Exact spectral decomposition of a time-dependent one-particle reduced density matrix

0.18 MB·

by  I. Nagy

Exact spectral decomposition of a time-dependent one-particle
reduced density matrix
I. Nagy,1,2 J. Pipek,1 and M. L. Glasser3,2
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1 1Department of Theoretical Physics, Institute of Physics,
0
2
Budapest University of Technology and Economics,

n
a H-1521 Budapest, Hungary
J
1 2Donostia International Physics Center, P. Manuel de Lardizabal 4,
3
E-20018 San Sebasti´an, Spain

]
h 3Department of Physics, Clarkson University, Potsdam,
c
e
New York 13699-5820, USA
m
-
t (Dated: February 1, 2017)
a
t
s Abstract
.
t
a
m We determine the exact time-dependent non-idempotemt one-particle reduced density matrix
-
d and its spectral decomposition for a harmonically confined two-particle correlated one-dimensional
n
o system whentheinteraction terms in theSchro¨dingerHamiltonian arechanged abruptly. Based on
c
[ this matrix in coordinate space we derive a precise condition for the equivalence of the purity and

1
v theoverlap-squareofthecorrelatedandnon-correlatedwavefunctionsasthesystemevolvesintime.
0
3 Thisequivalenceholdsonlyiftheinterparticleinteractionsareaffected,whiletheconfinementterms
0
9 areunaffectedwithinthestabilityrangeofthesystem. Underthisconditionwealsoanalyzevarious
0
1. time-dependent measures of entanglement and demonstrate that, depending on the magnitude of
0
7 the changes made in the Schro¨dinger Hamiltonian, periodic, logarithmically incresing or constant
1
: value behavior of the von Neumann entropy can occur.
v
i
X
r PACS numbers: 03.67.Bg,03.67.Mn, 03.75.Kk
a
1
I. MOTIVATION
A quantum quench is an abrupt change in the state of a system due to changes in the
potential energy terms of its Schro¨dinger Hamiltonian. For an intarcting system such a
quantum quench can occur in the external confinement term, or in the interparticle terms.
Both seem to be feasible in modern experiments [1] on trapped systems. Since, in general,
the interparticle interaction is responsible for dynamic correlation, beyond any statistics-
mediated (exchange) correlation, its tuning is of special interest.
Equally importantly, advances in optical trapping of cold atoms have allowed for an
unprecedented manipulation over the size of these quantum systems such that the number of
atoms beeing trapped can be precisely specified [2]. Clearly, exactly solvable entangled two-
particle models could serve as benchmark systems [3, 4] to test the capability of approximate
many-body methods in the time domain. Recently, motivated by the obvious theoretical
interest on the fine details of informations, proposals have been made [5, 6] about measuring
entanglement in cold atom systems based on the interference between copies.
The present paper is organized as follows. The next, introductory Section is devoted
to a summary of the stationary model in order to provide a useful background. Section
III contains the time-dependent extension. The particular case where the abrupt change is
made in the interparticle interaction is analyzed in details. Atomic units are used.
II. SUMMARY ON THE TIME-INDEPENDENT CASE
Motivated by the really challenging interplay among Hamiltonian-based quantities
and information-theory-based measures, we first summarize the stationary (i.e., time-
independent) background to our present study. As our unquenched system, we take the
interacting model system [7, 8] first introduced by Heisenberg as one of the really simplest
(”Das denkbar einfachste Mehrk¨orperproblem”) many-body models
1 d2 d2 1 1
Hˆ(x ,x ) = + + ω2(x2 +x2) λω2(x x )2, (1)
1 2 − 2 dx2 dx2 2 0 1 2 − 2 0 1 − 2
(cid:18) 1 2(cid:19)
where λ measures the strength of the interparticle interaction energy in terms of ω2. The
0
repulsive (r) and attractive (a) interactions refer to λ Λ [0,0.5] and λ Λ < 0. The
r a
≡ ∈ ≡
restricted range for Λ will be clarified below. Physically, for Λ > 0.5, both interacting
r r
particles cannot both remain in the confining external field.
