t)i = n |Φn (t)iΦn :Floquet eigenstate!

n :Quasi-energy

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Redefining the Floquet states

Energy and quasi-energy

Original Floquet formulation
Periodic HamiltonianĤ(t + T ) = Ĥ(t)First order periodic equation → Floquet

TD Schroedinger eq

Derived to satisfy Schrodinger equation[Ĥ(t) − i∂t ] |Ψ(t)i = 0

Analogous to Bloch states

Floquet Schrodinger equation

Ĥ F |Φn (t)i = n |Φn (t)i
Physical Floquet state|Ψn (t)i = e−in t |Φn (t)iΦn :Floquet eigenstate!

n :Quasi-energy

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Redefining the Floquet states

Energy and quasi-energy

Original Floquet formulation
Periodic HamiltonianĤ(t + T ) = Ĥ(t)First order periodic equation → Floquet

TD Schroedinger eq

Derived to satisfy Schrodinger equation[Ĥ(t) − i∂t ] |Ψ(t)i = 0

Analogous to Bloch states(Crystal momentum)

Ĥ F |Φn (t)i = n |Φn (t)i

Floquet Schrodinger equation
Physical Floquet state|Ψn (t)i = e−in t |Φn (t)iΦn :Floquet eigenstate!

n :Quasi-energy

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Redefining the Floquet states

Energy and quasi-energy

Original Floquet formulation
Periodic HamiltonianĤ(t + T ) = Ĥ(t)First order periodic equation → Floquet

TD Schroedinger eq

Derived to satisfy Schrodinger equation[Ĥ(t) − i∂t ] |Ψ(t)i = 0

Analogous to Bloch states(Crystal momentum)

Ĥ F |Φn (t)i = n |Φn (t)i

Quasi-energy is not an observable
No energy equivalent
No ordering of the states

Floquet Schrodinger equation
Physical Floquet state|Ψn (t)i = e−in t |Φn (t)iO(t) = hΨ(t)|Ô|Ψ(t)i 6= hΦ(t)|Ô|Φ(t)i
Φn :Floquet eigenstate!

n :Quasi-energy

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Redefining the Floquet states

Energy and quasi-energy

Original Floquet formulation
Periodic HamiltonianĤ(t + T ) = Ĥ(t)First order periodic equation → Floquet

TD Schroedinger eq

Derived to satisfy Schrodinger equation[Ĥ(t) − i∂t ] |Ψ(t)i = 0

Analogous to Bloch states(Crystal momentum)

Ĥ F |Φn (t)i = n |Φn (t)i

Quasi-energy is not an observable
No energy equivalent
No ordering of the states

Floquet Schrodinger equation
Physical Floquet state|Ψn (t)i = e−in t |Φn (t)iO(t) = hΨ(t)|Ô|Ψ(t)i 6= hΦ(t)|Ô|Φ(t)i! Only propagation information

Φn :Floquet eigenstate!

n :Quasi-energy

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Redefining the Floquet states

Energy and quasi-energy

Role of quasi-energy

Observable
Ordering
Oscillation freq.
Commutingoperators
Thermodynamics
Ground statecristian.le@mpsd.mpg.de

EnergyQuasi-energy

Yes
Yes
Yes
Static/conserved

No
No
Yes
Time-periodic

Fully determined
0K state

Allowed transitions
Not exist“Ground states” of periodically driven systems

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Redefining the Floquet states

Energy and quasi-energy

We need another quantum number!

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Redefining the Floquet states

Floquet average energy

A new Floquet picture

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Redefining the Floquet states

Floquet average energy

A new Floquet picture
Fundamental symmetry[Ĥ(t), T̂ ] = 0

From fundamental symmetry6T̂ Ô(t) = Ô(t + T )T̂

6Le, C. M. et al. (2022)
PhD thesis .
cristian.le@mpsd.mpg.de

Physical Review A, 105, 052213; Le, C. M. (2021)“Ground states” of periodically driven systems



University of Tokyo,
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Redefining the Floquet states

