Diatomicmolecule as a quantum entanglement switch
Adam Rycerz ∗
Marian Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30–059 Kraków, Poland
Abstract
We investigate a pair entanglement of electrons in diatomic molecule, modeled as a correlated double quantum dot attached to
the leads. The low-temperature properties are derived from the ground state obtained by utilizing the Rejec-Ramšak variational
technique within the framework of EDABI method, which combines exact diagonalization with ab initio calculations. The results
show, that single-particle basis renormalization modifies the entanglement-switch effectiveness significantly. We also found the
entanglement signature of a competition between an extended Kondo and singlet phases.
Key words: Correlated nanosystems, Entanglement manipulation, EDABI method
PACS: 73.63.-b, 03.67.Mn, 72.15.Qm
Quantum entanglement, as one of the most intriguing
features of quantum mechanics, have spurred a great deal
of scientific activity during the last decade, mainly because
it is regarded as a valuable resource in quantum commu-
nication and information processing [1]. The question on
entanglement between microscopic degrees of freedom in a
condensed phase have been raised recently [2], in hope to
shed new lights on the physics of quantum phase transitions
and quantum coherence [3]. In the field of quantum elec-
tronics, a pair entanglement appeared to be a convenient
tool to characterize the nature of transport through quan-
tum dot, since its vanish when the system is in a Kondo
regime [4]. The analogical behavior was observed for two
qubits in double quantum dot, for either serial and paral-
lel configuration [5]. The latter case is intriguing, since the
concurrence [6] at T = 0 changes abruptly from C ≈ 1 to
C = 0 when varying the interdot coupling, so a finite An-
derson system shows a true quantum phase transition.
Here we consider a nanoscale version of such an entangle-
ment switch, inspired by conductance measurements for a
single hydrogenmolecule [7]. A special attention is payed to
electron-correlation effects, in particular the wave-function
renormalization [8]. Recent experiment [9] shows the cur-
rent through a molecule is carried by a single conductance
channel, so serial configuration shown in Fig. 1 seems to be
the realistic one. The Hamiltonian of the system is
∗ Corresponding author. Tel: (+48 12) 663–55–68 Fax: (+48 12)
633–40–79
Email address: rycerz@th.if.uj.edu.pl (Adam Rycerz).
���������� �������� �������� ������������������������ ��������
Fig. 1. Diatomic molecule modeled as a double quantum dot attached
serially to the leads. A cross-section of the single-particle potential
along the main system axis is shown schematically.
H = HL + VL +HC + VR +HR, (1)
where HC models the central region, HL(R) describes the
left (right) lead, and VL(R) is the coupling between the lead
and the central region. Both HL(R) and VL(R) terms have a
tight–binding form, with the chemical potential in leads µ,
the hopping t, and the tunneling amplitude V , as depicted
schematically in Fig. 1. The central-region Hamiltonian
iσcjσ +
iσ 6=jσ′
Uijniσniσ′ + (Ze)
2/R (2)
(with i, j = 1, 2 and σ =↑, ↓) describes a double quan-
tum dot with electron-electron interaction. tij and Uij are
single-particle and interaction elements, the last term de-
scribes the Coulomb repulsion of the two ions at the dis-
tance R. Here we put Z = 1 and calculate all the param-
eters tij , Uij as the Slater integrals [10] for 1s-like hydro-
genic orbitals Ψ1s(r) =
α3/π exp(−α|r|), where α−1 is
the orbital size (cf. Fig. 1). The parameter α is optimized
to get a minimal ground-state energy for whole the system
Preprint submitted to Elsevier 15 November 2018
http://arxiv.org/abs/0704.0743v2
-3 -2 -1  0  1
chemical potential,   
 0  1  2  3  4
 average filling,   
〈n1+n2〉
Fig. 2. Entanglement and transport through the system in Fig. 1 as
a function of the chemical potential µ (top panel) and the average
filling 〈n1+n2〉 (bottom panel). Tick (thin) solid and dashed lines
shows the concurrence C (conductance G) for Γ = t/9 and t/4,
respectively. The limits Γ → 0 are depicted with dotted lines in the
bottom panel. The interatomic distance is R = 1.5a.