2
Introducing standard normal coordinates X (x +x )/√2 and X (x x )/√2,
1 1 2 2 1 2
≡ ≡ −
one can easily rewrite the unperturbed Hamiltonian into the form
1 d2 d2 1 1
Hˆ(X ,X ) = + + ω2X2 + ω2X2, (2)
1 2 − 2 dX2 dX2 2 1 1 2 2 2
(cid:18) 1 2(cid:19)
whereω ω andω ω √1 2λdenotethefrequenciesoftheindependent normalmodes.
1 0 2 0
≡ ≡ −
Based on Eq.(2), the normalized ground-state wave function Ψ(X ,X ) is the product
1 2
ω 1/4 1 ω 1/4 1
Ψ(X ,X ) = 1 exp ω X2 2 exp ω X2 . (3)
1 2 π −2 1 1 π −2 2 2
(cid:20) (cid:21) (cid:20) (cid:21)
(cid:16) (cid:17) (cid:16) (cid:17)
We stress at this point that the price of this normal-mode transformation is that one loses
the intuitive physical picture of real particles and, instead, operates with effective particles
representing the transformed coordinates. Since their frequencies are different one can not
reinterpret them as quasiparticles, at least within the famous Landau’s picture.
Before turning to the reduced one-particle density matrix, a key quantity in our analysis,
we introduce here a more conventional measure of interparticle correlation by calculating
the overlap-square O (λ) = < Ψ(X ,X ,λ) Ψ(X ,X ,λ = 0) > 2. The result becomes
1 1 2 1 2
| | |
2√ω ω
1 2
O (λ) = . (4)
1
ω +ω
1 2
With the separated form for our model one obtains the same expression, Q (λ) = Q (λ),
1 2
from an other overlap defined by interchanging (formally x x ) the coordinates of
2 2
⇒ −
effective particles in Eq.(3), i.e., from O (λ) := < Ψ(X ,X ,λ) Ψ(X ,X ,λ) > .
2 1 2 2 1
| | |
By rewriting the wave function Ψ(X ,X ) in terms of physical coordinates x and x , we
1 2 1 2
determine the one-matrix Γ (x ,x ) via the following nonlinear mapping
1 1 2
∞
Γ (x ,x ) = dx Ψ (x ,x )Ψ(x ,x ). (5)
1 1 2 3 ∗ 1 3 2 3
Z
−∞
From this, we arrive at a very informative, Jastrow-like [9], representation
Γ (x ,x ) = φ (x )φ (x ) e D[(x1 x2)/√2]2 (6)
1 1 2 s 1 s 2 − −
×
where, with ω 2ω ω /(ω +ω ), we introduced the following abbreviations
s 1 2 1 2
≡
ω 1/4
φs(x) = s e−21ωsx2 (7)
π
h i
1 (ω ω )2
1 2
D = − 0. (8)
4 ω +ω ≥
1 2
3
The diagonal of Γ (x ,x ) gives the one-particle probability density, n(x) = Γ(x,x), of
1 1 2
unit-norm, which is the basic quantity of the mean-field Density Functional Theory (DFT).
However, if we neglect the role of the relative coordinate by taking D = 0 in Eq.(6), we
get an idempotent one-matrix [10]. In DFT a density-optimal auxiliary orbital, φ (x), is
s
occupied with unit probability. The double Fourier transform [10] of the one-dimensional
one-matrix in Eq.(6) gives the momentum-dependent one-matrix. After integrations we get
Γ1(k1,k2) = 1 e−12(k12+k22)ωs(ωωss++D2D) e+ωsD(ωks1+k22D).
π(ω +2D)
s
The diagonal component of Γp(k,k) is the normalized one-body momentum distribution
1
functionanditsapplicationgives theexactkinetic energy ofthesystem: < K >= (1/4)(ω +
s
2D) = (1/4)(ω + ω ). Clearly, by using the virial theorem for bounded systems with
1 2
harmonic interactions, one gets the exact ground-state energy as well. Since one has access
toΓ (k,k)byComptonscattering, itisanobservableencodinginformationonentanglement.
