qualitative measurement of klauder coherent states using bohmian machanics, city december 3
TRANSCRIPT
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Qualitative measurement of Klauder coherent states usingBohmian Mechanics
Sanjib Dey
City University London
December 03, 2013
Based on Phys. Rev. A 88, 022116 (2013), with Prof. Andreas Fring
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What is a coherent state?
Superposition of large no of quantum states⇒ Classical particle.For example, Glauber coherent state :
|α〉= N (α)∞
∑n=0
αn√
n!|n〉, N (α)⇒ e−
|α|22
Sometimes called the minimum uncertainty wavepacket ∆x∆p≈ }/2
Canonical coherent states : ∆x = ∆p = }/√
2a|α〉= α|α〉|α〉= eαa†−α?a|0〉= D(α)|0〉,〈β|α〉 6= δ(α−β)
Squeezed coherent state : ∆x∆p = }/2
Applications : Quantum Optics, Quantum information, Laser Physics,Mathematical Physics etc.
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Generalised Klauder coherent state
For Hermitian H ; bounded below, nondegenerate eigenspectrum En = }ωen
and orthonormal eigenstate |φn〉 :
ψJ(x,γ) :=1
N (J)
∞
∑n=0
Jn/2e−iγen
√ρn
φn(x), J ∈ R+0
ρn := ∏nk=1 ek, N 2(J) := ∑
∞
k=0 Jk/ρk, ρ0 = 1
Properties1 Continuous in time and J.2
∫|ψJ〉〈ψJ| dµ = 1
3 Temporarily stable : e−iH tψJ(x,γ) = ψJ(x,γ+ωt), ω = Constant4 Satisfies action angle identity : 〈ψJ|H |ψJ〉= }ωJ
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Procedure
One can analyse all the properties mathematically. Which is notsufficient to realise the quality precisely.
How would you measure the precise quality?
Draw the classical trajectories by solving :
x =∂H∂p
, p =−∂H∂x
(1)
Draw the dynamics of the coherent states of the particle and compare.
How would you draw the trajectories of the coherent states?
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Bohmian mechanics
Quantum theory⇒ Solution of Schrodinger equation : ψ⇒ Probabilitiesof actual result.
Is it possible to find some other interpretation?
David Bohm(1952)⇒ Alternative trajectory based interpretation.
Undoubtedly successful : photodissociation problems, tunnellingprocess, atom diffraction by surfaces, high harmonic generation etc.
Bohmian mechanics =⇒ Still ongoing and controversial.Keeping interpretational issues aside =⇒ Apply it.
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Bohmian mechanics (real case)
Time dependent Schrodinger equation :
ih∂ψ(x, t)
∂t=− h2
2m∂2ψ(x, t)
∂x2 +V(x)ψ(x, t)
WKB polar decomposition :
ψ(x, t) = R(x, t)eih S(x,t), R(x, t),S(x, t) ∈ R
Substitute ψ(x, t) into Schrodinger equation and separate real and imaginarypart :
St +(Sx)
2
2m+V(x)− h2
2mRxx
R= 0 ⇐ Quantum Hamilton-Jacobi equation
mRt +RxSx +12
RSxx = 0 ⇐ Continuity equation
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Real Bohmian
∗ Velocity :
mv(x, t) = Sx =h2i
[ψ∗ψx−ψψ∗x
ψ∗ψ
]∗ Quantum potential :
Q(x, t) =− h2
2mRxx
R=
h2
4m
[(ψ∗ψ)2
x
2(ψ∗ψ)2 −(ψ∗ψ)xx
ψ∗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
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Bohmian mechanics (complex case)
∗ Decompose :ψ(x, t) = e
ih S(x,t), S(x, t) ∈ C
∗ Substitute ψ(x, t)⇒ time dependent Schrodinger equation :
St +(Sx)
2
2m+V(x)− ih
2mSxx = 0
∗ Velocity :
mv(x, t) = Sx =hi
ψx
ψ
∗ Quantum potential :
Q(x, t) =− ih2m
Sxx =−h2
2m
[ψxx
ψ− ψ2
x
ψ2
]
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Summarize
Solve canonical equations =⇒ Classical trajectoryCoherent state =⇒ Bohmian scheme =⇒ Trajectoriesof coherent stateCompare these two =⇒ Quality of coherent states
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Application : Poschl-Teller model (real case)
φn(x) =1√Nn
cosλ
( x2a
)sinκ
( x2a
)2F1
[−n,n+κ+λ;k+
12
;sin2( x
2a
)]Stationary state Bohmian :
v(t) = 0 ⇐ Not the behaviour of a classical particle.
Klauder coherent state :
ψJ(x,γ) :=1
N (J)
∞
∑n=0
Jn/2e−iγen
√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+√
γcos
(√2Em
ta
)], α, β, γ constant
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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
Qualitatively not identical with classical trajectories !!
Look at the uncertainty of X & P
Look at the behaviour of |ψ(x, t)|2 with time too.
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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!!
Need to localise the wavepacket !!How can we do that??
