Eternal life of entropy in non-Hermitian quantum systems

Introduction A Simple Model Three types of entropy evolution Conclusion

Eternal life of entropy in non-Hermitian quantum systems

Thomas Frith

City, University of London

7th International Workshop on New challenges in Quantum Mechanics: Integrability and Supersymmetry

Benasque

2nd September 2019

A. Fring and T. Frith, Phys. Rev. A ,100, 010,102 (2019).

Introduction A Simple Model Three types of entropy evolution Conclusion

Eternal life of entropy in non-Hermitian quantum systems

Thomas Frith

City, University of London

7th International Workshop on New challenges in Quantum Mechanics: Integrability and Supersymmetry

Benasque

2nd September 2019

A. Fring and T. Frith, Phys. Rev. A ,100, 010,102 (2019).

Introduction A Simple Model Three types of entropy evolution Conclusion

Outline

1 Introduction

Non-Hermitian QM Von Neumann entropy

2 A Simple Model PT Symmetry

Calculation of the metric

3 Three types of entropy evolution Eternal life of entropy

4 Conclusions

Introduction A Simple Model Three types of entropy evolution Conclusion

Introduction

Introduction A Simple Model Three types of entropy evolution Conclusion

Why Study Non-Hermitian Systems?

What happens in quantum mechanics when our Hamiltonian is non-Hermitian,H^†6=H?

Real eigenvalues are still possible!

Hamiltonians with PT-symmetry give real eigenvalues. PT :p→p, x → −x, i → −i

Figure:Bender and Boettcher, 1998 H=p²+ (ix)

Introduction A Simple Model Three types of entropy evolution Conclusion

Why Study Non-Hermitian Systems?

What happens in quantum mechanics when our Hamiltonian is non-Hermitian,H^†6=H?

Real eigenvalues are still possible!

Hamiltonians with PT-symmetry give real eigenvalues. PT :p→p, x → −x, i → −i

Figure:Bender and Boettcher, 1998 H=p²+ (ix)

Introduction A Simple Model Three types of entropy evolution Conclusion

Why Study Non-Hermitian Systems?

What happens in quantum mechanics when our Hamiltonian is non-Hermitian,H^†6=H?

Real eigenvalues are still possible!

Hamiltonians with PT-symmetry give real eigenvalues. PT :p→p, x → −x, i → −i

Figure:Bender and Boettcher, 1998 H=p²+ (ix)

Introduction A Simple Model Three types of entropy evolution Conclusion

Why Study Non-Hermitian Systems?

What happens in quantum mechanics when our Hamiltonian is non-Hermitian,H^†6=H?

Real eigenvalues are still possible!

Hamiltonians with PT-symmetry give real eigenvalues.

PT :p→p, x → −x, i → −i

Figure:Bender and Boettcher, 1998 H=p²+ (ix)

Introduction A Simple Model Three types of entropy evolution Conclusion

Why Study Non-Hermitian Systems?

What happens in quantum mechanics when our Hamiltonian is non-Hermitian,H^†6=H?

Real eigenvalues are still possible!

Hamiltonians with PT-symmetry give real eigenvalues.

PT :p→p, x → −x, i → −i

Figure:Bender and Boettcher, 1998 H=p²+ (ix)

Introduction A Simple Model Three types of entropy evolution Conclusion

Theory

h(t) =h^†(t), H(t)6=H^†(t)

h(t)φ(t) =i~∂_tφ(t), H(t) Φ (t) =i~∂_tΦ (t). Time dependent Dyson operator

φ(t) =η(t) Φ (t).

=⇒ Time dependent Dyson equation

h(t) =η(t)H(t)η(t)⁻¹+i~∂tη(t)η(t)⁻¹.

=⇒ Time dependent quasi-Hermiticity relation H^†(t)ρ(t)−ρ(t)H(t) =i~∂tρ(t). whereρ(t) =η^†(t)η(t)

Introduction A Simple Model Three types of entropy evolution Conclusion

Theory

h(t) =h^†(t), H(t)6=H^†(t)

h(t)φ(t) =i~∂_tφ(t), H(t) Φ (t) =i~∂_tΦ (t). Time dependent Dyson operator

φ(t) =η(t) Φ (t).