1
Next we calculate the purity [11] defined by
Π(λ) := Tr[Γ2] = Γ2(x,x)dx, (9)
1 1
Z
where, the operator-square Γ2(x ,x ) is written
1 1 2
Γ2(x ,x ) = Γ (x ,x )Γ (x ,x )dx . (10)
1 1 2 1 1 3 1 3 2 3
Z
The direct calculation of the purity, based on Eqs.(9-10) with Eq.(6), results in
1 2√ω ω ω
1 2 s
Π(λ) = = = O (λ). (11)
(1+2D/ω )1/2 ≡ ω +ω √ω ω 1
s 1 2 1 2
This is a remarkable equality. It says that two measures of probabilistic nature are equal
in our entangled model system. If one of them, say the overlap, could be accessible experi-
mentally [5, 6], we could characterize the other one as well.
The purity rests on an operator-square, and is calculable directly from knowledge of
the coordinate-representation of the one-particle reduced density matrix. However, for a
compact spectral analysis [12] of correlation one needs Tr[Γq] for noninteger q values as well.
1
Based on them one can calculate R´enyi’s and von Neumann’s entropies [13]. The possibility
of point-wise [14] direct decomposition of a two-variable function rests on the mathematical
observation that Mehler’s formula [15–17] gives
(ω¯/π)1/2e−ω2¯(cid:16)11−+ZZ22(cid:17)(x21+x22)eω¯1−2ZZ2 x1x2 = ∞ (1 Z2)1/2Zkφk(ω¯,x1)φk(ω¯,x2), (12)
−
k=0
X
4
where the parameter Z [0,1], and x ( , ). The φ (ω¯,x) decomposition-functions
i k
∈ ∈ −∞ ∞
form a complete set of orthonormal eigenfunctions of a one-dimensional harmonic oscillator
with potential energy ω¯2(x2/2) in the Schro¨dinger wave equation and are given by
ω¯ 1/4 1
φk(ω¯,x) = e−12ω¯x2Hk(√ω¯x). (13)
π √2kk!
(cid:16) (cid:17)
Comparison of exponentials in Eq.(6) and Eq.(12) results in the two constraints
1+Z2
(ω +D) = ω¯ (14)
s
1 Z2
−
2Z
D = ω¯ . (15)
1 Z2
−
One can solve the algebraic equations easily for Z and ω¯ in terms of D and ω . We get
s
1+2D/ω 1 √ω √ω 2
s 1 2
Z(λ) = − = − (16)
1+2D/ω +1 √ω +√ω
p s (cid:18) 1 2(cid:19)
p
ω¯ = ω 1+2D/ω √ω ω . (17)
s s 1 2
≡
p
It follows from Eq. (16) that Z(Λ ) = Z(Λ ), i.e., there is a duality [11, 18] under the
a r
constraint of Λ = Λ /(1 2Λ ). Thus, under this constraint at stationary condition,
a r r
− −
probabilistic measures alone can not reproduce the sign of the interparticle interaction.
Since ω = ω¯(1 Z)/(1+Z) from Eqs.(14-15), we obtain a point-wise, closed-shell-like
s
−
expansion for the one-particle reduced density matrix
∞
Γ (x ,x ,ω¯) = P (Z)φ (ω¯,x )φ (ω¯,x ), (18)
1 1 2 k k 1 k 2
k=0
X
where the occupation numbers of the so-called natural orbitals, φ (ω¯,x), are
k
P = (ω /ω¯)1/2(1 Z2)1/2Zk = (1 Z)Zk (19)
k s
− −
and we have ∞k=0Pk = 1. In the knowledge of occupation numbers, i.e., the eigenvalues
of the one-maPtrix [19], one can calculate R´enyi’s entropies [13] for 0 < q < , since in our
∞
case (P )q = (1 Z)q(Zq)k. Thus, a desirable [12] spectral analysis of information-theoretic
k
−
measures becomes feasible. For instance, as a useful check, we get
1 ∞ 1 Z(λ) ω 2(1 2λ)1/4
Π(λ) = O (λ) = (P )2 = − = s = − 1
1 k
1+2D/ω ≡ 1+Z(λ) ω¯ 1+√1 2λ ≤
s k=0 −
X
p
This series exhibits useful connections between physical and auxiliary variables.
