solvable models on noncommutative spaces with minimal length uncertainty relations
TRANSCRIPT
Solvable models on noncommutative spaces withminimal length uncertainty relations
Sanjib Dey
Based on:
J. Phys. A: Math. Theor. 45, 385302(2012), Phys. Rev. D 86, 064038(2012),
J. Phys. A: Math. Theor. 46, 335304(2013), Phys. Rev. D 87, 084033(2013),
Acta Polytechnica 53, 268− 270, (2013), Phys. Rev. A 88, 022116(2013),
Ann. Phys. 346, 28− 41(2014)1 / 47
Motivations
Drawback of QFT
Gravitational field is not renormalisable with ordinary QFT⇓
Quantum gravity remains open problem in 21st century
Noncommutative QFT play the role
Minimal observable length can be introduced as a cut-off torenormalise
⇓Minimal length is a natural prediction of the noncommutativetheory (Heisenberg, Snyder [1947] and Yang [1947]
⇓Did not become so popular at that time
Rebirth
String theory can be realised as field theory on noncommutativespace (Seiberg and Witten [1999])
2 / 47
Noncommutative spaces
• Flat (abelian) 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)].
3 / 47
q-deformed noncommutative spaces
• Deformation on Heisenberg’s canonical commutation relations
PX − qXP = i~
• Deformation on commutation relation between creation andannihilation operator
AA† − q2A†A = 1
Why deformation?
X = α(A† + A
), P = iβ
(A† − A
), α, β ∈ R
Constraints
α =~
2β, q = e2τβ2
, τ ∈ R+
Non-trivial limit β → 0
[X ,P] = i~(1 + τP2
).
4 / 47
Minimal lengthUncertainty 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
For the case:
[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〉25 / 47
Flat noncommutative spaces in 3D
[x0, y0] = iθ1, [x0, z0] = iθ2, [y0, z0] = iθ3,[x0, px0 ] = i~, [y0, py0 ] = i~, [z0, pz0 ] = i~ for θ1, θ2, θ3 ∈ R
Linear Ansatz:
ϕi =3∑
j=1
κij aj + λija†j , for ~ϕ = {x0, y0, z0, px0 , py0 , pz0}
⇒ 72 free parameters
Utilize PT -symmetry to reduce number ofparameters
6 / 47
PT -symmetric flat noncommutative spaces
PT ± : x0 → ±x0, y0 → ∓y0, z0 → ±z0, i → −i ,px0 → ∓px0 , py0 → ±py0 , pz0 → ∓pz0 , θ2 = 0
PT θ± : x0 → ±x0, y0 → ∓y0, z0 → ±z0, i → −i ,px0 → ∓px0 , py0 → ±py0 , pz0 → ∓pz0 , θ2 → −θ2
PT xz : x0 → z0, y0 → y0, z0 → x0, i → −i ,px0 → −pz0 , py0 → −py0 , pz0 → −px0
PT ±-symmetric Ansatz
a1 = α1x0 + iα2y0 + α3z0 + iα4px0 + α5py0 + iα6pz0 ,
a2 = α7x0 + iα8y0 + α9z0 + iα10px0 + α11py0 + iα12pz0 ,
a3 = α13x0 + iα14y0 + α15z0 + iα16px0 + α17py0 + iα18pz0
7 / 47
Solution:
x0 =α9α17 − α11α15
2 detM1a+
1 +α5α15 − α3α17
2 detM1a+
2 +α3α11 − α5α9
2 detM1a+
3 ,
y0 =α10α18 − α12α16
2i detM2a−1 +
α6α16 − α4α18
2i detM2a−2 +
α4α12 − α6α10
2i detM2a−3 ,
z0 =α11α13 − α7α17
2 detM1a+
1 +α1α17 − α5α13
2 detM1a+
2 +α5α7 − α1α11
2 detM1a+
3 ,
px0 =α12α14 − α8α18
2i detM2a−1 +
α2α18 − α6α14
2i detM2a−2 +
α6α8 − α2α12
2i detM2a−3 ,
py0 =α7α15 − α9α13
2 detM1a+
1 +α3α13 − α1α15
2 detM1a+
2 +α1α9 − α3α7
2 detM1a+
3 ,
pz0 =α8α16 − α10α14
2i detM2a−1 +
α4α14 − α2α16
2i detM2a−2 +
α2α10 − α4α8
2i detM2a−3
with
a±i = ai ± a†i ,
(Ml)jk = 6j + 2k + l − 8 for l = 1, 2.