=⇒ Time dependent Dyson equation

h(t) =η(t)H(t)η(t)⁻¹+i~∂tη(t)η(t)⁻¹.

=⇒ Time dependent quasi-Hermiticity relation H^†(t)ρ(t)−ρ(t)H(t) =i~∂tρ(t). whereρ(t) =η^†(t)η(t)

Introduction A Simple Model Three types of entropy evolution Conclusion

Theory

h(t) =h^†(t), H(t)6=H^†(t)

h(t)φ(t) =i~∂_tφ(t), H(t) Φ (t) =i~∂_tΦ (t). Time dependent Dyson operator

φ(t) =η(t) Φ (t).

=⇒ Time dependent Dyson equation

h(t) =η(t)H(t)η(t)⁻¹+i~∂tη(t)η(t)⁻¹.

=⇒ Time dependent quasi-Hermiticity relation H^†(t)ρ(t)−ρ(t)H(t) =i~∂tρ(t). whereρ(t) =η^†(t)η(t)

Introduction A Simple Model Three types of entropy evolution Conclusion

Theory

h(t) =h^†(t), H(t)6=H^†(t)

h(t)φ(t) =i~∂_tφ(t), H(t) Φ (t) =i~∂_tΦ (t). Time dependent Dyson operator

φ(t) =η(t) Φ (t).

=⇒ Time dependent Dyson equation

h(t) =η(t)H(t)η(t)⁻¹+i~∂tη(t)η(t)⁻¹.

=⇒ Time dependent quasi-Hermiticity relation H^†(t)ρ(t)−ρ(t)H(t) =i~∂tρ(t). whereρ(t) =η^†(t)η(t)

Introduction A Simple Model Three types of entropy evolution Conclusion

Theory

h(t) =h^†(t), H(t)6=H^†(t)

h(t)φ(t) =i~∂_tφ(t), H(t) Φ (t) =i~∂_tΦ (t). Time dependent Dyson operator

φ(t) =η(t) Φ (t).

=⇒ Time dependent Dyson equation

h(t) =η(t)H(t)η(t)⁻¹+i~∂tη(t)η(t)⁻¹.

=⇒ Time dependent quasi-Hermiticity relation H^†(t)ρ(t)−ρ(t)H(t) =i~∂tρ(t). whereρ(t) =η^†(t)η(t)

Introduction A Simple Model Three types of entropy evolution Conclusion

Observables

Observableso(t) in the Hermitian system must be self-adjoint for real eigenvalues.

ObservablesO(t) in the non-HermitianO(t) are quasi Hermitian

o(t) =η(t)O(t)η⁻¹(t). Then we have

hφ(t)|o(t)φ(t)i=hΨ(t)|ρ(t)O(t)Ψ(t)i. ρ(t) =η^†(t)η(t) is vital!

Introduction A Simple Model Three types of entropy evolution Conclusion

Observables

Observableso(t) in the Hermitian system must be self-adjoint for real eigenvalues.

ObservablesO(t) in the non-HermitianO(t) are quasi Hermitian

o(t) =η(t)O(t)η⁻¹(t). Then we have

hφ(t)|o(t)φ(t)i=hΨ(t)|ρ(t)O(t)Ψ(t)i. ρ(t) =η^†(t)η(t) is vital!

Introduction A Simple Model Three types of entropy evolution Conclusion

Observables

Observableso(t) in the Hermitian system must be self-adjoint for real eigenvalues.

ObservablesO(t) in the non-HermitianO(t) are quasi Hermitian

o(t) =η(t)O(t)η⁻¹(t). Then we have

hφ(t)|o(t)φ(t)i=hΨ(t)|ρ(t)O(t)Ψ(t)i. ρ(t) =η^†(t)η(t) is vital!