Floquet average energy

A new Floquet picture
Fundamental symmetry[Ĥ(t), T̂ ] = 0

From fundamental symmetry6

Effective commutation
ˆ ] =0[Ĥ F , H̄nDerived by Bloch analogy

Floquet eigenstates(F
Ĥ |Φna i = n |Φna i
ˆ |Φ i = Ē |Φ i
H̄
n

na

na

T̂ Ô(t) = Ô(t + T )T̂

6Le, C. M. et al. (2022)
PhD thesis .
cristian.le@mpsd.mpg.de

Physical Review A, 105, 052213; Le, C. M. (2021)“Ground states” of periodically driven systemsna

University of Tokyo,
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Redefining the Floquet states

Floquet average energy

A new Floquet picture
Fundamental symmetry[Ĥ(t), T̂ ] = 0
Bloch symmetry

From fundamental symmetry6
Derived by Bloch analogy~ˆ = 0[Ĥ, R]( Bloch eigenstatesˆ~ku= ~ku~kn~
kn
Ĥ~k u~kn = E~kn u~kn!
T̂ Ô(t) = Ô(t + T )T̂~ˆ Ô(~r) = Ô(~r + R)
~ R
~ˆ
R

6

Le, C. M. et al. (2022)
PhD thesis .
cristian.le@mpsd.mpg.de

Physical Review A, 105, 052213; Le, C. M. (2021)“Ground states” of periodically driven systems

University of Tokyo,
NQS-2022

9 / 22

Redefining the Floquet states

Floquet average energy

A new Floquet picture
Fundamental symmetry[Ĥ(t), T̂ ] = 0

From fundamental symmetry6
Derived by Bloch analogy
Complete eigenstate picture

Effective commutation
ˆ ] =0[Ĥ F , H̄nFloquet eigenstates(F
Ĥ |Φna i = n |Φna i
ˆ |Φ i = Ē |Φ i
H̄
n

na

na

na

Average energyˆ |Φ iĒna = hΦna |H̄
n na
Z
1 T
= lim
hΨna |Ĥ(t)|Ψna i dt
T →∞ T −T
T̂ Ô(t) = Ô(t + T )T̂
6Le, C. M. et al. (2022)
PhD thesis .
cristian.le@mpsd.mpg.de

Physical Review A, 105, 052213; Le, C. M. (2021)“Ground states” of periodically driven systems



University of Tokyo,
NQS-2022

9 / 22

Redefining the Floquet states

Floquet average energy

A new Floquet picture
Fundamental symmetry[Ĥ(t), T̂ ] = 0

From fundamental symmetry6
Derived by Bloch analogy
Complete eigenstate picture
Observable (energy equivalent)
Ordering quantum number
Lower-bounded

Effective commutation
ˆ ] =0[Ĥ F , H̄nFloquet eigenstates(F
Ĥ |Φna i = n |Φna i
ˆ |Φ i = Ē |Φ i
H̄
n

na

na

Average energyˆ |Φ iĒna = hΦna |H̄
n na
Z
1 T
= lim
hΨna |Ĥ(t)|Ψna i dt
T →∞ T −T
T̂ Ô(t) = Ô(t + T )T̂

6Le, C. M. et al. (2022)
PhD thesis .
cristian.le@mpsd.mpg.dena

Physical Review A, 105, 052213; Le, C. M. (2021)“Ground states” of periodically driven systems



University of Tokyo,
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Redefining the Floquet states

Floquet average energy

Role of average energy

Observable
Ordering
Oscillation freq.
Commutingoperators
Thermodynamics
Ground statecristian.le@mpsd.mpg.de

EnergyQuasi-energy

Average energy

Yes
Yes
Yes
Static/conserved

No
No
Yes

Yes
Yes
No

Fully determined
0K state

Time-periodic

Allowed transitions Overall transition≈ 0K state?