8 / 47
q-deformed noncommutative spaces
Deformed oscillator algebras
AiA†j − q2δijA†jAi = δij ,
[A†i ,A
†j
]= [Ai ,Aj ] = 0, i , j = 1, 2, 3; q ∈ R
The limit q → 1 gives standard Fock space Ai → ai :[ai , a
†j
]= δij , [ai , aj ] =
[a†i , a
†j
]= 0, i , j = 1, 2, 3
q-deformed Fock space representation (1D):
|n〉q =
(A†)n√
[n]q!|0〉, 〈0|0〉 = 1, A|0〉 = 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 .
9 / 47
Noncommutative spaces from oscillator algebras
PT ±-symmetric Ansatz:
X = κ1(A†1 + A1) + κ2(A†2 + A2) + κ3(A†3 + A3),
Y = i κ4(A†1 − A1) + i κ5(A†2 − A2) + i κ6(A†3 − A3),
Z = κ7(A†1 + A1) + κ8(A†2 + A2) + κ9(A†3 + A3),
Px = i κ10(A†1 − A1) + i κ11(A†2 − A2) + i κ12(A†3 − A3),
Py = κ13(A†1 + A1) + κ14(A†2 + A2) + κ15(A†3 + A3),
Pz = i κ16(A†1 − A1) + i κ17(A†2 − A2) + i κ18(A†3 − A3),
with κi = κi√
~/mω for i = 1, ...., 9κi = κi
√~mω for i = 10, ...., 18
Note: X ,Y ,Z ,PX ,PY ,PZ are non-Hermitian in usual space
10 / 47
Compute non-Hermitian commutators:
[X ,Y ] = 2i∑3
j=1κj κ3+j
[1 +
(q2 − 1
)]A†jAj ,
[Y ,Z ] = −2i∑3
j=1κ3+j κ6+j
[1 +
(q2 − 1
)]A†jAj ,
[X ,Px ] = 2i∑3
j=1κj κ9+j
[1 +
(q2 − 1
)]A†jAj ,
[Y ,Py ] = −2i∑3
j=1κ3+j κ12+j
[1 +
(q2 − 1
)]A†jAj ,
[Z ,Pz ] = 2i∑3
j=1κ6+j κ15+j
[1 +
(q2 − 1
)]A†jAj ,
[Px ,Py ] = −2i∑3
j=1κ9+j κ12+j
[1 +
(q2 − 1
)]A†jAj ,
[Py ,Pz ] = 2i∑3
j=1κ12+j κ15+j
[1 +
(q2 − 1
)]A†jAj ,
[X ,Pz ] = 2i∑3
j=1κj κ15+j
[1 +
(q2 − 1
)]A†jAj ,
[Z ,Px ] = 2i∑3
j=1κ6+j κ9+j
[1 +
(q2 − 1
)]A†jAj ,
[X ,Z ] = [Px ,Pz ] = [X ,Py ] = [Y ,Px ] = [Y ,Pz ] = [Z ,Py ] = 011 / 47
A particular PT ±-symmetric solution
κ1 = κ4 = κ5 = κ8 = κ10 = κ12 = κ13 = κ14 = κ17 = κ18 = 0
[X ,Y ] =2iθ1
1 + q2+ i
θ1
~L
(mω
2κ26
Y 2 +2κ2
6
mωP2y
),
[Y ,Z ] =2iθ3
1 + q2+ i
θ3
~L
(mω
2κ26
Y 2 +2κ2
6
mωP2y
),
[X ,Px ] =2i~
1 + q2+ i2mωL
(κ2
11X2 +
P2x /4
m2ω2κ211
+θ2
1κ211P
2y
~2+κ2
11XPy
~/(2θ1)
),
[Y ,Py ] =2i~
1 + q2+ i2mωL
(1
4κ26
Y 2 +κ2
6
m2ω2P2y
),
[Z ,Pz ] =2i~
1 + q2+ i2mωL
(Z 2
4κ27
+κ2
7
m2ω2P2z +
θ23
4~2κ27
P2y −
θ3
2~2κ27
ZPy
),
where L = (q2 − 1)/(q2 + 1), with constraints
κ2 =~
2κ11, κ3 =
θ1
2κ6, κ9 = − θ3
2κ6, κ15 = − ~
2κ6, κ16 =
~2κ7
.12 / 47
Reduced three dimensional solution for q → 1
• Set κ11 = κ6, κ7 = 1/2κ6, q = Exp(2τκ26), then κ6 → 0:
[X ,Y ] = iθ1
(1 + τY 2
), [Y ,Z ] = iθ3
(1 + τY 2
),
[X ,Px ] = i~(1 + τP2
x
), [Y ,Py ] = i~
(1 + τY 2
),
[Z ,Pz ] = i~(1 + τP2
z
),
where τ = τmω/~, τ = τ/(mω~)
• Representation in flat noncommutative spaces:
X = (1 + τp2x0
)x0 + θ1~
(τp2
x0− τy2
0
)py0 , Px = px0 ,
Z = (1 + τp2z0
)z0 + θ3~(τy2
0 − τp2z0
)py0 , Pz = pz0 ,
Py = (1 + τy20 )py0 , Y = y0.
