Quantum optical models in noncommutative spaces

Quantum optical models in noncommutativespaces

Sanjib Dey

Universite de Montreal & Concordia University

Seminar Physique Mathematique, September 15, 2015

S. Dey; Phys. Rev. D 91, 044024 (2015),

S. Dey, V. Hussin; Phys. Rev. D 91, 124017 (2015)

S. Dey, A. Fring, V. Hussin; arxiv: 1506.08901

1 / 25

Noncommutative spaces

Snyder’s noncommutative space

[xµ, xν ] = iθ (xµpν − xνpµ)

[xµ, pν ] = i~ (δµν + θpµpν)

[pµ, pν ] = 0

Lorentz covarient, but Poincare symmetry is violated [H. S.Snyder; Phys. Rev. 71, 38 (1947)].

Poincare symmetries were deformed to make the algebra compatiblewith Snyder’s version [R. Banerjee, S. Kulkarni, S. Samanta; JHEP2006, 077 (2006)].

Flat noncommutative space

[xµ, xν ] = iθµν , [xµ, pν ] = i~δµν and [pµ, pν ] = 0

θµν is constant antisymmetric tensor, which breaks Lorentz-Poincaresymmetry [N. Seiberg, E. Witten; JHEP 1999, 032 (1999)].

2 / 25

q-deformed noncommutative spaces

Deformed oscillator algebras in 3D

AiA†j − q2δijA†jAi = δij ,

[A†i ,A

†j

]= [Ai ,Aj ] = 0, q ∈ R

The limit q → 1 gives standard Fock space Ai → ai :[ai , a

†j

]= δij , [ai , aj ] =

[a†i , a

†j

]= 0.

Consider X = α(A†+A

)and P = iβ

(A†−A

), α, β ∈ R,

Deformed canonical commutation relation:

[X ,P] =4iαβ

1 + q2

[1 +

q2 − 1

4

(X 2

α2+

P2

β2

)]Constraints =⇒ α = ~

2β , q = e2τβ2, τ ∈ R+

Non-trivial limit β → 0

3 / 25

Physical consequences

[X ,P] = i~(1 + τP2

)Generalised uncertainty relation:

∆X∆P ≥ 1

2

∣∣∣ 〈[X ,P]〉∣∣∣

≥ ~2

[1 + τ (∆P)2 + τ〈P〉2

]

Standard case: [X ,P] = Constant; give up knowledge aboutP, for ∆X = 0

Noncommutative case: [X ,P] ≈ P2; give up knowledge alsoabout P, for ∆X 6= 0

4 / 25

Minimal lengths, areas and volumes

Minimal length

∆Xmin = ~√τ√

1 + τ〈P2〉,

from minimizing with (∆X )2 = 〈X 2〉 − 〈X 〉2[B. Bagchi, A. Fring; Phys. Lett. A 373, 4307–4310 (2009)]

2D&3D-versions are more complicated and lead to “minimal areas”and “minimal volumes” [S. Dey, A. Fring, L. Gouba; J. Phys. A:Math. Theor. 45, 385302 (2012)]

Hermitian representation of X = α(A† + A),P = iβ(A† − A):

A =i√

1− q2

(e−i x − e−i x/2e2τ p

), A† =

−i√1− q2

(e i x − e2τ pe i x/2

)with x = x

√mω/~ and p = p/

√mω~ , [x , p] = i~

X † = X , P† = P for q < 1PT : x → −x , p → p, i → −i

5 / 25

Why is Hermiticity a good property to have?

Hermiticity of H ensures real eigenvalues, Hψ = Eψ

〈ψ|H|ψ〉 = E 〈ψ|ψ〉〈ψ|H†|ψ〉 = E ∗〈ψ|ψ〉

}= 0 = (E − E ∗)〈ψ|ψ〉

Hermiticity ensures conservation of probability densities

|ψ(t)〉 = e−iHt |ψ(0)〉

〈ψ(t)|ψ(t)〉 =〈ψ(0)|e iH†te−iHt |ψ(0)〉 = 〈ψ(0)|ψ(0)〉

Hermiticity is not essential:Operators O which are left invariant under an antilinear involutionI and whose eigenfunctions φ also respect this symmetry,

[O, I] = 0 ∧ Iφ = φ,

have real eigenvalues [E. Wigner; J. Math. Phys. 1, 409 (1960)]6 / 25

PT -symmetry (an example)

Unbroken PT -symmetry guarantees real eigenvalues

PT -symmetry: PT : x → −x p → p i → −i(P : x → −x , p → −p; T : x → x , p → −p, i → −i)

PT is an anti-linear operator:

PT (λΦ + µΨ) = λ∗PT Φ + µ∗PT Ψ λ, µ ∈ C

Real eigenvalues from unbroken PT -symmetry:

[H,PT ] = 0 ∧ PT Φ = Φ ⇒ ε = ε∗ for HΦ = εΦ

Proof :

εΦ = HΦ = HPT Φ = PT HΦ = PT εΦ = ε∗PT Φ = ε∗Φ

-PT-symmetry is only an example of an antilinear involution

7 / 25

PT -symmetry (an example)

Unbroken PT -symmetry guarantees real eigenvalues

PT -symmetry: PT : x → −x p → p i → −i(P : x → −x , p → −p; T : x → x , p → −p, i → −i)

PT is an anti-linear operator:

PT (λΦ + µΨ) = λ∗PT Φ + µ∗PT Ψ λ, µ ∈ C

Real eigenvalues from unbroken PT -symmetry:

[H,PT ] = 0 ∧ PT Φ = Φ ⇒ ε = ε∗ for HΦ = εΦ

Proof :

εΦ = HΦ = HPT Φ = PT HΦ = PT εΦ = ε∗PT Φ = ε∗Φ

-PT-symmetry is only an example of an antilinear involution

7 / 25

PT -symmetry (an example)

Unbroken PT -symmetry guarantees real eigenvalues

PT -symmetry: PT : x → −x p → p i → −i(P : x → −x , p → −p; T : x → x , p → −p, i → −i)

PT is an anti-linear operator:

PT (λΦ + µΨ) = λ∗PT Φ + µ∗PT Ψ λ, µ ∈ C

Real eigenvalues from unbroken PT -symmetry:

[H,PT ] = 0 ∧ PT Φ = Φ ⇒ ε = ε∗ for HΦ = εΦ

Proof :

εΦ = HΦ = HPT Φ = PT HΦ = PT εΦ = ε∗PT Φ = ε∗Φ

-PT-symmetry is only an example of an antilinear involution

7 / 25

PT -symmetry (an example)

Unbroken PT -symmetry guarantees real eigenvalues

PT -symmetry: PT : x → −x p → p i → −i(P : x → −x , p → −p; T : x → x , p → −p, i → −i)

PT is an anti-linear operator:

PT (λΦ + µΨ) = λ∗PT Φ + µ∗PT Ψ λ, µ ∈ C

Real eigenvalues from unbroken PT -symmetry:

[H,PT ] = 0 ∧ PT Φ = Φ ⇒ ε = ε∗ for HΦ = εΦ

Proof :

εΦ = HΦ = HPT Φ = PT HΦ = PT εΦ = ε∗PT Φ = ε∗Φ

-PT-symmetry is only an example of an antilinear involution

7 / 25

PT -symmetry (an example)

Unbroken PT -symmetry guarantees real eigenvalues

PT -symmetry: PT : x → −x p → p i → −i(P : x → −x , p → −p; T : x → x , p → −p, i → −i)

PT is an anti-linear operator:

PT (λΦ + µΨ) = λ∗PT Φ + µ∗PT Ψ λ, µ ∈ C

Real eigenvalues from unbroken PT -symmetry:

[H,PT ] = 0 ∧ PT Φ = Φ ⇒ ε = ε∗ for HΦ = εΦ

Proof :εΦ = HΦ

= HPT Φ = PT HΦ = PT εΦ = ε∗PT Φ = ε∗Φ

-PT-symmetry is only an example of an antilinear involution

7 / 25

PT -symmetry (an example)

Unbroken PT -symmetry guarantees real eigenvalues

PT -symmetry: PT : x → −x p → p i → −i(P : x → −x , p → −p; T : x → x , p → −p, i → −i)

PT is an anti-linear operator:

PT (λΦ + µΨ) = λ∗PT Φ + µ∗PT Ψ λ, µ ∈ C

Real eigenvalues from unbroken PT -symmetry:

[H,PT ] = 0 ∧ PT Φ = Φ ⇒ ε = ε∗ for HΦ = εΦ

Proof :εΦ = HΦ = HPT Φ

= PT HΦ = PT εΦ = ε∗PT Φ = ε∗Φ

-PT-symmetry is only an example of an antilinear involution

7 / 25

PT -symmetry (an example)

Unbroken PT -symmetry guarantees real eigenvalues

PT -symmetry: PT : x → −x p → p i → −i(P : x → −x , p → −p; T : x → x , p → −p, i → −i)

PT is an anti-linear operator:

PT (λΦ + µΨ) = λ∗PT Φ + µ∗PT Ψ λ, µ ∈ C

Real eigenvalues from unbroken PT -symmetry:

[H,PT ] = 0 ∧ PT Φ = Φ ⇒ ε = ε∗ for HΦ = εΦ

Proof :εΦ = HΦ = HPT Φ = PT HΦ

= PT εΦ = ε∗PT Φ = ε∗Φ

-PT-symmetry is only an example of an antilinear involution

7 / 25

PT -symmetry (an example)

