Negativity of the wigner function as an indicator of nonclassicality. Wigner distribution, nonclassicality and decoherence of. One or two entangled twolevel atoms interact with a coherent state trapped in a cavity. The study of the wigner function of a decaying schro dinger cat state 4 is also within reach.
Cavity state manipulation using photonnumber selective. See below for a characterization of pure states whose wigner functions are nonnegative. The important difference is that the in the quantum case the wigner function. Your question has little to do with wigner functions, though. An extra source atom is then used to prepare a first photon in the cavity before the abovedescribed sequence starts. We have reconstructed the quantum state of optical pulses containing single photons using the method of phaserandomized pulsed optical homodyne tomography. Recent advances in wigner function approaches aip publishing.
Fock states play an important role in the second quantization formulation of quantum mechanics the particle representation was first treated in detail by paul dirac for. Squeezed and correlated states of quantum fields and multiplicity. Investigation of thewigner distribution functions at the output fields iscarried out, where it is found that the distributionspreads out as it rotates in the phase space. Sections v and vi treat ceqd like systems, namely the ionlaser and mirror led interactions, respectively. The input lightfield of the linear amplifier is assumed to be asqueezed displaced fock state. Quantum state reconstruction of the singlephoton fock. In the quantum number state representation, these multiples of three are responsible for the threefold phasespace symmetry seen. A quantum state in the fock space is called a fock state if it is an element of the occupancy number basis. The existence of negative quasiprobabilities is generally considered as a signature of nonclassical states.
The nonclassical character of the fock state shows up as a negativity of the wigner function at the origin. When plotting the wigner function of a coherent state using. Quantum state reconstruction of the singlephoton fock state core. Quantumtoclassical transition with singlephotonadded. Due to the fact that the evolution in an ideal 3 medium is periodic, we achieve the same shape of the wigner function for 2. These states are named after the soviet physicist vladimir fock. Fockstate stabilization and emission in superconducting. Wigner distribution function and wigner spectrum due to the singular commutation relations 1 and 5, for the continuousmode singlephoton fock state. Braasch et aldissipative phonon fock state production in strong nonlinear optomechanics g.
Coordinatemomentum versus coherent state representations. Approximate particle number distribution from direct. The wigner function of the measured state is clearly negative. Several articles on the wigner function alone without the. The wigner function is analogous to a classical joint probability function, but it can take negative values, which is precisely the benchmark of quantum effects.
This bypasses the requirements of working with complicated manifolds for calculating renyi entropies for many body systems. The wigner distribution function for the outputof a linear amplifier is discussed. It was introduced 1 by eugene wigner in 1932 to study quantum corrections to classical statistical mechanics. The wigner function is a quasiprobability distribution 57, which fully describes the state of a quantum system in phase space either the positionmomentum space for an harmonic oscillator or, equivalently, the space spanned by two orthogonal quadratures of the. A probability distribution of the phaseaveraged electric eld amplitudes with a strongly nongaussian shape is obtained with the total detection e ciency of 55 1%. We have measured the complete wigner function w of the vacuum and of a singlephoton state for a field stored in a highq cavity. In section iv more states of the harmonic oscillator are introduced. It is a little \smoother than the pdistribution, but can be negative for certain states like fock or number states so it has its own peculiarities. Any multimode gaussian state \\varrho\ is completely parametrized by its first and second moments, namely the vector of means \\vec\beta\ with components. Direct measurement of thewigner function of a onephoton fock. C and then into a fock like state after an interaction time. The blue spots indicate that the wigner function is negative at that point, but a coherent state should have a wigner function thats positive everywhere.
Nuclear physics a512 1990 253274 northholland large. The wigner quasiprobability distribution is a quasiprobability distribution. We provide an exact analytic formula for renyi and vonneumann entanglement entropies of noninteracting open quantum. To obtain the fock space state vector ket of a pure gaussian state with quadrature covariance matrix \\bmv\ and vector of means \\bmr\ use the function thewalrus. Jan 10, 2014 the wigner functions for the gbs and the rbs are obtained by using and shown in fig.
Wigner functions and weyl transforms for pedestrians statistical. Clearly there exist negative regions of wigner function in these figures and these negative regions illustrate that these two fsfss i. It is energetically very unfavorable to move any distance along the same state. We see two round hills at the top and at the bottom, associated with each coherent state involved. Wigner function evolution of quantum states in the presence. B to d same, but with an input coherent field of increasing amplitude. In quantum mechanics, a fock state or number state is a quantum state that is an element of a fock space with a welldefined number of particles or quanta. A reconstructed wigner function for the singlephoton fock state obtained without injection. The coherent state is just a rotation of the ground fock state ij,j away from the south pole. G p 3 and a second one in which the state of interest jci, such as a fock state j1i, is prepared. From this coherent state, a wigner function in phase space. Wigner function of the thermo number state wigner function of quantum states in phase space is an important topic in quantum mechanics, quantum statistics and quantum optics. Steadystate negative wigner functions of nonlinear. A twophoton fock state wigner function could also be measured easily.
