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terminated surfaces. The band features measured on Sb (marked by orange arrows in Fig. 4c-
4g) and Cr (marked by orange arrows in Fig. 4j-4n) surfaces exhibit negligible photon energy
dependence [momentum distribution curve (MDC) analysis of band features measured on Cr
terminated surface see Fig. S10 in SM], evidencing that it corresponds to a SS. Comparison
with the corresponding calculated SSs, shows that their band features (marked by orange | [
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] |
with the corresponding calculated SSs, shows that their band features (marked by orange
arrows in Fig. 4c-4g and Fig. 4j-4n) agree well with the SFAs (marked by orange arrows
in Fig. 4b and 4i), indicating that the SFAs are intrinsic and robust. Their traces are also
observed at the Fermi energy (Fig. 2l and 2m). Furthermore, the DFT calculations show
the kz location of WPs at an energy of -0.357 eV (round dots in Fig. 4a-4b and 4h-4i) is | [
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] |
the kz location of WPs at an energy of -0.357 eV (round dots in Fig. 4a-4b and 4h-4i) is
±0.27 π/c. This corresponds to a photon energy of about 82 eV. We show the WP in the
band dispersion panel measured with 82 eV (Fig. 4d) and indeed it is located right at the
intersection of the bands. In addition, we also measured the band dispersion along Γ−K with
photon energy of 85 eV on Sb (Fig. 4q) and Cr (Fig. 4t) surfaces (for more detailed photon | [
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] |
photon energy of 85 eV on Sb (Fig. 4q) and Cr (Fig. 4t) surfaces (for more detailed photon
energy dependent measurements along this direction see Fig. S11 in SM). Also along these
high-symmetry lines, SFAs (denoted by orange arrows in Fig. 4q and 4t) can be observed
for both surface terminations, reflecting the large portion of the surface BZ traversed by the
SFA, consistent with the calculations.
Thus our high-resolution and spin-resolved ARPES measurements quantify the altermag- | [
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] |
Thus our high-resolution and spin-resolved ARPES measurements quantify the altermag-
netic spin-spliting of bands in CrSb, establishing that it ranges from 1 eV to zero at high
symmetry lines in the BZ. This momentum-dependent splitting leads to the presence of two
6
types of topological nodal structures in the electronic structure, opposite-spin and same-
spin WPs. The SFAs and WPs, as well as the consistency with their calculated properties, | [
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] |
spin WPs. The SFAs and WPs, as well as the consistency with their calculated properties,
unequivocally establishes altermagnetic CrSb as a Weyl semimetal. Whereas opposite-spin
WPs can also be encountered in canonical antiferromagnets, the spin-full points are beyond
it.
They produce spin-polarized SFAs with a magnetization that depends on the spin-
projection of the WPs that they connect. The same-spin WPs carry two degrees of freedom: | [
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projection of the WPs that they connect. The same-spin WPs carry two degrees of freedom:
spin and topological charge. In altermagnets, crystal symmetries connecting the magnetic
sublattices and improper rotation relate the same-spin WPs with different spin-projection
and topological charge, respectively. These two degrees of freedom are locked, and conse-
quently the physical response to external perturbations (such as strain and electric field) | [
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] |
quently the physical response to external perturbations (such as strain and electric field)
of spin will also reveal in topological features, particularly the ensuing topological SSs, and
vice versa. This implies that the unique bulk altermagnetic spin-transport properties are
promoted by topology to be properties of the Fermi arcs at the surface as well, which may
render same-spin Weyl altermagnets interesting materials for spintronics interfaces. These | [
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] |
render same-spin Weyl altermagnets interesting materials for spintronics interfaces. These
insights not only spotlight the distinctive altermagnetic attributes of CrSb but also its ca-
pacity to induce novel physics and potential applications in the area of topological materials.
Methods
Sample The CrSb single crystals were grown by the chemical vapor transport (CVT)
method. A stoichiometric ratio of chromium and antimony powders, together with iodine | [
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] |
method. A stoichiometric ratio of chromium and antimony powders, together with iodine
of 2.5 mg/ml as the transport agent, were mixed and sealed in an evacuated quartz am-
poule. The ampoule was slowly heated and finally exposed to a temperature gradient of
925°C to 900°C where the CVT preceded for one week, then naturally cooled down to room
temperature. CrSb crystals in size of 5 mm with regular shapes and shiny surfaces were
obtained. | [
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] |
temperature. CrSb crystals in size of 5 mm with regular shapes and shiny surfaces were
obtained.
ARPES Measurements High-resolution ARPES measurements were performed at the
Bloch beamline of MAX IV and at the I05 beamline of the Diamond synchrotron light
source. The total energy resolution (analyzer and beamline) was set at 15∼20 meV for the
measurements. The angular resolution of the analyser was ∼0.1 degree. The beamline spot | [
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] |
measurements. The angular resolution of the analyser was ∼0.1 degree. The beamline spot
size on the sample was about 10 µm×12 µm at the Bloch beamline of MAX IV and about 70
µm×70 µm at the I05 beamline of the Diamond synchrotron. The samples were cleaved in
situ and measured at about 18 K at the Bloch beamline of MAX IV and about 10 K at the
7
I05 beamline of the Diamond synchrotron in ultrahigh vacuum with a base pressure better | [
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] |
7
I05 beamline of the Diamond synchrotron in ultrahigh vacuum with a base pressure better
than 1.0×10−10 mbar. The spin-resolved ARPES (SARPES) measurements were performed
at the Bloch B-branch beamline of MAX IV with photon energy of 102 eV. The samples were
cleaved in situ and measured at about 77 K at the Bloch B-batch beamline of MAX IV. The
total energy resolution (analyzer and beamline) was set at ∼50 meV for the measurements.