Unbroken PT -symmetry guarantees real eigenvalues

PT -symmetry: PT : x → −x p → p i → −i(P : x → −x , p → −p; T : x → x , p → −p, i → −i)

PT is an anti-linear operator:

PT (λΦ + µΨ) = λ∗PT Φ + µ∗PT Ψ λ, µ ∈ C

Real eigenvalues from unbroken PT -symmetry:

[H,PT ] = 0 ∧ PT Φ = Φ ⇒ ε = ε∗ for HΦ = εΦ

Proof :εΦ = HΦ = HPT Φ = PT HΦ = PT εΦ

= ε∗PT Φ = ε∗Φ

-PT-symmetry is only an example of an antilinear involution

7 / 25

PT -symmetry (an example)

Unbroken PT -symmetry guarantees real eigenvalues

PT -symmetry: PT : x → −x p → p i → −i(P : x → −x , p → −p; T : x → x , p → −p, i → −i)

PT is an anti-linear operator:

PT (λΦ + µΨ) = λ∗PT Φ + µ∗PT Ψ λ, µ ∈ C

Real eigenvalues from unbroken PT -symmetry:

[H,PT ] = 0 ∧ PT Φ = Φ ⇒ ε = ε∗ for HΦ = εΦ

Proof :εΦ = HΦ = HPT Φ = PT HΦ = PT εΦ = ε∗PT Φ

= ε∗Φ

-PT-symmetry is only an example of an antilinear involution

7 / 25

PT -symmetry (an example)

Unbroken PT -symmetry guarantees real eigenvalues

PT -symmetry: PT : x → −x p → p i → −i(P : x → −x , p → −p; T : x → x , p → −p, i → −i)

PT is an anti-linear operator:

PT (λΦ + µΨ) = λ∗PT Φ + µ∗PT Ψ λ, µ ∈ C

Real eigenvalues from unbroken PT -symmetry:

[H,PT ] = 0 ∧ PT Φ = Φ ⇒ ε = ε∗ for HΦ = εΦ

Proof :εΦ = HΦ = HPT Φ = PT HΦ = PT εΦ = ε∗PT Φ = ε∗Φ

-PT-symmetry is only an example of an antilinear involution

7 / 25

PT -symmetry (an example)

Unbroken PT -symmetry guarantees real eigenvalues

PT -symmetry: PT : x → −x p → p i → −i(P : x → −x , p → −p; T : x → x , p → −p, i → −i)

PT is an anti-linear operator:

PT (λΦ + µΨ) = λ∗PT Φ + µ∗PT Ψ λ, µ ∈ C

Real eigenvalues from unbroken PT -symmetry:

[H,PT ] = 0 ∧ PT Φ = Φ ⇒ ε = ε∗ for HΦ = εΦ

Proof :εΦ = HΦ = HPT Φ = PT HΦ = PT εΦ = ε∗PT Φ = ε∗Φ

-PT-symmetry is only an example of an antilinear involution

7 / 25

Pseudo-Hermiticity

H is Hermitian with respect to a new metric

∗ Assume pseudo-Hermiticity:

h = ηHη−1 = h† = (η−1)†H†η† ⇔ H†η†η = η†ηH

Φ = η−1φ η† = η

⇒ H is Hermitian with respect to the new metric

Proof :

〈Ψ|HΦ〉η = 〈Ψ|η2HΦ〉

= 〈η−1ψ|η2Hη−1φ〉 = 〈ψ|ηHη−1φ〉 =

〈ψ|hφ〉 = 〈hψ|φ〉 = 〈ηHη−1ψ|φ〉 = 〈HΨ|ηφ〉 = 〈HΨ|η2Φ〉= 〈HΨ|Φ〉η

⇒ Eigenvalues of H are real, eigenstates are orthogonal

M. Froissart; Nuovo Cim. 14, 197 (1959)

A. Mostafazadeh; J. Math. Phys. 43, 2814 (2002)8 / 25

Pseudo-Hermiticity

H is Hermitian with respect to a new metric

∗ Assume pseudo-Hermiticity:

h = ηHη−1 = h† = (η−1)†H†η† ⇔ H†η†η = η†ηH

Φ = η−1φ η† = η

⇒ H is Hermitian with respect to the new metricProof :

〈Ψ|HΦ〉η = 〈Ψ|η2HΦ〉

= 〈η−1ψ|η2Hη−1φ〉 = 〈ψ|ηHη−1φ〉 =

〈ψ|hφ〉 = 〈hψ|φ〉 = 〈ηHη−1ψ|φ〉 = 〈HΨ|ηφ〉 = 〈HΨ|η2Φ〉= 〈HΨ|Φ〉η

⇒ Eigenvalues of H are real, eigenstates are orthogonal

M. Froissart; Nuovo Cim. 14, 197 (1959)

A. Mostafazadeh; J. Math. Phys. 43, 2814 (2002)8 / 25

Pseudo-Hermiticity

H is Hermitian with respect to a new metric

∗ Assume pseudo-Hermiticity:

h = ηHη−1 = h† = (η−1)†H†η† ⇔ H†η†η = η†ηH

Φ = η−1φ η† = η

⇒ H is Hermitian with respect to the new metricProof :

