Relation between the entropy and the purity parameter in the ion-laser interaction
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Article
Relation between the entropy and the purity
parameter in the ion-laser interaction
Raúl Juárez-Amaro, Leonardo Moya Rosales, Jorge A. Anaya-Contreras 2 , Arturo
Zúñiga-Segundo 1 and Héctor M. Moya-Cessa 3
1 Universidad Tecnológica de la Mixteca, Apdo. Postal 71, 69000 Huajuapan de León, Oax., Mexico
2 Instituto Politécnico Nacional, ESFM, Departamento de Física. Edificio 9, Unidad Profesional “Adolfo López
Mateos,” CP 07738 CDMX, Mexico
3 Instituto Nacional de Astrofísica, Óptica y Electrónica, 72840 Sta. María Tonantzintla, Pue., Mexico
* Correspondence: hmmc@inaoep.mx; Tel.: +52-22-2266-3100
Received: 13 December 2019
Abstract: It is shown that, in the interaction of a field and a qubit, there exists a relation between the
linear entropy and the von Neumann entropy. The Cayley-Hamilton theorem is used to obtain such
relation. In the case we study, the qubit is given by the external degrees of freedom of an ion trapped
in a Paul trap and the field given by its internal (vibrational) degrees of freedom.
Keywords: entropy, entanglement, fluctuations
1. Introduction
Nonclassical states of quantum systems [1–12] are of great importance not only because of
fundamental aspects, such as the fact that they may present fluctuations that are below the quantum
standard limit defined by coherent states, or a variety of linear combination of discrete variable
systems and many-qubit system [13–22] but also because of practical reasons, for instance in metrology,
quantum information and quantum computation [23].
Trapped ions interacting with laser fields and quantized fields interacting with two-level atoms
have many common features as in both subjects it is possible to generate nonclassical states of the
vibrational motion of the ion and of the quantized field, respectively, and to realize interactions of
the Jaynes-Cummings [24–26] and anti-Jaynes-Cummings [27] type and multiphonon/multiphoton
transitions in those systems. Trapped ions interacting with laser fields in the Lamb-Dicke regime
may be described by the Jaynes-Cummings model, which in this case describes the interaction of an
electronic transition and the quantized center-of-mass motion, assisted by a laser beam, in the resolved
sideband regime [28,29]. The advantage in the ion-laser interaction, over the atom-field interaction, is
the fact that decoherence processes do not affect the ion-laser interaction as much as it does to cavities
[30].
Quantum systems such as the atom field and the ion-laser interactions and their non-linear
generalizations may be modelled by using classical interactions [31] such as propagation of light
through inhomogeneous media, namely waveguide arrays [32–35]
Although some information about an initial state of the vibrational motion of the ion and/or the
quantized field may extracted from atomic properties such as Rabi oscillations [36], for information
about its degree of purity or mixedness we need some other quantities, like entropy [37] or linear
entropy [38].
In next Section, we introduce the von Neumann entropy and the linear entropy. In Section 3
we present the ion-laser interaction Hamiltonian and give its solution. Still in Section 3 we use the
Cayley-Hamilton theorem to write the powers of the vibrational density matrix in terms of powers of
Entropy 2020, xx, 5; doi:10.3390/Entropyxx010005 www.mdpi.com/journal/entropyEntropy 2020, xx, 5 2 of 10
the spin density matrix. In Section 4 we give the relation between the von Neumann entropy and the
linear entropy and Section 5 is left for the conclusions.
2. Entropy and linear entropy
The Araki-Lieb inequality [37,38], |S A − SV | ≤ S AV ≤ S A + SV , may be of great help to obtain
the entropy of the one subsystem (for instance the vibrational motion of the ion) from the entropy of
another subsystem (ion’s electronic states) which is simpler to calculate. In the above expression, S AV
denotes the total entropy, while S A is the entropy for the ion and SV is the vibrational entropy.
From the above inequality one may note that, if the two subsystems are initially in pure states,
the total entropy of the system is zero, implying that both subsystems entropies are equal after after
both subsystems interact.