The angular resolution of the analyser was ∼1.3 degree. | [
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] |
The angular resolution of the analyser was ∼1.3 degree.
DFT calculations Full relativistic spin-polarized electronic structure and Fermi surface
calculations are done using the full potential local orbital code FPLO [43, 44], on a k-mesh
of 20 × 20 × 13 points. We use the generalized gradient approximation to the exchange and
correlation potential by Perdew, Burke, and Ernzerhof (PBE). The SS and WPs calcula-
tions are performed based on the DFT calculation from VASP [45] employing the projector | [
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] |
tions are performed based on the DFT calculation from VASP [45] employing the projector
augmented wave method [46]. The Brillouin zone is sampled with a 10×10×8, with Gamma-
centered k-point. The energy cutoff of the plane wave basis is set to 550 eV. The Hubbard
term was introduced and set to be 0.8 eV in the d-orbital of Cr-atom in the DFT framework
(DFT+U) in order to account for the electron-electron correlation. The Wannier Hamilto- | [
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] |
(DFT+U) in order to account for the electron-electron correlation. The Wannier Hamilto-
nian is symmetrized based on the maximally localized Wannier functions generated by the
WANNIER90 interface [47]. To locate the WPs and calculate the chirality, Wanniertools is
implemented [48].
Data Availability
The authors declare that all data supporting the findings of this study are available within
the paper and its Supplementary Information files. | [
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] |
the paper and its Supplementary Information files.
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with Nonrelativistic Spin and Crystal Rotation Symmetry. Phys. Rev. X 12, 031042 (2022).
[2] L. Smejkal et al., Emerging Research Landscape of Altermagnetism. Phys. Rev. X 12, 040501
(2022).
[3] Y. Guo et al., Spin-split collinear antiferromagnets: A large-scale ab-initio study. Materials
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] |
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] |
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Acknowledgements
The work presented here was financially supported by the Swedish Research council (2019-
00701) and the Knut and Alice Wallenberg foundation (2018.0104).
We thank Ulrike
Nitzsche for technical assistance. | [
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] |
We thank Ulrike
Nitzsche for technical assistance.
We acknowledge financial support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation), through SFB 1143 (Project
ID 247310070), project A05, Project No. 465000489, and the W¨urzburg-Dresden Cluster of
Excellence on Complexity and Topology in Quantum Matter, ct.qmat (EXC 2147, Project
ID 390858490). M.G.V. thanks support from the Deutsche Forschungsgemeinschaft (DFG, | [
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] |
ID 390858490). M.G.V. thanks support from the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) GA3314/1-1 -FOR 5249 (QUAST) and to the Spanish Min-
isterio de Ciencia e Innovacion grant PID2022-142008NB-I0. M.L.H. thanks the support
from the Alexander von Humboldt Foundation and the useful discussion and help of I˜nigo
Robredo. Z.L.L thanks the support from the Youth Innovation Promotion Association of
Chinese Academy of Sciences (No. 2021008). H.L. was supported by the Swedish Research
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] |
Chinese Academy of Sciences (No. 2021008). H.L. was supported by the Swedish Research
11
Council (VR, grant 2018-00313), the Wallenberg Academy Fellows program of the Knut
and Alice Wallenberg Foundation (2018.0460) and the G¨oran Gustafsson Foundation for
Research in Natural Sciences and Medicine. We acknowledge MAX IV Laboratory for time
on Beamline BLOCH under Proposal 20230262 and 20231119. Research conducted at MAX | [
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] |
on Beamline BLOCH under Proposal 20230262 and 20231119. Research conducted at MAX
IV, a Swedish national user facility, is supported by the Swedish Research council under con-
tract 2018-07152, the Swedish Governmental Agency for Innovation Systems under contract
2018-04969, and Formas under contract 2019-02496.
Author Contributions
C.L. and H.L. proposed the project.
C.L., M.L.H., H.L., O.T. and J.V.D.B. conceived
the project. C.L. carried out the ARPES experiments with the assistance from Y.W. and | [
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] |
the project. C.L. carried out the ARPES experiments with the assistance from Y.W. and
W.Y.C.. M.L.H. and O.J. contributed to the band structure calculations. Z.L.L. contributed
to CrSb crystal growth. C.L. contributed to software development for data analysis and
analyzed the data. C.L., M.L.H., H.L., O.T. and J.V.D.B. wrote the paper. B.T., M.L.,
C.P. and T.K. provided the beamline support. C.L, M.L.H., Y.W., H.L., C.F., M.V., O.J.,
O.T. and J.V.D.B. contributed to the scientific discussions. | [
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] |
O.T. and J.V.D.B. contributed to the scientific discussions.
All authors participated in
commented on the paper.
Competing Interests
The authors declare no competing interests.
12
FIG. 1. Crystal structure, WPs and calculated electronic structure of CrSb. (a) The crystal
structure of CrSb with the space group P63/mmc (no. 194). (b) The 3D BZ of the original unit cell of CrSb,
and the corresponding two-dimensional BZ projected on the (001) plane (green lines). The red, green, blue | [
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] |
and orange planes are the four mirror planes in the BZ on which the altermagnetic bands are spin-degenerate
in absence of SOC. (c) Calculated bulk Fermi surface in the first BZ. The colormap from blue to orange of
Fermi surface sheet represents the expectation value of spin along the z direction. (d-e) Illustration of the
two types of WPs: opposite-spin (d) and same-spin (e). Top (f) and side (g) views of the distribution of | [
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] |
same-spin WPs in the 3D BZ. Red dots represent nodes with chirality χ = +1, green dots χ = −1; the spin
polarizations are indicated by cyan and pink. The blue line represents spin-up and spin-down degeneracy.
(h-i) The calculated SS of (001) on Sb (h) and Cr (i) terminations along M1 − Γ − M2 − K2 − Γ − K1 − M1.
