SHOOTING QUASI PARTICLES FROM ANDREEV BOUND STATES IN A SUPERCONDUCTING CONSTRICTION
R.-P. Riwara, M. Houzeta, J. S. Meyera, Y. V. Nazarovh*
a Univ. Grenoble Alpes, INAC-SPSMS, F-3S000 Grenoble, France, CEA, INAC-SPSMS, F-3S000 Grenoble, France
b Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, NL-262S CJ, Delft, The Netherlands
Received May 19, 2014
A few-channel superconducting constriction provides a set of discrete Andreev bound states that may be populated with quasiparticles. Motivated by recent experimental research, we study the processes in an a.c. driven constriction whereby a quasiparticle is promoted to the delocalized states outside the superconducting gap and flies away. We distinguish two processes of this kind. In the process of ionization, a quasiparticle present in the Andreev bound state is transferred to the delocalized states leaving the constriction. The refill process involves two quasiparticles: one flies away while another one appears in the Andreev bound state. We notice an interesting asymmetry of these processes. The electron-like quasiparticles are predominantly emitted to one side of the constriction while the hole-like ones are emitted to the other side. This produces a charge imbalance of accumulated quasiparticles, that is opposite on opposite sides of the junction. The imbalance may be detected with a tunnel contact to a normal metal lead.
Contribution for the JETP special issue in honor of A. F. Andreev's 75th birthday
DOI: 10.7868/S0044451014120037
1. INTRODUCTION
Superconducting niososcopic structures arc among the most promising candidates to realize quantum computation devices in the solid state [1]. Apart from extrinsic sources of decoherence that might get in the way, quasiparticle poisoning constitutes one of the major obstacles inherent to superconductors [2]. In a Cooper pair box, the presence of quasiparticles leads to a coupling of even and odd charge modes, providing a channel of decoherence for the charge qubit [3,4]. In addition, quasiparticle excitations can break the fermion parity required for the protection of a Majorana state [5 7]. Naively, the superconducting gap A should ensure an exponentially suppressed quasiparticle population at sufficiently low temperature. However, various experiments indicate that a long-lived, non-equilibrium quasiparticle population persists in the superconductor, harming the desired operation of superconducting devices [8 13].
* E-mail: Y.V.Nazarov'ffltudelft.nl
This makes it important to develop the means of an active control of the quasiparticle population in bound states associated with a nano-dcvicc. Thus motivated, we theoretically investigate the control of the population of quasiparticles in the Andreev bound states at a superconducting constriction by means of pulses of microwave irradiation. We concentrate on the generic case of a few-channel superconducting constriction with highly transparent channels. Such constrictions are made on the basis of atomic break junctions [14]. The simplicity of their theoretical description enabled detailed theoretical research [15 17]. In the presence of a phase difference at the constriction, an Andreev bound state is formed in each channel [18,19]. In a recent experiment, the population of such a single bound state has been detected by its effect on the supcrcurrcnt in the constriction. The spectroscopy of Andreev states has also been successfully performed [20,21] in this setup.
In this work, we investigate the processes that switch the Andreev bound state population. We assume low temperatures that permit to neglect the population of delocalized quasiparticle states. Let us con-
sider a quasiparticlo with energy Ea < A, A being the superconducting gap edge in the leads. If we modulate the superconducting phase with the frequency-fin > A — Ea, we can transfer this quasiparticlo to the states of the delocalized spectrum. This is an ionization process. Suppose we start with no quasiparticlo in the constriction and wish to fill the bound state. This can be achieved by the absorption of a quantum of the high-frequency phase modulation, provided the quantum energy exceeds Ea + A. In the course of such a refill process, one quasiparticlo emerges in the Andreev level while another one is promoted to the delocalized states and leaves the constriction.
We will utilize a master equation approach to describe the corresponding transitions. As usual, this works if the transition rates in energy units are much smaller than the energies involved. In our situation, the energy scale is A while the transition rates due to the modulation with amplitude 6<j> can be estimated as A(6(f))2. Therefore, the master equation approach is justified if -C 1, that is, in the limit of small modulations.
We compute the rates of the ionization and refill processes in the lowest order in the phase modulation amplitude and shortly explain how to control the population by applying the a.c. pulses that initiate the process.
We find an interesting asymmetry of the quasiparticles emitted in the course of these processes. The quasiparticles fly with equal probability to both leads. However, more electron-like quasiparticles leave to one of the leads while more hole-like ones leave to the opposite one. This results in a net charge transfer per process and in principle can be regarded as a non-equilibrium addition to the supercurrent in the constriction. Similar to the supercurrent, the effect changes sign upon changing the sign of the superconducting phase.
