We study relations between bosonic quadrature squeezing and atomic spin squeezing, and find that the latter reduces to the former in the limit of a large number of atoms for even and odd states. We demonstrate this reduction by treating even and odd spin coherent states, for which analytical solutions are readily obtained, and prove that even spin coherent states always exhibit spin squeezing, whereas odd spin coherent states do not, analogous to the squeezing characteristic of even and odd bosonic coherent states. Finally, we examine the squeezing transfer between photons and atoms via the Dicke Hamiltonian, where a perfect transfer of squeezing is demonstrated in the limit of a large number of atoms.
We study relations between bosonic quadrature squeezing and atomic spin squeezing, and find that the latter reduces to the former in the limit of a large number of atoms for even and odd states. We demonstrate this reduction by treating even and odd spin coherent states, for which analytical solutions are readily obtained, and prove that even spin coherent states always exhibit spin squeezing, whereas odd spin coherent states do not, analogous to the squeezing characteristic of even and odd bosonic coherent states. Finally, we examine the squeezing transfer between photons and atoms via the Dicke Hamiltonian, where a perfect transfer of squeezing is demonstrated in the limit of a large number of atoms.
We study the photonic principle quadrature squeezing and atomic spin squeezing, and demonstrate that the latter reduces to the former in the limit of a large number of atoms for even and odd states. We exemplify this reduction by treating the even and odd spin coherent states, for which analytical solutions are readily obtained, and prove that even spin coherent states always exhibit spin squeezing, whereas odd spin coherent states do not, analogous to the squeezing characteristic of even and odd photonic coherent states. Finally, we examine the squeezing transfer between photons and atoms via the Dicke Hamiltonian, where a perfect transfer of squeezing is demonstrated in the limit of a large number of atoms.
Spin squeezing via atom-field interactions is considered within the context of the Tavis-Cummings model. An ensemble of N two-level atoms interacts with a quantized cavity field. For all the atoms initially in their ground states, it is shown that spin squeezing of both the atoms and the field can be achieved provided the initial state of the cavity field has coherence between number states differing by 2. Most of the discussion is restricted to the case of a cavity field initially in a coherent state, but initial squeezed states for the field are also discussed. Optimal conditions for obtaining squeezing are obtained. An analytic solution is found that is valid in the limit that the number of atoms is much greater than unity and is also much larger than the average number of photons, ${\ensuremath{\alpha}}^{2},$ initially in the coherent state of the cavity field. In this limit, the degree of spin squeezing increases with increasing $\ensuremath{\alpha},$ even though the field more closely resembles a classical field for which no spin squeezing could be achieved.
In this thesis I explore the use of collective spin angular momentum as a platform for quantum information processing. Such systems have several nice features that make them excellent choices for such protocols, especially ones where information is also stored in photonic variables. The longer coherence times of atoms makes it possible to store information from light in atoms for future use, and it is generally easier to couple atomic variables than to create nonlinear interactions of light. Above and beyond the advantages of atomic systems, collective atomic systems have additional strengths. In the limit of a large number of atoms, the collective variables of atomic systems have a natural connection to the bosonic algebra of light (known as the Holstein-Primakoff or HP approximation) where components of the collective spin angular momentum effectively act as quadratures, making them natural systems for coupling to light. Also, collectively addressing the atoms allows one to access a large state space without individual addressing of the atoms. A natural first step towards control of these collective variables is generating large amounts of atomic spin squeezing. In the HP approximation going from rotations to squeezing corresponds to going from linear to quadratic interactions in the atomic quadratures; for extremely large squeezing one should see the breakdown of the HP approximation and the ability to generate arbitrary collective atomic states. I have sought to improve previous schemes for the spin squeezing of atomic ensembles, such as the proposal of Takeuchi et. al. based on coherent quantum feedback. In this scheme a beam of linearly polarized light passes through the atomic ensemble (prepared in a coherent state), coupling to the atoms through a state-dependent index of refraction (the Faraday effect). The light is then passed through a wave-plate and reflected back through the atoms for a second pass. This double-pass scheme leads to an effective nonlinearity as the atomic fluctuations are mapped onto the light on the first pass and then back on to the atoms in the second pass. The light acts as a bus coupling each atom to each of the others. This nonlinear interaction forms a shearing of the atomic coherent state that results in squeezing. The light is entangled to the atoms through these interactions, and remains entangled as it escapes the system. This leads to decoherence of the atoms as