The WPs are marked by the red (χ = 1) and green (χ = −1) dots. Star, square and round dots represent | [
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] |
the opposite-spin WPs on the high symmetry line/plane (WP1), opposite-spin WPs on general momenta
(WP2) and same-spin WPs (WP3), respectively.
13
FIG. 2. Fermi surface of CrSb. (a-b) projected Fermi surfaces for Sb (a) and Cr (b) surface terminations.
(c-h) Calculated bulk Fermi surfaces at kz = 1 π/c (c), 0.8 π/c (d), 0.6 π/c (e), 0.4 π/c (f), 0.2 π/c (g)
and 0 π/c (h) planes, integrated over a ±0.1 π/c interval. (i) Photon energy dependent ARPES spectral | [
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] |
intensity map at Fermi level along M−Γ−M. (j) corresponding calculated kz-kx Fermi surface. (k) Photon
energy dependent Fermi surfaces measured with photon energies of 97 eV (i), 102 eV (ii), 107 eV (iii), 112
eV (iv), 117 eV (v), 122 eV (vi) on an uneven surface. (l-m) The same measurements on flat areas of Sb (l)
and Cr (m) terminated surfaces. The SSs are marked by orange arrows. The corresponding sample photos
are shown in the bottom left corner of k(i), l(i) and m(i).
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] |
are shown in the bottom left corner of k(i), l(i) and m(i).
14
FIG. 3. Band dispersions of CrSb. (a) The photon energy dependent ARPES spectra along M − Γ − M
measured with photon energies of 97 eV (i), 102 eV (ii), 107 eV (iii), 112 eV (iv), 117 eV (v) and 122 eV
(vi) on an uneven surface. (b-c) The same measurements on flat areas of Sb (b) and Cr (c) terminated
surfaces. (d-i) The DFT calculated spin polarized band structures along M − Γ − M direction at kz = 1 | [
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] |
π/c (d), 0.8 π/c (e), 0.6 π/c (f), 0.4 π/c (g), 0.2 π/c (h) and 0 π/c (i) planes. The red bands represent a
spin-up and the blue bands spin-down along the z direction. (j-k) Spin-polarized EDCs along the left (j)
and right (k) cut lines indicated in [a(ii)]. The spin-polarized EDCs were measured with photon energy of
102 eV, while the red (blue) curve corresponds to spin-up (spin-down) along z. The EDCs intensity are | [
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] |
already normalized in the non-spin polarized region. In order to obtain sufficient spin statistics, more than
30 hours were accumulated in total. (l) Spin-polarized intensity difference between spin-up and spin-down,
with the red (blue) colors indicating spin-up (spin-down) polarization for the left (right) bands.
15
FIG. 4. Weyl fermions and Fermi arcs in CrSb. (a) Calculated total spectral density along M−Γ−M | [
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] |
15
FIG. 4. Weyl fermions and Fermi arcs in CrSb. (a) Calculated total spectral density along M−Γ−M
for CrSb terminated by a Sb (001) surface. (b) surface electronic structure as in (a) but with bulk spectral
weight subtracted. The WPs together with the chirality are marked by red (χ = 1) and green (χ = −1) dots.
Square and round dots represent the opposite-spin WPs at general momenta (WP2) and same-spin WPs | [
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] |
Square and round dots represent the opposite-spin WPs at general momenta (WP2) and same-spin WPs
(WP3), respectively. (c-g) The photon energy dependent spectra on a Sb terminated surface with photon
energies of 77 eV (a), 82 eV (b), 85 eV (c), 89 eV (d), 93 eV (e) under LH polarization along M − Γ − M.
(h-n) The same calculations and measurements as (a-g) for the Cr terminal surface. In (j) on the left also | [
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] |
the bulk bands for kz = 0 π/c are shown. (o) Calculated total spectral density along K − Γ − K of the
Sb surface. (p) The same but with bulk contribution subtracted. (q) Measured band dispersion along the
K − Γ − K direction of a Sb terminated surface with photon energy of 85 eV. (r-t) The same calculations
and measurements as (o-q) but on a Cr terminated surface. The bulk calculations for kz = 0.4 π/c are | [
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also shown on the left side of (q) and (t). The SFAs are marked by orange arrows. The left sides in (c-g)
correspond to the EDC second derivative and (j-n) show the MDC second derivative, enhancing the visibility
of the bands.
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] |
Controlling dephasing of coupled qubits via shared-bath coherence
L. M. J. Hall,∗ L. S. Sirkina, A. Morreau, W. Langbein, and E. A. Muljarov†
School of Physics and Astronomy, Cardiff University, Cardiff CF24 3AA, United Kingdom
(Dated: May 24, 2024)
The interaction of a quantum system with its environment limits qubit coherence times and re-
stricts its utility in quantum information processing applications.
In this Letter, we show that | [
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] |
stricts its utility in quantum information processing applications.
In this Letter, we show that
the decoherence of a coupled qubit system can be minimized, or even eliminated by exploiting the
quantum coherence of the bath itself. We investigate the dephasing in a system of two spatially sep-
arated, electronically decoupled qubits, with direct or mediated coupling, interacting with a shared
bath. For illustration we treat F¨orster or cavity-mediated coupling between semiconductor quantum | [
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] |
bath. For illustration we treat F¨orster or cavity-mediated coupling between semiconductor quantum
dots interacting with acoustic phonons. Using the rigorous method of Trotter’s decomposition with
cumulant expansion, we demonstrate a reduction in the dephasing rates at specific distances. This
control is a coherent effect of the shared bath and is absent for independent baths. It can be un-
derstood in terms of phonon-assisted transitions between the entangled qubit states of the coupled
system. | [
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] |
system.
A quantum bit (qubit) is a two-level quantum-
mechanical system which is the basic unit of quantum
information, analogous to the classic binary bit.