5
III. RESULTS FOR THE TIME-DEPENDENT CASE
Motivated by the remarkable experimental possibilities, outlined in the first Section on
trapped systems with controllable numbers of constituents, we suppose abrupt changes are
made in Heisenberg’s Hamiltonian at t = 0 and have for t > 0 for the new (n) Hamiltonian
1 d2 d2 1 1
Hˆ (x ,x ) = + + Ω2(x2 +x2) λ Ω2(x x )2. (20)
n 1 2 − 2 dx2 dx2 2 0 1 2 − 2 ′ 0 1 − 2
(cid:18) 1 2(cid:19)
Ω = 0, is considered a complete quench. This results in a free propagation [20, 21] of the
0
initially entangled system. As above, we use this Hamiltonian in its separated form
1 d2 d2 1 1
Hˆ (X ,X ) = + + Ω2X2 + Ω2X2, (21)
n 1 2 − 2 dX2 dX2 2 0 1 2 2 2
(cid:18) 1 2(cid:19)
where Ω = Ω √1 2λ. The particular choice of Ω = ω and λ = 0 in Hˆ (x ,x ), but
2 0 ′ 0 0 ′ n 1 2
− 6
taking λ = 0 in Hˆ(x ,x ), refers to entanglement production addressed recently [22].
1 2
As we have a separated form for Hˆ (X ,X ) in the time-dependent Schro¨dinger equation
n 1 2
for t > 0, we proceed by propagating [23] independently both normalized stationary normal
(j = 1,2) modes, ψ (X ,ω ) and ψ (X ,ω ) from Ψ(X ,X ) = ψ (X )ψ (X ) of Eq.(3), by
1 1 1 2 2 2 1 2 1 1 2 2
using the corresponding (Ω2 0, i.e., the λ > 0.5 explosion-case is excluded) propagators
j ≥ ′
1/2
Ω iΩ
G (X ,X ,Ω ,t) = j exp j [(X2 +X 2)cosΩ t 2X X ]
j j j′ j 2πisinΩ t 2sinΩ t j j′ j − j j′
(cid:18) j (cid:19) (cid:26) j (cid:27)
which reproduces the free-propagation case when one first takes Ω 0 at fixed t. With
j
→
this textbook G (t), we have to perform the convolution
j
∞
ψ (X ,t) = dX G (X ,X ,Ω ,t)ψ (X ), (22)
j j j′ j j j′ j j j′
Z
−∞
to derive Ψ (X ,X ,t) = ψ (X ,t)ψ (X ,t). We arrive at the normalized solution
n 1 2 1 1 2 2
1/4
B 1
ψ (X ,t) = j exp X 2B (1 iC ) (23)
j j j j j
π −2 −
(cid:18) (cid:19) (cid:20) (cid:21)
where the coefficients, B (ω ,Ω ,t) and C (ω ,Ω ,t), are given by
j j j j j j
Ω 2
j
B (t) = ω
j j Ω 2cos2Ω t+ω 2sin2Ω t
j j j j
1 ω 2 Ω 2
j j
C (t) = − sin(2Ω t)
j j
2 ω Ω
j j
6
In the complete-quench case we get C (t) = ω t and B (t) = ω /(1 + ω2t2), and thus
j j j j j
B (t)[1 + C2(t)] = ω . Here one has a simple free expansion without oscillation. In other
j j j
quenches, weexpectcharacteristiclyoscillatingwavefunctionsandcorrespondingprobability
measures. However, and this is very important from physical point of view, the period of
these oscillations will now depend on the sign of the interparticle interactions. For the
overlap-square defined by O (t) = < Ψ (X ,X ,λ,t) Ψ(X ,X ,λ = 0) > 2, one gets
1 n 1 2 ′ 1 2
| | |
2√ω B 2√ω B
0 1 0 2
O (t) = , (24)
1 [(ω +B )2 +(B C )2]1/2 [(ω +B )2 +(B C )2]1/2
0 1 1 1 0 2 2 2
which reproduces the stationary result in Eq.(4) at t = 0, Ω = ω and λ = λ. We stress
0 0 ′
that we evaluate this overlap with the stationary noninteracting state, as at Eq.(4).