〈Ψ|HΦ〉η = 〈Ψ|η2HΦ〉 = 〈η−1ψ|η2Hη−1φ〉

= 〈ψ|ηHη−1φ〉 =

〈ψ|hφ〉 = 〈hψ|φ〉 = 〈ηHη−1ψ|φ〉 = 〈HΨ|ηφ〉 = 〈HΨ|η2Φ〉= 〈HΨ|Φ〉η

⇒ Eigenvalues of H are real, eigenstates are orthogonal

M. Froissart; Nuovo Cim. 14, 197 (1959)

A. Mostafazadeh; J. Math. Phys. 43, 2814 (2002)8 / 25

Pseudo-Hermiticity

H is Hermitian with respect to a new metric

∗ Assume pseudo-Hermiticity:

h = ηHη−1 = h† = (η−1)†H†η† ⇔ H†η†η = η†ηH

Φ = η−1φ η† = η

⇒ H is Hermitian with respect to the new metricProof :

〈Ψ|HΦ〉η = 〈Ψ|η2HΦ〉 = 〈η−1ψ|η2Hη−1φ〉 = 〈ψ|ηHη−1φ〉

=

〈ψ|hφ〉 = 〈hψ|φ〉 = 〈ηHη−1ψ|φ〉 = 〈HΨ|ηφ〉 = 〈HΨ|η2Φ〉= 〈HΨ|Φ〉η

⇒ Eigenvalues of H are real, eigenstates are orthogonal

M. Froissart; Nuovo Cim. 14, 197 (1959)

A. Mostafazadeh; J. Math. Phys. 43, 2814 (2002)8 / 25

Pseudo-Hermiticity

H is Hermitian with respect to a new metric

∗ Assume pseudo-Hermiticity:

h = ηHη−1 = h† = (η−1)†H†η† ⇔ H†η†η = η†ηH

Φ = η−1φ η† = η

⇒ H is Hermitian with respect to the new metricProof :

〈Ψ|HΦ〉η = 〈Ψ|η2HΦ〉 = 〈η−1ψ|η2Hη−1φ〉 = 〈ψ|ηHη−1φ〉 =

〈ψ|hφ〉

= 〈hψ|φ〉 = 〈ηHη−1ψ|φ〉 = 〈HΨ|ηφ〉 = 〈HΨ|η2Φ〉= 〈HΨ|Φ〉η

⇒ Eigenvalues of H are real, eigenstates are orthogonal

M. Froissart; Nuovo Cim. 14, 197 (1959)

A. Mostafazadeh; J. Math. Phys. 43, 2814 (2002)8 / 25

Pseudo-Hermiticity

H is Hermitian with respect to a new metric

∗ Assume pseudo-Hermiticity:

h = ηHη−1 = h† = (η−1)†H†η† ⇔ H†η†η = η†ηH

Φ = η−1φ η† = η

⇒ H is Hermitian with respect to the new metricProof :

〈Ψ|HΦ〉η = 〈Ψ|η2HΦ〉 = 〈η−1ψ|η2Hη−1φ〉 = 〈ψ|ηHη−1φ〉 =

〈ψ|hφ〉 = 〈hψ|φ〉

= 〈ηHη−1ψ|φ〉 = 〈HΨ|ηφ〉 = 〈HΨ|η2Φ〉= 〈HΨ|Φ〉η

⇒ Eigenvalues of H are real, eigenstates are orthogonal

M. Froissart; Nuovo Cim. 14, 197 (1959)

A. Mostafazadeh; J. Math. Phys. 43, 2814 (2002)8 / 25

Pseudo-Hermiticity

H is Hermitian with respect to a new metric

∗ Assume pseudo-Hermiticity:

h = ηHη−1 = h† = (η−1)†H†η† ⇔ H†η†η = η†ηH

Φ = η−1φ η† = η

⇒ H is Hermitian with respect to the new metricProof :

〈Ψ|HΦ〉η = 〈Ψ|η2HΦ〉 = 〈η−1ψ|η2Hη−1φ〉 = 〈ψ|ηHη−1φ〉 =

〈ψ|hφ〉 = 〈hψ|φ〉 = 〈ηHη−1ψ|φ〉

= 〈HΨ|ηφ〉 = 〈HΨ|η2Φ〉= 〈HΨ|Φ〉η

⇒ Eigenvalues of H are real, eigenstates are orthogonal

M. Froissart; Nuovo Cim. 14, 197 (1959)

A. Mostafazadeh; J. Math. Phys. 43, 2814 (2002)8 / 25

Pseudo-Hermiticity

H is Hermitian with respect to a new metric

∗ Assume pseudo-Hermiticity:

h = ηHη−1 = h† = (η−1)†H†η† ⇔ H†η†η = η†ηH

Φ = η−1φ η† = η

⇒ H is Hermitian with respect to the new metricProof :