Co-
herent superpositions of qubit states have application
in quantum information processing and quantum cryp-
tography [1].
Two-qubit logic gates, in particular, are
a fundamental building block in any quantum comput-
ing architecture [2, 3]. Such gates require a controlled
long-range interaction between isolated qubits, which can | [
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] |
long-range interaction between isolated qubits, which can
be mediated by their strong coupling to a photonic cav-
ity [4]. While there are many possible physical imple-
mentations of a qubit, we will focus on semiconductor
quantum dots (QDs), which are often called “artificial
atoms” and are promising candidates for qubits. In fact,
quantum interference of single photons emitted by re-
mote GaAs quantum dots (QDs) has been experimentally
demonstrated [5–7], with QDs coupled to optical cavities | [
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] |
demonstrated [5–7], with QDs coupled to optical cavities
shown to enhance this effect [8].
The interaction of a quantum system with its environ-
ment limits qubit coherence times and restricts their ap-
plication [9, 10]. As cavities historically were of poorer
quality, the qubit-cavity coupling was a prominent de-
phasing mechanism and consequently, the cavity was re-
garded as the bath. Therefore, previous works exploiting
bath coherent properties to reduce dephasing have fo- | [
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] |
bath coherent properties to reduce dephasing have fo-
cused on decoherence-free subspaces of subradiant quan-
tum superpositions [11, 12].
Here, the introduction of
a second qubit coupled to the same cavity gives rise to
a subradiant superposition state that is decoupled from
the lossy cavity. It has also been shown that the strong
coupling of distant qubits to a cavity induces temporal
oscillations and revivals of qubit entanglement [13], ow-
ing to the bath memory depth. | [
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] |
oscillations and revivals of qubit entanglement [13], ow-
ing to the bath memory depth.
Although the quality factor of optical cavities dramat-
ically increased over the past decade, the environment,
typically described by a thermal bath, still remains a
major source of decoherence in solid-state qubits. In par-
ticular, acoustic phonons present an intrinsic source of
dephasing in semiconductor QDs. Even at low tempera-
tures, acoustic phonons induce a rapid non-Markovian | [
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] |
tures, acoustic phonons induce a rapid non-Markovian
decay of the QD coherence [14, 15], also known as a
phonon broad band (BB) in the QD spectrum, followed
by a nearly Markovian long-time decay of the zero-
phonon line (ZPL) due to real or virtual phonon-assisted
transitions to other QD levels [16, 17].
The QD interaction with a phonon bath fundamen-
tally differs from the bilinear QD-cavity coupling, so the
above mentioned idea of decoherence-free subspaces of
qubit states is not applicable here. | [
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] |
above mentioned idea of decoherence-free subspaces of
qubit states is not applicable here.
Nevertheless, suc-
cessful attempts have been made to reduce QD deco-
herence, although none of them were based on coherent
properties of the bath. In particular, using a controlled
off-resonant optical pulse with the laser pulse frequency
tuned to the BB allows one to prepare almost pure qubit
states by using phonon assisted transitions [18, 19]. No-
tably, this only works for low enough temperatures to | [
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] |
tably, this only works for low enough temperatures to
neglect phonon absorption. The idea has been general-
ized to a phonon-assisted two-photon excitation scheme
to create indistinguishable entangled photon pairs from
remote QDs [20]. The Purcell effect helps to reduce the
phonon-induced decoherence by a resonant weak coupling
of a QD exciton to a cavity mode that results in reduction
of the relative weight of the BB and enhancement of the
ZPL emission [21]. Moreover, in the QD-cavity strong | [
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] |
ZPL emission [21]. Moreover, in the QD-cavity strong
coupling regime, the BB is almost entirely eliminated in
the cavity excitation scheme [22, 23]. However, the ZPL
gains an additional dephasing [24] which can be under-
stood and quantified in terms of phonon-assisted tran-
sitions between the polariton states of the system [25].
Such a ZPL dephasing can be even stronger in coupled
qubits where all parts of the system are interacting with
the bath.
In this Letter, we demonstrate a reduction or even | [
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] |
the bath.
In this Letter, we demonstrate a reduction or even
a complete elimination of the ZPL dephasing in a sys-
tem of two QD qubits coupled to each other directly or
via an optical cavity.
We show that, while the inter-
action of entangled qubits with a shared environment
usually causes dephasing of qubit states, the coherent
arXiv:2405.14685v1 [cond-mat.mes-hall] 23 May 2024
2
FIG. 1. Schematic of the system with a pair of dipole-coupled
qubits separated by a distance vector d, coupled to an op- | [
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] |
qubits separated by a distance vector d, coupled to an op-
tical cavity, and interacting with three-dimensional acoustic
phonons described by a wave vector q and angle θ.
properties of the bath can help to reduce this decoher-
ence. As a practical example, we consider the decoher-
ence of electronically decoupled qubits separated by a
distance d.
The qubits may interact with each other
directly through dipolar F¨orster-type coupling [26–28],
or indirectly through cavity-mediated coupling [29–31], | [
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] |
or indirectly through cavity-mediated coupling [29–31],
while both qubits interact with a shared bath. We use
a semiconductor QD interacting with a bath of three-
dimensional (3D) acoustic phonons, as a candidate for a
qubit, well studied in the literature [14–17, 32].
