Quantum magnetometry uses quantum resources to measure magnetic fields with a precision and accuracy that cannot be achieved by its classical counterparts. In this paper, we propose a scheme for quantum magnetometry using discrete-time quantum walk (DTQW) where multipath interference plays a central role. The dynamics of a spin-half particle implementing DTQW on a one-dimensional lattice gets affected by magnetic fields and the controlled dynamics of DTQW help in estimating the fields' strength. To gauge the effects of the field, we study the variance of the particle's position probability distribution (PD) and use it to determine the direction of the magnetic field maximally affecting the quantum walk. We then employ statistical tools like quantum Fisher information (QFI) and Fisher information (FI) of the particle's position and spin measurements to assess the system's sensitivity to the magnetic fields. We find that one can use the position and spin measurements to estimate the strengths of the magnetic fields. Calculations for an electron implementing quantum walk of 50 time steps show that the estimate has a root-mean-square error of the order of 0.1 picoTesla. Moreover, the sensitivity of our system can be tuned to measure any desired magnetic field. Our results indicate that the system can be used as a tool for optimal quantum magnetometry.
Quantum magnetometry uses quantum resources to measure magnetic fields with a precision and accuracy that cannot be achieved by its classical counterparts. In this paper, we propose a scheme for quantum magnetometry using discrete-time quantum walk (DTQW) where multipath interference plays a central role. The dynamics of a spin-half particle implementing DTQW on a one-dimensional lattice gets affected by magnetic fields and the controlled dynamics of DTQW help in estimating the fields' strength. To gauge the effects of the field, we study the variance of the particle's position probability distribution (PD) and use it to determine the direction of the magnetic field maximally affecting the quantum walk. We then employ statistical tools like quantum Fisher information (QFI) and Fisher information (FI) of the particle's position and spin measurements to assess the system's sensitivity to the magnetic fields. We find that one can use the position and spin measurements to estimate the strengths of the magnetic fields. Calculations for an electron implementing quantum walk of 50 time steps show that the estimate has a root-mean-square error of the order of 0.1 picoTesla. Moreover, the sensitivity of our system can be tuned to measure any desired magnetic field. Our results indicate that the system can be used as a tool for optimal quantum magnetometry.
Discrete-time quantum walk (DTQW) represents a convenient mathematical framework for describing the motion of a particle on a discrete set of positions when this motion is conditioned by the values of certain internal degrees of freedom, which are usually referred to as the coin of the particle. As such, and owing to the inherent dependence of the position distribution on the coin degrees of freedom, DTQWs naturally emerge as promising candidates for quantum metrology. In this paper, we explore the use of DTQWs as quantum probes in scenarios where the parameter of interest is encoded in the internal degree of freedom of the walker and investigate the role of the topology of the walker's space on the attainable precision. In particular, we start considering the encoding of the parameter by rotations for a walker on the line and evaluate the quantum Fisher information (QFI) and the position Fisher information (FI), explicitly determining the optimal initial state in position space that maximizes the QFI across all encoding schemes. This allows us to understand the role of interference in the position space and to introduce an optimal topology, which maximizes the QFI of the coin parameter and makes the position FI equal to the QFI.
We put forward the idea of lattice quantum magnetometry, i.e., quantum sensing of magnetic fields by a charged (spinless) particle placed on a finite two-dimensional lattice. In particular, we focus on the detection of a locally static transverse magnetic field, either homogeneous or inhomogeneous, by performing ground-state measurements. The system turns out to be of interest as a quantum magnetometer, since it provides non-negligible quantum Fisher information (QFI) in a large range of configurations. Moreover, the QFI shows some relevant peaks, determined by the spectral properties of the Hamiltonian, suggesting that certain values of the magnetic fields may be estimated better than others, depending on the value of other tunable parameters. We also assess the performance of coarse-grained position measurement, showing that it may be employed to realize nearly optimal estimation strategies.
In discrete-time quantum walk (DTQW) the walker's coin space entangles with the position space after the very first step of the evolution. This phenomenon may be exploited to obtain the value of the coin parameter $\ensuremath{\theta}$ by performing measurements on the sole position space of the walker. In this paper, we evaluate the ultimate quantum limits to precision for this class of estimation protocols and use this result to assess measurement schemes having limited access to the position space of the walker in one dimension. We find that the quantum Fisher information (QFI) of the walker's position space ${H}_{w}(\ensuremath{\theta})$ increases with $\ensuremath{\theta}$ and with time, which, in turn, may be seen as a metrological resource. We also find a difference in the QFI of bounded and unbounded DTQWs and provide an interpretation of the different behaviors in terms of interference in the position space. Finally, we compare ${H}_{w}(\ensuremath{\theta})$ to the full QFI ${H}_{f}(\ensuremath{\theta})$, i.e., the QFI of the walkers position plus coin state, and find that their ratio is dependent on $\ensuremath{\theta}$, but saturates to a constant value, meaning that the walker may probe its coin parameter quite faithfully.
We address the scattering of a quantum particle by a one-dimensional barrier potential over a set of discrete positions. We formalize the problem as a continuous-time quantum walk on a lattice with an impurity and use the quantum Fisher information as a means to quantify the maximal possible accuracy in the estimation of the height of the barrier. We introduce suitable initial states of the walker and derive the reflection and transmission probabilities of the scattered state. We show that while the quantum Fisher information is affected by the width and central momentum of the initial wave packet, this dependency is weaker for the quantum signal-to-noise ratio. We also show that a dichotomic position measurement provides a nearly optimal detection scheme.
We develop a general theory to estimate magnetic field gradients in quantum metrology. We consider a system of N particles distributed on a line whose internal degrees of freedom interact with a magnetic field. Usually gradient estimation is based on precise measurements of the magnetic field at two different locations, performed with two independent groups of particles. This approach, however, is sensitive to fluctuations of the offset field determining the level splitting of the particles and results in collective dephasing. In this work, we use the framework of quantum metrology to assess the maximal accuracy for gradient estimation. For arbitrary positioning of particles, we identify optimal entangled and separable states, allowing the estimation of gradients with maximal accuracy, quantified by the quantum Fisher information. We also analyze the performance of states from the decoherence-free subspace (DFS), which are insensitive to the fluctuations of the magnetic offset field. We find that these states allow us to measure a gradient directly, without the necessity of estimating the magnetic offset field. Moreover, we show that DFS states attain a precision for gradient estimation comparable to the optimal entangled states. Finally, for the above classes of states, we find simple and feasible measurements saturating the quantum Cram\'er-Rao bound.
A new family of 2D discrete-time quantum walks (DTQWs) is presented and shown to coincide, in the continuous limit, with the Dirac dynamics of a spin 1/2 fermion coupled to a constant and uniform magnetic field. Landau levels are constructed, not only in the continuous limit, but also for the DTQWs i.e. for finite non-vanishing values of the time- and position-step, by a perturbative approach in the step. Numerical simulations support the above results and suggest that the magnetic interpretation is valid beyond the scope of the continuous limit. The possibility of quantum simulation of condensed-matter systems by DTQWs is also discussed.