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Coherent oscillations in a single electron spin at room temperature

Rabi nutations of a single electron spin in a single defect center have been detected. The coherent evolution of the spin quantum state is followed via optical detection of the spin state. Coherence times up to several microseconds at room temperature have been measured. Optical readout of the spin states leads to decoherence, equivalent to the so-called Zeno effect. Quantum beats between electron spin transitions in a single spin Hahn echo experiment are observed. A closer analysis reveals that beats also result from the hyperfine coupling of the electron spin to a single 14N nuclear spin. The results are analysed in terms of a density matrix approach of an electron spin interacting with two oscillating fields.

Detection and manipulation of single electron spin states in solids have recently received much attention in the context of quantum computing [1][2][3][4]. Mainly, this is because localized electron spins in solids show long relaxation [5] and coherence times and their states can be easily manipulated via microwave or radio frequency pulses. Yet reading out a single spin state, a prerequisite for most quantum algorithms, has remained a major challenge up to now.
Although single spin detection has been reported previously [6] [7], most of the systems and techniques have not shown the ability to measure the quantum state of a single electron spin.
The only solid state system where this has been successful is the nitrogen-vacancy (NV) defect center in diamond. The defect is known to have a strongly dipole allowed optical transition between its electron spin triplet ground state ( 3 A) and a first excited spin triplet state ( 3 E) ( Fig.1) [8]. Fluorescence emission strongly depends on the electron spin quantum state, which has lead to optically detected magnetic resonance on single defect centers [9]. Electron spin relaxation times (T 1 ) range from 2ms at room temperature to seconds at low temperature.
Recently, electron spin quantum jumps have been detected [10], thus showing that the spin state of a single electron spin could be determined in this defect center. In this Letter we report on the generation of coherence among single electron spin states in the NV center. We analyse how spin coherence is influenced by readout and coupling to additional degrees of freedom, like single 14 N nuclear spins. Single spin coherence in solids has been observed before [11,12]. However, the NV center in diamond is particularly interesting because of its electron paramagnetic ground state. Contrary to other systems, like organic molecules, spin coherence time is thus not limited by the electronic lifetime of the state. Above all, the electron spin state can be readout directly via optical excitation and subsequent fluorescence relaxation. Previously this has been the basis of a number of ingenious studies on NV center ensembles, like the observation of electromagnetic induced transparency [13] and dressed state nutation [14].
Experiments have been carried out with a home-build confocal microscope operating in a temperature range between 2K and 300K. Microwaves are coupled to the sample by a miniaturized short cut loop connected to a 40W travelling wave tube amplifier to generate microwave H fields large enough for short microwave pulses. As sample material, synthetic nanocrystalline diamond ( Type 1b) with an average crystal size of 20nm was used. To allow for comparison with a density matrix based theory, all experiments have been time averaged over 10 5 to 10 7 cycles to obtain smooth curves.
The ground and excited electron states of the NV defect are spin triplets (S=1) with spin sublevels m s =0,±1 ( Fig. 1) [15]. The degeneracy among the sublevels is lifted due to either the mutual dipolar interaction of the two unpaired electron spins or an external magnetic field.
Most of the experiments in this paper have been performed without applying an external magnetic field. Because of the C 3v symmetry of the defect, the m s =±1 sublevels remain degenerated (E=0) but are split from the m s =0 sublevel by roughly 2.9 GHz. Fluorescence of a single defect center is visible only when the spin is in the m s =0 spin sublevel. Optical excitation leads to a strong spin polarization, such that, averaged in time, the electron spin is found with at least 70% probability in the m s =0 sublevel. The application of a resonant microwave pulse causes a transition of the system between the spin levels and thus modulates the fluorescence intensity. When time averaged, the nutation of the electron spin due to coherent interaction of the spin with the microwave H-field is described as , where ∆ω is the detuning between microwave and transition frequency and , with ω 1 as microwave Rabi frequency [16]. r 3 (t) is the difference between the probability to find the system in the m s =0 and m s =±1 sublevel. Figure 2 shows the nutation of a single NV center electron spin. Since coupling to the microwave and optical field has to be considered, the above mentioned equation has to be extended and experiments are compared with microwave-optical Bloch equations starting from the semiclassical Hamiltonian In this equation ħω I is the energy of the ground and excited states, Ω=E 0 d 13 /ħ is the Rabi frequency corresponding to the amplitude of the laser electric field E 0 , at frequency ν L =ω L /2π, interacting with the defect center transition dipole moment d 13  hyperfine coupling to the nitrogen nuclear spin. In both cases, the dephasing rate is strongly distance dependent (1/r 3 ). We found a substantial difference among the T 2 values of different NV centers, possibly due to a change in the distance between the electron spin of the center and the next nearest neighbour nitrogen in the lattice.
