Temperature dependence of long coherence times of oxide charge qubits

www.nature.com/scientificreports OPEN Received: 31 August 2017 Accepted: 9 February 2018 Published: xx xx xxxx Temperature dependence of long coherence times of oxide charge qubits A. Dey & S. Yarlagadda The ability to maintain coherence and control in a qubit is a major requirement for quantum computation. We show theoretically that long coherence times can be achieved at easily accessible temperatures (such as boiling point of liquid helium) in small (i.e., ~10 nanometers) charge qubits of oxide double quantum dots when only optical phonons are the source of decoherence. In the regime of strong electron-phonon coupling and in the non-adiabatic region, we employ a duality transformation to make the problem tractable and analyze the dynamics through a non-Markovian quantum master equation. We find that the system decoheres after a long time, despite the fact that no energy is exchanged with the bath. Detuning the dots to a fraction of the optical phonon energy, increasing the electron-phonon coupling, reducing the adiabaticity, or decreasing the temperature enhances the coherence time. Although semiconductors are the most widely used functional materials for electronic applications so far, nevertheless, semiconductor devices have some limitations: i) the characteristic length scales are sizeable so that further scaling down the existing system size is quite difficult; and ii) only the charge and spin degrees of freedom are utilized. On the other hand, owing to significantly smaller extent of the wavefunction, transition metal oxides can meet the miniaturization demands much better than semiconductors. Furthermore, oxides offer a vastly richer physics involving diverse spin, charge, lattice, and orbital correlations1–7. Low-dimensional oxides present new opportunities for devices where these diverse correlations can be optimized by engineering many-body interactions, fields, geometries, disorder, strain, etc. Therefore, oxides may be viewed as one of the best candidates to replace semiconductors in future electronic devices. Construction of scalable quantum computers has motivated identification of coherent two-level solid-state systems. A simple solid-state two-level system is the charge qubit. Charge qubit holds promise for high-speed manipulation due to strong coupling of the electron to electric field. On the other hand, large coherence times have not been achieved so far in the semiconductor double quantum dot (DQD) systems studied as charge qubits8–18. Decoherence, due to system-environment interactions, degrades the precious resource of quantum mechanical superpositions19,20 and is one of the main obstacles in quantum information processing. The temperature of the environment affects the reduced system dynamics and introduces additional relaxation channel for the system. Furthermore, the semiconductor quantum dots employed in the decoherence studies had a relatively large diameter (i.e., ~200 nm) with the corresponding electron temperatures being low (i.e., ~100 mK). Thus concomitant realization of fast operation and large coherence times in small solid state qubits at easily accessible temperatures (such as boiling points of liquid helium and liquid nitrogen and room temperature), although very useful for quantum computation, has been elusive so far. Here we show that, compared to a semiconductor DQD, an oxide DQD (modeled with phonons as the only source of decoherence) may yield significantly longer coherence times (i.e., longer by at least an order of magnitude) at higher temperatures (i.e., higher by an order of magnitude) and in smaller sized systems (i.e., smaller by an order of magnitude). Thus we illustrate the device potential of low-dimensional oxides through our analysis of a manganite-based DQD as a charge qubit. We employ a duality transformation, that is potentially widely applicable, to map a strong-coupling electron-phonon problem to a weak-coupling problem (where the small parameter is the inverse of that in the strong-coupling case); we consider a system that is initially decoupled from the phonon bath in the polaronic frame of reference and solve the non-Markovian quantum master equation. CMP Div., Saha Institute of Nuclear physics, 1/AF Salt Lake, Kolkata, 700064, India. Correspondence and requests for materials should be addressed to S.Y. (email: ) Scientific REPOrTS | (2018) 8:3487 | DOI:10.1038/s41598-018-21767-2 1 www.nature.com/scientificreports/ Formulation DQD model with environment. We consider a laterally coupled DQD system for our two-level qubit. The charge in the DQD system is denoted (N1, N2) with N1 and N2 being the number of electrons on dots 1 and 2, respectively. The quantum dots are taken to be identical with the same charging energy EC = e2/C where e is the charge of an electron and C is the capacitance between the dot and its surroundings. The capacitance C can be conservatively approximated by the self-capacitance C0 = 4εmε0D21 which for a manganite dot with dielectric constant εm = 10 and diameter D = 10 nm yields EC ~ 0.05 eV. We analyze situations where the thermal energy kBT as well as the the detuning Δε ≡ ε1 − ε2 (between the lowest energy levels in the two dots) are both smaller than EC so that the dynamics of a single electron can be studied when |N1 − N2| = 1. Consequently, we define the relevant charge states as |10〉 ≡ (N + 1, N) and |01〉 ≡ (N, N + 1). The coupled dots are described by the following Hamiltonian of a single electron tunneling between them: HDQD = ε1m1 + ε2m2 − J⊥ † (c1 c 2 + c 2†c1) + J m1m2 , 2 (1) where the electron destruction operator in dot i is defined as ci and mi ≡ ci†ci . Furthermore, the energies εi and the interdot tunnel coupling J⊥ are adjusted by external gates; the nearest neighbor repulsion J is due to Coulomb 2 interaction. J m1m2 is included for generality although its value is zero here. The total Hamiltonian is expressed as H = HDQD + HP + HEP where the additional term HP = ∑ i , k ωkai†, kai , k is due to the optical phonon environment while HEP = 1 ∑ i , k gkωk mi − 1 (ai , k + ai†, k) is due to the electron-phonon interaction; here, aj,k is the destruc2 N tion operator of mode k phonons at site j, gk is the electron-phonon coupling strength, and ωk is the optical phonon frequency with weak dispersion. The role played by the acoustic phonons in decoherence will be presented in the discussion section. In the strong coupling regime, to perform perturbation theory effectively, we locally displace the harmonic oscillators by Lang-Firsov (LF) transformation22 HL ≡ eS He−S with S = − 1 ∑ i , k gk mi − 1 (ai , k − ai†, k). In the 2 N LF/polaronic frame, the electron is clothed with phonons reducing the tunneling term J ⊥ in Eq. (1) to βωk 2 1 J⊥mf ≡ J⊥e−N ∑k gk coth 2 . This reduction of the polaronic tunneling at enhanced temperatures occurs for the same reason as that in a polaron band. Therefore, in the DQD, the single particle energy (~ J⊥mf ) is much smaller than (...truncated)