“Ground states” of periodically driven systems

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Redefining the Floquet states

Floquet average energy

Example: Two-level systemĤ(t) =

ω0
2
V iωt
e
2

V −iωt
2e
− ω20



1.5

Quasi-energy 

+
−
1

0.5

0

0

1

2
3
Driving strength V

45

Figure: Original quasi-energy picture
cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Redefining the Floquet states

Floquet average energy

Example: Two-level systemĤ(t) =

ω0
2
V iωt
e
2

V −iωt
2e
− ω20

P+
P−

2

Energy E



0

−2

0

1

2
3
Driving strength V

45

Figure: Energy spectra
cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Redefining the Floquet states

Floquet average energy

Example: Two-level systemĤ(t) =

ω0
2
V iωt
e
2

V −iωt2e− ω20

P ∞ (E)

Ē ∞

2

Ψ+
Ψ−

1

1.5
Energy E

1
0

0
0
−1

−1.5
−1
−2

0

1 V1

2

3

4

5

Driving strength V

Figure: Average energy
cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Redefining the Floquet states

Floquet average energy

Example: Particle in a box
∂2
πxĤ(t) = − 12 ∂x
2 + v1 sin 2a cos(ωt)

Quasi-energy 

8

6

4

2

0

0

10

20
30
Driving strength v1

40

50

Figure: Original quasi-energy picture
cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Redefining the Floquet states

Floquet average energy

Example: Particle in a box
∂2
πxĤ(t) = − 12 ∂x
2 + v1 sin 2a cos(ωt)
400

6Average energy Ē

Quasi-energy 

8

4

2

0

0

10

20
30
Driving strength v1

40

50

300

200

100

0

0

10

20
30
Driving strength v1

40

50

Figure: Quasi-energy + Average energy
cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Redefining the Floquet states

Floquet average energy

Example: Particle in a box
∂2
πxĤ(t) = − 12 ∂x
2 + v1 sin 2a cos(ωt)
8

6

Average energy Ē

Quasi-energy 

100

4

2

80
60
40
20

0

0

10

20
30
Driving strength v1

40

50

0

0

10

20
30
Driving strength v1

40

50

Figure: Lowest energy eigenstatescristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet ground state

Contents

1

Floquet physics

2

Redefining the Floquet states

3

Floquet ground state
Steady state
Floquet Hartree-Fock4

Conclusion

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet ground state

Steady state

Static open quantum system
Open quantum systemĤ = ĤS + ĤB + V̂SB
System coupled to thermal bathcristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet ground state

Steady state

Static open quantum system
Open quantum systemĤ = ĤS + ĤB + V̂SB
System coupled to thermal bath
Depends on the master equation

Lindladian
X master equation∂t ρn (t) = 2

Simplest form: Lindbladianρ(t) =

cristian.le@mpsd.mpg.de[−Γnm ρn + Γmn ρm ]m“Ground states” of periodically driven systemsXnρn (t) |Ψn (0)ihΨn (0)|







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Floquet ground state

Steady state

Static open quantum system
Open quantum systemĤ = ĤS + ĤB + V̂SB
System coupled to thermal bath
Depends on the master equation

Lindladian
X master equation∂t ρn (t) = 2[−Γnm ρn + Γmn ρm ]m

Simplest form: Lindbladian
I Weak coupling
I Markov approx., etc.

ρ(t) =

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systemsXnρn (t) |Ψn (0)ihΨn (0)|







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Floquet ground state

Steady state

Static open quantum system
Open quantum systemĤ = ĤS + ĤB + V̂SB
System coupled to thermal bath
Depends on the master equation
Simplest form: Lindbladian
I Weak coupling
I Markov approx., etc.

Lindladian
X master equation∂t ρn (t) = 2[−Γnm ρn + Γmn ρm ]mDissipation strength2Γmn = Vmnπν(|ωmn |)NB (−ωmn )ρ(t) =

X

ρn (t) |Ψn (0)ihΨn (0)|nωmn = En − Em







Vmn = hΨm |V̂SB |Ψn icristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet ground state

Steady state

Static open quantum system
Open quantum systemĤ = ĤS + ĤB + V̂SB
System coupled to thermal bath
Depends on the master equation
Simplest form: Lindbladian
I Weak coupling
I Markov approx., etc.