By rewriting the evolving wave function Ψ (X ,X ,t) in terms of original coordinates x
n 1 2 1
and x , we determine the one-matrix Γ (x ,x ,t) from the following nonlinear mapping
2 1 1 2
∞
Γ (x ,x ,t) = dx Ψ (x ,x ,t)Ψ (x ,x ,t), (25)
1 1 2 3 ∗n 1 3 n 2 3
Z
−∞
After a long, but quite straightforward, calculation we obtain
Γ1(x1,x2,t) = φs(x1,t)φ∗s(x2,t)e−21D(t)(x1−x2)2, (26)
in which we introduced a time-dependent [c.f., Eq.(7)] auxiliary function
1/4
ω (t)
φs(x,t) = s e−21ωs(t)x2ei21ωs(t)Cs(t)x2 (27)
π
(cid:20) (cid:21)
where ω (t) = 2B (t)B (t)/[B (t) + B (t)] and C (t) = [C (t) + C (t)]/2. Clearly, the
s 1 2 1 2 s 1 2
individual normal-mode characters (B and C ) influence the auxiliary φ (x,t) state via
j j s
their differently weighted forms. The differences of characters will appear in
1 [B (t) B (t)]2 +[B (t)C (t) B (t)C (t)]2
1 2 1 1 2 2
D(t) = − − 0. (28)
4 B (t)+B (t) ≥
1 2
By taking D(t) 0, i.e., neglecting the role of the relative coordinate, one arrives at the
≡
idempotent one-matrix of Time-Dependent Density-Functional Theory [24]). The diagonal
(x = x = x) part of Γ (x ,x ,t) gives the exact probability density n(x,t) = [φ (x,t)]2,
1 2 1 1 2 s
the basic variable of TDDFT, a mean-field theory. The other variable in a physically consis-
tent approximation in TDDFT is the probabilty current j(x,t), needed in the fundamental
continuity equation ∂ n(x,t)+∂ j(x,t) = 0. The current is defined, and is given by
t x
∂
∞
j(x,t) := Re Ψ (x,x,t) Ψ (x,x,t)dx = n(x,t)[xω (t)]C (t).
∗n ′ i∂x n ′ ′ s s
Z
−∞
7
It is easy to show, after substitution, that in the complete-quench case we get 2D(t)/ω (t) =
s
2D/ω for t > 0, heralding rigidity in the entropic-measures for free-propagation. Such a
s
memory is expected on physical grounds. Furthermore, as we will discuss for the most
important case of abrupt changes in the interparticle interaction, there could be finite t
values at which we get [2D(t )/ω (t )] = 2D/ω or [2D(t )/ω (t )] = 0, both periodically.
1 s 1 s 2 s 2
The one-matrix in Eq.(26) is self-adjoint Γ (x ,x ,t) = Γ (x ,x ,t) and positive. By its
1 1 2 ∗1 2 1
application in the defining Eqs.(9-11) we get directly
1
Π(t) = Tr[Γ2(t)] = . (29)
1 [1+2D(t)/ω (t)]1/2
s
Using this at t = 0, where B (t = 0) = ω and C (t = 0) = 0, the stationary result of
j j j
the previous Section is recovered, since 1+2D/ω = [1+Z(λ)]/[1 Z(λ)]. Thus, in the
s
−
complete quench case the purity remainspconstant. However, the overlap O (t) tends to zero,
1
as expected, and for free propagation O (t ) ω /ω [2/(tω )]2 = 4/(ω ω¯t2). The
1 0 2 0 0
→ ∞ ≃
overlap O (t), defined at Eq.(4) for the stationary casep, behaves with respect to time as
2
1
O (t) := < Ψ (X ,X ,λ,t) Ψ(X ,X ,λ,t) > = Π(t).