〈Ψ|HΦ〉η = 〈Ψ|η2HΦ〉 = 〈η−1ψ|η2Hη−1φ〉 = 〈ψ|ηHη−1φ〉 =

〈ψ|hφ〉 = 〈hψ|φ〉 = 〈ηHη−1ψ|φ〉 = 〈HΨ|ηφ〉

= 〈HΨ|η2Φ〉= 〈HΨ|Φ〉η

⇒ Eigenvalues of H are real, eigenstates are orthogonal

M. Froissart; Nuovo Cim. 14, 197 (1959)

A. Mostafazadeh; J. Math. Phys. 43, 2814 (2002)8 / 25

Pseudo-Hermiticity

H is Hermitian with respect to a new metric

∗ Assume pseudo-Hermiticity:

h = ηHη−1 = h† = (η−1)†H†η† ⇔ H†η†η = η†ηH

Φ = η−1φ η† = η

⇒ H is Hermitian with respect to the new metricProof :

〈Ψ|HΦ〉η = 〈Ψ|η2HΦ〉 = 〈η−1ψ|η2Hη−1φ〉 = 〈ψ|ηHη−1φ〉 =

〈ψ|hφ〉 = 〈hψ|φ〉 = 〈ηHη−1ψ|φ〉 = 〈HΨ|ηφ〉 = 〈HΨ|η2Φ〉

= 〈HΨ|Φ〉η

⇒ Eigenvalues of H are real, eigenstates are orthogonal

M. Froissart; Nuovo Cim. 14, 197 (1959)

A. Mostafazadeh; J. Math. Phys. 43, 2814 (2002)8 / 25

Pseudo-Hermiticity

H is Hermitian with respect to a new metric

∗ Assume pseudo-Hermiticity:

h = ηHη−1 = h† = (η−1)†H†η† ⇔ H†η†η = η†ηH

Φ = η−1φ η† = η

⇒ H is Hermitian with respect to the new metricProof :

〈Ψ|HΦ〉η = 〈Ψ|η2HΦ〉 = 〈η−1ψ|η2Hη−1φ〉 = 〈ψ|ηHη−1φ〉 =

〈ψ|hφ〉 = 〈hψ|φ〉 = 〈ηHη−1ψ|φ〉 = 〈HΨ|ηφ〉 = 〈HΨ|η2Φ〉= 〈HΨ|Φ〉η

⇒ Eigenvalues of H are real, eigenstates are orthogonal

M. Froissart; Nuovo Cim. 14, 197 (1959)

A. Mostafazadeh; J. Math. Phys. 43, 2814 (2002)8 / 25

Pseudo-Hermiticity

H is Hermitian with respect to a new metric

∗ Assume pseudo-Hermiticity:

h = ηHη−1 = h† = (η−1)†H†η† ⇔ H†η†η = η†ηH

Φ = η−1φ η† = η

⇒ H is Hermitian with respect to the new metricProof :

〈Ψ|HΦ〉η = 〈Ψ|η2HΦ〉 = 〈η−1ψ|η2Hη−1φ〉 = 〈ψ|ηHη−1φ〉 =

〈ψ|hφ〉 = 〈hψ|φ〉 = 〈ηHη−1ψ|φ〉 = 〈HΨ|ηφ〉 = 〈HΨ|η2Φ〉= 〈HΨ|Φ〉η

⇒ Eigenvalues of H are real, eigenstates are orthogonal

M. Froissart; Nuovo Cim. 14, 197 (1959)

A. Mostafazadeh; J. Math. Phys. 43, 2814 (2002)8 / 25

Coherent states

Glauber coherent states:

a|α〉 = α|α〉 or |α〉 = D(α)|0〉, D(α) = eαa†−α∗a

⇒ |α〉 =1

N (α)

∞∑n=0

αn

√n!|n〉, α ∈ C

⇒ Very close to classical objects

Nonlinear coherent states:

(a, a†)⇒ (A,A†) :

{A = af (n) = f (n + 1)aA† = f (n)a† = a†f (n + 1)

A|α, f 〉 = α|α, f 〉 ⇒ |α, f 〉 =1

N (α, f )

∞∑n=0

αn

√n!f (n)!