Lindladian
X master equation∂t ρn (t) = 2[−Γnm ρn + Γmn ρm ]mDissipation strength2Γmn = Vmnπν(|ωmn |)NB (−ωmn )Steady state

Focus on the steady state
Ignore coherence, Lamb shift, etc.

∂t ρn (t) = 0
ρ(t) =

X

ρn (t) |Ψn (0)ihΨn (0)|nωmn = En − Em







Vmn = hΨm |V̂SB |Ψn icristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet ground state

Steady state

Floquet open quantum system
Floquet Open quantum system

Ĥ(t) = ĤS (t) + ĤB + V̂SB
Lindladianhmaster equationiX(k)∂t ρn (t) = 2−Γ(k)nm ρn + Γmn ρm
m,k

Γ(k)
mn =

Dissipation strength2πν(|ωmnk |)NB (−ωmnk )

(k)
Vmn

Steady state∂t ρn (t) = 0








ρ(t) =Xρn (t) |Φn (0)ihΦn (0)|



n




ωmnk = n − m + kω

E
XD
(k)
(l)
(k+l) Vmn =Φm V̂SB Ψnlcristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet ground state

Steady state

Floquet open quantum system
Floquet Open quantum system

Ĥ(t) = ĤS (t) + ĤB + V̂SB

Solved like the static system

Lindladianhmaster equationiX(k)∂t ρn (t) = 2−Γ(k)nm ρn + Γmn ρm
m,k

Γ(k)
mn =

Dissipation strength2πν(|ωmnk |)NB (−ωmnk )

(k)
Vmn

Steady state∂t ρn (t) = 0








ρ(t) =Xρn (t) |Φn (0)ihΦn (0)|



n




ωmnk = n − m + kω

E
XD
(k)
(l)
(k+l) Vmn =Φm V̂SB Ψnlcristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet ground state

Steady state

Floquet open quantum system
Floquet Open quantum system

Ĥ(t) = ĤS (t) + ĤB + V̂SB

Solved like the static system
Multiple dissipation channels

Lindladianhmaster equationiX(k)∂t ρn (t) = 2−Γ(k)nm ρn + Γmn ρm
m,k

Γ(k)
mn =

Dissipation strength2πν(|ωmnk |)NB (−ωmnk )

(k)
Vmn

Steady state∂t ρn (t) = 0








ρ(t) =Xρn (t) |Φn (0)ihΦn (0)|



n




ωmnk = n − m + kω

E
XD
(k)
(l)
(k+l) Vmn =Φm V̂SB Ψnlcristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet ground state

Steady state

Floquet open quantum system
Floquet Open quantum system

Ĥ(t) = ĤS (t) + ĤB + V̂SB

Solved like the static system
Multiple dissipation channels→ No longer thermal distribution→ SS depends on bath coupling

Lindladianhmaster equationiX(k)∂t ρn (t) = 2−Γ(k)nm ρn + Γmn ρm
m,k

Γ(k)
mn =

Dissipation strength2πν(|ωmnk |)NB (−ωmnk )

(k)
Vmn

Steady state∂t ρn (t) = 0








ρ(t) =Xρn (t) |Φn (0)ihΦn (0)|



n




ωmnk = n − m + kω

E
XD
(k)
(l)
(k+l) Vmn =Φm V̂SB Ψnlcristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet ground state

Steady state

Example: Two-level systemĤ(t) =

ω0
2
V iωt
e
2

V −iωt
2e
− ω20





0 1
+ VSB
(b + b† ) + ωB b† b
1 0
1

2

0.5

1.5
Energy E

1
0

(0)

0

(−2)

Γ+−

0

Γ−+

0
−1

−0.5

−1.5

SS density matrix ρz

1

−1
−2

0

1

2
3
Driving strength V

4

5

−1

z
z
Figure: Steady state ρSS = 1+ρ|Φ+ ihΦ+ | + 1−ρ|Φ− ihΦ− |22cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet ground state

Steady state

Example: Two-level systemĤ(t) =

ω0
2
V iωt
2e

V −iωt
2e
− ω20






0 1
+ VSB
(b + b† ) + ωB b† b
1 0

Notice1
2∃ non-overlapping states

0

(0)

