We present detailed measurements of the discrete electron-tunneling level spectrum within nanometer-scale cobalt particles as a function of magnetic field and gate voltage, in this way probing individual quantum many-body eigenstates inside ferromagnetic samples. Variations among the observed levels indicate that different quantum states within one particle are subject to different magnetic anisotropy energies. Gate-voltage studies demonstrate that the low-energy tunneling spectrum is affected dramatically by the presence of nonequilibrium spin excitations.
In the spin boson model, the properties of the oscillator bath are fully characterized by the spectral density of oscillators J(ω). We study the case when this function is of Breit-Wigner shape and has a sharp peak at a frequency Ω with width Γ ≪ Ω. We use a number of approaches such as the weak-coupling Bloch-Redfield equation, the non-interacting blip approximation (NIBA) and the flow-equation renormalization scheme. We show, that if Ω is much larger than the qubit energy scales, the dynamics corresponds to an Ohmic spin boson model with a strongly reduced tunnel splitting. We also show that the direction of the scaling of the tunnel splitting changes sign when the bare splitting crosses Ω. We find good agreement between our analytical approximations and numerical results. We illuminate how and why different approaches to the model account for these features and discuss the interpretation of this model in the context of an application to quantum computation and read-out.
We propose a simple phenomenological model for an ultrasmall ferromagnetic grain, formulated in terms of the grain's discrete energy levels. We compare the model's predictions with recent measurements of the discrete tunneling spectrum through such a grain. The model can qualitatively account for the observed features if we assume (i) that the anisotropy energy varies among different eigenstates of one grain, and (ii) that nonequilibrium spin accumulation occurs.PACS numbers: 73.23. Hk, 75.50.Cc, 73.40.Gk What are the properties of individual quantum states in the electronic excitation spectrum of a nanometerscale ferromagnetic particle? This is becoming an increasingly important question, since the size of memory elements in magnetic storage technologies is decreasing extremely rapidly [1], and particles as small as 4 nm are coming under investigation [2]. In this size regime, the excitation spectrum becomes discrete; indeed, Guéron, Deshmukh, Myers and Ralph (GDMR) [3] have recently succeeded to resolve individual quantum states in the spectrum of ferromagnetic Cobalt nanograins, using single-electron tunneling spectroscopy. They found complex nonmonotonic and hysteretic energy level shifts in an applied magnetic field and an unexpected abundance of low-energy excitations, which could not be fully understood within the simple models used for ferromagnetic nanograins so far [3,4].In this Letter, we propose a phenomenological model for ferromagnetic nanograins that is explicitly formulated in terms of the discrete states occupied by the itinerant conduction electrons and capable of qualitatively explaining the observed features. The model is similar in spirit to that advanced independently by Canali and MacDonald [4], but our analysis includes two further ingredients beyond theirs: (i) mesoscopic fluctuations of the anisotropy energy (i.e. it may vary among different eigenstates), and (ii) nonequilibrium spin accumulation.Experimental Results.-GDMR studied Co-particles 1-4 nm in diameter. Assuming a hemispherical shape, the number of atoms in such grains is in the range N a ≈ 20-1500, and the total spin, s 0 ≈ 0.83N a [5], thus is s 0 ≈ 17 − 1250. In GDMR's devices, a grain is connected to two aluminum electrodes via aluminum oxide barriers. Its tunneling conductance consists of a series of distinct peaks (see Fig. 2 in [3]), whose positions yield a set of tunneling energies of the form [6] ∆E ± f i ≡ E N ±1 f − E N i each corresponding to the energy cost of some rate-limiting electron tunneling process |i N → |f N ±1 onto or off the grain. Here |i N denotes a discrete eigenstate, with eigenenergy E N i , of a grain with N electrons, etc. As the magnetic field is swept, the resonances for Cograins undergo energy shifts and crossings (Fig. 3 in [3], [7]). The resulting tunneling spectra have several properties that differ strikingly from those of previously-studied nonmagnetic Al and Au grains [8,9]: (P1): Many more low-energy excitations are observed than expected: For all values of magnetic field, the mean l...
We discuss dephasing times for a two-level system (including bias) coupled to a damped harmonic oscillator. This system is realized in measurements on solid-state Josephson qubits. It can be mapped to a spin-boson model with a spectral function with an approximately Lorentzian resonance. We diagonalize the model by means of infinitesimal unitary transformations (flow equations), and calculate correlation functions, dephasing rates, and qubit quality factors. We find that these depend strongly on the environmental resonance frequency $\Omega$; in particular, quality factors can be enhanced significantly by tuning $\Omega$ to lie below the qubit frequency $\Delta$.Comment: 5 psges, 5 figure
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