 1  2  3  4  5  6
interatomic distance,   
Fig. 3. Concurrence (tick lines) and conductance (thin lines) at
the half-filled sector 〈n1+n2〉 = 2 as a function of the interatomic
distance R. The remaining parameters are the same as in Fig. 2.
described by the Hamiltonian (1). Thus, following the idea
of EDABI method [8], we reduce the number of physical
parameters of the problem to just a three: the interatomic
distance R, the lead-molecule hybridization Γ = V 2/t, and
the chemical potential µ (we put the lead hopping t =
1 Ry = 13.6 eV to work in the wide–bandwidth limit).
The entanglement between electrons placed on two atoms
can be characterized by the charge concurrence [4]
C = 2max
0, |〈c
iσcjσ〉| −
〈niσnjσ〉〈n̄iσ n̄jσ〉
where n̄iσ ≡ 1−niσ. We also discuss the conductivity cal-
culated from the formula G = G0 sin
2(E+−E−)/4tN [11],
where G0 = 2e
2/h̄, and E± are the ground-state ener-
gies of the system with periodic and antiperiodic boundary
conditions, respectively. Either the energies E± or correla-
tion functions in Eq. (3) are calculated within the Rejec–
Ramšak variational method [11], complemented by the or-
bital size optimization, as mentioned above. We use up to
N = 104 sites to reach the convergence.
In Fig. 2 we show the concurrence and conductance for
R = 1.5a0 (where a0 is the Bohr radius) and two values
of the hybridization Γ = t/9 and t/4. The conductance
spectrum asymmetry, caused by wave-function renormal-
ization [8], is followed by an analogical effect on entangle-
ment, which changes significantly faster for the upper con-
duction band, where the average filling is 〈n1+n2〉 ≈ 3
(one extra electron). The asymmetry vanish when analyz-
ing the system properties as a function of 〈n1+n2〉, show-
ing it originates from varying charge compressibility χc =
∂〈n1+n2〉/∂µ ≈ 2/(U11 + U12) ∼ 1/α. We also note the
convergence of discussed quantities with Γ → 0 to C ≈
1− |〈n1+n2〉 − 2|/2 and G ≈ G0 sin
2(π〈n1+n2〉/2).
Entanglement evolution with R is illustrated in Fig. 3,
where we focus on the charge neutral section 〈n1+n2〉 =
2. The abrupt entanglement drop follows the sharp con-
ductance peak for Γ = t/9, which is associated with the
competition between double Kondo and spin/charge sin-
glet phases [12]. For Γ = t/4 both C andG dependence onR
become smooth, but the switching behavior is still present.
Earlier, we have shown that Γ = t/4 is large enough to
cause molecule instability and therefore may allow the in-
dividual atom manipulation [8].
In conclusion, we analyzed a pair entanglement of elec-
trons in diatomic molecule attached serially to the leads.
Entanglement evolution with the chemical potential speeds
up remarkably for the negatively charged system, due to
electron correlation effects. The switching behavior was also
observed when changing the interatomic distance.
The work was supported by Polish Science Foundation
(FNP), and Ministry of Science Grant No. 1 P03B 001 29.
References
[1] See review by C.H. Bennet and D.P. Divincenzo, Nature 404, 247
(2000); M.A. Nielsen and I. L. Chuang, Quantum Computation
and Quantum Information (Cambridge, 2000).
[2] A. Osterloh et al., Nature 416, 608 (2002); T. J. Osborne and
M. A. Nielsen, Phys. Rev. A 66, 032110 (2002); S.–J. Gu et al.,
Phys. Rev. Lett. 93, 086402 (2004).
[3] J. van Wezel, J. van den Brink, J. Zaanen, Phys. Rev. Lett. 94,
230401 (2005); cond-mat/0606140.
[4] A. Rycerz, Eur. Phys. J. B 52, 291 (2006); S. Oh, J. Kim, Phys.
Rev. B 73, 052407 (2007).
[5] A. Ramšak, J. Mravlje, R. Žitko, J. Bonča, Phys. Rev. B 74,
241305(R) (2006); R. Žitko, J. Bonča, ibid. 74, 045312 (2006).
[6] W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
[7] R.H.M. Smit et al., Nature 419, 906 (2002).
[8] J. Spa lek et al., cond-mat/0610815.
[9] D. Djukic, J.M. van Ruitenbeek, Nano. Lett. 6, 789 (2006); M.
Kiguchi et al., cond-mat/0612681.
[10] J. C. Slater, Quantum Theory of Molecules and Solids, McGraw–
Kill (New York, 1963), Vol. 1, p. 50.
[11] T. Rejec, A. Ramšak, Phys. Rev. B 68, 033306 (2003).
[12] P.S. Cornaglia, D.R. Grempel, Phys. Rev. B 71, 075305 (2005);
J. Mravlje, A. Ramšak, T. Rejec, ibid. 73, 241305(R) (2006).
	References

  We investigate a pair entanglement of electrons in diatomic molecule, modeled
as a correlated double quantum dot attached to the leads. The low-temperature
properties are derived from the ground state obtained by utilizing the
Rejec-Ramsak variational technique within the framework of EDABI method, which
combines exact diagonalization with ab initio calculations. The results show,
that single-particle basis renormalization modifies the entanglement-switch
effectiveness significantly. We also found the entanglement signature of a
competition between an extended Kondo and singlet phases.