2 | ∗ 1 2 | 2 1 | ≡ [1+2D(t)/ω (t)]1/2
s
Thus, inthetwo-bodymodelinvestigated, theinterchangeofcoordinatesofeffectiveparticles
can result in O (t) = O (t) for t = 0. A related analysis will be given at Eq.(34), below.
1 2
6 6
Following closely our arguments behind a point-wise decomposition of the stationary one-
matrix in the previous Section, we turn now to the time-dependent case. Thus, we introduce
time-dependent ω¯(t) and Z(t), via the two constraints
1+Z2(t)
[ω (t)+D(t)] = ω¯(t)
s
1 Z2(t)
−
2Z(t)
D(t) = ω¯(t)
1 Z2(t)
−
One can solve these equations for ω¯(t) and Z(t) in terms of D(t) and ω (t) to obtain
s
1+2D(t)/ω (t) 1
s
Z(t) = − (30)
1+2D(t)/ω (t)+1
p s
p
ω¯(t) = ω (t) 1+2D(t)/ω (t). (31)
s s
p
In terms of these solutions we get the point-wise spectral decomposition
∞
Γ (x ,x ,t) = P (t)φ (x ,t)φ (x ,t), (32)
1 1 2 k k 1 ∗k 2
k=0
X
8
where P (t) = ([1 Z(t)][Z(t)]k, and φ (x,t) are the so-called natural orbitals
k k
−
1/4
ω¯(t) 1
φk(x,t) = e−12ω¯(t)x2 Hk( ω¯(t)x) ei21ωs(t)Cs(t)x2. (33)
π √2kk!
(cid:20) (cid:21) (cid:20) (cid:21)
p
Knowing P (t), one can easily calculate different time-dependent measures for entangle-
k
ment such as Π(t), R´enyi’s [S (q,t)] and von Neumann [S (t) = S (q 1,t)] entropies
R N R
→
1 Z(t)
Π(t) = −
1+Z(t)
1 [1 Z(t)]q
S (q,t) = ln −
R
1 q 1 [Z(t)]q
− −
d 1 q Z(t)
S (t) = q2 − S (q,t) = ln[1 Z(t)] ln[Z(t)].
N R
− dq q − − − 1 Z(t)
(cid:20) (cid:18) (cid:19)(cid:21)q=1 −
Remember, that we obtained for the stationary case the remarkable equality in Eq.(11),
Π(λ) = O (λ). Is there a similar, possibly very useful, equality in the time-dependent case?
1
The answer is yes, if and only if we make abrupt changes in the interparticle interaction
only. For such changes, where Ω = ω but λ = λ, instead of Eq.(28) we have
0 0 ′
6
1 [ω B (t)]2 +[B (t)C (t)]2
0 2 2 2
D(t) = − 0.
4 ω +B (t) ≥
0 2
since B (t) = ω and C (t) = 0. From Eq.(24) [ω (t) = 2ω B (t)/(ω +B (t)] we obtain
1 0 1 s 0 2 0 2
1 2√ω B
0 2
Π(t) = O (t) = = O (t). (34)
2 [1+2D(t)/ω (t)]1/2 ≡ [(ω +B )2 +(B C )2]1/2 1
s 0 2 2 2
We stress, that this is true when we manipulate, solely, the interparticle interaction. Re-
markably, such a (gedanken) case was addressed first by Einstein, Podolsky, and Rosen
(EPR) in [25]. Now, this seems to be feasible in modern experiments on trapped systems.
For our EPR-case, we derive expressions exhibiting the role of λ λ changes.
