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arXiv:quant-ph/0609054 · Physical Review A

How to measure the quantum state of collective atomic spin excitation

Koji Usami, Jun-ichi Takahashi, Mikio Kozuma

Published Oct 19, 2006• 4 citations
arXivHTMLPDF

Abstract

The spin state of an atomic ensemble can be viewed as two bosonic modes, i.e., a quantum signal mode and a $c$-numbered ``local oscillator'' mode when large numbers of spin-1/2 atoms are spin polarized along a certain axis and collectively manipulated within the vicinity of the axis. We present a concrete procedure, which determines the spin-excitation-number distribution, i.e., the diagonal elements of the density matrix in the Dicke basis for the collective spin state. By seeing the collective spin state as a statistical mixture of the inherently entangled Dicke states, the physical picture of its multiparticle entanglement is made clear.

DOI
10.1103/physreva.74.043815

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The spin state of an atomic ensemble can be viewed as two bosonic modes, i.e., a quantum signal mode and a ccc-numbered ``local oscillator'' mode when large numbers of spin-1/2 atoms are spin polarized along a certain axis and collectively manipulated within the vicinity of the axis. We present a concrete procedure, which determines the spin-excitation-number distribution, i.e., the diagonal elements of the density matrix in the Dicke basis for the collective spin state. By seeing the collective spin state as a statistical mixture of the inherently entangled Dicke states, the physical picture of its multiparticle entanglement is made clear.
We study quantum trajectories of collective atomic spin states of NNN effective two-level atoms driven with laser and cavity fields. We show that interesting ``entangled-state cycles'' arise probabilistically when the (Raman) transition rates between the two atomic levels are set equal. For odd (even) NNN, there are (N+1)/2(N+1)/2(N+1)/2 (N/2)(N/2)(N/2) possible cycles. During each cycle the NNN-qubit state switches, with each cavity photon emission, between the states $(|N/2,m⟩\ifmmode\pm\else\textpm\fi{}|N/2,\ensuremath{-}m⟩)/\sqrt{2}$, where ∣N/2,m⟩|N/2,m⟩∣N/2,m⟩ is a Dicke state in a rotated collective basis. The quantum number mmm (>0)(>0)(>0), which distinguishes the particular cycle, is determined by the photon counting record and varies randomly from one trajectory to the next. For even NNN it is also possible, under the same conditions, to prepare probabilistically (but in steady state) the Dicke state ∣N/2,0⟩|N/2,0⟩∣N/2,0⟩, i.e., an NNN-qubit state with N/2N/2N/2 excitations, which is of particular interest in the context of multipartite entanglement.
We report on the observation of quantum interference of a collective single spin excitation with a spin ensemble of ${N}_{\mathrm{a}}\ensuremath{\approx}{10}^{5}$ atoms. Detection of a single photon scattered from the atoms creates the single spin excitation, a Fock state embedded in the collective spin of the ensemble. The state of the atomic ensemble is then detected by a quantum nondemolition measurement of the collective spin. A macroscopic difference of the order of Na{\sqrt{N}}_{\mathrm{a}}N​a​ in the marginal distribution of the collective spin state arises from the interference between the single excited spin and Na{N}_{\mathrm{a}}Na​ atoms. These hybrid discrete-continuous manipulation and measurement procedures of collective spin states in an atomic ensemble pave the road towards generation of even more exotic ensemble states for quantum information processing, precision measurements, and communication.
We study trajectories of collective spin states of an ensemble of spinors. The spinors considered here are either trapped ions in free space or atoms confined in a cavity, both systems of which are engineered through their interactions with light fields to obey an effective Dicke model. In an appropriate limit of the Dicke model, one obtains one-axis twisting dynamics of the collective spin and evolution after a finite time to a spin cat state or, in the long-time limit, the Dicke state ${|S,0\ensuremath{\rangle}}_{x}$, conditioned upon there being no photon emissions from the system (i.e., no quantum jumps). If there is a jump, however, the system evolves probabilistically into one of a finite number of entangled-state cycles, where the system then undergoes a persistent sequence of jumps between two Dicke-state superpositions in a rotated basis. The different cycles can be distinguished by the frequency at which jumps occur.
We consider the optical generation and characterization of entanglement in atomic ensembles under nonuniform interaction between the ensemble and an optical mode. We show that for a wide range of parameters a system of nonuniformly coupled atomic spins can be described as an ensemble of uniformly coupled spins with a reduced effective atom-light coupling and a reduced effective atom number, with a reduction factor of order unity given by the ensemble-mode geometry. This description is valid even for complex entangled states with arbitrary phase-space distribution functions as long as the average total spin remains large, and the detection does not resolve single spins. Furthermore, we derive an analytic formula for determining the observable entanglement in the case, of relevance in practice, where the ensemble-mode coupling differs between state generation and measurement.
We present a method to detect bipartite entanglement based on number-phase-like uncertainty relations in split spin ensembles. First, we derive an uncertainty relation that plays the role of a number-phase uncertainty for spin systems. It is important that the relation is given with well-defined and easily measurable quantities, and that it does not need assuming infinite dimensional systems. Based on this uncertainty relation, we show how to detect bipartite entanglement in an unpolarized Dicke state of many spin-1/2 particles. The particles are split into two subensembles, then collective angular momentum measurements are carried out locally on the two parts. First, we present a bipartite Einstein-Podolsky-Rosen (EPR) steering criterion. Then, we present an entanglement condition that can detect bipartite entanglement in such systems. We demonstrate the utility of the criteria by applying them to a recent experiment given in K. Lange et al. [Science 360, 416 (2018)] realizing a Dicke state in a Bose-Einstein condensate of cold atoms, in which the two subensembles were spatially separated from each other. Our methods also work well if split spin-squeezed states are considered. We show in a comprehensive way how to handle experimental imperfections, such as the nonzero particle number variance including the partition noise, and the fact that, while ideally BECs occupy a single spatial mode, in practice the population of other spatial modes cannot be fully suppressed.
We describe a new experimental approach to probabilistic atom-photon (signal) entanglement. Two qubit states are encoded as orthogonal collective spin excitations of an unpolarized atomic ensemble. After a programmable delay, the atomic excitation is converted into a photon (idler). Polarization states of both the signal and the idler are recorded and are found to be in violation of the Bell inequality. Atomic coherence times exceeding several microseconds are achieved by switching off all the trapping fields--including the quadrupole magnetic field of the magneto-optical trap--and zeroing out the residual ambient magnetic field.