The system Hamiltonian can be written as a sum of
two exactly solvable parts,
H = H0 + HIB ,
(1)
where H0 describes the coupling between the qubits and
the cavity, and HIB is a generalized independent boson | [
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] |
the cavity, and HIB is a generalized independent boson
(IB) model Hamiltonian describing the coupling of the
qubits to the shared environment. For the system illus-
trated in Fig. 1 of two remote QDs coupled to an optical
cavity, H0 takes the form (ℏ = 1)
H0 = Ω1d†
1d1 + Ω2d†
2d2 + ΩCa†a + g(d†
1d2 + d†
2d1)
+ g1(d†
1a + a†d1) + g2(d†
2a + a†d2) ,
(2)
where d†
j is the fermionic exciton creation operator in QD
j (j = 1, 2), a† is the cavity photon creation operator, | [
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] |
j (j = 1, 2), a† is the cavity photon creation operator,
Ωj (ΩC) is the exciton (cavity photon) frequency, g and
gj are, respectively, the coupling strengths between the
QD excitons and between the exciton in QD j and the
cavity photon. The IB model Hamiltonian describes the
interaction of the QD excitons with a shared acoustic-
phonon bath,
HIB = Hph + d†
1d1V1 + d†
2d2V2 ,
(3)
where
Hph =
�
q
ωqb†
qbq
and
Vj =
�
q
λq,j(bq + b†
−q)
(4)
are, respectively, the free phonon bath Hamiltonian and | [
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] |
ωqb†
qbq
and
Vj =
�
q
λq,j(bq + b†
−q)
(4)
are, respectively, the free phonon bath Hamiltonian and
the QD coupling to the bath, via the bosonic creation op-
erator b†
q of a phonon in the bulk mode with the momen-
tum q and frequency ωq (where q = |q|). The coupling
of the exciton in QD j to the phonon mode q is given
by the matrix element λq,j, which depends on the ma-
terial parameters, exciton wave function, and position of
the QD [33]. For identical QD qubits separated by a dis- | [
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] |
the QD [33]. For identical QD qubits separated by a dis-
tance vector d, the matrix elements satisfy an important
relation
λq,2 = eiq·dλq,1 .
(5)
In the following, we focus on the linear optical polar-
ization [33] of the qubits, Pj(t) = Tr{ρ(t)dj} (with ρ(t)
being the full density matrix), allowing us to study the
coherence of the system as a function of the distance be-
tween the qubits. Taking advantage of the two exactly
solvable parts of the Hamiltonian Eq. (1), we apply the | [
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] |
solvable parts of the Hamiltonian Eq. (1), we apply the
rigorous method of Trotter’s decomposition with cumu-
lant expansion [25, 33], allowing us to take into account
the effect of the phonon environment exactly.
Below we consider two special cases: Case 1 of directly
coupled qubits with interaction strength g but without
coupling to the cavity (g1 = g2 = 0) and Case 2 of qubits
without direct coupling (g = 0) but with indirect cou-
pling mediated by the cavity via g1 and g2. To clearly | [
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] |
pling mediated by the cavity via g1 and g2. To clearly
show the effect of the shared environment on the system
coherence and its dependence on the distance d = |d|
between the qubits, we assume here that the coupling
constants g1, g2, and g are distance independent.
We
also choose without loss of generality that the first QD
is instantaneously excited (e.g. by an ultrashort optical
pulse), creating an excitonic polarization with P1(0) = 1.
In Case 1, the time evolution of P1(t) for a system | [
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] |
In Case 1, the time evolution of P1(t) for a system
of dipolar-coupled (g = 0.5 meV) identical isotropic QDs
separated by the center-to-center distance d = 5 nm is
shown in Fig. 2(a) by a blue dotted line, demonstrating
decay and oscillations. This behavior is qualitatively ex-
plained by the energy level diagram in the right inset of
Fig. 2(a) showing hybridized states |±⟩ = (|1⟩ ± |2⟩)/
√
2
of the two-qubit coupled system at zero detuning (Ω1 =
Ω2), where |1⟩ and |2⟩ are the isolated-qubit states. The | [
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] |
Ω2), where |1⟩ and |2⟩ are the isolated-qubit states. The
energy levels are separated by the Rabi splitting 2g de-
termining the beat frequency in |P1(t)| which physically
expresses the quantum information exchange between
the qubits. The temporal decay of the linear polariza-
tion expresses the decoherence in this two-qubit system
as a consequence of the interaction of the qubits with
a shared bath.
For these QD qubits, the decoherence
is due to phonon-assisted transitions between the hy-
bridized states. | [
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] |
is due to phonon-assisted transitions between the hy-
bridized states.
With this picture in mind, we have applied to the long-
time dynamics of P1(t) a complex biexponential fit of the
form �
j Cje−iωjt, extracting the complex amplitudes
Cj, energies Re ωj, and dephasing rates Γj = − Im ωj
of the phonon-dressed mixed states. The fit, applied af-
ter the phonon-memory cut-off [33] [dashed green vertical
line in Fig. 2(a)], shows a remarkable agreement with the | [
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] |
line in Fig. 2(a)], shows a remarkable agreement with the
full calculation with a relative error below 10−12. At ear-
lier times, the deviation is due to the formation of a po-
3
0
5
10
15
20
25
time (ps)
0.0
0.5
1.0
|P1(t)|
d = 5nm
0
10
20
30
40
50
0
10
20
30
40
+
exact
FGR
indep. bath
0
10
20
30
40
50
distance between qubits, d (nm)
0
10
20
30
40
50
dephasing rate, ( eV)
+
(a)
(b)
(c)
FIG. 2.
(a) Linear optical polarization P1(t) (blue dots) | [
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0.3418038785457611
] |
50
dephasing rate, ( eV)
+
(a)
(b)
(c)
FIG. 2.
(a) Linear optical polarization P1(t) (blue dots)
and its complex bi-exponential fit (red lines) for dipolar cou-
pled isotropic QD qubits (left inset) at zero detuning, sep-
arated by a distance d = 5 nm, with excitation and mea-
surement in QD 1.
Right inset: energy level diagram for
the mixed qubit states, with real phonon-assisted transitions
(red and blue arrows).