The effect leads to charge imbalance [22, 23] of the non-equilibrium quasiparticles that are accumulated in the leads on the spatial scale set by the inelastic relaxation of the quasiparticles [24]. This charge imbalance can be measured with a normal-metal voltage probe attached to the superconductor: the method proposed in [24] and widely applied in recent years [25,26].
This paper is organized as follows. We formulate the model in Sec. 2 and we give results for the rates in Sec. 3. Section 4 is dedicated to the estimations of the charge imbalance effect in the voltage-probe setup.
A(x)
V
yv
R
0 L
Fig. 1. Id model of the superconducting constriction
2. MODEL
We model the superconducting weak link with a Id quantum Hamiltonian corresponding to a single transport channel, x being the coordinate (see also Fig. 1). The constriction of the length L is modelled by a scattering potential V(x). In addition, a finite vector potential A(x) on a local support provides a phase bias between the left and right contact, <f> = 2« J dxA(x), e being the elementary charge. We focus on the regime where the excitation energy is much smaller than the Fermi energy, E -C Ep- such that the spectrum can be linearised. The pseudo spin \L,R) thus signifies a left/right moving electron with the Fermi wave vector =FA\f, where a- = \L)(L\ — \R)(R\. In the linearized regime, the current density operator is represented as j = —vpa-. The Bogoliubov-de Gennes Hamiltonian is then given as (h = 1)
H = [—ivFdrcr~+V(x)crr] t- ^evpA(x)a~+Arr, (1)
where the Pauli matrices n represent the Nanibu space. The potential V provides the reflection, as ax = = \L)(R.\ + \R)(L\. Both V and .4 are real functions and have a finite support in the interval x € [0,L],
Let us first deal with a stationary phase <f>. We diagonalize the Hamiltonian, Eq. (1), in the limit of a short constriction, E.A -C vp/L. There is one Andreev bound state solution |ipyi(.i;)), with a subgap
eigenenergy Ea = il^/l — T0sin2(0/2), T0 being the normal state transmission coefficient characterizing the transport channel under consideration. The Andreev bound state is responsible for the supercurrent in the constriction. Since the levels are spin-degenerate, the Andreev level can host n = 0,1,2 quasiparticles, the supercurrent being Js( 1 — n), where Js = —2edlPEA■ In addition, there are the extended scattering eigen-states with eigenenergies E > A. They have the
DCS density of states v{E) = 0(E^A)E/VE2^A2v0, where v0 is the density of states in the normal metal. The indices a: I.. I! and // = < . h indicate the scattering state with an //-like quasiparticlo outgoing to the
contact a. The outgoing scattering states correspond to the solution of the advanced propagator. This set of states is related to the incoming scattering states (retarded propagator) via the scattering matrix =
= {<P?1
^ai]
-a' ty l1^«»?)- Our scattering matrix coincides with the one found in Rcf. [17].
To describe an a.c. driven system, we assume <f>(t) = = <f> + S<f>sm(i}t) and treat the phase modulation amplitude as a perturbation. We compute the rates of various processes in the lowest order when they are proportional to (6<f))2.
In addition, the constriction may be subject to quantum phase fluctuations, i.e., the phase modulation becomes an operator, 6<j>(t) —¥ <j>, whose dynamics is determined by the electromagnetic environment of the junction. The phase noise spectrum is S(ui) = f dt e~tul{6<j> (0) 6$ (t))cnv, where the expectation value is with respect to the environment degrees of freedom. If the environment is in thermal equilibrium, the noise can be related to the impedance Z(ui) felt by the constriction via the fluctuation dissipation theorem, S(ui) = 4ttGqZ(ui)/ui, where u> > 0, Gq = e2/irh. The rates of the inelastic processes are readily computed with this.
3. THE TRANSITION RATES AND MANIPULATION
r/(A(^)2/16) 0.5
fi/A
Fig. 2. Ionization and refill rates for T0 = 0.5 and 6 = = 7t, when Ea « 0.7A. The ionization rate appears at the threshold Q « 0.3A, while the threshold for the refill is a 1.7A
Substituting the wave functions into Eqs. (2), (3), and (4), we arrive at the following expressions:
16
EA
EaQ + A2[cos(0) + 1]
(5)
To compute the rates, we apply Fermi's Golden rule. The advantage of the model and the gauge in use is that the matrix elements of the perturbation only depend on
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