• Bopp shift to standard canonical variables:x0 → xs − θ1
~ pys , y0 → ys , z0 → zs + θ3~ pys ,
px0 → pxs , py0 → pys , pz0 → pzs
13 / 47
• Dyson map: η = ηy0ηPx0ηPz0
ηy0 =(1 + τy2
0
)−1/2, ηPx0
=(1 + τp2
x0
)−1/2, ηPz0
=(1 + τp2
z0
)−1/2
• Hermitian variables:
x := ηXη−1 = η−1Px0
(x0 +
θ1
~
)η−1Px0− θ1
~η−1y0
Py0η−1y0
= x†
y := ηY η−1 = y0 = y †
z := ηZη−1 = η−1Pz0
(z0 −
θ3
~
)η−1Pz0
+θ3
~η−1y0
Py0η−1y0
= z†
px := ηPxη−1 = px0 = p†x
py := ηPyη−1 = η−1
y0py0η
−1y0
= p†y
pz := ηPzη−1 = pz0 = p†z
• Isospectral Hermitian counterpart:
H (X ,Y ,Z ,Px ,Py ,Pz) 6= H† (X ,Y ,Z ,Px ,Py ,Pz)⇒ h = ηHη−1 = h†
• Metric: ρ = η2
14 / 47
Harmonic oscillators on a noncommutative space1D harmonic oscillator:
H =P2
2m+
mω2
2X 2 = H1D
ho +mω2
2
(τp2
s x2s + τxsp
2s xs + τ2p2
s xsp2s xs)
⇒ Non-Hermitian but PT ±-symmetricSolved exactly in p-space, i.e. xs → i~∂ps :
ψ2n−i (z) = c1
2n−i∑k=i
zk(z2 − 1)ν−2n−1+i
2
k+i−22∏
l=i
2(n − l)(2n + 2ν − 2l + 1)
k!(−1)1+i (1− z2)ν
,
En = ω~(
1
2+ n
)√1 +
τ2
4+ τ
ω~4
(1 + 2n + 2n2) for n ∈ N0
ψ2n−i (z) vanishes as |z | → ∞, if ν > −1, guaranteed for τmω > 0ψ2n−i (z)→ square integrable and forms orthonormal basis.
η =(1 + τP2
)−1/2
15 / 47
Exact treatment is always difficultPerturbation theory:
E(p)n = E
(0)n + E
(1)n + E
(2)n +O(τ3) with
E(0)n = ω~
(n +
1
2
),
E(1)n =
τω~4
(1 + 2n + 2n2
)+τ2ω~
16
(3 + 8n + 6n2 + 4n3
),
E(2)n = −1
8τ2ω~
(1 + 3n + 3n2 + 2n3
)Upto order τ3, the energy matches exactly with the exact case
Higher dimensional models have also been treated similarly, forinstance in 2D and in 3D
16 / 47
Minimal lengths, areas and volumes
Uncertainty relation:
∆A∆B ≥ 1
2|〈[A,B]〉ρ|
For special solution with q → 1:
∆Xmin = |θ1|√τ + τ2 〈Y 〉2ρ, ∆Ymin = 0, ∆Zmin = |θ3|
√τ + τ2 〈Y 〉2ρ,
∆Xmin = ~√τ + τ τ 〈Y 〉2ρ, ∆Ymin = 0, ∆Zmin = ~
√τ + τ τ 〈Y 〉2ρ,
∆ (Px)min = 0, ∆ (Py )min = ~√τ + τ2 〈Y 〉2ρ, ∆ (Pz)min = 0.
17 / 47
Before taking the limit:
∆Ymin = |κ6|
√1
2(q2 − q−2) + (q − q−1)2
(1
4κ26
〈Y 〉2ρ +κ2
6
~2
⟨P2y
⟩ρ
),
Similar expressions can be found for X and Z .Absolute minimal lengths:
∆X0 =1
2√
2
∣∣∣∣ θ1
κ6
∣∣∣∣Q, ∆Y0 =|κ6|√
2Q, ∆Z0 =
1
2√
2
∣∣∣∣ θ3
κ6
∣∣∣∣Q,where Q :=
√q2 − q−2
Absolute minimal uncertainty volume:
∆V0 =1√2
∣∣∣∣θ1θ3
κ6
∣∣∣∣ (q2 − q−2)3/2
.
