quantum optical models in noncommutative spaces
TRANSCRIPT
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