′
→
2D(t) 2D 1 Ω2 ω2 Ω2 ω2
1+ = 1+ + 2 − 1 2 − 2 sin2(Ω t). (35)
2
ω (t) ω Ω ω Ω 4ω
s (cid:18) s (cid:19) 2 (cid:18) 1 2 (cid:19) 2
Based on this equation we arrive, after substitutions, at the following
2 2
1+Z(λ,t) 1+Z(λ) λ(λ λ)
= + ′ ′ − sin2(ω t√1 2λ). (36)
1 Z(λ,t) 1 Z(λ) √1 2λ(1 2λ) 0 − ′
(cid:20) − (cid:21) (cid:20) − (cid:21) − − ′
which shows that at t 0 the stationary result will vary quadratically in time. In that
→
limit, the associated changes in entropic measures will exhibit a similar, t2, scaling.
∝
9
Notice that the long-time behavior in the repulsive case needs some care. When we tune,
frombelow, therepulsive λ < 0.5 coupling to itscritical limit λ 0.5, we arrive at Ω 0+
′ 2
→ →
in the parameters given at Eq.(23), i.e., we have B (t) = ω /[1+ C2(t)] and C (t) = ω t.
2 2 2 2 2
The corresponding overlap O (t), and thus Π(t), tend to zero at t . In that asymptotic
1
→ ∞
limit for critical coupling (λ 0.5) our Z(t) tends to unity as
′
→
2(1 2λ)1/4 1 4
Z(t ) 1 − = 1 .
→ ∞ ≃ − 0.5(0.5 λ) tω − tω (1 2λ)1/4
0 0
− −
This gives for the sign-dependent, limiting behavior in the von Neumann entropy
p
S (tω¯ 1) ln(tω¯).
N
≫ ≃
Slow, i.e., logarithmic, growth of entanglement is a hot topic in nonequilibrium many-body
systems [26, 27] as well. Because of such similarity, we are tempted to think in terms of a
kindofuniversality. Ifthisturnsouttobetruerigorously, thencritical transitionsintrapped
and solid-state disordered systems could get a common probabilistic characterization.
Based on Eq.(36), we turn to illustrative cases which are exhibited in Figures 1-2. We
stress, that only the EPR-like situation, i.e., a λ λ change, is considered. When λ = 0,
′ ′
→
i.e., whenweturnoffabruptlytheinterparticleinteractions, therearenochangesinentropies
and purity since Z(λ,t) = Z(λ). Similar, memory-like character in entanglement was found
recently in [28] on pre-thermalization in a one-dimensional Bose system. There, further
arguments are given about a possible generic phenomenon of keeping entanglement.
InFiguere1showingΠ(t), wetakeΛ = 0.49asbefore-quenchvalueforrepulsive coupling
r
(λ). The dotted curve refers to a change λ λ where, after quench, one has Λ =
→ ′ ′r
(1 √1 2Λ )/2. Thus we tune the interaction in order to get Ω ω¯, i.e., the frequency
r 2
− − ≡
in stationary natural orbitals. As expected for the physical consequence of minimization (m)
on the rhs of Eq.(36), we get Z(t ) = 0 and thus exactly zero entropies, at times t where
m
t Ω = t ω¯ = (2m+ 1)π/2. The solid curve refers to a change prescribed by λ = 1.01λ,
m 2 m ′
i.e., to a 1% enhancement in coupling.
Figure 2 on Π(t) is devoted to the attractive case. We use Λ = 24.5, at which
a
−
Z(Λ ) = Z(Λ ), i.e., we have, before changes, a duality in entropic measures. Similarly
r a
to Figure 1, dotted and solid curves refer to Λ = (1 √1 2Λ )/2 and λ = 1.01λ,
′a − − a ′
respectively. Comparison of the illustrative Figures shows that, in the evolving cases,
the oscillation-periods reflect the sign of interparticle interaction. Oscillations in a time-
dependent measure were found in [22], for an entanglement production where λ = 0 initially,
10