-Similarly for the momenta
18 / 47
Different types of representations
[X ,P] = i~(1 + τP2
)non-Hermitian: X(1) = (1 + τp2)x , P(1) = p,
Hermitian: X(2) = (1 + τp2)1/2x(1 + τp2)1/2, P(2) = p,
Hermitian: X(3) = x , P(3) =1√τ
tan(√
τp),
non-Hermitian: X(4) = ix(1 + τp2)1/2, P(4) = −ip(1 + τp2)−1/2,
Incorrect: X(4′) = x(1 + τp2)1/2, P(4) = p(1 + τp2)−1/2
[P. G. Castro, R. Kullock and F. Toppan:J. Math. Phys. 52, 062105 (2011)]
How are these representations related?
19 / 47
Construction of solvable non-Hermitian potentials
H(p)ψ(p) = Eψ(p) ⇔ −f (p)ψ′′(p)+g(p)ψ′(p)+h(p)ψ(p) = Eψ(p)
Transformation:
ψ(p) = eχ(p)φ(p), χ(p) =
∫f ′(p) + 2g(p)
4f (p)dp, q =
∫ √f (p)dp,
H(q)ψ(q) = Eψ(q) ⇔ −φ′′(q) + V (q)φ(q) = Eφ(q),
so that the potential, V (q) =4g2 + 3 (f ′)2 + 8gf ′
16f− f ′′
4− g ′
2+ h
∣∣∣∣∣q
.
Factorize the wave function further:
φ(q) = v(q)F [w(q)],
where F [w(q)] is introduced as a special function20 / 47
The Ansatz converts the potential equation back into
F ′′(w) + Q(w)F ′(w) + R(w)F (w) = 0,
where, Q(w) :=2v ′
vw ′+
w ′′
(w ′)2and R(w) :=
E − V (q)
(w ′)2+
v ′′
v (w ′)2.
1st relation gives:
v(q) =(w ′)−1/2
exp
[1
2
∫ w(q)
Q(w)dw
],
Eliminating v(q) in the 2nd relation:
E−V (q) =w ′′′
2w ′−3
4
(w ′′
w ′
)2
+(w ′)2
R(w)−(w ′)2 Q ′(w)
2−(w ′)2 Q2(w)
4,
with known Q(w),R(w),F [w(q)]⇒ compute w(q)General formula for the metric:
ρ(p) = %(p)e−2Reχ(p) |v(p)|−2 dw
dp.
21 / 47
Nc harmonic oscillator in different representations
Representation 1:
H(1)(p) =p2
2m+
mω2
2
(x2 + τp2x2 + τxp2x + τ2p2xp2x
),
In momentum space (x → i~∂p)
f (p) =mω2~2
2(1 + τp2)2, g(p) = −τ~ωp(1 + τp2), h(p) =
p2
2m.
Transformations:
ψ(p) = φ(p), q =
√2
τω~arctan
(√τp),
Such that
V (q) =~ω2τ
tan2
(√τω~
2q
).
22 / 47
Assume F (w)⇒ associated Legendre polynomial Pµν (w)
Q(w) =2w
w2 − 1and R(w) =
ν(ν + 1)
1− w2− µ2
(1− w2)2
so that,
E − ~ω2τ
tan2
(√τω~
2q
)
=(w ′)2
(ν2 + ν + 1
1− w2+
w2 − µ2
(1− w2)2
)− 3 (w ′′)2
4 (w ′)2+
w ′′′
2w ′
Assume first term on R.H.S constant, (w ′)2 /(1− w2) = c ∈ R+
w(q) = sin(√cq), so that
ψn(p) =1√Nn
1
(1 + τp2)1/4Pµ−n−µ−
( √τp√
1 + τp2
),
En = ω~(
1
2+ n
)√1 +
τ2
4+τω~
4(1 + 2n + 2n2).