0

(−2)

Γ+−

0

Γ−+

0
−1

−0.5

−1.5

SS density matrix ρz

0.5

1Energy E

Steady state ≈ Ground state

1

1.5

−1
−2

0

1

2
3
Driving strength V

4

5

−1

z
z
Figure: Steady state ρSS = 1+ρ|Φ+ ihΦ+ | + 1−ρ|Φ− ihΦ− |22cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet ground state

Steady state

Role of Floquet ground states
Energyqspectra

E

(k)

Φ(k)
=
n

Pn

Φ̃(k)
n

E

Separate the amplitude of the energy spectracristian.le@mpsd.mpg.de“Ground states” of periodically driven systems|Φn (t)i =Xe−ikωt Φ(k)
n

k

D

(k)
Φ̃(k)
n Φ̃n

E

E



=1
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Floquet ground state

Steady state

Role of Floquet ground states
EnergyqspectraEE(k)Pn Φ̃(k)nX2(k)(l) (k+l)
Γ(k)
∝
Pm
Pn
mn ∝ Vmn
Φ(k)
=
n

l

Separate the amplitude of the energy spectra
Dissipation is proportional to spectracristian.le@mpsd.mpg.de“Ground states” of periodically driven systems|Φn (t)i =Xe−ikωt Φ(k)
n

k

D

(k)
Φ̃(k)
n Φ̃n

E

E



=1
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Floquet ground state

Steady state

Role of Floquet ground states
EnergyqspectraEE(k)Pn Φ̃(k)nX2(k)(l) (k+l)
Γ(k)
∝
Pm
Pn
mn ∝ Vmn
Φ(k)=nSeparate the amplitude of the energy spectra
Dissipation is proportional to spectra
Define: spectra overlap measurelSpectranXoverlap measureo(l) (k+l) n +kω<mSmn ≈ maxPm Pn
Ēn >Ēmklcristian.le@mpsd.mpg.de“Ground states” of periodically driven systems|Φn (t)i =Xe−ikωt Φ(k)
n

k

D

(k)
Φ̃(k)
n Φ̃n

E

E



=1
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Floquet ground state

Steady state

Role of Floquet ground states
EnergyqspectraEE(k)Pn Φ̃(k)nX2(k)(l) (k+l)
Γ(k)
∝
Pm
Pn
mn ∝ Vmn
Φ(k)=nSeparate the amplitude of the energy spectra
Dissipation is proportional to spectra
Define: spectra overlap measure
If weakly overlapping states:
I Dissipation: high AE → low AE
I Steady state: AE ground statelSpectranXoverlap measureo(l) (k+l) n +kω<mSmn ≈ maxPm Pn
Ēn >ĒmklApprox. master equation(

Ēn > Ēm →cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems∂t ρn = −Γnm ρn + O(Smn )∂t ρm = −O(Smn ) + Γnm ρm|Φn (t)i =Xe−ikωt Φ(k)
n

k

D

(k)
Φ̃(k)
n Φ̃n

E

E



=1
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Floquet ground state

Steady state

Example: Two-level systemĤ(t) =

ω0
2
V iωt
2e

V −iωt
2e
− ω20






0 1
+ VSB
(b + b† ) + ωB b† b
1 0

1
2

0.5

Energy E

1
0

(0)

0

(−2)

Γ+−

0

Γ−+

0
−1

−0.5

−1.5

SS density matrix ρz

1

1.5

−1
−2

0

1

2
3
Driving strength V

4

5

−1

Figure: Steady state and spectra overlap

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet ground state

Floquet Hartree-Fock:
Possible because of Ritz variation
Similar derivation to static HF

Floquet Hartree-FockFloquet ground state

Floquet Hartree-Fock

Hartree-Fock ansatz

Floquet Hartree-Fock:
Possible because of Ritz variation
Similar derivation to static HF
Ansatz: Many-body Floquet ground state is close
to a Slater determinant

φ1 (~r1 t) φ2 (~r1 t) . . .
1 φ (~r t) φ (~r t) . . .
1 2
2 2
Φ0 (~r1~r2 . . . t) ≈ √
..
..
N!
..
.
.
.