The von Neumann entropy may be defined as the expectation value of the entropy operator,
Ŝ = − ln ρ,
S = Tr {ρŜ}, (1)
where ρ is the density matrix of the quantum system.
Linear entropy
Linear entropy, originally called purity parameter, is a much simpler function of the density
matrix, compared to entropy, and therefore much easier to calculate. It is given as
ξ = 1 − Tr {ρ2 }. (2)
The linear entropy is always lower than the entropy, being its limiting case. In this contribution
we give a relationship between them.
We next consider the ion-laser interaction and show that, for certain parameters, it may describe
the atom-field interaction. We solve it and write the total density matrix in order to find the reduced
density matrices for the vibrational and the internal degrees of freedom. We show that powers of
the vibrational density matrix may be obtained from powers of the internal degrees of freedom
density matrix, which, being a 2 × 2 matrix, its powers are easily obtained. This allows us to write a
relation between the entropy and the linear entropy in the case we consider initial pure states for the
wavefunctions associated to the vibrational motion and the internal degrees of freedom of the ion.
3. Ion-laser interaction
We consider the Hamiltonian of a single ion trapped in a harmonic potential in interaction with
laser light in the rotating wave approximation, which reads
Ĥ = ν ↠â + ωeg Âee + [λE(−) ( x̂, t)  ge + H.c.], (3)
where â and  ab are the annihilation operator of a quantum of the ionic vibrational motion and the
electronic (two-level) flip operator for the |bi → | ai transition of frequency ωeg , respectively. The
frequency of the trap is ν, λ is the electronic coupling matrix element, and E(−) ( x̂, t) the negative part
of the classical electric field of the driving field.
We assume the ion driven by a laser field tuned to the mth lower sideband, we may write E(−) ( x̂, t)
as
E(−) ( x̂, t) = Ee−i(k x̂−ωeg +mν)t , (4)
where k is the wave vector of the driving field. If m = 0 it would correspond to the driving field being
on resonance with the electronic transition. The operator x̂ may be written as
k x̂ = η ( â + ↠), (5)Entropy 2020, xx, 5 3 of 10
with â and ↠are the annihilation and creation operators of the vibrational motion, respectively, and η
is the so-called Lamb-Dicke parameter.
In the resolved sideband limit, the vibrational frequency ν is much larger than other characteristic
frequencies and the interaction of the ion with the laser may be treated using a nonlinear Hamiltonian
[1,8]. The Hamiltonian (3) in the interaction picture can then be written as
Figure 1. We plot the atomic inversions as a function of time with Ω = 1 and α = 4 and for (a) η = 0.2,
(b) η = 0.1 and (c) η = 0.
2 /2 n̂! (m)
Ĥ I = Âeg Ωe−η Ln̂ (η 2 ) âm + H.c., (6)
(n̂ + m)!
(m)
where Ln̂ (η 2 ) are the associated Laguerre polynomials that depend on the number operator, n̂ = ↠â,
and Ω is the Rabi frequency.
By solving the Schrödinger equation for the Hamiltonian we find the wavefunction [6], ψ(t), and
from it the the total system’s density matrix, ρ̂(t) = |ψ(t)ihψ(t)|
ρ̂(t) = |eih| ⊗ |cihc| + |eih g| ⊗ |cihs| (7)
+ | gih| ⊗ |sihc| + | gih g| ⊗ |sihs|
where the unnormalized wavefunctions of the vibrational motion of the ion are given by
√ √
|ci = cos λt n̂ + 1|αi , |si = −iV̂ † sin λt n̂ + 1|αi , (8)
and we have used m = 1 and have considered an initial vibrational state given by a coherent state, |αi,
and the ion in its excited state, |ei. The operator
1
V= √ â (9)
n̂ + 1
is the so-called London phase operator [39,40].Entropy 2020, xx, 5 4 of 10
We find the vibrational reduced density matrix by tracing over the ion’s external degrees of
freedom
ρ̂V = |cihc| + |sihs| , (10)
while the atomic density matrix is found by tracing over the internal degrees of freedom such that we
obtain
ρ̂ A = |eihe|hc|ci + |eih g|hs|ci (11)
+ | gihe|hc|si + | gih g|hs|si
= |eihe|ρee + |eih g|ρeg + | gihe|ρ ge + | gih g|ρ gg .