23 / 47
Representation 2:
As H(2) = ρ1/2H(1)ρ−1/2, ψn(2)
= ρ−1/2ψn(1), En(2)
= En(1)
Representation 3:
H(3)(p) = H(1)(q = p√
2/m/~ω), ∴ φ(q = p√
2/m/~ω)
Representation 4’:
E = ~ω/2τ − c/4(1 + 2ν)2 ⇐ Not bounded from below
Representation 4 makes it physical.For all representations:⟨
ψ(i)
∣∣F (P(i),X(i)
)ψ(i)
⟩ρ(i)
=1
N
∫ 1
−1F
[z√
τ(1− z2), i~√τ(1− z2)∂z
] ∣∣∣Pµ−m−µ− (z)∣∣∣2 dz
24 / 47
Klauder coherent states
|J, γ, φ〉 =1
N (J)
∞∑n=0
Jn/2e−iγen√ρn
|φn〉, J ∈ R+0 , γ ∈ R.
with
h|φn〉 = ~ωen|φn〉, ρn =n∏
k=1
ek and N 2(J) =∞∑k=0
Jk
ρk,
Basis properties:
• Continuous in J, γ
• Provide a resolution of the identity
• Temporarily stable
• Satisfy action identity
〈J, γ, φ|H|J, γ, φ〉η = 〈J, γ, φ|h|J, γ, φ〉 = ~ωJ
J. R. Klauder, Annals Phys. 237, 147− 160 (1995)
25 / 47
Generalised Heisenberg’s uncertainty relation
For a measurement of two observables A and B we have:
∆A∆B ≥ 1
2
∣∣∣〈J, γ,Φ| [A,B] |J, γ,Φ〉η∣∣∣
Uncertainties:
∆A = 〈J, γ,Φ|A2 |J, γ,Φ〉η − 〈J, γ,Φ|A |J, γ,Φ〉2η
Ehrenfest theorem
i~d
dt〈J, γ + tω,Φ|A |J, γ + tω,Φ〉η = 〈J, γ + tω,Φ| [A,H] |J, γ + tω,Φ〉η
Time evolution
e iHt/~|J, γ,Φ〉 = |J, γ + tω,Φ〉
26 / 47
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
First order perturbation theory
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)
ρn =1
2nτnn!
(2 +
2
τ
)n
, N 2(J) = eJ(
1− τJ − τ
4J2)
+O(τ2)
27 / 47
The K-states saturate the generalized uncertainty relation
∆X 2 = 〈J, γ,Φ|X 2 |J, γ,Φ〉η − 〈J, γ,Φ|X |J, γ,Φ〉2η
=~
2mω[1 + τ (1 + J(2− 2γ sin 2γ − cos 2γ))]
∆P2 = 〈J, γ, φ| p2 |J, γ, φ〉 − 〈J, γ, φ| p |J, γ, φ〉2
=~mω
2[1− τJ(cos 2γ − 2γ sin 2γ)]
Therefore
∆X∆P =~2
[1 +
τ
2
(1 + 4J sin2 γ
)]=
~2
(1 + τ 〈J, γ,Φ|P2 |J, γ,Φ〉
)⇒ Squeezed coherent states
Strong disagreement with
[S. Gosh, P. Roy; Phys. Lett. B711, 423 (2012)]
28 / 47
Ehrenfest theorem
i~d
dt〈J, γ + tω,Φ|X |J, γ + tω,Φ〉η = 〈J, γ + tω,Φ| [X ,H] |J, γ + tω,Φ〉η ,
= 〈J, γ + tω,Φ| i~m
(P + τP3) |J, γ + tω,Φ〉η
= −i~3/2
√2Jω
m
[sin γ + τ
[(J + 1)γ cos γ +
1
2sin γ(2 + J − 3J cos 2γ)
]]
i~d
dt〈J, γ + tω,Φ|P |J, γ + tω,Φ〉η = 〈J, γ + tω,Φ| [P,H] |J, γ + tω,Φ〉η ,
= 〈J, γ + tω,Φ| − im~ω2
(X +
τ
2XP2 +
τ
2P2X
)|J, γ + tω,Φ〉η
= −i√
2Jm~3/2ω3/2[cos γ +
τ
4[(3J + 2) cos γ − 4(J + 1)γ sin γ − 3J cos 3γ]
]29 / 47
1D non-perturbative noncommutative harmonic oscillator
Now consider deformed canonical commutation relations:
[X ,P] = i~ + iq2 − 1
q2 + 1
(mωX 2 +
1
mωP2
)Take
X = α(A† + A
)and P = iβ
(A† − A
)with α = 1
2
√1 + q2
√~
mω and β = 12
√1 + q2
√~mω
Non-perturbative NCHO
H = ~ω(A†A + 1
), En = ~ωen = ~ω[n]q, |φn〉 = |n〉q
H =2
(1 + q2)2
[mω2X 2 +
1
mP2 +
~ω8
(q4 − 2q2 − 3
)]
30 / 47
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~
Non-Hermitian representation:
A =1
1− q2Dq, and A† = (1− x)− x(1− q2)Dq
Jackson derivatives Dqf (x) := [f (x)− f (q2x)]/[x(1− q2)]
Generalised uncertainty relation for Hermitian representation:
∆X∆P||J,γ〉q ≥1
2
∣∣∣∣(q〈J, γ| [X ,P] |J, γ〉q)η
∣∣∣∣are shown to hold, but are saturated only for t = 0
Ehrenfest theorem are also shown to hold for both X and P31 / 47
Revival structureDynamics of classical systems with equidistant energy lavels:
Tcl =2π~∆En
Non-equidistant energy lavels: Trev ⇒ regains its original structure
Revival structure requires
• Large number of waves from highly excited discrete states
• Strong localisation of the wave packet
Localisation
Given: |ψ(t)〉 =∑n≥0
cne−iEnt/~|φn〉 with
∞∑n=0
|cn|2 = 1
Compare: |cn|2 ⇐⇒ 〈n〉ke−〈n〉k!