(b,c) Dephasing rates Γ± of the | [
-0.0366717204451561,
-0.019265705719590187,
0.4501148760318756,
-0.014935326762497425,
0.01840497925877571,
0.37758222222328186,
0.06404231488704681,
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0.12457266449928284,
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0.40077826380729675,
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0.22349365055561066,
0.39072906970977783,
0.8580057621002197,
0.6059857606887817,
0.2540035545825958,
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0.24030357599258423,
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0.5107875466346741,
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0.00792640820145607,
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0.084470234811306,
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0.4482220411300659,
0.3768042325973511,
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0.5330607295036316,
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0.2179136723279953,
0.7774221301078796,
0.06297928094863892,
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0.12992966175079346,
0.6968889832496643,
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0.19012786448001862,
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0.1814420372247696,
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0.02697966992855072,
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0.09240949153900146,
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0.09525933116674423,
0.16007466614246368,
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0.6026210188865662,
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0.07618161290884018,
0.9904171824455261,
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0.6225525736808777,
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0.09258123487234116,
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-0.4239802956581116,
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0.49089914560317993,
0.41930750012397766,
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-0.11367601901292801,
0.6976661682128906,
0.21322372555732727,
-0.1922260820865631,
0.9848979711532593,
0.8440974354743958,
0.7580054998397827,
0.6579383015632629,
-0.2448195070028305,
-0.14690886437892914,
-0.7539442181587219,
0.20687885582447052,
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-0.043367791920900345,
-0.005292677320539951,
0.052869804203510284,
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-0.4565647840499878,
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-0.07880667597055435,
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0.5243096351623535,
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-0.5379182696342468,
0.2781793475151062,
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0.43379268050193787,
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] |
(red and blue arrows).
(b,c) Dephasing rates Γ± of the
mixed states |±⟩ as a function of d, calculated exactly (solid
lines) and via FGR (dashed lines) for (b) isotropic QDs with
the confinement length of l = 5.6 nm and (c) anisotropic
QDs with l = 7.5 nm and l⊥ = 2.5 nm along the separa-
tion. The rates for independent phonon baths are shown by
thin dashed lines. The phonon bath and coupling parame-
ters are [16, 17] Dc − Dv = −6.5 eV, vs = 4.6 × 103 m/s, | [
-0.023823173716664314,
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0.29808303713798523,
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0.34228286147117615,
-0.713603138923645,
0.5742323994636536,
0.30518051981925964,
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0.029540667310357094,
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0.31268617510795593,
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0.38738712668418884,
0.2900013327598572,
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0.21518342196941376,
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0.42944109439849854,
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1.191049337387085,
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0.2708873152732849,
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0.17731493711471558,
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0.2807064950466156,
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0.03842909634113312,
0.20676207542419434,
0.13409659266471863
] |
ters are [16, 17] Dc − Dv = −6.5 eV, vs = 4.6 × 103 m/s,
ρm = 5.65 g/cm3, dipolar coupling g = 0.5 meV and bath
temperature T = 20 K.
laron cloud around the optically excited QD, responsible
for non-Markovian dephasing and the BB [15, 23, 25].
The dephasing rates Γj extracted from the fit as func-
tions of the QD separation d are shown by solid lines
in Fig. 2(b) for isotropic and in Fig. 2(c) for anisotropic
QDs.
They are the dephasing rates of the states |±⟩, | [
0.045315470546483994,
-0.5715327262878418,
0.7538361549377441,
-0.32457706332206726,
-0.4148164391517639,
0.33133837580680847,
0.5120789408683777,
-0.7793511152267456,
0.19142219424247742,
0.00958996545523405,
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0.6321563720703125,
-0.061958763748407364,
0.5889695882797241,
-0.08958781510591507,
1.0923036336898804,
0.6307135224342346,
0.10343590378761292,
-0.32299208641052246,
-0.7255298495292664,
0.29391103982925415,
-0.03001951053738594,
0.7734991908073425,
-0.42196518182754517,
0.46645495295524597,
0.016098352149128914,
-0.1460782140493393,
0.10374288260936737,
0.1551826298236847,
0.23925337195396423,
-0.20528116822242737,
0.4959597587585449,
-0.3747352659702301,
-0.10130337625741959,
0.6804300546646118,
0.08415625989437103,
0.35604169964790344,
-0.1314731389284134,
0.06690169870853424,
-0.21992549300193787,
0.27076226472854614,
-0.2967805862426758,
0.31746411323547363,
0.3615368604660034,
-1.0349645614624023,
-0.25408148765563965,
-1.1446270942687988,
0.1929457187652588,
-0.2144717276096344,
-0.20808391273021698,
-0.18748535215854645,
-0.2353227585554123,
0.6416476368904114,
0.42090409994125366,
-0.262996643781662,
0.30370450019836426,
-0.6750717759132385,
-0.10045403987169266,
0.6282708644866943,
-0.45375940203666687,
0.15819178521633148,
0.11346534639596939,
-0.23818446695804596,
0.41845113039016724,
-0.1412852704524994,
0.36114832758903503,
0.3289423882961273,
-0.0009917545830830932,
-0.5760575532913208,
0.16747541725635529,
0.7037091851234436,
0.31424394249916077,
-0.3987041711807251,
-0.021964048966765404,
-0.08612485229969025,
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] |
QDs.
They are the dephasing rates of the states |±⟩,
denoted by Γ±, and can be understood as being due to
phonon assisted transitions between the states. At short
distances we observe a dramatic increase of the dephas-
ing rates, followed by oscillatory behaviour at larger dis-
tances. Importantly, the minima of the dephasing rates
are lower than the rates for independent baths (thin hor-
izontal lines) [33], and are thus a manifestation of shared
bath coherence. | [
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] |
izontal lines) [33], and are thus a manifestation of shared
bath coherence.