Diatomicmolecule as a quantum entanglement switch
Adam Rycerz ∗
Marian Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30–059 Kraków, Poland
Abstract
We investigate a pair entanglement of electrons in diatomic molecule, modeled as a correlated double quantum dot attached to
the leads. The low-temperature properties are derived from the ground state obtained by utilizing the Rejec-Ramšak variational
technique within the framework of EDABI method, which combines exact diagonalization with ab initio calculations. The results
show, that single-particle basis renormalization modifies the entanglement-switch effectiveness significantly. We also found the
entanglement signature of a competition between an extended Kondo and singlet phases.
Key words: Correlated nanosystems, Entanglement manipulation, EDABI method
PACS: 73.63.-b, 03.67.Mn, 72.15.Qm
Quantum entanglement, as one of the most intriguing
features of quantum mechanics, have spurred a great deal
of scientific activity during the last decade, mainly because
it is regarded as a valuable resource in quantum commu-
nication and information processing [1]. The question on
entanglement between microscopic degrees of freedom in a
condensed phase have been raised recently [2], in hope to
shed new lights on the physics of quantum phase transitions
and quantum coherence [3]. In the field of quantum elec-
tronics, a pair entanglement appeared to be a convenient
tool to characterize the nature of transport through quan-
tum dot, since its vanish when the system is in a Kondo
regime [4]. The analogical behavior was observed for two
qubits in double quantum dot, for either serial and paral-
lel configuration [5]. The latter case is intriguing, since the
concurrence [6] at T = 0 changes abruptly from C ≈ 1 to
C = 0 when varying the interdot coupling, so a finite An-
derson system shows a true quantum phase transition.
Here we consider a nanoscale version of such an entangle-
ment switch, inspired by conductance measurements for a
single hydrogenmolecule [7]. A special attention is payed to
electron-correlation effects, in particular the wave-function
renormalization [8]. Recent experiment [9] shows the cur-
rent through a molecule is carried by a single conductance
channel, so serial configuration shown in Fig. 1 seems to be
the realistic one. The Hamiltonian of the system is
∗ Corresponding author. Tel: (+48 12) 663–55–68 Fax: (+48 12)
633–40–79
Email address: rycerz@th.if.uj.edu.pl (Adam Rycerz).
���������� �������� �������� ������������������������ ��������
Fig. 1. Diatomic molecule modeled as a double quantum dot attached
serially to the leads. A cross-section of the single-particle potential
along the main system axis is shown schematically.
H = HL + VL +HC + VR +HR, (1)
where HC models the central region, HL(R) describes the
left (right) lead, and VL(R) is the coupling between the lead
and the central region. Both HL(R) and VL(R) terms have a
tight–binding form, with the chemical potential in leads µ,
the hopping t, and the tunneling amplitude V , as depicted
schematically in Fig. 1. The central-region Hamiltonian
iσcjσ +
iσ 6=jσ′
Uijniσniσ′ + (Ze)
2/R (2)
(with i, j = 1, 2 and σ =↑, ↓) describes a double quan-
tum dot with electron-electron interaction. tij and Uij are
single-particle and interaction elements, the last term de-
scribes the Coulomb repulsion of the two ions at the dis-
tance R. Here we put Z = 1 and calculate all the param-
eters tij , Uij as the Slater integrals [10] for 1s-like hydro-
genic orbitals Ψ1s(r) =
α3/π exp(−α|r|), where α−1 is
the orbital size (cf. Fig. 1). The parameter α is optimized
to get a minimal ground-state energy for whole the system
Preprint submitted to Elsevier 15 November 2018
http://arxiv.org/abs/0704.0743v2
-3 -2 -1  0  1
chemical potential,   
 0  1  2  3  4
 average filling,   
〈n1+n2〉
Fig. 2. Entanglement and transport through the system in Fig. 1 as
a function of the chemical potential µ (top panel) and the average
filling 〈n1+n2〉 (bottom panel). Tick (thin) solid and dashed lines
shows the concurrence C (conductance G) for Γ = t/9 and t/4,
respectively. The limits Γ → 0 are depicted with dotted lines in the
bottom panel. The interatomic distance is R = 1.5a.