Introduction A Simple Model Three types of entropy evolution Conclusion

Observables

Observableso(t) in the Hermitian system must be self-adjoint for real eigenvalues.

ObservablesO(t) in the non-HermitianO(t) are quasi Hermitian

o(t) =η(t)O(t)η⁻¹(t).

Then we have

hφ(t)|o(t)φ(t)i=hΨ(t)|ρ(t)O(t)Ψ(t)i. ρ(t) =η^†(t)η(t) is vital!

Introduction A Simple Model Three types of entropy evolution Conclusion

Observables

Observableso(t) in the Hermitian system must be self-adjoint for real eigenvalues.

ObservablesO(t) in the non-HermitianO(t) are quasi Hermitian

o(t) =η(t)O(t)η⁻¹(t). Then we have

hφ(t)|o(t)φ(t)i=hΨ(t)|ρ(t)O(t)Ψ(t)i. ρ(t) =η^†(t)η(t) is vital!

Introduction A Simple Model Three types of entropy evolution Conclusion

Von Neumann entropy

Hermitian

%_h=X

i

p_i|φ_ii hφ_i|, i∂_t%_h= [h, %_h], S_h=−tr[%_hln%_h].

Substitute time-dependent Dyson equation into von Neumann equation,

h =ηHη⁻¹+i~∂_tηη⁻¹→i∂_t%_h= [h, %_h].

Introduction A Simple Model Three types of entropy evolution Conclusion

Von Neumann entropy

Hermitian

%_h=X

i

p_i|φ_ii hφ_i|, i∂_t%_h= [h, %_h], S_h=−tr[%_hln%_h].

Substitute time-dependent Dyson equation into von Neumann equation,

h =ηHη⁻¹+i~∂_tηη⁻¹→i∂_t%_h= [h, %_h].

Introduction A Simple Model Three types of entropy evolution Conclusion

Von Neumann entropy

Hermitian

%_h=X

i

p_i|φ_ii hφ_i|,

i∂_t%_h= [h, %_h], S_h=−tr[%_hln%_h].

Substitute time-dependent Dyson equation into von Neumann equation,

h =ηHη⁻¹+i~∂_tηη⁻¹→i∂_t%_h= [h, %_h].

Introduction A Simple Model Three types of entropy evolution Conclusion

Von Neumann entropy

Hermitian

%_h=X

i

p_i|φ_ii hφ_i|, i∂_t%_h= [h, %_h],

S_h=−tr[%_hln%_h].

Substitute time-dependent Dyson equation into von Neumann equation,

h =ηHη⁻¹+i~∂_tηη⁻¹→i∂_t%_h= [h, %_h].

Introduction A Simple Model Three types of entropy evolution Conclusion

Von Neumann entropy

Hermitian

%_h=X

i

p_i|φ_ii hφ_i|, i∂_t%_h= [h, %_h], S_h=−tr[%_hln%_h].

Substitute time-dependent Dyson equation into von Neumann equation,

h =ηHη⁻¹+i~∂_tηη⁻¹→i∂_t%_h= [h, %_h].

Introduction A Simple Model Three types of entropy evolution Conclusion

Von Neumann entropy

Hermitian

%_h=X

i

p_i|φ_ii hφ_i|, i∂_t%_h= [h, %_h], S_h=−tr[%_hln%_h].

Substitute time-dependent Dyson equation into von Neumann equation,

h =ηHη⁻¹+i~∂_tηη⁻¹→i∂_t%_h= [h, %_h].

Introduction A Simple Model Three types of entropy evolution Conclusion

Von Neumann entropy

Hermitian

%_h=X

i

p_i|φ_ii hφ_i|, i∂_t%_h= [h, %_h], S_h=−tr[%_hln%_h].

Substitute time-dependent Dyson equation into von Neumann equation,

h =ηHη⁻¹+i~∂_tηη⁻¹→i∂_t%_h= [h, %_h].

Introduction A Simple Model Three types of entropy evolution Conclusion

Von Neumann entropy

%_h=η%_Hη⁻¹ Non-Hermitian

%H =X

i

pi|ψ_ii hψ_i|ρ, i∂t%_H = [H, %_H], S_H =−tr[%_Hln%_H].