|n〉

f (n) can be associated with the eigenvalues of a Hamiltonian

H ∼ A†A = f (n)a†af (n) = f 2(n)a†a ∼ f 2(n)n = En

9 / 25

Coherent states

Glauber coherent states:

a|α〉 = α|α〉 or |α〉 = D(α)|0〉, D(α) = eαa†−α∗a

⇒ |α〉 =1

N (α)

∞∑n=0

αn

√n!|n〉, α ∈ C

⇒ Very close to classical objects

Nonlinear coherent states:

(a, a†)⇒ (A,A†) :

{A = af (n) = f (n + 1)aA† = f (n)a† = a†f (n + 1)

A|α, f 〉 = α|α, f 〉 ⇒ |α, f 〉 =1

N (α, f )

∞∑n=0

αn

√n!f (n)!

|n〉

f (n) can be associated with the eigenvalues of a Hamiltonian

H ∼ A†A = f (n)a†af (n) = f 2(n)a†a ∼ f 2(n)n = En

9 / 25

Coherent states

Glauber coherent states:

a|α〉 = α|α〉 or |α〉 = D(α)|0〉, D(α) = eαa†−α∗a

⇒ |α〉 =1

N (α)

∞∑n=0

αn

√n!|n〉, α ∈ C

⇒ Very close to classical objects

Nonlinear coherent states:

(a, a†)⇒ (A,A†) :

{A = af (n) = f (n + 1)aA† = f (n)a† = a†f (n + 1)

A|α, f 〉 = α|α, f 〉 ⇒ |α, f 〉 =1

N (α, f )

∞∑n=0

αn

√n!f (n)!

|n〉

f (n) can be associated with the eigenvalues of a Hamiltonian

H ∼ A†A = f (n)a†af (n) = f 2(n)a†a ∼ f 2(n)n = En

9 / 25

Coherent states

Glauber coherent states:

a|α〉 = α|α〉 or |α〉 = D(α)|0〉, D(α) = eαa†−α∗a

⇒ |α〉 =1

N (α)

∞∑n=0

αn

√n!|n〉, α ∈ C

⇒ Very close to classical objects

Nonlinear coherent states:

(a, a†)⇒ (A,A†) :

{A = af (n) = f (n + 1)aA† = f (n)a† = a†f (n + 1)

A|α, f 〉 = α|α, f 〉 ⇒ |α, f 〉 =1

N (α, f )

∞∑n=0

αn

√n!f (n)!

|n〉

f (n) can be associated with the eigenvalues of a Hamiltonian

H ∼ A†A

= f (n)a†af (n) = f 2(n)a†a ∼ f 2(n)n = En

9 / 25

Coherent states

Glauber coherent states:

a|α〉 = α|α〉 or |α〉 = D(α)|0〉, D(α) = eαa†−α∗a

⇒ |α〉 =1

N (α)

∞∑n=0

αn

√n!|n〉, α ∈ C

⇒ Very close to classical objects

Nonlinear coherent states:

(a, a†)⇒ (A,A†) :

{A = af (n) = f (n + 1)aA† = f (n)a† = a†f (n + 1)

A|α, f 〉 = α|α, f 〉 ⇒ |α, f 〉 =1

N (α, f )

∞∑n=0

αn

√n!f (n)!

|n〉

f (n) can be associated with the eigenvalues of a Hamiltonian

H ∼ A†A = f (n)a†af (n)

= f 2(n)a†a ∼ f 2(n)n = En

9 / 25

Coherent states

Glauber coherent states:

a|α〉 = α|α〉 or |α〉 = D(α)|0〉, D(α) = eαa†−α∗a

⇒ |α〉 =1

N (α)

∞∑n=0

αn

√n!|n〉, α ∈ C

⇒ Very close to classical objects

Nonlinear coherent states:

(a, a†)⇒ (A,A†) :

{A = af (n) = f (n + 1)aA† = f (n)a† = a†f (n + 1)

A|α, f 〉 = α|α, f 〉 ⇒ |α, f 〉 =1

N (α, f )

∞∑n=0

αn

√n!f (n)!

|n〉

f (n) can be associated with the eigenvalues of a Hamiltonian

H ∼ A†A = f (n)a†af (n) = f 2(n)a†a

∼ f 2(n)n = En

9 / 25

Coherent states

Glauber coherent states:

a|α〉 = α|α〉 or |α〉 = D(α)|0〉, D(α) = eαa†−α∗a

⇒ |α〉 =1

N (α)

∞∑n=0

αn

√n!|n〉, α ∈ C

⇒ Very close to classical objects

Nonlinear coherent states:

(a, a†)⇒ (A,A†) :

{A = af (n) = f (n + 1)aA† = f (n)a† = a†f (n + 1)

A|α, f 〉 = α|α, f 〉 ⇒ |α, f 〉 =1

N (α, f )

∞∑n=0

αn

√n!f (n)!

