LETTERS Complete quantum control of a single quantum dot spin using ultrafast optical pulses David Press 1 , Thaddeus D. Ladd 1,2 , Bingyang Zhang 1 & Yoshihisa Yamamoto 1,2 A basic requirement for quantum information processing systems is the ability to completely control the state of a single qubit 1–6 . For qubits based on electron spin, a universal single-qubit gate is realized by a rotation of the spin by any angle about an arbitrary axis. Driven, coherent Rabi oscillations between two spin states can be used to demonstrate control of the rotation angle. Ramsey interference, pro- duced by two coherent spin rotations separated by a variable time delay, demonstrates control over the axis of rotation. Full quantum control of an electron spin in a quantum dot has previously been demonstrated using resonant radio-frequency pulses that require many spin precession periods 7–10 . However, optical manipulation of the spin allows quantum control on a picosecond or femtosecond timescale 11–18 , permitting an arbitrary rotation to be completed within one spin precession period 6 . Recent work in optical single- spin control has demonstrated the initialization of a spin state in a quantum dot 19–22 , as well as the ultrafast manipulation of coherence in a largely unpolarized single-spin state 17 . Here we demonstrate complete coherent control over an initialized electron spin state in a quantum dot using picosecond optical pulses. First we vary the intensity of a single optical pulse to observe over six Rabi oscillations between the two spin states; then we apply two sequential pulses to observe high-contrast Ramsey interference. Such a two-pulse sequence realizes an arbitrary single-qubit gate completed on a pico- second timescale. Along with the spin initialization and final pro- jective measurement of the spin state, these results demonstrate a complete set of all-optical single-qubit operations. Coherent control of a single qubit is often accomplished by driving the qubit at its resonant frequency. For a qubit composed of a single electron spin in a magnetic field, resonant coherent control requires the use of radio-frequency pulses of at least nanosecond duration 7–10 . One way to reduce this timescale is to construct qubits of multiple, coupled particles and to rapidly manipulate their coupling potential, as exemplified by the several-hundred-picosecond gate times of exchange-coupled electron pairs in electrically controlled quantum dots 23 . For an isolated, optically controlled quantum dot spin such as the one studied here, even shorter operation times may be achieved using ultrafast optical pulses 11–18 . Using such optical pulses, the axis of rotation of the qubit is determined by the arrival time of the pulse with respect to the qubit oscillation period 6 . A single-qubit gate con- sisting of an arbitrary rotation about any axis may thereby be com- pleted in a single Larmor period. For electron spin qubits, a large magnetic field is therefore necessary to increase the speed of a single- qubit gate, and quantum information processing with clock speeds in excess of 10 GHz may be possible 6 . Our scheme to rotate a single electron spin using a picosecond pulse is shown in Fig. 1a. A single electron is confined in the quantum dot. The electron spin states j#æ and j"æ are split by an externally applied magnetic field B ext 5 7 T, aligned parallel to the z axis (Voigt geometry; see Fig. 1d), to provide a large Larmor precession frequency of d e /2p 5 26.3 GHz. The lowest energy interband transi- tions are to the two trion states consisting of a pair of electrons in a spin singlet and an unpaired heavy hole 24 , denoted j"#, Yæ and j"#, Xæ, which are split by a frequency d h . Each trion state forms an inde- pendent L system with the two metastable states j#æ and j"æ. Optical selection rules dictate that the vertical and cross transitions in Fig. 1a couple to orthogonal linear polarizations of light, denoted H and V, and are p/2 out of phase with each other. The exact orientations of H and V are determined by the shape and strain of the quantum dot 25 . Each transition has a Rabi frequency V H 5 mE H /B or V V 5 mE V /B, where m is the transition’s dipole strength and E H and E V are the complex electric field amplitudes of the rotation pulse in the corres- ponding polarization basis. A circularly polarized rotation pulse ensures that the probability amplitudes from the two L systems add constructively, and a large detuning D minimizes undesired population in the excited states. Hence, a single broadband rotation pulse will coherently change the spin from j#æ to j"æ and back through a stimulated Raman transition. The dynamics may be qualitatively described by the condition that V H = D and V V = D, under which the upper levels can be adiabatically eliminated. Doing so, we expect to find two-state Rabi oscillations with an effective Rabi frequency V eff < V H V V /D between states j#æ and j"æ. The spin rotation may alternatively be described in terms of an optical Stark shift 17 . In addition to rotations, a complete set of single-qubit operations also requires initialization and measurement. We perform both of these tasks by optical pumping (Fig. 1b). A narrowband, continuous- wave laser optically drives the j#æ« j"#, Yæ transition with rate V P . The optical pumping laser has negligible effect on the spin rotation because V P = V eff . Spontaneous decay into the two spin states at half the trion’s total spontaneous emission rate, denoted C, quickly initi- alizes the electron into the j"æ state. After spin rotation, the popu- lation in the j#æ state is measured using the same optical pumping process. If the spin is rotated to j#æ, the quantum dot will emit a single photon from the j"#, Yæ R j"æ transition, which can be detected using a single-photon counter. Our single-spin measurement technique has been proposed for use in quantum computation 1 , and offers the experimental convenience of including measurement and initialization in the same step. However, the fidelity of a single-shot readout is limited by the photon collection efficiency. An optical microcavity would boost the mea- surement scheme’s efficiency, and could also enable coherent con- version of spin qubits into photon qubits for quantum networking 26 . Resonant absorption measurements 19–21 offer similar advantages, but also require a microcavity-enhanced absorption cross-section to enable single-shot readout. Quantum non-demolition measure- ments based on dispersive Kerr rotation 27 , Faraday rotation 28 or a recycling transition 29 use many photons to measure the spin and are therefore more robust to photon loss, but they require a separate initialization step. 1 E. L. Ginzton Laboratory, Stanford University, Stanford, California 94305, USA. 2 National Institute of Informatics, Hitotsubashi 2-1-2, Chiyoda-ku, Tokyo 101-8403, Japan. Vol 456 | 13 November 2008 | doi:10.1038/nature07530 218 ©2008 Macmillan Publishers Limited. All rights reserved