We report results of a high precision phase estimation based on a weak measurements scheme using commercial light-emitting diode. The method is based on a measurement of the imaginary part of the weak value of a polarization operator. The imaginary part of the weak value appeared due to the measurement interaction itself. The sensitivity of our method is equivalent to resolving light pulses of order of attosecond and it is robust against chromatic dispersion.High precision phase measurements play a significant role in modern physics. The standard tool is an interferometer with a balanced homodyne detection [1]. It requires a coherent source and the precision is dominated by the intrinsic quantum noise [2]. To reduce the influence of the noise, quantum metrology technologies [3] including N00N states [4] and squeezed states [5] have been exploited, while white light is usually deemed to be useless in quantum metrology. Recently it has been proposed that white light can be used for a very precise phase estimation [6,7], when weak measurements are performed. Here we experimentally demonstrate such a sensitive method utilizing white light from a commercial light-emitting diode (LED). This opens a new avenue for a high-resolution phase estimation.As in other weak measurement experiments in which the Aharonov-Vaidman-Albert (AAV) amplification effect [8] was demonstrated, we measure the photon polarization operator A with eigenvalues 1 and -1 for the two orthogonal polarizations. The polarization can be preand post-selected with a very good precision. The role of the measuring device is played by the spatial degree of freedom of light. In most weak measurement experiments, the relevant spatial degree of freedom is the position in the transverse direction, i.e. perpendicular to the direction of the light propagation. Here we consider, instead, the longitudinal direction [6,7].The interaction Hamiltonian iswhere g(t) is the coupling strength satisfying g(t)dt = k and P is a component of the momentum of the photon. In the first realization of weak measurement the transversal shift was created by a tilted plate of a birefringent material [9]. In our experiment, the plate is placed perpendicularly to the photon's velocity and leads to a longitudinal shift, see Fig. 1a. We consider a very thin birefringent plate which leads to a time delay of a few attoseconds between the wave packets with different polarizations. The AAV effect with the proper pre-and post-selection of polarization can increase the time delay significantly, but a truly dramatic advantage is obtained for the measurement of the imaginary part of the weak Weak measurement of the photon polarization. Photons emitted from the source are preselected by a polarization beamsplitter (PBS) in a state |ψpre , undergo weak measurement interaction by passing through birefringent plate and are post-selected at a nearly orthogonal state |φpost by a second PBS. a). The wave packets with orthogonal polarizations are delayed after birefringent plate one relative to the othe...

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The advantages of weak measurements, and especially measurements of imaginary weak values, for precision enhancement, are discussed. A situation is considered in which the initial state of the measurement device varies randomly on each run, and is shown to be in fact beneficial when imaginary weak values are used. The result is supported by numerical calculation and also provides an explanation for the reduction of technical noise in some recent experimental results. A connection to quantum metrology formalism is made.In 1988 Aharonov, Albert and Vaidman (AAV) [1] discovered that the measured value of an observable can be 100 times bigger than its biggest eigenvalue, provided the measurement interaction is weak and a postselection is employed. They showed that a system which is coupled weakly to another, pre-and postselected system, described by the two-state vector Φ| |Ψ , via an observable C, is effectively coupled to the weak value of the observable [2]The replacement of the interaction operator with its weak value, which is a complex number [3], is known as the AAV effect and the procedure in which the weak value is measured is referred to as a weak measurement. In this letter we will analyze the process of weak measurement as a method for precision measurements. Furthermore, we will present a concrete model for technical noise affecting the preparation of the measurement device (meter), and show that in the presence of such a noise the precision is enhanced.We start with an overview of known results regarding the precision achievable by weak measurements. Consider a physical interaction:where C is an observable on a system, P is an operator on a meter and g(t) is a coupling function satisfying g(t)dt = k. Our concern is estimating the size of k, or in some cases simply observing the interaction. A straight forward approach is to put the system in an eigenstate of C having some eigenvalue c, and the meter in a Gaussian state:where Q is a variable conjugate to P , and ∆ is its quantum uncertainty. An estimate of k can be obtained from the shift in Q due to the interaction, Q = kc , and its precision is determined by the standard deviation∆. In the case kc ∆, little information is acquired from a single measurement, but by repeating the procedure N times and averaging the results, the precision is enhanced. Strictly speaking, the amount of information gathered, regarding k, is measured by the Fisher information [15], but for our purposes we can use the more intuitive concept of signal to noise ratio (S/N ) [17], which in this case isSince our interest is in the regime where kc ∆, which is a condition for the AAV effect [18], we will, for now, assume that the AAV effect occurs and later examine its validity in more detail. Thus, we will consider the system to be initially in a state |Ψ and take into account the meter results only when the system was found in a state |Φ , after the interaction, which implies a replacement C → C w in (2) [19]. The shift in Q is given by Q Φ = kReC w [1], andwhere N Φ ∼ N | Φ|Ψ | ...

The concept of a modular value of an observable of a pre- and postselected quantum system is introduced. It is similar in form and in some cases has a close connection to the weak value of an observable, but instead of describing an effective interaction when the coupling is weak, it describes a coupling of any strength but only to qubit meters. The generalization of the concept for a coupling of a composite system to a multiqubit meter provides an explanation of some current experiments.