Non-Hermitian noncommutative models in quantum optics and their superiorities

Non-Hermitian noncommutative models inquantum optics and their superiorities

Sanjib Dey

University of Montreal, Canada

PHHQP-XV, University of Palermo, Italy, 18-23 May, 2015

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

S. Dey, V. Hussin; arXiv: 1505.04801, to appear in PRD

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Noncommutative spaces

• Flat noncommutative space

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

Nonvanishing θµν breaks Lorentz-Poincare symmetry

• Snyder’s Lorentz covariant version

[xµ, xν ] = iθ (xµpν − xνpµ)

[xµ, pν ] = i~ (δµν + θpµpν)

[pµ, pν ] = 0

However, Poincare symmetry is still violated

• Poincare symmetries were deformed to make the algebracompatible with Snyder’s version [R. Banerjee, S. Kulkarni, S.Samanta; JHEP 2006, 077 (2006)].

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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.

q-deformed Fock space representation (1D):

|n〉q =

(A†)n√

[n]q!|0〉q, q〈0|0〉q = 1, A|0〉q = 0,

A†|n〉q =√

[n + 1]q |n + 1〉q, A|n〉q =√

[n]q |n − 1〉q

⇒ [n]q :=1− q2n

1− q2, where [n]q! =

n∏k=1

[k]q .

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Physical reality

Deformed oscillator algebra in 1D

AA† − q2A†A = 1, q ≤ 1

Consider X = α(A†+A

)and P = iβ

(A†−A

), α, β ∈ R,

Hermitian representation:

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~

PT : x → −x , p → p, i → −i

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Physical consequencesDeformed 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

[X ,P] = i~(1 + τP2

)Generalised uncertainty relation:

∆A∆B ≥ 1

2

∣∣∣ 〈[A,B]〉∣∣∣

• Standard case: [A,B] = Constant; give up knowledge aboutB, for ∆A = 0

• Noncommutative case: [A,B] ≈ B2; give up knowledge alsoabout B, for ∆A 6= 0

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Minimal lengths, areas and volumes

• In 1D, [X ,P] = i~(1 + τP2

):

∆X∆P ≥ ~2

[1 + τ (∆P)2 + τ〈P〉2

]⇒ minimal length

∆Xmin = ~√τ√

1 + τ〈P2〉,

from minimizing with (∆A)2 = 〈A2〉 − 〈A〉2[B.Bagchi, A. Fring; Phys. Lett. A 373, 4307–4310 (2009)]

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

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Nonlinear coherent states

|α, φ〉 =1

N (α)

∞∑n=0

αn

√ρn|φn〉, α ∈ C

with

h|φn〉 = ~ωen|φn〉, ρn =n∏

k=1

ek and N 2(α) =∞∑k=0

|α|2k

ρk

q-deformed nonlinear coherent states:

en = [n]q =1− q2n

1− q2, ρn = [n]q!, |φn〉 = |n〉q .

Uncertainties of X = (A + A†)/2, Y = (A− A†)/2i :

(∆X )2 = (∆Y )2 =1

2

∣∣∣ q〈α, φ|[X ,Y ]|α, φ〉q∣∣∣ =

1

4

[1 + (q2 − 1)|α|2

]∗ Generalised uncertainty relation is saturated.∗ Uncertainties of quadratures X and Y are identical.∗ coherent states produce equal optical noise as vacuum states.

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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 = (q2 − 1)|α|2

• In the limit q = 1 (ordinary harmonic oscillator), Q = 0

• In |q| < 1 (deformed harmonic oscillator), Q < 0

• Number squeezing is a strong evidence of nonclassicality.

q-deformed nonlinear coherent states show classical like behaviour,as well as nonclassicality !!

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Schrodinger cat states

|α, φ〉± =1

N (α)±

(|α, φ〉 ± | − α, φ〉

)with

N 2(α)± = 2± 2

N 2(α)

∞∑k=0

(−1)k |α|2k

[k]q!

Uncertainties:

(∆X )2± = Gq +

1

4

(α2 + α∗2 + 2|α|2F±

); F± :=

1∓ Eq(−2|α|2)

1± Eq(−2|α|2)

(∆Y )2± = Gq −

1

4

(α2 + α∗2 − 2|α|2F±

); Eq(|α|2) :=

∞∑k=0

|α|k

[k]q!

and

1

4

∣∣∣ q,±〈α, φ|[X ,Y ]|α, φ〉q,±∣∣∣2 =

1

16

[1 + (q2 − 1)|α|2F±

]2= G 2

q

Quadrature Y is squeezed for even cat states!!