 1  2  3  4  5  6
interatomic distance,   
Fig. 3. Concurrence (tick lines) and conductance (thin lines) at
the half-filled sector 〈n1+n2〉 = 2 as a function of the interatomic
distance R. The remaining parameters are the same as in Fig. 2.
described by the Hamiltonian (1). Thus, following the idea
of EDABI method [8], we reduce the number of physical
parameters of the problem to just a three: the interatomic
distance R, the lead-molecule hybridization Γ = V 2/t, and
the chemical potential µ (we put the lead hopping t =
1 Ry = 13.6 eV to work in the wide–bandwidth limit).
The entanglement between electrons placed on two atoms
can be characterized by the charge concurrence [4]
C = 2max
0, |〈c
iσcjσ〉| −
〈niσnjσ〉〈n̄iσ n̄jσ〉
where n̄iσ ≡ 1−niσ. We also discuss the conductivity cal-
culated from the formula G = G0 sin
2(E+−E−)/4tN [11],
where G0 = 2e
2/h̄, and E± are the ground-state ener-
gies of the system with periodic and antiperiodic boundary
conditions, respectively. Either the energies E± or correla-
tion functions in Eq. (3) are calculated within the Rejec–
Ramšak variational method [11], complemented by the or-
bital size optimization, as mentioned above. We use up to
N = 104 sites to reach the convergence.
In Fig. 2 we show the concurrence and conductance for
R = 1.5a0 (where a0 is the Bohr radius) and two values
of the hybridization Γ = t/9 and t/4. The conductance
spectrum asymmetry, caused by wave-function renormal-
ization [8], is followed by an analogical effect on entangle-
ment, which changes significantly faster for the upper con-
duction band, where the average filling is 〈n1+n2〉 ≈ 3
(one extra electron). The asymmetry vanish when analyz-
ing the system properties as a function of 〈n1+n2〉, show-
ing it originates from varying charge compressibility χc =
∂〈n1+n2〉/∂µ ≈ 2/(U11 + U12) ∼ 1/α. We also note the
convergence of discussed quantities with Γ → 0 to C ≈
1− |〈n1+n2〉 − 2|/2 and G ≈ G0 sin
2(π〈n1+n2〉/2).
Entanglement evolution with R is illustrated in Fig. 3,
where we focus on the charge neutral section 〈n1+n2〉 =
2. The abrupt entanglement drop follows the sharp con-
ductance peak for Γ = t/9, which is associated with the
competition between double Kondo and spin/charge sin-
glet phases [12]. For Γ = t/4 both C andG dependence onR
become smooth, but the switching behavior is still present.
Earlier, we have shown that Γ = t/4 is large enough to
cause molecule instability and therefore may allow the in-
dividual atom manipulation [8].
In conclusion, we analyzed a pair entanglement of elec-
trons in diatomic molecule attached serially to the leads.
Entanglement evolution with the chemical potential speeds
up remarkably for the negatively charged system, due to
electron correlation effects. The switching behavior was also
observed when changing the interatomic distance.
The work was supported by Polish Science Foundation
(FNP), and Ministry of Science Grant No. 1 P03B 001 29.
References
[1] See review by C.H. Bennet and D.P. Divincenzo, Nature 404, 247
(2000); M.A. Nielsen and I. L. Chuang, Quantum Computation
and Quantum Information (Cambridge, 2000).
[2] A. Osterloh et al., Nature 416, 608 (2002); T. J. Osborne and
M. A. Nielsen, Phys. Rev. A 66, 032110 (2002); S.–J. Gu et al.,
Phys. Rev. Lett. 93, 086402 (2004).
[3] J. van Wezel, J. van den Brink, J. Zaanen, Phys. Rev. Lett. 94,
230401 (2005); cond-mat/0606140.
[4] A. Rycerz, Eur. Phys. J. B 52, 291 (2006); S. Oh, J. Kim, Phys.
Rev. B 73, 052407 (2007).
[5] A. Ramšak, J. Mravlje, R. Žitko, J. Bonča, Phys. Rev. B 74,
241305(R) (2006); R. Žitko, J. Bonča, ibid. 74, 045312 (2006).
[6] W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
[7] R.H.M. Smit et al., Nature 419, 906 (2002).
[8] J. Spa lek et al., cond-mat/0610815.
[9] D. Djukic, J.M. van Ruitenbeek, Nano. Lett. 6, 789 (2006); M.
Kiguchi et al., cond-mat/0612681.
[10] J. C. Slater, Quantum Theory of Molecules and Solids, McGraw–
Kill (New York, 1963), Vol. 1, p. 50.
[11] T. Rejec, A. Ramšak, Phys. Rev. B 68, 033306 (2003).
[12] P.S. Cornaglia, D.R. Grempel, Phys. Rev. B 71, 075305 (2005);
J. Mravlje, A. Ramšak, T. Rejec, ibid. 73, 241305(R) (2006).
	References
