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ARTICLE
OPEN
Tunable phonon-induced steady-state coherence in a
double-quantum-dot charge qubit
Archak Purkayastha
1✉
, Giacomo Guarnieri1, Mark T. Mitchison
1
, Radim Filip2 ✉ and John Goold1 ✉
Charge qubits can be created and manipulated in solid-state double-quantum-dot (DQD) platforms. Typically, these systems are
strongly affected by quantum noise stemming from coupling to substrate phonons. This is usually assumed to lead to decoherence
towards steady states that are diagonal in the energy eigenbasis. In this article, we show, to the contrary, that due to the presence
of phonons the equilibrium steady state of the DQD charge qubit spontaneously exhibits coherence in the energy eigenbasis with
high purity. The magnitude and phase of the coherence can be controlled by tuning the Hamiltonian parameters of the qubit. The
coherence is also robust to the presence of fermionic leads. In addition, we show that this steady-state coherence can be used to
drive an auxiliary cavity mode coupled to the DQD.
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npj Quantum Information (2020)6:27 ; https://doi.org/10.1038/s41534-020-0256-6
INTRODUCTION
Preparation and coherent control of qubit states is at the heart of
many quantum technologies1. Undoubtedly, quantum coherence
is the most primordial non-classical effect and is at the root of
many advantages displayed by quantum technologies over their
classical equivalents. One of the major challenges for applications
is to prepare a qubit in a state that has a controllable amount of
coherence with stability in the long-time limit2. In real physical
systems, quantum coherence is usually a fragile property, which is
eventually destroyed by the presence of a surrounding environment3,4. It is therefore no surprise that a plethora of strategies to
preserve coherence have been conceived, such as quantum error
correction5, dynamical decoupling6 or feedback control7. All of
these schemes are to some degree an inevitable battle against
decoherence. Rather than fighting this battle, here we highlight a
different counter-intuitive route to generate and preserve
coherence using quantum noise.
We specifically consider a semiconductor double-quantum-dot
(DQD) embedded on a substrate8–14 that realizes a charge qubit
coupled to a phononic bath15,16. It has been previously demonstrated that properties of the DQD may be used to extract
information about the phononic bath8,9. In a recent experiment10,11, the DQD and substrate were coupled to an auxiliary
optical cavity, which was in turn used to experimentally
characterize the spectral density of the substrate phonons. In this
article, we model the dynamical evolution of these platforms and
focus on their steady-state properties. Remarkably, we find that
the presence of phonons autonomously drives the DQD charge
qubit to a steady state that has coherence in the energy
eigenbasis while retaining a significant degree of purity. This
surprising result finds its explanation in the particular structure of
the system–bath interaction17. Furthermore, the magnitude of
steady-state coherence can be controlled by changing the
experimentally tunable parameters of the qubit Hamiltonian, the
detuning and the hopping. This is proven through an explicit
calculation and characterization of the steady-state Bloch vector of
the charge qubit as a function of the controllable Hamiltonian
parameters. We also show that the coherence is robust to the
presence of fermionic leads.
In addition to the obvious importance of generating coherence
for quantum information processing, there is currently significant
interest in harnessing coherence and exploiting it as a resource in
other contexts18. In particular, coherence in the energy eigenbasis
has been identified as one of the key features distinguishing
quantum thermodynamics from its classical counterpart19–21. For
example, coherence may enhance the performance of quantum
refrigerators22–25 and heat engines26–28 or be directly converted
into work29–31. To address this point, we conclude the article by
showing that, in our set-up, the above phonon-induced steadystate coherence of the DQD can in fact be exploited to drive a
mode of the surrounding cavity.
RESULTS
Autonomous generation of steady-state coherence
The main theoretical idea behind the autonomous generation of
steady-state coherence was first introduced and explored in ref. 17.
In that work, sufficient conditions concerning the structure of the
interaction Hamiltonian between a qubit and a bosonic bath were
identified that lead to steady-state coherence. In particular, it was
shown that a spin-boson model with a Hamiltonian of the form
X
X y
^y b
^
^ þb
^k
^ ¼ ωq σ^ z þ
^ z þ f 2 σ^ x Þ λk b
H
Ωk b
(1)
k k þ ðf 1 σ
k
2
k
k
autonomously leads to a non-zero steady-state value for hσ^ x i. Here
σ^x;y;z denote the usual Pauli spin operators, f1, f2 ≠ 0 two generic
^k is the bosonic annihilation operator of
coupling constants and b
the kth bath mode. The results presented in this paper stem from
the crucial observation that a semiconductor DQD in contact with
a phononic substrate is described exactly by a Hamiltonian of the
form of Eq. (1).
The DQD charge qubit
The set-up we consider is depicted schematically in Fig. 1. The
DQD comprises two fermionic modes with strong repulsive
1
School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland. 2Department of Optics, Palacký University, 17. listopadu 1192/12, 771 46 Olomouc, Czech Republic.
✉email: ; fi
[email protected];
Published in partnership with The University of New South Wales
A. Purkayastha et al.
2
In the transformed basis, we have
^ S ¼ ωq ðN
^ 2 ; ωq ¼
^1 N
^2Þ þ V N
^ 1N
H
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ε2 þ 4t2c ;
(5)
h
i
^ SE ¼ cos θðN
^
^1 N
^ 2 Þ sin θðA
^y A
^y ^ ^
H
1 2 þ A2 A1 Þ B;
Fig. 1 Schematic depiction of two quantum dots. The two
quantum dots are detuned in energy by ε with inter-dot tunnelling
tc and Coulomb repulsion V interacting with a substrate supporting
phononic excitations.
interaction between them. Substrate phonons are coupled
to the electric dipole moment of the DQD. The full Hamiltonian
^ ¼H
^S þ H
^ SE þ H
^ E with
(refs 11,13,32) is then given by H
^ SE ¼ ε þ B
^ ðn
^S þH
^2 ;
^1 n
^1 n
^2 Þ þ t c ^cy1^c2 þ ^cy2^c1 þ V n
H
2
P ^y ^
P ^y ^
^ E ¼ Ωk b bk ;
^ ¼ λk b þ bk ; H
B
k
k
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k
k
^ SE ¼ B
^ ðn
^ S ¼ ε ðn
^2 ; H
^2 Þ þ tc ^cy1^c2 þ ^cy2^c1 þ V n
^1 n
^2 Þ:
^1 n
^1 n
H
2
(2)
^‘ ¼ ^cy‘ ^c‘ , ^c‘ is the fermionic annihilation operator of the ℓth
Here n
^k is the phononic annihilation operator of the kth mode
site and b
of the bath. The experimentally controllable parameters of the
DQD are the detuning ε and the hopping tc. The repulsive
interaction between the two sites is given by V, which is usually
much larger than any other energy scale in the regime of
^ embodies the noisy detuning due to the
operation. The operator B
fluctuating phonon bath. Assuming that the latter is in a th (...truncated)
