Environment Assisted Precision Measurement G. Goldstein $^{1}$, P. Cappellaro1 , 2 , $^{3}$, J.R. Maze$^{1}$, J. S. Hodges1 , $^{3}$, L. Jiang1 , $^{5}$, A. S. Sørensen$^{4}$, M. D. Lukin1 , 2 $^{1}$Department of Physics, Harvard University, $^{2}$Harvard-Smithsonian Center for Astrophysics, Cambridge MA 02138 USA $^{3}$Nuclear Science and Engineering Dept., Massachusetts Institute of Technology, Cambridge MA 02139 USA 4 QUANTOP, Niels Bohr Institute, Copenhagen University, DK 2100, Denmark and $^{5}$Institute of Quantum Information, California Institute of Technology, Pasadena CA 91125 USA We describe a method to enhance the sensitivity of precision measurements that takes advantage of a quantum sensor’s environment to amplify its response to weak external perturbations. An individual qubit is used to sense the dynamics of surrounding ancillary qubits, which are in turn affected by the external field to be measured. The resulting sensitivity enhancement is determined by the number of ancillas that are coupled strongly to the sensor qubit; it does not depend on the exact values of the coupling strengths and is resilient to many forms of decoherence. The method achieves nearly Heisenberg-limited precision measurement, using a novel class of entangled states. We discuss specific applications to improve clock sensitivity using trapped ions and magnetic sensing based on electronic spins in diamond. Precision measurement is among the most important applications of resonance methods in physics. For exam- ple, quantum control of atomic systems forms the physi- cal basis of the world’s best clocks. Ideas from quantum information science have been used to demonstrate that quantum entanglement can enhance these measurements [1, 2]. At the same time a wide range of quantum systems have been recently developed aimed at novel realizations of solid state qubits. Potentially such systems can be used as quantum measurement devices such as magnetic sensors with a unique combination of sensitivity and spa- tial resolution [3, 4, 12, 33]. In this Letter, we describe a novel technique that makes use of the sensor’s local en- vironment as a resource to amplify its response to weak perturbations. We shall use solid state sensors and ion clocks as examples. ized and collectively controlled, but cannot be directly detected. The system is described by the Hamiltonian H = H$_{b}$ + H$_{int}$ , with The purpose of quantum metrology is to detect a small external field, coupled to the sensor by a Hamiltonian: H S b = b ( t ) κS$_{z}$ , where S$_{z}$ is the spin operator of the quan- tum sensor. Here, b ( t ) can be an external magnetic field or the detuning of a laser from a clock transition while κ is the spin’s coupling to the field. The working principle of almost any quantum metrology scheme can be reduced to a Ramsey experiment [18, 19], where the field is mea- sured via the induced phase difference between two states of the quantum sensor. The figure of merit for quantum sensitivity is the smallest field δb$_{min}$ that can be read out during a total time T . For a single spin 1/2, if the sensing time is limited to τ (e.g. by environmental decoherence) then: δb$_{min}$ ≃ 1 $_{κ}$√$_{T τ}$ . where λ$_{i}$ are the couplings between sensor and environ- ment spins, while κ and ξ are couplings to the external field of the central and dark spins respectively. Here | 0 〉 , | 1 〉 and S$_{z}$ refer to the central spin while |↑〉 , |↓〉 , I i z to the dark spins (and we set ℏ = 1). We consider two cases. In the first one H$_{int}$ can be turned on and off at will and is much larger than any other interaction in the sys- tem (e.g. a laser-mediated ion interaction). In the second case, H$_{int}$ is intrinsic to the system and of the same order of magnitude as the relevant sensing time (e.g. dipole- dipole interactions between solid state spins). In all cases we will assume collective control over the dark spins. To illustrate the sensing method we consider first the idealized case where the couplings between the central and the dark spins can be turned on and off at will and the dark spins are initialized in a pure state |↑↑ ... ↑〉 . In many cases the external field also acts on the sen- sor’s environment, which normally only induces decoher- ence and limits the sensitivity. Here we show that in some cases the environment can instead be used to en- hance the sensitivity. For generality we will illustrate the key ideas using the central spin model (Fig. 1a). In this model a central spin (which can be prepared in a well de- fined initial state, coherently manipulated and read out) is coupled to a bath of dark spins that can be polar- Consider the circuit in Fig. 1(b). First, the central spin is prepared in an equal superposition of the two internal states ∼ | 0 〉 + | 1 〉 . Then H$_{int}$ is rotated to the x -axis and H$_{b}$ = b ( t ) ( κS$_{z}$ + ξ ∑ I i z ) , H$_{int}$ = | 1 〉〈 1 | ∑ λ$_{i}$I i $_{z}$, (1) arXiv:1001.0089v1 [quant-ph] 31 Dec 2009 FIG. 1: Ideal Model. (a) A central spin coupled to a spin bath. (b) Ideal measurement procedure. Gates labeled λI$_{x}$ represent controlled rotations e − iλI x of the dark spins, ob- tained via the Hamiltonian H$_{int}$ . The gates bI$_{z}$ represent rotations e − ibI z due to the external field b . The central spin undergoes a spin-echo before measurement. | GLYPH<a2up> 〉 | GLYPH<a2up> 〉 b) Scrambling  Phase Acquistion  Echo a) X λ$_{1}$ I$_{x}$ |0 〉+ |1 〉 λ$_{n}$ I$_{x}$ λ$_{1}$ I$_{x}$ λ$_{n}$ I$_{x}$ bI$_{z}$ bS$_{z}$ bI$_{z}$ −i τ H$_{int}$ −itH$_{b}$ x e e −i τ H$_{int}$ x e GLYPH<radical_old>2 ...