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 ...

Abstract. We discuss the theory and experimental considerations of a quantum feedback scheme for producing deterministically reproducible spin squeezing. Continuous nondemolition atom number measurement from monitoring a probe field conditionally squeezes the sample. Simultaneous feedback of the measurement results controls the quantum state such that the squeezing becomes unconditional. We find that for very strong cavity coupling and a limited number of atoms, the theoretical squeezing approaches the Heisenberg limit. Strong squeezing will still be produced at weaker coupling and even in free space (thus presenting a simple experimental test for quantum feedback). The measurement and feedback can be stopped at any time, thereby freezing the sample with a desired amount of squeezing.

A continuous atom laser will almost certainly have a linewidth dominated by the effect of the atomic interaction energy, which turns fluctuations in the condensate atom number into fluctuations in the condensate frequency. These correlated fluctuations mean that information about the atom number could be used to reduce the frequency fluctuations, by controlling a spatially uniform potential. We show that feedback based on a physically reasonable quantum non-demolition measurement of the atom number of the condensate in situ can reduce the linewidth enormously. 03.75.Fi, 42.50.Lc An atom laser is a continuous source of coherent atom waves, analogous to an ordinary laser which is a continuous source of coherent photon waves (light) [1,2]. Ideas for creating an atom laser were published independently by a number of authors [3][4][5][6], shortly after the first achievement of Bose-Einstein condensation (BEC) of gaseous atoms [7][8][9]. There have since been some important advances in the coherent release of pulses [10,11] and beams [12,13] of atoms from a condensate. Since the condensate is not replenished in these experiments, the output coupling cannot continue indefinitely [14]. Nevertheless they represent major steps towards achieving a continuously operating atom laser.The coherence of an atom laser beam can be defined analogously to that of an optical laser beam: the atoms should have a relatively small longitudinal momentum spread, they should ideally be restricted to a single transverse mode and internal state, and their flux should be relatively constant [2]. A fourth condition, rarely considered for optical lasers because it is so easily satisfied, is that the laser beam be Bose degenerate. This requires that the atom flux be much larger than the linewidth (the reciprocal of the coherence time) [2]. A crucial contributor to the linewidth of an atom laser is the collisional interaction of atoms, which is negligible for photons. This is difficult to avoid because it is the collisions between atoms that enables BEC by evaporative cooling, at present the only method for achieving an atom laser.In this Letter we show that using a feedback mechanism can reduce the effect of atomic interactions on the atom laser linewidth by a factor as large as the square root of the atom number. For a single-mode condensate the dominant effect of collisions is to turn atom number fluctuations in the condensate into fluctuations in the energy, which are equivalent to frequency fluctuations. By monitoring the number fluctuations, it is possible using feedback to largely compensate for the linewidth caused by these frequency fluctuations. The key practical points are that the measurement does not rely upon any external condensate phase reference, and that the control requires only the ability to change the energy of the atoms, which could be done with a spatially uniform optical or magnetic field.We begin by deriving the standard laser linewidth (for non-interacting bosons) using a simple method which is applied to all later ca...