If our ensemble were an incoherent, statistical mixture of these two states, its wigner function would just feature these two hills. Some significant quantum statistical properties of the gsts are investigated, such as the photonnumber distribution. Entanglement entropy of fermions from wigner functions. Excited states and open quantum systems saranyo moitra1, and rajdeep sensarma1 1department of theoretical physics, tata institute of fundamental research, mumbai 400005, india. We analyze this quantity for fock states, squeezed displaced fock states and catlike states defined as coherent superposition of two gaussian wave packets. Wigner spectrum and coherent feedback control of continuous. A quantum state is called gaussian if its wigner function is gaussian. We analyze this quantity for fock states, squeezed displaced fock states and cat like states defined as coherent superposition of two gaussian wave packets. Summary and outlook to sum up, we have derived simple analytical solutions for a quantum state and its wigner function at the output ports of a general passive and lossless optical four port like a beam splitter, when a coherent state and a singlephoton fock state are incident on the input ports. In particular, if the filter is considered to be in a vacuum state then the.
Experimental state tomographyof itinerant single microwave. Quantum state reconstruction of the singlephoton fock state. The wigner function is analogous to a classical joint probability function, but it can take negative values, which. Apr 26, 2012 in this paper, we present the density operator of the generalized squeezed thermal state gsts and obtain its normal ordering form by virtue of the technique of integration within an ordered product of operators and the weyl ordering invariance under similarity transformations. Linear amplifier and quasiprobability distribution. The angleaveraged wigner function recon structed from this distribution shows a strong dip reaching classically impossible negative values. The wigner quasiprobability distribution also called the wigner function or the wigner ville distribution after eugene wigner and jeanandre ville is a quasiprobability distribution. Direct measurement of thewigner function of a onephoton fock state in a cavity p. It is hard to generate experimentally, but the wigner function is instructive to look at. In d, the reconstructed wigner function for both the spacs and the unexcited seed coherent state wireframe surface are shown. A probability distribution of the phaseaveraged electric field amplitudes with a strongly. Photonnumber distribution and wigner function of generalized. The singlephoton fock state \1 was prepared using conditional measurements on photon pairs born in the process of parametric downconversion.
Nonclassicality of fdeformed photonaddedthensubtracted. The phasenumberwigner function for the quasiphase state 16 for m 20 and. A recently developed highfrequency homodyne tomography technique has been used to completely characterize the fock state by means of a pulsetopulse analysis of the detectors difference photocurrent. You are seeking to represent the generic density matrix, so, really, state, in terms of the complete basis of coherent states. It turns out that all pure states have singular p distributions. Jun 14, 2004 a singlephoton fock state has been generated by means of conditional preparation from a twophoton state emitted in the process of spontaneous parametric downconversion. This classically impossible phenomenon is a signature of highly nonclassical character of a quantum state. Tomographic reconstruction of the singlephoton fock state by. In practice both histograms are accumulated in an interleaved fashion, changing betweenthetwocasesevery25 stoavoidsystematicerrors due to drifts. In section 6 the formalism of wigner function and weyl. Oct 22, 2004 experimental wigner functions for the spacs. Towards a complete, continuous, wigner function for an.
But because the superposition is coherent, the wigner function exhibits an additional, highly nonclassical feature in the middle. Tomographic reconstruction of the singlephoton fock state. Quantum homodyne tomographyof a twophoton fock state. Color online the wigner function for 5 and 0 is a gaussian function for the coherent state left. Reconstruction on wigner functions on different observation. Pdf direct measurement of the wigner function of a one. The field, represented here by its wigner function, evolves from a coherent state to a macroscopic superposition after an interaction time. The development of a quasiprobability distribution associated with a phase operator has previously been impossible because of the absence of a unique hermitian phase operator defined on the hilbert space.
Phasespace formulation of quantum mechanics and quantum state. The supplemental material 19 contains wigner functions showing the free evolution as well as the kerrcorrected state from 1 to. This state is an analog of the squeezed state, but is generated in theoreticians notebooks by means of 3photon, rather than 2photon. Pdf quantum interference between a singlephoton fock.
This experiment implements the direct lutterbach and davidovich. Counting two photons in one beam projects the other beam in the desired twophoton fock state, which is analyzed by using a pulsed homodyne detection. Fock states are also eigenstates of the number operator. In particular, if the filter is considered to be in a vacuum state. Dirac function 14, resulting in the distribution d. Quantum interference between a singlephoton fock state and a.
Fock state k1a, which exhibits negative values of the wigner function 3, 4. The wigner function was formulated in 1932 by eugene paul wigner, at a time when quantum. Wigner function at the origin of phase space coincides with the expectation value of the photonnumber parity operator 19. Measurement of a negative value for the wigner function of. Recently, wigner functions of some quantum states have been calculated by using the coherent entangled state representation of wigner operator.
This yields an expression of the wigner function in the fock basis which is easy to reconstruct, as was. Quantum state reconstruction and tomography using phase. Quantum interference between a singlephoton fock state. Introduction the wigner function, or the wigner quasiprobability. This state is obtained from a pulsed nondegenerate parametric ampli. A comparison of two nonclassical measures, entanglement.
710 1154 1576 1134 1644 362 433 1322 142 487 711 1175 258 1135 1057 109 1547 1400 30 1509 1335 1589 1039 1385 229 1110 279 545