In the experiment, we observed a decrease in the decay time of the spin nutation upon increased optical excitation (readout) intensity. Figure 3 shows experimental data together with theoretically predicted curves. In the microwave optical Bloch equations, the differential equation for the coherence between the spin sublevels ρ 12 does not directly comprise optical pumping. Rather the effect of optical pumping enters via ρ 11, which is a function of the optical In order to probe how well the single spin state can be coherently manipulated by microwave pulses, the spin state evolution under a Hahn echo pulse sequence has been investigated.
Adapted to optically detected magnetic resonance, the Hahn echo pulse sequence consists of the well-known 90 o -τ-180 o -τ´ microwave pulses and waiting times followed by a 90 o pulse to convert the spin echo phenomenon into populations, measurable by fluorescence detection [18]. Figure 4 shows the single spin Hahn echo together with echo decay measurements. The and -1 state of the particular defect studied. This splitting is the result of a slight deviation from C 3v symmetry of the defect. Since the microwave Rabi frequency ω 1 is substantially larger than the frequency splitting ∆ω between the two states, both transitions are excited simultaneously. To confirm our interpretation a small external magnetic field H 0 was applied.
The Hamiltonian which now describes the system is H = SDS+g e β e SH 0 +SAI-g n β n IH 0 , where D and A are the fine structure and hyperfine splitting (hfs) tensors, and g e,n and β e,n are the electron and nuclear g-factors and Bohr magnetons, respectively. The electron Zeeman term further splits the m s =±1 levels. For an H 0 field of 0.02T, the splitting is roughly 150 MHz, such that the microwave pulse only excites the m s =0 to -1 transition. As a result the beating in the Hahn echo due to the interference between the two electronic transitions disappears.
Instead, a modulation with much lower frequency appears. The Fourier transform now shows frequency components between 5 and 10 MHz. Following a previous analysis of Hahn echo decay data on NV center ensembles [20], we believe that this envelope modulation arises from hyperfine coupling of the electron spin to 14 N nuclear spins. Due to the low spin density of the electrons at the location of the nucleus, the hyperfine coupling of 14 N usually is not resolved in the ESR spectrum [21]. Although the frequency components caused by 14 N are also visible in the echo decay data for H 0 = 0 T, they are considerably more pronounced when a small magnetic field is applied. In order to detect the nuclear modulation effect allowed (∆m I =0) as well as forbidden (∆m I =±1), ESR transitions have to be excited. The modulation depth is largest when the nuclear Zeeman splitting ω I equals the hyperfine frequency, i.e., ω I =|m s |A iso , where A iso is the isotropic hyperfine coupling. Under this condition, the nuclear spin is no longer quantized along the external magnetic field. The allowed and forbidden ESR transitions have same magnitude, and the modulation depth is at maximum [19]. This condition is met for the experiment shown (Fig. 4b) where H 0 =0.02T. Hyperfine coupling to 14 N also explains the beating seen in Fig.4a. If ω M is detuned by ∆ω from the spin resonance transition, the echo is expected to be modulated by cos(∆ω(τ-τ´)), where ∆ω is the hyperfine coupling. Indeed, the observed modulation frequencies correspond well with those found in the Fourier transform of the Hahn echo decay in an external magnetic field.
In summary, we report on the first experimental investigation of coherent oscillations among spin sublevels of a ground state single electron spin at room temperature. The measured dephasing times range from 1µs to 2µs. They strongly depend on the optical readout frequency, which is in good agreement with microwave optical Bloch equations. The coherence times are long enough to allow for a coherent manipulation of the single spin.
These experiments reveal the hyperfine interaction of the electron with the 14 N nuclear spin of the defect center. Since our experiments demonstrate a coherent evolution of electron spin transitions for different nitrogen nuclear spin configurations, they may be a first step towards conditional non-trivial two single spin quantum logical operations.