non-hermitian noncommutative models in quantum optics and their superiorities
TRANSCRIPT
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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