SH =−X

i

λilnλi =Sh.

Introduction A Simple Model Three types of entropy evolution Conclusion

Von Neumann entropy

%_h=η%_Hη⁻¹

Non-Hermitian

%H =X

i

pi|ψ_ii hψ_i|ρ, i∂t%_H = [H, %_H], S_H =−tr[%_Hln%_H].

SH =−X

i

λilnλi =Sh.

Introduction A Simple Model Three types of entropy evolution Conclusion

Von Neumann entropy

%_h=η%_Hη⁻¹ Non-Hermitian

%H =X

i

pi|ψ_ii hψ_i|ρ, i∂t%_H = [H, %_H], S_H =−tr[%_Hln%_H].

SH =−X

i

λilnλi =Sh.

Introduction A Simple Model Three types of entropy evolution Conclusion

Von Neumann entropy

%_h=η%_Hη⁻¹ Non-Hermitian

%_H =X

i

pi|ψ_ii hψ_i|ρ,

i∂t%_H = [H, %_H], S_H =−tr[%_Hln%_H].

SH =−X

i

λilnλi =Sh.

Introduction A Simple Model Three types of entropy evolution Conclusion

Von Neumann entropy

%_h=η%_Hη⁻¹ Non-Hermitian

%_H =X

i

pi|ψ_ii hψ_i|ρ, i∂t%_H = [H, %_H],

S_H =−tr[%_Hln%_H].

SH =−X

i

λilnλi =Sh.

Introduction A Simple Model Three types of entropy evolution Conclusion

Von Neumann entropy

%_h=η%_Hη⁻¹ Non-Hermitian

%_H =X

i

pi|ψ_ii hψ_i|ρ, i∂t%_H = [H, %_H], S_H =−tr[%_Hln%_H].

SH =−X

i

λilnλi =Sh.

Introduction A Simple Model Three types of entropy evolution Conclusion

Von Neumann entropy

%_h=η%_Hη⁻¹ Non-Hermitian

%_H =X

i

pi|ψ_ii hψ_i|ρ, i∂t%_H = [H, %_H], S_H =−tr[%_Hln%_H].

SH =−X

i

λilnλi =Sh.

Introduction A Simple Model Three types of entropy evolution Conclusion

A Simple Model

Introduction A Simple Model Three types of entropy evolution Conclusion

System/bath coupled harmonic oscillators

H=νa^†a+ν

N

X

n=1

q^†_nqn+ (g +κ)a^†

N

X

n=1

qn+ (g−κ)a

N

X

n=1

q^†_n, ν,g,κ∈ R are time-independent parameters.

PT symmetry

PT : i → −i, a→ −a, a^†→ −a^†, qn→ −q_n, q_n^†→ −q^†_n E_m,N^± =m

ν±√

Np

g²−κ²

. PT symmetry spontaneously broken wheng < κ

Introduction A Simple Model Three types of entropy evolution Conclusion

System/bath coupled harmonic oscillators

H=νa^†a+ν

N

X

n=1

q_n^†qn+ (g +κ)a^†

N

X

n=1

qn+ (g−κ)a

N

X

n=1

q^†_n, ν,g,κ∈ R are time-independent parameters.

PT symmetry

PT : i → −i, a→ −a, a^†→ −a^†, qn→ −q_n, q_n^†→ −q^†_n E_m,N^± =m

ν±√

Np

g²−κ²

. PT symmetry spontaneously broken wheng < κ

Introduction A Simple Model Three types of entropy evolution Conclusion

System/bath coupled harmonic oscillators

H=νa^†a+ν

N

X

n=1

q_n^†qn+ (g +κ)a^†

N

X

n=1

qn+ (g−κ)a

N

X

n=1

q^†_n, ν,g,κ∈ R are time-independent parameters.