In Figure 1 we plot the atomic inversions,
W (t) = ρee − ρ gg , (12)
for different values of the Lamb-Dicke parameter. The quantities ρee and ρ gg are defined in equation
(11). Figure 1 shows that, as the Lamb-Dicke parameter gets smaller, the ion-laser interaction becomes
similar to the interaction between a two-level atom and a quantized field. In particular, in Figure 1 (c)
the common revivals of oscillations [25] may be clearly observed.
3.1. Relation of the powers of reduced density matrices
We use the Cayley-Hamilton theorem, this is, the fact that all square matrices obey their eigenvalue
equation, to show that powers of the vibrational density matrix may be related to powers of the spin
system. In order to be more specific, Cayley-Hamilton’s theorem states that, given an N × N matrix A,
whose characteristic equation reads
x N − q N −1 x N −1 − q N −2 x N −2 − · · · − q1 x − q0 = 0, (13)
the matrix A also obeys such equation, namely
A N − q N −1 A N −1 − q N −2 A N −2 − · · · − q1 A − q0 1̂ = 0, (14)
with 1̂ the N × N unit matrix.
It may be proved that for two subsystems initially in pure states, after interaction, it may be found
a relation between the powers of the reduced density matrices [41]
n +1
ρ̂V = Tr A {ρ̂ρ̂nA } , (15)
where the reduced (atomic) density matrix should be taken in tensor product with vibrational identity
operator, that we have obviated. We prove the relation (15) in the appendix. The expression above
helps to calculate arbitrary functions of any of the density matrices, and, in particular, the entropy
operator.
3.2. Qubit entropy operator
In order to calculate the entropy operator, we follow Ref. [42]. For this we first need to find ρ̂nA , so
we write
| gih g| + |eihe| 1̂
ρ̂ A = + R̂ = + R̂ , (16)
2 2Entropy 2020, xx, 5 5 of 10
where
δ δ
R̂ = |eihe| + |eih g|ρeg + | gihe|ρ ge − | gih g| , (17)
2 2
with δ = ρee − ρ gg , 1̂ is the 2 × 2 unit density matrix and we have used the fact that ρee + ρ gg = 1. The
powers of ρ̂ A are then given by
n n
!
1̂ n 1
ρ̂nA =
2
+ R̂ = ∑ m 2n − m
R̂m . (18)
m =0
We split the above sum into two sums, one with odd powers of R̂ and one with even powers
! !
[n/2] [n/2]
n 1 n 1
ρ̂nA = ∑ 2m 2n−2m
R̂ 2m
+ ∑ 2m + 1 2n−2m−1
R̂2m+1 , (19)
m =0 m =0
and use also the relations
R̂ 2m+1
R̂2m = e2m 1̂, R̂2m+1 = e , (20)
e
In terms of ρ A the above equation is written as
ρ̂nA = G(n)ρ̂ A − |ρ̂ A |G(n − 1)1 (21)
where
n n
1 1 1
G(n) = +e − −e , (22)
2e 2 2
1/2
δ2 1
where we have defined e = 4 + |ρ ge |2 [41] and the determinant |ρ̂ A (t)| = 4 − e2 .
1̂−ρ̂ A
By using the fact that ρ̂− 1
A = |ρ̂ A |
, i.e., Cayley-Hamilton’s theorem we may write the entropy
operator as
Ŝ A = ln ρ̂− 1
A = ln(1̂ − ρ̂ A ) − 1̂ ln | ρ̂ A ( t )| . (23)
From the Taylor series ln(1 − x ) = − ∑∞
n =1
xn
n we write
∞ ρ̂nA
Ŝ A = − ∑ n
− 1̂ ln |ρ̂ A (t)| . (24)
n =1
and from equation (21) we may find a relation between the atomic entropy operator and the atomic
density matrix
Ŝ A = F1 ρ̂ A + F0 1̂ , (25)
where the functions associated to the powers of the density matrix are
1 λ−
F1 = ln , (26)
2e λ+
and
1 1 λ−
F0 = − ln |ρ A | + ln . (27)
2 2e λ+Entropy 2020, xx, 5 6 of 10
Figure 2. We plot the vibrational entropies as a function of time with Ω = 1 and α = 4 and for (a)
η = 0.2, (b) η = 0.1 and (c) η = 0.
