Spin squeezing via quantum feedback

Abstract

We propose a quantum feedback scheme for producing deterministically reproducible spin squeezing. The results of a continuous nondemolition atom number measurement are fed back to control the quantum state of the sample. For large samples and strong cavity coupling, the squeezing parameter minimum scales inversely with atom number, approaching the Heisenberg limit. Furthermore, ceasing the measurement and feedback when this minimum has been reached will leave the sample in the maximally squeezed spin state. [4,6], and direct coupling to an entangled state through intermediate states such as collective motional modes for ions [7] or molecular states for atoms [8].Other proposals create spin squeezing via quantum nondemolition (QND) measurements [9,10,11]. A striking recent achievement of QND measurements is the entanglement of two macroscopic atomic samples [12]. These QND schemes produce conditional squeezed states that are dependent on the measurement record. On the other hand, unconditional squeezing would ensure that the state is deterministically reproducible. Mølmer [13] has shown that alternating QND measurements and incoherent feedback can produce sub-Poissonian number correlations. However, that work does not treat the quantum effects of the measurement back action or the feedback on the mean spin (which is assumed to be zero). Hence it cannot predict the strength of the entanglement.In this Rapid Communication, we suggest achieving spin squeezing via feedback that is coherent and continuous. We consider a continuous QND measurement of the total population difference of an atomic sample. The results of the measurement, which conditionally squeeze the atomic sample, are used to drive the system into the desired, deterministic, squeezed spin state. This involves amplitude modulation of a radio-frequency (rf) magnetic field, where the feedback strength varies in time such that the mean number difference is kept at zero.An ensemble of N two-level atoms can be described by a spin-J system [14], i.e., a collection of 2J = N spin- where n i (i = 1, 2, 3) are orthogonal unit vectors. Systems with ξ 2 n < 1 are spin squeezed in the direction n and also have multiparticle entanglement [4].Let the internal states, |1 and |2 , of each atom be the degenerate magnetic sublevels of a J = 1 2 state, e.g., an alkali ground state. Each atom is prepared in an equal superposition of the two internal states, thus giving a CSS of length J in the x-direction. The atomic sample is placed in a strongly driven, heavily damped, optical cavity, as shown in Fig. 1(a). The cavity field is assumed to be far off resonance with respect to transitions probing state |2 , see Fig. 1(b). This dispersive interaction causes a phase shift of the cavity field proportional to the number of atoms in |2 . Thus, the QND measurement of J z (since N is conserved) is effected by the homodyne detection of the light exiting the cavity [15]. This interaction is defined by the HamiltonianhχJ z b † b where b,b † are the cavity field operators and χ = g 2 ...

Teleportation and spin squeezing utilizing multimode entanglement of light with atoms

“…The collective spin of the ensemble is then probed by an off resonant pulse which propagates along z and is linearly polarized along x. Thorough descriptions of this interaction and the final state of light and atoms after the scattering can be found in [14,15,16,17,18] and especially in [19,20,21,22] for the specific system we have in mind. We derive the final state here with a special focus on the effects of Larmor precession and light propagation in order to identify the light modes which are actually populated in the scattering process.…”

Quantum-noise-limited interferometric measurement of atomic noise: Towards spin squeezing on the Cs clock transition

“…So far, we have ignored the electronic noise in the above calculations. It is however important if feedback schemes should be applied in order to either enhance the spin squeezing [20], or if we wish, to rotate the mean spin direction according to our measurement with the goal of obtaining a specific spin squeezed state. The latter is relevant if the spin state should be employed in, e.g., atomic clocks, where the a Bloch vector in the equatorial plane is desired [21].…”

“…Additionally, we must update the atomic state based on the result of the QND measurement [20]. Depending on the outcome of the measurement the mean value of the pseudo spin vector will be shifted away from  z = 0.…”

Decoherence dynamics of a single spin versus spin ensemble

“…5,6 Also, the studies of single spins allow detailed exploration of fundamental questions related to complex quantum dynamics in spin systems. During the last few years, rapid progress in this direction leads to implementation of measurement and control of various single-spin systems, such as single electron spins localized in quantum dots, [7][8][9][10] spins of impurity centers in diamond, [11][12][13][14] or electron spins in SiO 2 ͑Refs.…”