PT symmetry

PT : i → −i, a→ −a, a^†→ −a^†, qn→ −q_n, q_n^†→ −q^†_n E_m,N^± =m

ν±√

Np

g²−κ²

. PT symmetry spontaneously broken wheng < κ

Introduction A Simple Model Three types of entropy evolution Conclusion

System/bath coupled harmonic oscillators

H=νa^†a+ν

N

X

n=1

q_n^†qn+ (g +κ)a^†

N

X

n=1

qn+ (g−κ)a

N

X

n=1

q^†_n, ν,g,κ∈ R are time-independent parameters.

PT symmetry

PT : i → −i, a→ −a, a^†→ −a^†, qn → −q_n, q_n^†→ −q^†_n

E_m,N^± =m

ν±√ Np

g²−κ²

. PT symmetry spontaneously broken wheng < κ

Introduction A Simple Model Three types of entropy evolution Conclusion

System/bath coupled harmonic oscillators

H=νa^†a+ν

N

X

n=1

q_n^†qn+ (g +κ)a^†

N

X

n=1

qn+ (g−κ)a

N

X

n=1

q^†_n, ν,g,κ∈ R are time-independent parameters.

PT symmetry

PT : i → −i, a→ −a, a^†→ −a^†, qn → −q_n, q_n^†→ −q^†_n E_m,N^± =m

ν±√

Np

g²−κ²

.

PT symmetry spontaneously broken wheng < κ

Introduction A Simple Model Three types of entropy evolution Conclusion

System/bath coupled harmonic oscillators

H=νa^†a+ν

N

X

n=1

q_n^†qn+ (g +κ)a^†

N

X

n=1

qn+ (g−κ)a

N

X

n=1

q^†_n, ν,g,κ∈ R are time-independent parameters.

PT symmetry

PT : i → −i, a→ −a, a^†→ −a^†, qn → −q_n, q_n^†→ −q^†_n E_m,N^± =m

ν±√

Np

g²−κ²

. PT symmetry spontaneously broken wheng < κ

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

Generators N_A =a^†a, N_Q=

N

X

n=1

q_n^†q_n, N_AQ =N_A− 1

NN_Q− 1 N

X

n6=m

q^†_nq_m,

A_x = 1

√N a^†

N

X

n=1

q_n+a

N

X

n=1

q_n^†

!

, A_y = i

√N a^†

N

X

n=1

q_n−a

N

X

n=1

q_n^†

! .

Algebra

[N_A,N_Q] = 0, [N_A,N_AQ] = 0, [N_A,A_x] =−iAy, [N_A,A_y] =iA_y,

[NQ,Ax] =iAy, [NQ,Ay] =−iAx, [NAQ,Ax] =−2iAy, [NAQ,Ay] = 2iAx.

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

Generators

N_A =a^†a, N_Q=

N

X

n=1

q_n^†q_n, N_AQ =N_A− 1

NN_Q− 1 N

X

n6=m

q^†_nq_m,

A_x = 1

√N a^†

N

X

n=1

q_n+a

N

X

n=1

q_n^†

!

, A_y = i

√N a^†

N

X

n=1

q_n−a

N

X

n=1

q_n^†

! .

Algebra

[N_A,N_Q] = 0, [N_A,N_AQ] = 0, [N_A,A_x] =−iAy, [N_A,A_y] =iA_y,

[NQ,Ax] =iAy, [NQ,Ay] =−iAx, [NAQ,Ax] =−2iAy, [NAQ,Ay] = 2iAx.

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

Generators N_A =a^†a, N_Q =

N

X

n=1

q_n^†q_n, N_AQ =N_A− 1

NN_Q− 1 N

X

n6=m

q^†_nq_m,

A_x = 1

√ N a^†

N

X

n=1

q_n+a

N

X

n=1

q_n^†

!

, A_y = i

√ N a^†

N

X

n=1

q_n−a

N

X

n=1

q_n^†

! .

Algebra

[N_A,N_Q] = 0, [N_A,N_AQ] = 0, [N_A,A_x] =−iAy, [N_A,A_y] =iA_y,

[NQ,Ax] =iAy, [NQ,Ay] =−iAx, [NAQ,Ax] =−2iAy, [NAQ,Ay] = 2iAx.