Deviation measured by: Mandel parameter Q = (∆n)2
〈n〉 − 1Q = 0⇒ Poissonian, Q < 0⇒ Sub, Q > 0⇒ Super
32 / 47
Considering the wave packet strongly weighted around 〈n〉:Expand energy eigenvalue in Taylor series:
En ' En + E ′n (n − n) +1
2!E ′′n (n − n)2 +
1
3!E ′′′n (n − n)3 + .... ,
so that
Tcl =2π~|E ′n|
, Trev =2π~× 2!
|E ′′n |Tsuprev =
2π~× 3!
|E ′′′n |.
Classical description
Evidence: Revival structure (non-equidistant energy lavels)Visualised by autocorrelation function:
A(t) = 〈ψ(0)|ψ(t)〉 =∞∑n=0
|cn|2e−iEnt/~ .
Numerically |A(t)|2 varies between 0 and 1
33 / 47
Fractional revival of perturbative NCHO
Klauder coherent states: |J, γ, φ〉 =∑∞
n=0 cn(J)e−iEnt/~|φn〉Weighting function: cn(J) = Jn/2
N (J)√ρn
〈n〉 = J−τ(J +
J2
2
)+O(τ2),
⟨n2⟩
= J+J2−τ(J + 3J2 + J3
)+O(τ2)
∆n2 =⟨n2⟩− 〈n〉2 = J − τ
(J + J2
)+O(τ2)
Mandel parameter:
Q :=∆n2
〈n〉− 1 = −Jτ
2+O(τ2) < 0,
⇒ Sub-Poissonian statistics (Strong localisation)
34 / 47
Weighting function
0.0 2.5 5.0 7.5 10.00.0
0.1
0.2
0.3
0.4
J = 1.5 J = 3
------- J = 4.5
(a)
|c(J
)|2
n 0 5 10 15 20 250.00
0.06
0.12
0.18
0.24 J = 3 J = 6
- - - - J = 15
(b)
|c(J
)|2
n
(a) τ = 0.1 with 〈n〉 = 1.24, 2.25, 3.04(b) τ = 0.01 with 〈n〉 = 2.93, 5.76, 13.72
35 / 47
Autocorrelation function
0 50 100 150 200 2500.00
0.25
0.50
0.75
1.00
J = 1.5(a)A(t)
t 0 500 1000 1500 2000 25000.00
0.25
0.50
0.75
1.00
Trev/3
Trev/4
Trev/8
Trev/6
J = 6(b)A(t)
t
Trev/2
(a) J = 1.5, τ = 0.1, ω = 0.5, ~ = 1, γ = 0,Tcl = 10.05,Trev = 251.32(b) J = 6, τ = 0.01, ω = 0.5, ~ = 1, γ = 0,Tcl = 11.74,Trev = 2513.27
Tcl =2π
ω− τ
ω(1 + 2J)π, and Trev =
4π
ωτ
36 / 47
Fractional super-revival of non-perturbative NCHO
En = ~ω[n]q, [n]q =1− q2n
1− q2⇒ dkEn
dnk6= 0 for k = 1, 2, 3..