4. Relation between the entropy and the linear entropy
Just as the atomic entropy operator may be related to its zeroth and first powers, as seen in
equation (25), the vibrational entropy operator may be related to its first and second power
F0 F0 2
ŜV = F1 + ρ̂V − ρ̂ . (28)
|ρ A | |ρ A | V
From equation (15) it may be easily shown that any function, Q, of the vibrational density matrix obeys
the relation
ρ̂V Q(ρ̂V ) = Tr A {ρ̂Q(ρ̂ A )} , (29)
such that the the operator ρ̂V ŜV my be written from equation (23) as
ρ̂V ŜV = F0 ρ̂V + F1 ρ̂2V , (30)
whose trace is the vibrational entropy
SV = F0 + F1 Tr {ρ̂2V }. (31)
We can use then equation (2), for the linear entropy, and the above equation for the von Neumann
entropy to find the relation
1 − 2e 1 − 2e
1 1
SV = ln ξV − ln . (32)
2e 1 + 2e 4e 1 + 2e
In Figures 2 and 3 we plot the entropies and linear entropies, respectively, for the same set of parameters
as the inversions in Figure 1. A very similar behaviour may be found in both figures and the tendencyEntropy 2020, xx, 5 7 of 10
to recover the well-known result as for the Jaynes-Cummings model my be seen in Figures 2(c) and 3
(c).
Figure 3. We plot the vibratonal linear entropies as a function of time with Ω = 1 and α = 4 and for (a)
η = 0.2, (b) η = 0.1 and (c) η = 0.
5. Conclusion
We have shown a relation between the linear entropy and the von Neumann entropy in the ion
laser interaction. By using the Cayley-Hamilton theorem we showed that the atomic entropy could be
written in terms of the density matrix while the vibrational entropy could be related to the square of
the vibrational density matrix that in turn allowed us to relate it to the linear entropy. Because the
form to relate the von Neumann and linear entropies is by using the Cayley-Hamilton theorem, the
result shown here could be scaled to bigger systems, i.e., not restricted to qubit systems. For instance,
the relation could be extended to interactions between multilevel atoms with fields and the Dicke
model. Of course, being higher dimensional systems, more powers of the atomic density matrices
would play a role.
Author Contributions: R.J.-A. and L.M.R. conceived the idea and developed it under the
supervision of J.A.A.-C., A.Z.-S. and H.M.M.-C. The manuscript was written by all authors, who have
read and approved the final manuscript.
Appendix A
We may show that
n +1
ρ̂V = Tr A {ρ̂ρ̂nA } , (A1)Entropy 2020, xx, 5 8 of 10
by using Schmidt decomposition [43]. As we assumed the initial states of the two subsystems to be in
pure states, the total evolved wavefunction reads
|ψ(t)i = |ci|ei + |si| gi, (A2)
while the atomic density matrix is found by tracing over the internal degrees of freedom such that we
obtain p p
|ψ(t)i = λ+ |ψ+ i|+i + λ− |ψ− i|−i, (A3)
such that, the total density matrix is written as
p
ρ(t) = λ+ ρ+ |+ih+| + λ− ρ− |−ih−| + λ+ λ− (|ψ− ihψ+ ||−ih+| + H.c). (A4)
From the above equation we may easily find the reduced density matrices
ρV ( t ) = λ + ρ + + λ − ρ − , ρ A (t) = λ+ |+ih+| + λ− |−ih−|, (A5)
and their powers
n +1
ρV (t) = λn++1 ρ+ + λn−+1 ρ− , ρnA (t) = λn+ |+ih+| + λn− |−ih−|. (A6)
By multiplying ρnA (t) by the total density matrix (A4) and tracing over the internal degrees of freedom
we obtain equation (A1).
Funding: This research received no external funding.
Acknowledgments: We thank CONACYT for support.
Conflicts of Interest: The authors declare no conflict of interest.
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