|n〉

f (n) can be associated with the eigenvalues of a Hamiltonian

H ∼ A†A = f (n)a†af (n) = f 2(n)a†a ∼ f 2(n)n

= En

9 / 25

Coherent states

Glauber coherent states:

a|α〉 = α|α〉 or |α〉 = D(α)|0〉, D(α) = eαa†−α∗a

⇒ |α〉 =1

N (α)

∞∑n=0

αn

√n!|n〉, α ∈ C

⇒ Very close to classical objects

Nonlinear coherent states:

(a, a†)⇒ (A,A†) :

{A = af (n) = f (n + 1)aA† = f (n)a† = a†f (n + 1)

A|α, f 〉 = α|α, f 〉 ⇒ |α, f 〉 =1

N (α, f )

∞∑n=0

αn

√n!f (n)!

|n〉

f (n) can be associated with the eigenvalues of a Hamiltonian

H ∼ A†A = f (n)a†af (n) = f 2(n)a†a ∼ f 2(n)n = En

9 / 25

1D perturbative noncommutative harmonic oscillator

H =P2

2m+

mω2

2X 2 − ~ω

(1

2+τ

4

),

defined on the noncommutative space

[X ,P] = i~(1 + τP2

), X = (1 + τp2)x , P = p

Reality of spectrum, h = ηHη−1, with η = (1 + τp2)−1/2

h =p2

2m+

mω2x2

2+ωτ

4~(x2p2 + p2x2 + 2xp2x)− ~ω

(1

2+τ

4

)+O(τ2)

Eigenvalues and eigenfunctions:

En = ~ωen = ~ωn[1 +

τ

2(1 + n)

]+O(τ2)

|φn〉 = |n〉 − τ

16

√(n − 3)4 |n − 4〉+

τ

16

√(n + 1)4 |n + 4〉+O(τ2)

Pochhammer function (x)n := Γ(x + n)/Γ(x)

10 / 25

Nonlinear coherent states as superposition of Fock states:

|α, f 〉 =1

N (α, f )

∞∑n=0

C(α, n)√n!f (n)!

|n〉

C(α, n) =

{αn − τ

16αn+4 f (n)!

f (n+4)! , 0 ≤ n ≤ 3

αn − τ16α

n+4 f (n)!f (n+4)! + τ

16αn−4 n!

(n−4)!f (n)!

f (n−4)! , n ≥ 4

Uncertainties of X = (A + A†)/√

2, Y = i(A† − A)/√

2:

(∆X )2 = R + τ

(1

4+|α|2

2

), (∆Y )2 = R − τ

(1

4+|α|2

2

)R =

1

2

∣∣〈α, f |[X ,Y ]|α, f 〉η∣∣ =

1

4

[2 + τ − τ(α− α∗)2

]Generalised Uncertainty Relation:

∆X∆Y ≥ 1

2

∣∣〈α, f |[X ,Y ]|α, f 〉η∣∣

∗ ∆X∆Y = R, with Y being squeezed ⇒ ideal squeezed state11 / 25

Photon number squeezing

Number squeezing ⇒ photon number distribution is narrowerthan the average number of photons, (∆n)2 < 〈n〉Mandel parameter:

Q =(∆N)2

〈N〉− 1 = −τ |α|

2

2

In the limit τ = 0, Q = 0 (ordinary harmonic oscillator)

Coherent states for NCHO⇓

Quadrature squeezed+

Photon number squeezed⇓

Nonclassical

12 / 25

Quantum beam splitter

Input: X → a, Y → b,Output: W : c → BaB†, Z : d → BbB†, [c , c†] = [d , d†] = 1

B = eθ2

(a†be iφ−ab†e−iφ) ⇐ Beam splitter operator

Output states are entangled, when at least one of the input statesis nonclassical

13 / 25

Entanglement measureFock state |n〉 at input X and vacuum state |0〉 at input Y :

B|n〉X |0〉Y =n∑

q=0

(nq

)1/2

tqrn−q |q〉W |n − q〉Z

Noncommutative coherent states at input X and vacuum at Y :

|out〉 = B|α, f 〉X |0〉Y =1

N (α, f )

∞∑n=0

C(α, n)√n!f (n)!

B|n〉X |0〉Y

=1

N (α, f )

∞∑q=0

∞−q∑m=0

C(α,m + q)√m!q!f (m + q)!

tqrm |q〉W |m〉Z

Partial trace: ρA =

1

N 2(α, f )

∞∑q=0

∞∑s=0

∞−max(q,s)∑m=0

C(α, ζ,m + q)C∗(α, ζ,m + s)

m!√q!s!f (m + q)!f (m + s)!

tqts |r |2m |q〉〈s|

14 / 25

Linear entropy

S = 1− Tr(ρ2A)

= 1− 1

N 4(α, f )

∞∑q=0

∞∑s=0

∞−max(q,s)∑m=0

∞−max(q,s)∑n=0

|t|2(q+s)|r |2(m+n)

× C(α,m + q)C∗(α,m + s)C(α, n + s)C∗(α, n + q)

q!s!m!n!f (m + q)!f (m + s)!f (n + s)!f (n + q)!