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

Generators N_A =a^†a, N_Q =

N

X

n=1

q_n^†q_n, N_AQ =N_A− 1

NN_Q− 1 N

X

n6=m

q^†_nq_m,

A_x = 1

√ N a^†

N

X

n=1

q_n+a

N

X

n=1

q_n^†

!

, A_y = i

√ N a^†

N

X

n=1

q_n−a

N

X

n=1

q_n^†

! .

Algebra

[N_A,N_Q] = 0, [N_A,N_AQ] = 0, [N_A,A_x] =−iAy, [N_A,A_y] =iA_y,

[NQ,Ax] =iAy, [NQ,Ay] =−iAx, [NAQ,Ax] =−2iAy, [NAQ,Ay] = 2iAx.

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

Generators N_A =a^†a, N_Q =

N

X

n=1

q_n^†q_n, N_AQ =N_A− 1

NN_Q− 1 N

X

n6=m

q^†_nq_m,

A_x = 1

√ N a^†

N

X

n=1

q_n+a

N

X

n=1

q_n^†

!

, A_y = i

√ N a^†

N

X

n=1

q_n−a

N

X

n=1

q_n^†

! .

Algebra

[N_A,N_Q] = 0, [N_A,N_AQ] = 0, [N_A,A_x] =−iAy, [N_A,A_y] =iA_y,

[NQ,Ax] =iAy, [NQ,Ay] =−iAx, [NAQ,Ax] =−2iAy, [NAQ,Ay] = 2iAx.

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

H=νN_A+νN_Q +

√

NgA_x−i

√ NκA_y. ρ(t) =η^†(t)η(t)

h(t) =η(t)Hη⁻¹(t) +i~∂_tη(t)η⁻¹(t) Ansatz

η(t) =e^β(t)Aye^α(t)NAQ, Coupled differential equations

˙

α=−tanh (2β)h√

Ngcosh (2α) +√

Nκsinh (2α)i , β˙ =√

Nκcosh (2α) +√

Ngsinh (2α).

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

H=νN_A+νN_Q +

√

NgA_x−i

√ NκA_y.

ρ(t) =η^†(t)η(t)

h(t) =η(t)Hη⁻¹(t) +i~∂_tη(t)η⁻¹(t) Ansatz

η(t) =e^β(t)Aye^α(t)NAQ, Coupled differential equations

˙

α=−tanh (2β)h√

Ngcosh (2α) +√

Nκsinh (2α)i , β˙ =√

Nκcosh (2α) +√

Ngsinh (2α).

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

H=νN_A+νN_Q +

√

NgA_x−i

√ NκA_y. ρ(t) =η^†(t)η(t)

h(t) =η(t)Hη⁻¹(t) +i~∂_tη(t)η⁻¹(t) Ansatz

η(t) =e^β(t)Aye^α(t)NAQ, Coupled differential equations

˙

α=−tanh (2β)h√

Ngcosh (2α) +√

Nκsinh (2α)i , β˙ =√

Nκcosh (2α) +√

Ngsinh (2α).

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

H=νN_A+νN_Q +

√

NgA_x−i

√ NκA_y. ρ(t) =η^†(t)η(t)

h(t) =η(t)Hη⁻¹(t) +i~∂_tη(t)η⁻¹(t)

Ansatz

η(t) =e^β(t)Aye^α(t)NAQ, Coupled differential equations

˙

α=−tanh (2β)h√

Ngcosh (2α) +√

Nκsinh (2α)i , β˙ =√

Nκcosh (2α) +√

Ngsinh (2α).

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

H=νN_A+νN_Q +

√

NgA_x−i

√ NκA_y. ρ(t) =η^†(t)η(t)

h(t) =η(t)Hη⁻¹(t) +i~∂_tη(t)η⁻¹(t) Ansatz

η(t) =e^β(t)Aye^α(t)NAQ, Coupled differential equations

˙

α=−tanh (2β)h√

Ngcosh (2α) +√

Nκsinh (2α)i , β˙ =√

Nκcosh (2α) +√

Ngsinh (2α).