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Quadrature squeezing

0.0 0.4 0.8 1.2 1.6 2.00.00

0.06

0.12

0.18

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(Y)2

| |

q = 0.9 q = 0.7 q = 0.1 q = -1.5 q = -1.9 q = -1.1

(a)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

(X)2

| |

q = 0.9 q = 0.7 q = 0.1 q = -1.5 q = -1.9 q = -1.1

(b)

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.00.00

0.04

0.08

0.12

(Y)2

q

• Even cat ⇒ quadraturesqueezing

• q adds an extra degree offreedom in squeezing

• Odd cat ⇒ no squeezing

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Number squeezing

Mandel parameter:

Q± =(∆n)2

〈n〉− 1 =

1

F±− 1 + (q2 − F±)|α|2

0.4 0.8 1.2 1.6 2.0 2.4 2.8

-1

0

1

2

3

Q+

| |

q = 0.9 q = 0.8 q = 0.1 Undeformed

Case (q = 1)

(a)

0.4 0.8 1.2 1.6 2.0 2.4 2.8-6

-5

-4

-3

-2

-1

0

Q-

| |

q = 0.9 q = 0.8 q = 0.5 Undeformed

case (q = 1)

(b)

Even cat states Odd cat states

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Photon distribution

Photon distribution function:

Pn,± =∣∣∣〈n|α, φ〉±∣∣∣2 =

∣∣∣ 1

N (α)N (α)±

( αn√[n]q!

± (−1)nαn√[n]q!

)∣∣∣2

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300.00

0.04

0.08

0.12

0.16

Coherent Even cat

P(n)

n

(a)

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300.00

0.06

0.12

0.18

0.24

0.30

Coherent Even cat

P(n)

n

(b)

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Coherent states

Classicality Nonclassicality

Ordinary HO X ×Noncommutative HO X X

Even cat states

Quadrature squeezing Number squeezing

Ordinary X ×Noncommutative X X

Odd cat states

Quadrature squeezing Number squeezing

Ordinary × XNoncommutative × X

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

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Squeezed states

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

12

(ζa†a†−ζ∗aa)

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• Alternative definition of squeezed states:

(a + ζa†)|α, ζ〉 = α|α, ζ〉, α, ζ ∈ C

• Generalisation is done by replacing a, a† by A,A†:

A|n〉 =√k(n)|n − 1〉, A†|n〉 =

√k(n + 1)|n + 1〉

Alternative approach of generalising ladder operators:

A†f = f (n)a†, Af = af (n)

Two approaches are equivalent for k(n) = nf 2(n)

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• Now consider:

|α, ζ〉 =1

N (α, ζ)

∞∑n=0

I(α, ζ, n)√ρn

|φn〉

Eigenvalue equation definition yields

I(α, ζ, n + 1) = α I(α, ζ, n)− ζ k(n) I(α, ζ, n − 1)

• Special case: k(n) = n ⇒ squeezed states of ordinary HO:

|α, ζ〉ho =1

N (α, ζ)

∞∑n=0

1√n!

(ζ2

)n/2Hn(

α√2ζ

)|n〉

• Special case: ζ = 0 ⇒ coherent states:

|α〉 =1

N (α)

∞∑n=0

αn

√ρn|φn〉

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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 = ~ω(An + Bn2

)+O(τ2), A = 1 +

τ

2,B =

τ

2

|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)

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Noncommutative squeezed states

|α, ζ〉 =1

N (α, ζ)

∞∑n=0

I(α, ζ, n)√ρn

|φn〉

=1

N (α, ζ)

∞∑n=0

S(α, ζ, n)√ρn

|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

]

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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

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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 squeezed states at input X and vacuum at Y :

|out〉 = B|α, ζ〉X |0〉Y =1

N (α, ζ)

∞∑n=0

S(α, ζ, n)√k(n)!

B|n〉X |0〉Y

=1

N (α, ζ)

∞∑q=0

∞−q∑m=0

S(α, ζ,m + q)√m!q!f (m + q)!

tqrm |q〉W |m〉Z

Partial trace: ρA =

1

N 2(α, ζ)

∞∑q=0

∞∑s=0

∞−max(q,s)∑m=0

S(α, ζ,m + q)S∗(α, ζ,m + s)

m!√q!s!f (m + q)!f (m + s)!

tqts |r |2m |q〉〈s|

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Linear entropy

S = 1− Tr(ρ2A)

= 1− 1

N 4(α, ζ)

∞∑q=0

∞∑s=0

∞−max(q,s)∑m=0

∞−max(q,s)∑n=0

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

× S(α, ζ,m + q)S∗(α, ζ,m + s)S(α, ζ, n + s)S∗(α, ζ, n + q)

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

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

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Entangled noncommutative squeezed states

0.2

0.3

0.4

0.5

0.6

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Entangled noncommutative coherent states

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

0

0.2

0.4

0.6

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Conclusions• Squeezed states of a perturbative NCHO have been

constructed.

• Coherent states in noncommutative spaces are dual in nature.

• 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

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