Floquet ground state

Floquet Hartree-Fock

Hartree-Fock ansatz

Floquet Hartree-Fock:

φ1 (~r1 t) φ2 (~r1 t) . . .
1 φ (~r t) φ (~r t) . . .
1 2
2 2
Φ0 (~r1~r2 . . . t) ≈ √
..
..
N!
..
.
.
.

Possible because of Ritz variation

HF self-consistency

Similar derivation to static HF[F̂ (t) − i∂t ] |φi (t)i = εi |φi (t)i

Ansatz: Many-body Floquet ground state is close
to a Slater determinant
Constrained minimization problem:
I Constraints: Orthonormal orbitals, Many-bodyquasi-energy eigenstateF̂ [φ] |φi i = ĥ |φi i +X

hφj |V̂ee |φj φi iFloquet ground state

Floquet Hartree-Fock

Hartree-Fock ansatz

φ1 (~r1 t) φ2 (~r1 t) . . .
1 φ (~r t) φ (~r t) . . .
1 2
2 2
Φ0 (~r1~r2 . . . t) ≈ √
..
..
N!
..
.
.
.

Floquet Hartree-Fock:
Possible because of Ritz variation

HF self-consistency

Similar derivation to static HF
Ansatz: Many-body Floquet ground state is close
to a Slater determinant
Constrained minimization problem:[F̂ (t) − i∂t ] |φi (t)i = εi |φi (t)i
HF minimizationnXĒ0 = min Ē[φ] +λj hϕj |F̂ − i∂t − εi |φi i{φ}I Constraints: Orthonormal orbitals, Many-bodyquasi-energy eigenstate
I Objective: Many-body average energy

+

ij

X



µij hφj |φi i − δij

o

ij

X

F̂ [φ] |φi i = ĥ |φi i +hφj |V̂ee |φj φi iX1Ē[φ] =
hφi |ĥ|φi i + hφi φj |V̂ee |φi φj i2cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet ground state

Floquet Hartree-Fock

Example: Hubbard dimerĤ(t) =v(t)(n̂1 − n̂2 ) − t(ĉ†1 ĉ2 + ĉ†2 ĉ1 ) + U (n̂1↑ n̂1↓ + n̂2↑ n̂2↓ )2
Φ Exact
φ HF

Average Energy Ē

4

2

0

−2

0

0.5

1

1.5

2

2.5

3

Coulomb interaction U

Figure: Average energy
cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet ground state

Floquet Hartree-Fock

Example: Hubbard dimerĤ(t) =v(t)(n̂1 − n̂2 ) − t(ĉ†1 ĉ2 + ĉ†2 ĉ1 ) + U (n̂1↑ n̂1↓ + n̂2↑ n̂2↓ )2
Φ Exact
φ HF
NSCF|Φ(0)i = |φ↑θ (0)φ↓θ (0)i


cos θ
|φθ (0)i =sin θ
NSCF: Static HF → Floquet

Parameter θ

1

0

−1

0

0.5

1

1.5

2

2.5

3

Coulomb interaction U

Figure: Initial state parameter
cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Floquet ground state

Floquet Hartree-Fock

Advantages
Self-consistent driving and MB interaction
Guarantees time-periodicity (cf. TD-HF)
Does not require accurate propagation

Disadvantages
No neat single equation form (cf. static HF)
No self-consistent loop approach
Exist self-consistent artifact solutions that break the ansatz

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Conclusion

Conclusion

New Floquet formalism
Ordered states with lower bounded ground state
Resolves adiabatic continuity
Has Ritz variational principle and more

Ground state application
It is the steady state if weakly overlapping spectraFirst principles methods can be adapted

cristian.le@mpsd.mpg.de“Ground states” of periodically driven systems

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Conclusion

Things to comeOpen quantum system
Apply to quasi-energy (near) degenerate states as wellFinite temperature approximation
General applicability

First principles methods
Floquet Hartree-Fock for real systems
Post Hartree-Fock methods
Density Functional Theory

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