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

H=νN_A+νN_Q +

√

NgA_x−i

√ NκA_y. ρ(t) =η^†(t)η(t)

h(t) =η(t)Hη⁻¹(t) +i~∂_tη(t)η⁻¹(t) Ansatz

η(t) =e^β(t)Aye^α(t)NAQ,

Coupled differential equations

˙

α=−tanh (2β)h√

Ngcosh (2α) +√

Nκsinh (2α)i , β˙ =√

Nκcosh (2α) +√

Ngsinh (2α).

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

H=νN_A+νN_Q +

√

NgA_x−i

√ NκA_y. ρ(t) =η^†(t)η(t)

h(t) =η(t)Hη⁻¹(t) +i~∂_tη(t)η⁻¹(t) Ansatz

η(t) =e^β(t)Aye^α(t)NAQ, Coupled differential equations

˙

α=−tanh (2β)h√

Ngcosh (2α) +√

Nκsinh (2α)i , β˙ =√

Nκcosh (2α) +√

Ngsinh (2α).

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

H=νN_A+νN_Q +

√

NgA_x−i

√ NκA_y. ρ(t) =η^†(t)η(t)

h(t) =η(t)Hη⁻¹(t) +i~∂_tη(t)η⁻¹(t) Ansatz

η(t) =e^β(t)Aye^α(t)NAQ, Coupled differential equations

˙

α=−tanh (2β)h√

Ngcosh (2α) +√

Nκsinh (2α)i , β˙ =√

Nκcosh (2α) +√

Ngsinh (2α).

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

Solution forα

tanh (2α) =

−Ngκ+ ˙β

qβ˙²+N(g²−κ²) Ng²+ ˙β² . Solution forβ

β¨+ 2 tanh (2β) h

Ng²−Nκ²+ ˙β² i

= 0. sinh (2β) =σ

¨

σ+ 4N g²−κ² σ = 0,

σ = c1

pg²−κ²sin

2

√ Np

g²−κ²(t+c2)

,

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

Solution forα

tanh (2α) =

−Ngκ+ ˙β

qβ˙²+N(g²−κ²) Ng²+ ˙β² . Solution forβ

β¨+ 2 tanh (2β) h

Ng²−Nκ²+ ˙β² i

= 0. sinh (2β) =σ

¨

σ+ 4N g²−κ² σ = 0,

σ = c1

pg²−κ²sin

2

√ Np

g²−κ²(t+c2)

,

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

Solution forα

tanh (2α) =

−Ngκ+ ˙β

qβ˙²+N(g²−κ²) Ng²+ ˙β² .

Solution forβ

β¨+ 2 tanh (2β) h

Ng²−Nκ²+ ˙β² i

= 0. sinh (2β) =σ

¨

σ+ 4N g²−κ² σ = 0,

σ = c1

pg²−κ²sin

2

√ Np

g²−κ²(t+c2)

,

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

Solution forα

tanh (2α) =

−Ngκ+ ˙β

qβ˙²+N(g²−κ²) Ng²+ ˙β² . Solution forβ

β¨+ 2 tanh (2β) h

Ng²−Nκ²+ ˙β² i

= 0. sinh (2β) =σ

¨

σ+ 4N g²−κ² σ = 0,

σ = c1

pg²−κ²sin

2

√ Np

g²−κ²(t+c2)

,

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

Solution forα

tanh (2α) =

−Ngκ+ ˙β

qβ˙²+N(g²−κ²) Ng²+ ˙β² . Solution forβ

β¨+ 2 tanh (2β) h

Ng²−Nκ²+ ˙β² i

= 0.

sinh (2β) =σ

¨

σ+ 4N g²−κ² σ = 0,

σ = c1

pg²−κ²sin

2

√ Np

g²−κ²(t+c2)