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Mandel parameter
ψJ(x,γ) := 1N (J)
∞
∑n=0
Jn/2e−iγen√ρn
φn(x)
ψJ(x,γ) :=∞
∑n=0
cn(J)e−iγen |φn〉, cn =Jn/2
N (J)√
ρn⇐ weighting function
ψJ(x,γ) needs to be well localised.
To examine : check weighting probability, |cn|2⇒ Poissonian.
Deviation of |cn|2 from Poissonian is captured by Mandel parameter, Q .
If ψJ is strongly weighted around 〈n〉, Q = ∆n2
〈n〉 −1 = J ddJ ln d
dJ lnN 2
Q = 0 ⇒ Pure Poissonian, Q > 0 ⇒ Super-Poissonian.Q < 0 ⇒ Sub-Poissonian, |Q | � 1 ⇒ Quasi-Poissonian.
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Sub-Poissonian regime
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)
Q =−0.307593,−0.149523,−0.042555
We are in sub-Poissonian regime !!!What happens in the quasi-Poissonian, Q→ 0 regime??
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Q(J,κ+λ) = J2+κ+λ
0F1(3+κ+λ;J)0F1(2+κ+λ;J) −
J1+κ+λ
0F1(2+κ+λ;J)0F1(1+κ+λ;J)
One can control κ, λ and J, so that Q→ 0
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 = 2,0.5,0.1 andκ = 2, λ = 3, J = 2,0.5,0.1
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Quasi-Poissonian regime
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)
t
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Poschl-Teller potential (complex case)
H =p2
2m+
V0
2
[λ(λ−1)
cos2(x/2a)+
κ(κ−1)sin2(x/2a)
]− V0
2(λ+κ)2 for 0≤ x≤ aπ
Complexify : x⇒ xr + ixi, p⇒ pr + ipi
Real and imaginary part
Hr =p2
r −p2i
2m− V0
2(λ+κ)2
+V0
[(λ2−λ)
[cosh
( xia
)cos( xr
a
)+1][
cosh( xi
a
)+ cos
( xra
)]2
−(κ2−κ)
[cosh
( xia
)cos( xr
a
)−1][
cos( xr
a
)− cosh
( xia
)]2
]
Hi =pipr
m+V0
[(λ2−λ)sinh
( xia
)sin( xr
a
)[cosh
( xia
)+ cos
( xra
)]2 − (κ2−κ)sinh( xi
a
)sin( xr
a
)[cos( xr
a
)− cosh
( xia
)]2]
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PT-symmetry and non-hermitian Hamiltonian
Hamiltonian→ non-hermitian 6= real eigenvalues.Bender et al [Phys. Rev. Lett. 80, 5243-5246 (1998)]
P T symmetric non-hermitian Hamiltonian⇒ Real eigenvalues.P → Parity transformation, T → Time reversal
In our case P T : xr→−xr, xi→ xi, pr→ pr, pi→−pi, i→−i
Solve canonical equations of motion :
xr =12
(∂Hr
∂pr+
∂Hi
∂pi
), xi =
12
(∂Hi
∂pr− ∂Hr
∂pi
),
pr = −12
(∂Hr
∂xr+
∂Hi
∂xi
), pi =
12
(∂Hr
∂xi− ∂Hi
∂xr
)
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Classical trajectory : Poschl-Teller potential
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(a)xi
xr
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0
1
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0
0
500
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500
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(b)xi
xr
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- 3
- 2
- 1
0
1
2
3
Blue : x0 = 4.5, p0 = 41.8376i, E =−31.7564Black : x0 = 3+1.5i, p0 =−30.1922+0.385121i, E =−6.55991−13.5182i
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Stationary state : complex case
ψn(x) =1√Nn
cosλ
( x2a
)sinκ
( x2a
)2F1
[−n,n+κ+λ;k+
12
;sin2( x
2a
)]
-6 -4 -2 0 2 4 6
-2
-1
0
1
2x
0= ±0.1
x0= ±1.5
x0= ±2.0
x0= ±2.45
x0= 5.0
x0= 5.5
x0(t)
t
(a)
-6 -4 -2 0 2 4 6-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
x0= ±0.1x0= ±0.3x0= ±0.9x0= ±1.5x0= ±2.7x0= ±3.6x0= ±4.5x0= ±5.0x0= ±5.5
x5(t)
t
(b)
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Classical and Klauder state
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xi
xr
(a)
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- 3
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- 1
0
1
2
3
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0.8
xr
xi (b)
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- 3
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- 1
0
1
2
3
Sub-Poissonian regime, Q < 0
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Classical and Klauder state
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xr
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1
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(b)xi
xr
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- 1
0
1
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3
Quasi-Poissonian regime, Q → 0Perfect matching : Classical⇐⇒ Klauder coherent state
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ConclusionOne must draw the trajectories of classical case and coherent states andcompare them to study the behaviour of the coherent states.
We have found an extra parameter Q which governs the behaviour of thecoherent states.
Q→ 0, Klauder state is a perfect coherent state for both real andcomplex cases.
Must take Klauder state for generalised models, instead of Glauber state.
Thank you for your attention
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