Download my poster

Phonon mediated spin relaxation in a moving quantum dot: Doppler shift,
Cherenkov radiation, and spin relaxation boom
Xinyu Zhao1, Peihao Huang1,2, and Xuedong Hu1
1
2
Department of Physics, University at Buffalo, SUNY, Buffalo, New York 14260, USA and
Department of Physics, California State University Northridge, Northridge, California 91330, USA
The Cherenkov radiation of phonons
Abstract
What we study: Spin relaxation of a moving quantum dot (QD)
Results: 1. Doppler effect on emitted phonons.
2. Phonons making dominant contribution to spin relaxation is concentrated in two
particular directions, which is similar to the classical Cherenkov radiation.
3. There is a peak in the relaxation curve near the sound barrier, we term this as
“spin relaxation boom” in analogy to the classical sonic boom.
Introduction
In a large-scale electron-spin-based quantum information processor, it is inevitable that
information is transferred over finite distances frequently. One straightforward way to achieve
such communication is to move the electron spin qubits themselves directly. For example, the
qubits can be carried by a surface acoustic wave (SAW). However, a moving spin qubit may
suffer decoherence due to the spin-orbit interaction. In this work, we investigate the influences
of motion on spin relaxation. In particular, we show that the Doppler effect is integral to the
spin relaxation process, and leads to interesting features similar to the classical Cherenkov
radiation and the sonic boom.
Model and solution
interacting with phonon reservoir and the resulting
Doppler effect.
Relaxation rate:
1
𝑇1
=
Kernel function: 𝑓 =
𝐴𝐵
Cutoff functions: 𝐹𝑧 = exp
, 𝐹𝑥𝑦 = exp
𝜆2 𝑤𝑧2
− 2 sin2 𝜃
2𝑣𝑗
,
Different from the static QD case discussed in Ref. [1], we have a shifted phonon frequency
𝑤𝑧 =
𝜔𝑍
1−𝜉𝑗
,
𝜉𝑗 =
𝑣0
sin 𝜃 cos(𝜙
𝑣𝑗
− 𝜙𝑣 )
which reflects the Doppler effect.
Directional phonon emission: Cherenkov radiation
Transition from subsonic regime to supersonic regime
The overall behavior of the angular distribution for the phonon emission is shown in Fig. 2.
(A). In the subsonic regime, the frequency of the phonon is shifted. But no directional
emission is observed. The extra energy of the phonon with shifted frequency is provided by
the classical force maintaining the linear motion of the electron.
(B). In the transonic regime, a shock wave is
formed in the forward direction, the angular
distribution is concentrated around 𝜙 = 0∘ .
In the cross-section plot, the higher peak at
the center is produced by LA phonons while
lower one by TA phonons.
(C). In the supersonic regime, the angular
distribution is split into two branches.
Fig. 2
𝑣1
𝜔𝑍 𝜆
−1
= cos
−
𝑣0
2𝑣0
angle 𝜙𝐶′ ≈ ±40∘ as shown
𝜙𝐶′
It gives the accurate Cherenkov
in Fig. 3.
Spin relaxation boom
Classically, the drag force on an aircraft increases sharply when the aircraft velocity
approaches the sound barrier. This is the so-called sonic boom. We find a similar behavior in
the relaxation rate for a moving spin qubit. In Fig. 4, the total spin relaxation rate (black, dotdashed) peaks at the two sound barriers due to TA and LA phonons. Each peak for a single
type of phonon is similar to the Prandtl-Glauert singularity for the classical “sonic boom”.
Accordingly, we name these peaks “spin-relaxation boom”.