,

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

Solution forα

tanh (2α) =

−Ngκ+ ˙β

qβ˙²+N(g²−κ²) Ng²+ ˙β² . Solution forβ

β¨+ 2 tanh (2β) h

Ng²−Nκ²+ ˙β² i

= 0.

sinh (2β) =σ

¨

σ+ 4N g²−κ² σ = 0,

σ = c1

pg²−κ²sin

2

√ Np

g²−κ²(t+c2)

,

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

Solution forα

tanh (2α) =

−Ngκ+ ˙β

qβ˙²+N(g²−κ²) Ng²+ ˙β² . Solution forβ

β¨+ 2 tanh (2β) h

Ng²−Nκ²+ ˙β² i

= 0.

sinh (2β) =σ

¨

σ+ 4N g²−κ² σ = 0,

σ = c1

pg²−κ²sin

2

√ Np

g²−κ²(t+c2)

,

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

Solution forα

tanh (2α) =

−Ngκ+ ˙β

qβ˙²+N(g²−κ²) Ng²+ ˙β² . Solution forβ

β¨+ 2 tanh (2β) h

Ng²−Nκ²+ ˙β² i

= 0.

sinh (2β) =σ

¨

σ+ 4N g²−κ² σ = 0,

σ = c1

pg²−κ²sin

2

√ Np

g²−κ²(t+c2)

,

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

Hermitian Hamiltonian

h(t) =νN_A+νN_Q +µ(t)Ax,

µ(t) =

g²−κ²√ N

q

c₁²+g²−κ² c₁²+ 2 (g²−κ²)−c₁²cos

4√

Np

g²−κ²(t+c2) .

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

Hermitian Hamiltonian

h(t) =νN_A+νN_Q +µ(t)Ax,

µ(t) =

g²−κ²√ N

q

c₁²+g²−κ² c₁²+ 2 (g²−κ²)−c₁²cos

4√

Np

g²−κ²(t+c2) .

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

Hermitian Hamiltonian

h(t) =νN_A+νN_Q +µ(t)Ax,

µ(t) =

g²−κ²√ N

q

c₁²+g²−κ² c₁²+ 2 (g²−κ²)−c₁²cos

4√

Np

g²−κ²(t+c2) .

Introduction A Simple Model Three types of entropy evolution Conclusion

Calculating the metric

Hermitian Hamiltonian

h(t) =νN_A+νN_Q +µ(t)Ax,

µ(t) =

g²−κ²√ N

q

c₁²+g²−κ² c₁²+ 2 (g²−κ²)−c₁²cos

4√

Np

g²−κ²(t+c2) .

Introduction A Simple Model Three types of entropy evolution Conclusion

Three types of entropy evolution

Introduction A Simple Model Three types of entropy evolution Conclusion

Von Neumann entropy

Initial State

|φ(0)i= sinγ|1_a0_qi+cosγ

√ N

N

X

i=1

|0_a1_ii,

State at timet

|φ(t)i=e^−iνt(sinγsinµ_I(t) + cosγcosµ_I(t))|1_a0_qi +e^−iνt

√

N (sinγcosµ_I(t)−cosγsinµ_I(t))

N

X

i=1

|0_a1ii.

µI(t) = Z t

µ(s)ds= 1 2arctan





pc₁²+g²−κ²tan

2√ Np

g²−κ²(t+c2) pg²−κ²



.

Introduction A Simple Model Three types of entropy evolution Conclusion

Von Neumann entropy

Initial State

|φ(0)i= sinγ|1_a0_qi+cosγ

√ N

N

X

i=1

|0_a1_ii,

State at timet

|φ(t)i=e^−iνt(sinγsinµ_I(t) + cosγcosµ_I(t))|1_a0_qi +e^−iνt

√

N (sinγcosµ_I(t)−cosγsinµ_I(t))

N

X

i=1

|0_a1ii.

µI(t) = Z t

µ(s)ds= 1 2arctan





pc₁²+g²−κ²tan

2√ Np

g²−κ²(t+c2) pg²−κ²



.