the light is lost to the environment, reducing the amount of spin squeezing achieved. The first step towards improving the double-pass scheme was to add a quantum eraser step in which the light is disentangled from the squeezed atoms. By first measuring one quadrature of the light, and then performing a measurement-dependent rotation on the atomic ensemble, it is possible to decouple the atoms and light so that the loss of the light does not reduce the atomic squeezing. This results in an improvement of the rate of atomic spin squeezing. The nonlinear shearing interaction that remains still falls short of the exponential squeezing seen in optical parametric amplification. The reason for this can be seen by decomposing the shearing interaction into squeezing along a given quadrature followed by a rotation. The rotation leads to a constantly changing axis along which the squeezing occurs, and thus the squeezing builds up more slowly. This can be corrected for by breaking up the light pulse into a sequence of small pulses, and performing a small phase matching rotation after each pulse. The squeezing then adds up along a constant direction, rather than along a constantly varying direction. This results in still further improvement of the rate of atomic spin squeezing. A complete model includes the effects of photon-atom scattering and other noise and loss effects on the overall rate of squeezing. Previous derivations of noise due to photon-atom scattering have started with the unjustified assumptions that the atomic and photonic decoherence channels were both Gaussian and independent of correlations between the two subsystems. They then proceed with very general statistical arguments that rely upon these simplifications. My work begins with the more fundamental master equation picture in which I find the Linblad jump operators. I find that in general the photonic and atomic loss channels are not independent, with the intensity of the light dictating the details of this dependence, and find the conditions under which the Gaussian approximation holds. Squeezing and loss is initially treated assuming ensembles of spin-1/2 atoms, but this work is further extended to higher dimensional subsystems. For a higher spin case, preparing the atoms in spin coherent states is not optimal. One can engineer a stronger interaction by preparing the atoms in an atomic "cat state", i.e., a superposition of the two stretch states along the direction of propagation of the light beam. The fluctuations of such a state are more strongly coupled to the light, resulting in a stronger nonlinearity. This leads to strong correlations between the atoms, but they are not immediately useful for squeezing; the cat state must be coherently mapped to a coherent state to achieve atomic spin squeezing. The state created in this manner is ultimately more squeezed than that achieved with the same interaction but prepared initially in a coherent state.
The generated magnitude of quadrature squeezing in a cavity-coupled ensemble, which is continuously driven using a coherent off-axis field, is theoretically explored. Using a truncated set of equations of motion derived from a Dicke Hamiltonian, steady-state quadrature squeezing of the cavity field is numerically calculated to approach a limit of $\ensuremath{-}3\phantom{\rule{0.16em}{0ex}}\mathrm{dB}$, while frequency-modulated quadrature squeezing approaches a limit of $\ensuremath{-}14\phantom{\rule{0.16em}{0ex}}\mathrm{dB}$, in the absence of pure dephasing and as a function of the ensemble's size and detuning. The impact of pure dephasing on steady-state quadrature squeezing is shown to be mitigated by increased detuning of the driving field, while frequency-modulated squeezing is only shielded in a regime where the cumulative coupling and driving rates are in excess of the pure-dephasing rate. Spin-squeezed entanglement is also calculated to occur simultaneously with weakly driven frequency-modulated quadrature squeezing.
We study squeezing and entanglement in superposition of a class of nonlinear spin coherent states. We analyze these properties as a function of the coherence and superposition parameters, number of qubits and the Hamiltonians involved, by presenting analytical and numerical results. We specifically observe that a subclass of these states, the even non-linear cat states are maximally entangled and squeezed, while another related subclass, the odd non-linear cat states, are entangled but not squeezed at all.
This paper reviews quantum spin squeezing, which characterizes the sensitivity of a state with respect to SU(2) rotations, and is significant for both entanglement detection and high-precision metrology. We first present various definitions of spin squeezing parameters, explain their origin and properties for typical states, and then discuss spin-squeezed states produced with nonlinear twisting Hamiltonians. Afterwards, we explain pairwise correlations and entanglement in spin-squeezed states, as well as the relations between spin squeezing and quantum Fisher information, where the latter plays a central role in quantum metrology. We also review the applications of spin squeezing for detecting quantum chaos and quantum phase transitions, as well as the influence of decoherence on spin squeezing. Finally, we review several experimental realizations of spin squeezing, as well as their corresponding theoretical backgrounds, including: producing spin-squeezed states via particle collisions in Bose-Einstein condensates, transferring photon squeezing to atomic ensembles, and generating spin squeezing via quantum non-demolition measurements.