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Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum

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Abstract

We have measured probability distributions of quadrature-field amplitude for both vacuum and quadrature-squeezed states of a mode of the electromagnetic field. From these measurements we demonstrate the technique of optical homodyne tomography to determine the Wigner distribution and the density matrix of the mode. This provides a complete quantum mechanical characterization of the measured mode.
... In a complementary way, the fidelity between an evolved state and its initial state provides insights into the periodic behavior of the system [18]. Likewise, the Wigner function [19], a quasiprobability distribution, offers a perspective on the distribution of coherent states in phase space, shedding light on their classical or quantum nature [20][21][22][23]. Initially, the Wigner function arose from a quantum mechanics formulation in phase space [24,25] in which, under certain requirements, it is possible to associate an integrable function defined on n 2  to an operator in a Hilbert space  through the so-called Weyl transform [26][27][28]. ...
... To build now the coherent states associated with this system, we need to introduce an appropriate 2 × 2 annihilation operator A − for those eigenfunctions in equation (23). For that reason we will define A − as follows [18]: ...
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... Determining the DM accurately is a central task in quantum science. Traditionally, this problem is tackled through Quantum State Tomography (QST) [2,3], which involves performing informationally complete (IC) measurements [4,5] and postprocessing the data to estimate the quantum state. In a d-dimensional Hilbert space, a general density matrix has d 2 − 1 independent parameters. ...
... When designing each eigenbasis, orthogonality and completeness are the key constraints, precisely, each pair of elements in an eigenbasis should be orthogonal, and there should be d elements in each basis. Thus we introduce elements {|ϕ − jk ⟩} and {|ψ − jk ⟩} into the set A d in Eq.(2). Consequently, there are 2d 2 − d elements. ...
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... The Wigner distribution, a quasiprobability distribution for the two quadratures of the harmonic mode, demonstrates energy pumping through an in-spiral analogous that of a classical harmonic oscillator with constant rate of energy loss. Furthermore, the Wigner distribution is routinely measured in circuit QED and cavity QED experiments [34][35][36]. An illustration of this topological in-spiral is in Fig. 4(c)-(h), showing topological pumping that eventually deviates from quantization when the system reaches the topological phase boundary. ...
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... Its potential applications have been discussed in several areas ranging from a quantum key distribution [3][4][5][6], entanglement and resources theory [6][7][8][9][10][11][12], quantum metrology and states discrimination [13][14][15][16][17], to quantum communication and state transfers [13,[18][19][20][21][22][23]. One of the advantages of this program is the ability to encode a qubit into continuous degrees of freedom that are intact to disturbances from the environment, by only using traditional sources of quantum systems and coherence, especially light. ...
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... However, the distributions of measured quadratures are visibly distinct, just not in their mean values, but instead in their second-order moments or covariances. To estimate such second-order moments (routinely required for example for Gaussian state tomography [45,52,53]), the standard approach is to obtain S shots and estimate the variance of the measurements over the dataset yielding readout features y = F NL [x] = 1 S S s (I 2 1 , Q 2 1 ) (shot s dependence of quadratures I k , Q k is implied). For Task I, we plot distributions of these nonlinear readout features in the second panel of Fig. 2(c). ...
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... Experimental reconstructions of the Wigner function rely on homodyne detection measurements [42] in which the optical state we want to characterize is combined with a well-characterized coherent state with a controllable phase, typically referred to as the local oscillator (LO), using a 50 : 50 beam splitter, as shown at the right of Fig. 1. The intensities of the two output modes of the beam splitter are then measured and subtracted, leaving us with ...
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