To understand the dependence on the distance be-
tween the qubits, we introduce fermionic operators p†
± =
D∓d†
1 ± D±d†
2, creating excitations in the mixed qubit
states and diagonalizing exactly the system Hamiltonian
H0 in the absence of the bath. The interaction with the
bath HIB then results in off-diagonal elements,
D+D−(V1 − V2)(p†
+p− + p†
−p+) ,
(6)
where D± =
�
(1 ± ∆/R)/2 with ∆ = Ω2 − Ω1 and
R =
� | [
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] |
D+D−(V1 − V2)(p†
+p− + p†
−p+) ,
(6)
where D± =
�
(1 ± ∆/R)/2 with ∆ = Ω2 − Ω1 and
R =
�
∆2 + 4g2 being, respectively, the detuning and
the Rabi splitting [33]. The off-diagonal terms Eq. (6)
enable phonon-assisted transitions between the hybrid
qubit states. The transition rates can be evaluated via
FGR [25, 33]: Γ− = NR Γph and Γ+ = (NR + 1) Γph,
where NR is the Bose distribution function taken at the
energy R and
Γph = Γ0
�
1 − sin(Rd/vs)
Rd/vs
�
(7)
for an isotropic model of the QDs.
In Eq.(7), Γ0 =
D2 | [
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] |
Γph = Γ0
�
1 − sin(Rd/vs)
Rd/vs
�
(7)
for an isotropic model of the QDs.
In Eq.(7), Γ0 =
D2
+D2
−R3(Dc − Dv)2/(2πρmv5
s)e−l2R2/v2
s,
with linear
phonon dispersion, ωq = vsq, where vs is the sound ve-
locity, ρm is the mass density, Dc (Dv) is the conduction
(valence) band deformation potential, and l is the Gaus-
sian localization length of the QD exciton. The FGR de-
phasing rates Eq. (7) are shown in Fig. 2(b,c) as dashed
lines, reproducing the main features of the exact calcu- | [
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] |
lines, reproducing the main features of the exact calcu-
lation, but showing discrepancies due to multi-phonon
processes not present in FGR. The single-phonon transi-
tions dominate at short distances as it is clear from the
excellent agreement between the two results.
The initial quadratic growth with distance, the oscil-
lations, and the reduction of Γ± at certain distances are
all caused by the coherent properties of the phonon bath.
According to Eq. (6), the phonon-assisted coupling be- | [
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] |
According to Eq. (6), the phonon-assisted coupling be-
tween the mixed qubit states is given by V1 − V2 which
is proportional to 1 − eiq·d [see Eq. (5)] and is vanishing
at q · d = 2πn, where n is an integer. This does not lead
to a vanishing dephasing though, apart from d = 0, ow-
ing to the 3D nature of the phonon momentum q of the
bath modes. However, in a 1D model of phonons with the
same dispersion and same coupling (which is not the case
of e.g. nanowire-based QDs [34] or QDs in carbon nan- | [
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] |
of e.g. nanowire-based QDs [34] or QDs in carbon nan-
otubes [35, 36]), the dephasing rate Eq. (7) would mod-
ify [33] to just Γph = Γ0[vs/(Rl) sin(Rd/2vs)]2, strictly
vanishing at Rd/vs = 2πn for all n.
To understand this phenomenon in 1D, let us take the
two-qubit state just before the event of phonon emission
or absorption as a superposition α|1⟩ + β|2⟩ with some
complex amplitudes α and β. Since the qubits are entan-
gled, they coherently emit or absorb the same phonon. | [
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0.42458146810531616,
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0.14692971110343933,
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1.0429569482803345,
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0.6109430193901062,
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0.7660781741142273,
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0.47412607073783875,
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0.8141169548034668,
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0.09537234902381897,
0.024539057165384293,
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0.5377596020698547,
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0.05530957132577896,
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0.20564799010753632,
0.03229408338665962,
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0.2986229658126831,
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0.006850346922874451,
0.19581478834152222,
-0.06844405829906464,
0.22589576244354248,
0.08343121409416199
] |
gled, they coherently emit or absorb the same phonon.
This changes their phases (which is usually the source of
decoherence) by φ1 and φ2, respectively, so that the two-
qubit wave function becomes αeiφ1|1⟩ + βeiφ2|2⟩, with
φ2 − φ1 = ±qd, according to Eq. (5) and energy conser-
vation requiring R = vsq. However, if Rd/vs = 2πn, the
4
0
20
40
60
80
100
distance between qubits, d (nm)
0.0
0.2
0.4
0.6
0.8
dephasing rate, ( eV)
+
101
102
103
10
3
10
2
10
1
100
Amplitudes
1/d
FIG. 3. | [
-0.14608319103717804,
-0.2963734269142151,
0.7696053385734558,
-0.1289781779050827,
-0.2996116578578949,
0.10918785631656647,
0.10309357196092606,
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-0.34755364060401917,
-0.1569867581129074,
-0.057443976402282715,
0.46726560592651367,
-0.1230962872505188,
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0.21760617196559906,
0.9390560984611511,
0.7083175778388977,
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-0.5881402492523193,
0.122942715883255,
0.18899638950824738,
0.5410547852516174,
0.2488718330860138,
0.08854984492063522,
-0.2253042757511139,
0.3248036801815033,
0.13151904940605164,
0.04343311861157417,
0.27536794543266296,
-0.22409619390964508,
0.6819795966148376,
0.18859970569610596,
0.5163772702217102,
0.5437602996826172,
0.3177499771118164,
0.0017310023540630937,
-0.2365739345550537,
-0.007725092116743326,
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0.005796313285827637,
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0.2807545065879822,
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0.275032103061676,
0.050163380801677704,
0.014722688123583794,
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0.03906848281621933,
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0.22259755432605743,
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0.17730624973773956,
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1.1056079864501953,
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] |
0.0
0.2
0.4
0.6
0.8
dephasing rate, ( eV)
+
101
102
103
10
3
10
2
10
1
100
Amplitudes
1/d
FIG. 3.