⇒ Infinite many revival
Tcl =π
ω
∣∣∣∣ q2 − 1
q2n ln q
∣∣∣∣ , Trev =π
ω
∣∣∣∣ q2 − 1
q2n ln2 q
∣∣∣∣ , Tsuprev =3π
2ω
∣∣∣∣ q2 − 1
q2n ln3 q
∣∣∣∣
0 1 2 3 4 5 60.00
0.25
0.50
0.75
1.00
J = 6(a)A(t)
t 0 250 500 750 1000 12500.00
0.25
0.50
0.75
1.00
Trev/3
Trev/4
Trev/8
Trev/6
J = 6(b)A(t)
t
Trev/2
0 100000 200000 300000 4000000.00
0.25
0.50
0.75
1.00 Tsuprev/3Tsuprev/6
J = 6(c)A(t)
t
Tsuprev/2
q = e−0.005, J = 6, n = 6.1875(a) Tcl = 6.65 (b) Trev = 1330.19 (c) Tsuprev = 3999056
37 / 47
Yet another efficient method
Quality of coherent states
• Solve canonical equations of motion ⇒ Draw the dynamicsof classical particles
• Utilize Bohmian mechanics ⇒ Draw the dynamics ofcoherentt states
• Compare directly ⇒ Study quality of the coherent states
Bohmian mechanics (real case):
Time dependent Schrodinger equation :
i~∂ψ(x , t)
∂t= − ~2
2m
∂2ψ(x , t)
∂x2+ V (x)ψ(x , t)
WKB polar decomposition :
ψ(x , t) = R(x , t)ei~S(x ,t), R(x , t),S(x , t) ∈ R
Substitute ψ(x , t) in Schrodinger equation ⇒ separate real andimaginary part :
38 / 47
Real Bohmian
St +(Sx)2
2m+ V (x)− ~2
2m
Rxx
R= 0 ⇐ Quantum Hamilton-Jacobi equation
mRt + RxSx +1
2RSxx = 0 ⇐ Continuity equation
∗ Velocity :
mv(x , t) = Sx =~2i
[ψ∗ψx − ψψ∗x
ψ∗ψ
]∗ Quantum potential :
Q(x , t) = − ~2
2m
Rxx
R=
~2
4m
[(ψ∗ψ)2
x
2 (ψ∗ψ)2−
(ψ∗ψ)xxψ∗ψ
]∗ Effective potential Veff(x , t) = V (x) + Q(x , t).∗ Two options to compute quantum trajectories :
1. Solve ⇒ v(x , t)2. Solve ⇒ mx = −∂Veff/∂x
39 / 47
Bohmian mechanics (complex case)
∗ Decompose :
ψ(x , t) = ei~ S(x ,t), S(x , t) ∈ C
∗ Substitute ψ(x , t)⇒ time dependent Schrodinger equation :
St +(Sx)2
2m+ V (x)− i~
2mSxx = 0
∗ Velocity :
mv(x , t) = Sx =~i
ψx
ψ
∗ Quantum potential :
Q(x , t) = − i~2m
Sxx = − ~2
2m
[ψxx
ψ− ψ2
x
ψ2
]∗ Less explored in the literature
40 / 47
Application : Poschl-Teller model (real case)
φn(x) =1√Nn
cosλ( x
2a
)sinκ
( x
2a
)2F1
[−n, n + κ+ λ; k +
1
2; sin2
( x
2a
)]Stationary state Bohmian :
v(t) = 0 ⇐ Not the behaviour of a classical particle.Klauder coherent state :
ψJ(x , t) :=1
N (J)
∞∑n=0
Jn/2 exp(−iωten)√ρn
φn(x)
ρn = n!(n + κ+ λ)n, N 2(J) = 0F1 (1 + κ+ λ; J)Classical solution :
x(t) = a arccos
[α− β
2+√γ cos
(√2E
m
t
a
)], α, β, γ constant
41 / 47
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 02 . 0 0
2 . 0 1
2 . 0 2
2 . 0 3
2 . 0 4
2 . 0 5
x ( t )
t
( a )
0 5 10 15 20 252
3
4
5
6
(c)
J = 20 J = 10 J = 2 J = 20.2846
x(t)
t
(a) Classical trajectories (c) Trajectories of coherent states
• Qualitatively not identical !!• Let us look at the uncertainty• Let us look at the behaviour of |ψ(x , t)|2 with time too
42 / 47
0 5 1 0 1 5 2 0 2 50
1
2
3
4
5
6
7
Q = - 0 . 3 0 7 5 9 3 Q = - 0 . 1 4 9 5 2 3 Q = - 0 . 0 4 2 5 5 5
∆x ∆p
t
( a )
0 1 2 3 4 5 60.0
0.2
0.4
0.6
0.8
t = 0t = 1t = 10t = 20t = 30
|(x
,t)|2
x
(b)
