The Tavis-Cummings model for more than one qubit interacting with a common oscillator mode is extended beyond the rotating wave approximation (RWA). We explore the parameter regime in which the frequencies of the qubits are much smaller than the oscillator frequency and the coupling strength is allowed to be ultra-strong. The application of the adiabatic approximation, introduced by Irish, et al. (Phys. Rev. B 72, 195410 (2005)), for a single qubit system is extended to the multi-qubit case. For a two-qubit system, we identify three-state manifolds of close-lying dressed energy levels and obtain results for the dynamics of intra-manifold transitions that are incompatible with results from the familiar regime of the RWA. We exhibit features of two-qubit dynamics that are different from the single qubit case, including calculations of qubit-qubit entanglement. Both number state and coherent state preparations are considered, and we derive analytical formulas that simplify the interpretation of numerical calculations. Expressions for individual collapse and revival signals of both population and entanglement are derived.
We find an algebraic formula for the N -partite concurrence of N qubits in an X-matrix. X-matrices are density matrices whose only non-zero elements are diagonal or anti-diagonal when written in an orthonormal basis. We use our formula to study the dynamics of the N -partite entanglement of N remote qubits in generalized N -party Greenberger-Horne-Zeilinger (GHZ) states. We study the case in which each qubit interacts with a local amplitude damping channel. It is shown that only one type of GHZ state loses its entanglement in finite time; for the rest, N -partite entanglement dies out asymptotically. Algebraic formulas for the entanglement dynamics are given in both cases. We directly confirm that the half-life of the entanglement is proportional to the inverse of N . When entanglement vanishes in finite time, the time at which entanglement vanishes can decrease or increase with N depending on the initial state. In the macroscopic limit, this time is independent of the initial entanglement.
The direct measurement of a complex wavefunction has been recently realized by using weakvalues. In this paper, we introduce a method that exploits sparsity for compressive measurement of the transverse spatial wavefunction of photons. The procedure involves a weak measurement in random projection operators in the spatial domain followed by a post-selection in the momentum basis. Using this method, we experimentally measure a 192-dimensional state with a fidelity of 90% using only 25 percent of the total required measurements. Furthermore, we demonstrate measurement of a 19200 dimensional state; a task that would require an unfeasibly large acquiring time with the conventional direct measurement technique.
The Laguerre-Gaussian (LG) modes constitute a complete basis set for representing the transverse structure of a paraxial photon field in free space. Earlier workers have shown how to construct a device for sorting a photon according to its azimuthal LG mode index, which describes the orbital angular momentum (OAM) carried by the field. In this paper we propose and demonstrate a mode sorter based on the fractional Fourier transform to efficiently decompose the optical field according to its radial profile. We experimentally characterize the performance of our implementation by separating individual radial modes as well as superposition states. The reported scheme can, in principle, achieve unit efficiency and thus can be suitable for applications that involve quantum states of light. This approach can be readily combined with existing OAM mode sorters to provide a complete characterization of the transverse profile of the optical field. DOI: 10.1103/PhysRevLett.119.263602 In recent years, the transverse structure of optical photons has been established as a resource for storing and communicating quantum information [1]. In contrast to the twodimensional Hilbert space of polarization, it takes an unbounded Hilbert space to provide a mathematical representation for the transverse structure of the optical field. The large information capacity of structured photons has been recently utilized to enhance quantum key distribution [2][3][4][5] and a multitude of other applications [6][7][8][9][10]. The orbital angular momentum (OAM) modes have become increasingly popular for implementing multidimensional quantum states due to the relative ease in generation [11], manipulation [12], and characterization of these modes [13,14].Although the OAM modes provide a basis set for representing the azimuthal structure of photons, they cannot completely span the entire transverse state space, which encompasses an extra (radial) degree of freedom. The Laguerre-Gaussian (LG) mode functions provide a basis to fully represent the spatial structure of the transverse field [15][16][17]. These modes are characterized by two numbers, the radial mode index p ∈ f0; 1; 2; …g and the azimuthal mode index l ∈ f0; AE1; AE2; …g. While the azimuthal number l is well studied due to its association with the OAM of light [16]; the radial index p has so far remained relatively unexplored. The quantum coherence of photons in a superposition of orthogonal radial modes has been recently demonstrated in the context of quantum communication and high-dimensional entanglement [17][18][19]. The radial LG modes also hold a number of promising features, and have been studied in the contexts of self-healing [20], super-resolution [21], and hyperbolic momentum charge [22]. Despite the growing theoretical interest in utilizing the radial structure of photons, the experimental realizations have thus far been impeded because of the difficulty of measuring these modes.The initial step in characterization of the radial degree of freedom of light is to find a radial ...
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