As Fig. 2(c) but for g = 2 meV and FGR only (solid
lines) Inset: amplitude of the oscillations in the dephasing
rates versus distance.
resulting wave function only acquires a common phase
factor eiφ1, and thus is not changing the state, making
such a process forbidden. In fact, in order for the tran-
sition to occur between the initial and final states [e.g. | [
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] |
sition to occur between the initial and final states [e.g.
between |+⟩ and |−⟩, see the inset in Fig. 2(a)], a change
of the two-qubit state is required.
For 3D phonons and spherical QDs, the dephasing
is absent only at d = 0 and according to Eq. (7) and
Fig. 2(b) has minima around Rd/vs = 2πn + π/2 (n =
1, 2, . . . ). The π/2 phase shift compared to the 1D case
and non-vanishing dephasing at the minima are due to
phonons (with fixed energy R) that are absorbed or emit- | [
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] |
phonons (with fixed energy R) that are absorbed or emit-
ted at different angles θ to the QD distance vector d
(Fig. 1), resulting in a variation of the phase difference
φ2 − φ1 between the dots. However, the reduction of the
decoherence is enhanced in anisotropic QDs, playing the
role of directional phonon emitters or absorbers. In fact,
in oblate QDs separated along their short axis [Fig. 2(c)],
directional coupling of phonons along the short axis ef- | [
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] |
directional coupling of phonons along the short axis ef-
fectively makes the system 1D in the limit l ≫ l⊥, where
l and l⊥ are, respectively, the lateral and perpendicular
(along d) exciton localization lengths [33]. The analogy
with pure 1D phonons becomes striking for d < l and
stronger coupled QDs (g = 2 meV) as shown in Fig. 3,
where the shorter phonon wavelength of the transition
provides fast oscillations versus d, allowing for minima
at short distances with near-vanishing dephasing. The | [
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] |
at short distances with near-vanishing dephasing. The
scaling of the oscillation amplitude with distance, given
in the inset, shows the constant 1D behaviour for d ≪ l,
while for d ≫ l the 1/d dependence of the 3D regime is
restored.
In Case 2 of QDs indirectly coupled via a cavity, the
dephasing is also controlled by bath coherence, though
in a more complex scenario.
The resulting three cou-
pled states |+, ±⟩ = (|1⟩ + |2⟩)/2 ± |C⟩/
√
2 and |−⟩ =
(|1⟩ − |2⟩)/
√
2 require a triexponential fit of P1(t) [33] | [
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] |
√
2 and |−⟩ =
(|1⟩ − |2⟩)/
√
2 require a triexponential fit of P1(t) [33]
to extract the dephasing rates Γj = − Im ωj, which are
shown in Fig. 4 across a range of distances. We observe
0
10
20
30
40
50
60
distance between qubits, d (nm)
0
10
20
30
40
50
dephasing rate, ( eV)
+,+
+,
exact
FGR
indep. bath
FIG. 4.
Dephasing rates for cavity-mediated coupled
anisotropic qubits (left inset) with interaction strength g1 =
g2 = ¯g = 0.5 meV, at zero detuning. Other parameters as in
Fig. 2(c). | [
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] |
g2 = ¯g = 0.5 meV, at zero detuning. Other parameters as in
Fig. 2(c).
oscillations versus distance d, as in Figs. 2(b) and (c),
however, now with two periods contributing to dephas-
ing rates Γ+,± of the states |+, ±⟩. This is because the
dephasing involves transitions at two distinct frequen-
cies, as seen in the right inset, with one twice the other
for zero detuning (Ω1 = Ω2 = ΩC). The dephasing rate
Γ− of the middle state |−⟩ is typically the largest, and | [
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] |
Γ− of the middle state |−⟩ is typically the largest, and
at zero detuning the two transitions between neighbor-
ing levels involve phonons of almost the same energy.
Therefore only one frequency is observed in the oscil-
lations, analogous to Case 1, with vanishing dephasing
rate at d = 0. Due to the nature of the mixed QD-cavity
states, the exciton-phonon matrix elements contributing
to FGR are now proportional to V1 ±V2, with + (−) cor-
responding to the transitions between distant (neighbor- | [
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] |
responding to the transitions between distant (neighbor-
ing) levels [33]. Since transitions between distant levels
contribute to Γ+,± and thus involve V1 + V2, there is a
non-vanishing contribution, even at d = 0. This is be-
cause the states involved in such transitions both have
a cavity contribution, and as cavity does not couple to
phonons, the reduction of the dephasing rate to zero is
not observed. However, these transition have typically
lower impact on decoherence due to the larger phonon | [
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] |
lower impact on decoherence due to the larger phonon
energy involved [33].
In conclusion, we have presented an asymptotically ex-
act solution for the linear optical response of a system of
two coupled qubits interacting with a shared bath which
causes decoherence, using semiconductor quantum dots
coupled to 3D acoustic phonons as illustration. We have
shown that the coherence of the bath can be exploited
to reduce decoherence of this two-qubit system, by con- | [
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] |
to reduce decoherence of this two-qubit system, by con-
trolling the distance between the qubits relative to the
wavelength of the bath modes interacting with the qubits.
Furthermore, for a 1D bath or for anisotropic qubits in
a 3D bath, decoherence can be eliminated entirely. This
concept can be generalized to multiple qubit systems.
L.H. acknowledges support from the EPSRC under
5
grant no.
EP/T517951/1.
L.S. acknowledges support
from the EPSRC under grant no. EP/R513003/1.
∗ [email protected] | [
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] |
L.S. acknowledges support
from the EPSRC under grant no. EP/R513003/1.
∗ [email protected]
† [email protected]
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