𝑣0
1
𝐻𝑒𝑓𝑓 = 𝑔𝜇𝐵 𝐵0 + Δ𝐵 + 𝛿𝐵 𝑡 ⋅ 𝜎
2
𝜋
2𝜋
𝑑𝜃 0 𝑑𝜙𝑓(𝜃, 𝜙)
0
ℏ𝜔𝑍 𝐹𝑆𝑂 (2𝑁𝑤𝑧 +1) 4
3 𝜃 cos 2 𝜙 𝐶 𝐹 𝐹
𝑤
sin
𝑧
𝑒𝑝 𝑧 𝑥𝑦
𝑗
2 2
8𝜋𝜌𝑐 𝑣𝑗4 𝑚∗ 𝜔𝑑
𝑑 2 𝑤𝑧2
− 2 cos2 𝜃
𝑣𝑗
Quantum correction on Cherenkov angle induced by confinement effect
The equation for classical Cherenkov angle predicts an angle of 𝜙𝐶 = ±38∘ , which is
slightly different from the numerical simulation. Besides, 𝜉𝑗 = 1 leads to a diverging phonon
frequency 𝑤𝑧 → ∞ which is unphysical. A more accurate description needs to include the
confinement effects. With the cutoff functions, the peak of the kernel function is modified to be
2𝑣1
at 𝑤𝑧 = . The Cherenkov angle is then
𝜆
We consider a spin qubit moving at a constant
speed in a nanostructure. Classically, the
waves emitted by a moving object experience a
Doppler shift in frequency. In particular, when
𝑣0 = 𝑣𝑠 , a shock wave is formed. When 𝑣0 >
𝑣𝑠 , superposition of the spherical waves
emitted by a moving object at different
moments forms a straight-line wavefront BC,
and the wavefront propagates in the AC
direction, at the Cherenkov angle from the
Fig. 1 A schematic diagram of a moving spin qubit motion direction: cos 𝜙 = 𝐴𝐶 = 𝑣𝑠 .
𝐶
Effective Hamiltonian:
Fig. 3
In Fig. 3, the angular distribution in the
supersonic regime is plotted. The dominant
contribution to spin relaxation is concentrated
along two particular directions (𝜙 ≈ ±40∘
and 𝜃 ≈ 90∘ ). This is a characteristic feature
of the Cherenkov radiation. Besides, noticing
that the peaks should appear at 𝜉𝑗 = 1, we
could obtain the peak angle as
𝑣𝑗
sin 𝜃 cos(𝜙 − 𝜙𝑣 ) =
𝑣0
which is identical to the classical relation of the
Cherenkov angle.
Here, the quantum confinement plays an
important role again. It modifies these peaks
by eliminating the singularities and
broadening them. The peak positions are
also shifted downward from 𝑣1 and 𝑣2
slightly.
Fig.4
Conclusion and discussion
In this work, we have studied decoherence of a moving spin qubit caused by phonon noise
through spin-orbit interaction. We find that the QD motion leads to Doppler shifts in the
emitted/absorbed phonons. When the moving velocity is larger than the sound velocity, spin
relaxation is dominated by phonon emission/absorption in certain directions, which is similar to
the classical Cherenkov radiation. We derive an explicit formula for the quantum confinement
correction to the Cherenkov angle. We also find a “spin-relaxation boom” when the moving QD
break the sound barriers, in analogy to the classical sonic boom. The current experimental
technology using the interference of two orthogonal SAWs could allow direct observation of
the TA boom, since it happens at a lower moving speed.
Our results may be used in various potential applications. For example, the narrowly
directional phonon emission from the moving spin qubit may also be used as a source of
phonons, even stimulated emission or lasing of phonons. Besides, knowing the phonon
emission angle may allow better continuous monitoring of the phonon environment, which
could in turn provide more accurate information to possible feedback operations.
Acknowledgements – We acknowledge financial support by US ARO (W911NF0910393) and
NSF PIF (PHY-1104672).
References:
[1] V. N. Golovach, A. Khaetskii, and D. Loss, Phys. Rev. Lett. 93, 016601 (2004).
[2] P. Huang and X. Hu, Phys. Rev. B 88, 075301 (2013).
[3] X. Zhao, P. Huang, X. Hu, arXiv:1503.00014.
www.buffalo.edu