τ = 2.0 (a)

τ = 0.6

τ = 1.5

τ = 1.0

0.5 1.0 1.5α

0.1

0.2

0.3

0.4

Entropy S

15 / 25

Schrodinger cat states

|α, f 〉± =1

N (α, f )±

(|α, f 〉 ± | − α, f 〉

)with

N 2(α, f )± = 2± 2

N 2(α, f )

∞∑k=0

(−1)k |α|2k

n!f 2(n)!

Uncertainties:

(∆X )2± = R±+U±, (∆Y )2

± = R±− U± R± ⇒ RHS of GUR

with U+ =

α2 + α∗2

2+|α|2 tanh(|α|2)+

τ

4

[1−(α2−α∗2)2+2|α|2 tanh(|α|2)− 4|α|4

cosh2(|α|2)

]Quadrature Y is squeezed for even and odd cat states!!

16 / 25

Quadrature squeezing and photon distribution function

0 2 4 6 8 1 0 1 2 1 40 . 0

0 . 1

0 . 2

0 . 3

0 . 4 C o h e r e n t E v e n c a t

P(n)

n

( a )

8 0 9 0 1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 00 . 0

5 . 0 x 1 0 4 7

1 . 0 x 1 0 4 8

1 . 5 x 1 0 4 8

2 . 0 x 1 0 4 8

2 . 5 x 1 0 4 8

3 . 0 x 1 0 4 8

C o h e r e n t O d d c a t

P(n)

n

( b )

17 / 25

Photon number squeezing

18 / 25

Entanglement

NCHO

HO (a)

0.5 1.0 1.5α

0.1

0.2

0.3

0.4

0.5

0.6

Entropy (S+ )

NCHO

HO

(a)

0.5 1.0 1.5α

0.2

0.4

0.6

Entropy (S- )

19 / 25

Coherent states

Classicality Nonclassicality

Ordinary HO X ×Noncommutative HO × X

Even cat states

Quadrature squeezing Number squeezing

Ordinary × ×Noncommutative X X

Odd cat states

Quadrature squeezing Number squeezing

Ordinary × XNoncommutative × X

Order of squeezing and/or nonclassicality is/are higher for NCHO

20 / 25

Squeezed states

|α, ζ〉 = D(α)S(ζ)|0〉, D(α) = eαa†−α∗a, S(ζ) = e

12

(ζa†a†−ζ∗aa)

21 / 25

Alternative definition of squeezed states:

(A + ζA†)|α, f , ζ〉 = α|α, f , ζ〉, α, ζ ∈ C, |ζ| < 1

Consider:

|α, f , ζ〉 =1

N (α, f , ζ)

∞∑n=0

I(α, ζ, n)√n!f (n)!

|n〉

Eigenvalue equation definition yields

I(α, ζ, n + 1) = α I(α, ζ, n)− ζ nf 2(n) I(α, ζ, n − 1)

Special case: f (n) = 1 ⇒ squeezed states of ordinary HO:

|α, ζ〉ho =1

N (α, ζ)

∞∑n=0

1√n!

(ζ2

)n/2Hn(

α√2ζ

)|n〉

22 / 25

Noncommutative squeezed states

|α, ζ〉 =1

N (α, ζ)

∞∑n=0

I(α, ζ, n)√n!f (n)!

|φn〉

=1

N (α, ζ)

∞∑n=0

S(α, ζ, n)√n!f (n)!

|n〉,

where S(α, ζ, n) ={I(α, ζ, n)− τ

16f (n)!

f (n+4)!I(α, ζ, n + 4), 0 ≤ n ≤ 3

I(α, ζ, n)− τ16

f (n)!f (n+4)!I(α, ζ, n + 4) + τ

16n!

(n−4)!f (n)!

f (n−4)!I(α, ζ, n − 4), n ≥ 4

and

I(α, ζ, n) = in (ζB)n/2

(1 +

A

B

)(n)

2F1

[− n,

1

2+

A

2B+

iα

2√ζB

; 1 +A

B; 2

]

23 / 25

Entangled noncommutative squeezed states

NCHO

HO

(a)

1 2 3 4α

0.1

0.2

0.3

0.4

0.5

0.6

Entropy S

NCHO

HO

(b)

1 2 3 4α

0.05

0.10

0.15

Entropy S

24 / 25

Conclusions

Coherent states in noncommutative spaces are nonclassical

Noncommutative cat states are found to be more nonclassicalthan the ordinary case

Noncommutative squeezed states are more entangled than theHO squeezed states

Thank you for your attention

25 / 25