• Not a squeezed coherent state, ∆x∆p ≫ }/2 !!• Shape of the wave packet changes with time, i.e. not a classical particle!!• Q = −0.307593,−0.149523,−0.042555⇐ Sub-Poissonian• Q(J, κ+ λ) = J
2+κ+λ0F1(3+κ+λ;J)
0F1(2+κ+λ;J) −J
1+κ+λ0F1(2+κ+λ;J)
0F1(1+κ+λ;J)• Adjust κ, λ and J, so that Q→ 0⇐ Poissonian
43 / 47
0 5 10 15 20 25
0.5100
0.5103
0.5106
0.5109
0.5112
x p
t
Q= -0.000054529 Q= -0.000013634 Q= -0.000002726
(a)
0.0 0.4 0.8 1.2
0.5000055
0.5000070
0 1 2 3 4 5 60
1
2
3
t = 0 , J = 0 . 0 0 2 2 9 0 6 t = 0 . 6 5 , J = 0 . 0 0 2 2 9 0 6 t = 0 , J = 2 t = 4 , J = 2|Ψ
(x,t)|2
x
( b )
Two sets: κ = 90, λ = 100, J = 2, 0.5, 0.1κ = 2, λ = 3, J = 2, 0.5, 0.1
0.0 0.2 0.4 0.6 0.8 1.02.00
2.01
2.02
2.03
2.04
2.05 (a)
J = 2.0 J = 0.5 J = 0.1
x(t)
t 0 5 10 15 20 25 302.00
2.01
2.02
2.03
2.04
2.05
2.06 (b)
J = 0.0022906 J = 0.00057265 J = 0.000114531
x(t)
t44 / 47
Classical and Klauder state in complex plane
- 0.9
- 0.9 - 0.9
- 0.9 - 0.9
- 0.9
- 0.9
- 0.9
- 0.9
- 0.9
- 0.8 - 0.8
- 0.8 - 0.8- 0.8
- 0.8
- 0.8- 0.8
- 0.8- 0.8
- 0.7
- 0.7
- 0.6
- 0.6
- 0.5
- 0.5
- 0.4
- 0.4
- 0.3- 0.3- 0.3 - 0.3 - 0.3
- 0.3- 0.2- 0.2
- 0.2- 0.2
- 0.2 - 0.1- 0.1- 0.1- 0.1 - 0.1
xi
xr
(a)
- 20 - 15 - 10 - 5 0 5
- 3
- 2
- 1
0
1
2
3
- 0.8
- 0.8
- 0.8
- 0.8
- 0.8- 0.8
- 0.8
- 0.8
- 0.8
- 0.6- 0.6
- 0.6- 0.6
- 0.6
- 0.6
- 0.6
- 0.6
- 0.6
- 0.4- 0.4- 0.4
- 0.4 - 0.4
- 0.4
- 0.4- 0.4
- 0.4
- 0.2
- 0.2
- 0.2 - 0.2
- 0.2- 0.2
- 0.2 - 0.2
- 0.2
00
00
0
0
00 0
0
0
0
0
0
0.2
0.20.2
0.2
0.20.2
0.2 0.2
0.2
0.4
0.4
0.4
0.4
0.4 0.4
0.4
0.4
0.4
0.6
0.60.6
0.6
0.6
0.6
0.6
0.6
0.6
0.80.80.8
0.8
0.8
0.8
0.8
0.8
0.8
xr
xi (b)
- 20 - 15 - 10 - 5 0 5
- 3
- 2
- 1
0
1
2
3
Sub-Poissonian regime, Q < 0
45 / 47
- 2500
- 2500- 2500
- 2500- 2500
- 2500
- 2000
- 2000
- 2000
- 2000
- 2000
- 2000
- 1500
- 1500
- 1500
- 1500
- 1500
- 1500
- 1000 - 1000
- 1000 - 1000
- 1000
- 1000
- 500
- 500
0
00
500
500500
(a)xi
xr
- 6 - 4 - 2 0 2 4 6
- 3
- 2
- 1
0
1
2
3
- 2500
- 2500
- 2500
- 2500
- 2000
- 2000- 2000
- 2000- 1500
- 1500- 1500
- 1500- 1000
- 1000
- 1000
- 1000
- 500
- 500
- 500
- 500
00
0
0 0
0
0
500
500
500
500
(b)xi
xr
- 6 - 4 - 2 0 2 4 6
- 3
- 2
- 1
0
1
2
3
Quasi-Poissonian regime, Q → 0Perfect matching : Classical ⇐⇒ Klauder coherent state
46 / 47
Conclusions• Constructed noncommutative models both perturbatively and
exactly.
• PT -symmetry has been utilized to obtain Hermitiancounterparts of non-Hermitian models.
• Analytical expression of the metric is found for a class ofnon-Hermitian models.
• Computed minimal volume.
• Constructed Klauder coherent states for both perturbative andnonperturbative noncommutative models.
• Utilised revival structure to study the qualities of the coherentstates.
• Qualities have been investigated furthur by using Bohmianmechanics.
Thank you for your attention47 / 47