Direct Measurement Of The Quantum Wave Function And Its Application

The wave function is at the foundation of quantum theory,which is assumed to give a complete description of a quantum system.An indirect method known as quantum state tomography(QST)has been developed to reconstruct the complete quantum wave function of a system.QST is based on the measurement of complementary variables of several copies of the same quantum system,followed on an estimation of the wave function that better reproduce the results obtained.To characterize a d-dimensional system through QST,the number of the independent measurements required is d2-1.Both the measurement time and the postprocessing of the reconstruction suffer a rapid growth as d grows.Recently,an alternative method known as the direct measurement(DM)has been proposed for estimation of a quantum state.Direct measurement is suggested as an efficient means for characterizing high-dimensional states due to its simplicity of realization without further time-consuming calculation process to fit on the measurement outcomes.Currently,some important works have proved the validity of the direct measure-ment in degrees of freedom of the photon:experimental demonstration of directly mea-suring the transverse spatial wave function of a single photon,full characterization of polarization states of light via direct measurement,direct measurement of a high-dimensional orbital-angular-momentum state vector,direct measurement of the density matrix of a qubit system and scan-free direct measurement of a high-dimensional pho-tonic state.Direct measurement relies on measurement of the ’weak value’,which is usually closely related to the result of the reconstruction state.As originally proposed,the weak value is obtained via weak measurement,where the interaction between the pointer(the measurement apparatus)and the system is weak.Weak measurements occur when the coupling strength between the pointer and the system is much less than the pointer width.However,the use of weak values implies that the measurement based on weak measurement schemes are approximated and affected by systematic errors and statistical errors.For practical situations,a trade-off between the validity of the approximation(improved by weakening the interaction)and the uncertainties(improved by increasing the interaction)should be made.Nevertheless,it has been shown that for pure states,the weak measurement is not necessary for direct measurement.The quantum wave function can be obtained by the same scheme used in direct weak measurement(DWM)but using measurements with arbitrary coupling strength.Hereafter in all discussion the term ’weak measurement’or’strong measurement’ refers to measurement characterized by a weak or strong cou-pling between the system and the pointer.Moreover,the use of strong measurements in most cases gives a better estimation of the quantum wave function,outperforming direct weak measurement in both accuracy and precision.My work focuses on direct strong measurement(DSM)of the quantum wave function.To improve the versatility and the scalability of direct strong measurement,we have developed several experimental setups:1.To extend the DSM method to the transverse spatial degree of light,the tech-nique of direct strong measurement of the two-dimensional spatial quantum wave func-tion has been developed,and 625-dimensioanl photonic states are reconstructed.2.The time-consuming scanning process is used to obtain the complete wave func-tion of the state in the previous method,we futher developed a scan-free method to measure a high-dimensional transverse spatial wave function with dimensionality of 105.3.Since the transverse spatial wave function of the light is analog to the transverse electric field of light,our technique is ready for appications in classical wavefront sens-ing.We perform quantitative phase imaging by direct measurement,and we hope our method will be used in diverse fields such as biology and medical science.