This chapter describes the use of squeezed states to improve the sensitivity in the estimation of the relative phase accumulated in a Bose--Einstein condensate (BEC) beyond the standard quantum limit. We begin by examining the specific example of a two-component BEC, where atoms in two different hyperfine levels interact with each other. By evolving an initial superposition of states under an effective Hamiltonian, a squeezed state is realized. Next, the squeezing is visualized using the Q-function, giving the mean spin direction and shape of the BEC distribution. We discuss the basic operation of Ramsey interferometry, which is a classic example of two-slit atom diffraction. We then calculate the error in estimating the relative phase shift using the error propagation formula that allows us to define the squeezing parameter. This measures the amount of metrological gain using squeezed states over unsqueezed states. We give an example of the metrological gain in the estimation of the relative phase shift using a squeezed state of the two-component BEC as an input state to an atom Mach--Zehnder interferometer. The Fisher information is used to find which rotation axis gives the optimal information from a given initial state. Finally, we parameterize the nonlinear phase per atom responsible for squeezing and discuss how this parameter is controlled in different experiments.
This chapter describes the use of squeezed states to improve the sensitivity in the estimation of the relative phase accumulated in a Bose--Einstein condensate (BEC) beyond the standard quantum limit. We begin by examining the specific example of a two-component BEC, where atoms in two different hyperfine levels interact with each other. By evolving an initial superposition of states under an effective Hamiltonian, a squeezed state is realized. Next, the squeezing is visualized using the Q-function, giving the mean spin direction and shape of the BEC distribution. We discuss the basic operation of Ramsey interferometry, which is a classic example of two-slit atom diffraction. We then calculate the error in estimating the relative phase shift using the error propagation formula that allows us to define the squeezing parameter. This measures the amount of metrological gain using squeezed states over unsqueezed states. We give an example of the metrological gain in the estimation of the relative phase shift using a squeezed state of the two-component BEC as an input state to an atom Mach--Zehnder interferometer. The Fisher information is used to find which rotation axis gives the optimal information from a given initial state. Finally, we parameterize the nonlinear phase per atom responsible for squeezing and discuss how this parameter is controlled in different experiments.
Reduction of quantum noise in one spin component is a significant tool for enhancing the sensitivities of interferometers and atomic clocks. It has been recently implemented for ultra-cold atomic Bose-Einstein condensate (BEC) interferometer. This type of quantum noise reduction reduces the measurement noise near some predetermined phase. However, if the phase is completely unknown prior to measurement, then it is not known which phase quadrature should be in a squeezed state. We introduce a novel planar squeezing uncertainty relation for spin variance in a plane, and analyze how to obtain such a planar quantum squeezed (PQS) state by using a double-well single component BEC, through the use of local nonlinear S-wave scattering interaction between trapped atoms. Here, we consider the PQS that is generated by using two hyperfine states in a two components BEC system, which is useful for quantum metrology. By comparison with the case of two spatial wells, the Hamiltonian parameters can be controlled in a more efficient way. The spin component can be measured by detecting the occupation number difference between the two internal modes, while one needs to observe a spatial interference pattern in the double well BEC case. This is the major difference between the internal and external cases. Another difference is that one can use the Rabi frequency Ω instead of the Josephson parameters to switch the Hamiltonian parameters through using a diabatic technique. Therefore the coupling could be switched off or on to study the different evolutions. PQS simultaneously reduces the quantum noises of two orthogonal spin projections below the standard quantum limit, while increases the noise in the third dimension. This allows the improvement in phase measurement at any phase-angle. PQS states that reductions of fluctuations everywhere in a plane have potential utility in "one-shot" phase measurement, where iterative or repeated measurement strategies cannot be utilized. The improved interferometric phase measurements and planar uncertainty relations are useful for detecting the entanglement in mesoscopic system between two distinguished modes regardless of the third component.
Phase estimation has a wide range of applications. Over the years, several strategies have been studied to improve precision in phase estimation. These strategies include using exotic quantum states to quantum detection schemes. This dissertation summarizes my effort in improving the precision of phase estimation with a linear and nonlinear interferometer. Chapter 1 introduces quantum optics and quantum metrology. I introduce all relevant quantum states of light used. We also look into tools and terminologies of quantum metrology such as Fisher information, shot-noise limit, Heisenberg limit, etc., along with examples of phase estimation with a Mach-Zehnder interferometer. In Chapter 2, I discuss multiple phase estimation using a multimode interferometer. Building upon previous work, our scheme consists of a multimode interferometer with single-photon inputs. By using a quantum Fisher information analysis, we show that our scheme gives a constant improvement over other schemes. We also show that our scheme with photon-number-resolving detection approaches the quantum Cram\'er-Rao bound. Moreover, we also consider the probabilistic nature of photon emission at the input, and we study its effect on phase sensitivity. I discuss phase estimation with SU(1,1) interferometer in Chapter 3. We look at phase sensitivity in this interferometer with different input states. Namely, we consider two different phase estimation scheme, one using thermal and squeezed states, and others using coherent and displaced squeezed states with parity and on-off as a detection scheme. We also look into the effect of photon loss inside the interferometer. In Chapter 4, we revisit phase estimation in SU(1,1) interferometer from the perspective of quantum Fisher information. I discuss in detail a longstanding confusion regarding the use of quantum Fisher information in SU(1,1) interferometer. We show that phase averaging or quantum Fisher information matrix method is needed in general for calculating the phase sensitivity which resolves inconsistencies reported in previously published articles.
We investigate the prospect of enhancing the phase sensitivity of atom interferometers in the Mach-Zehnder configuration with squeezed light. Ultimately, this enhancement is achieved by transferring the quantum state of squeezed light to one or more of the atomic input beams, thereby allowing operation below the standard quantum limit. We analyze in detail three specific schemes that utilize (1) single-mode squeezed optical vacuum (i.e., low-frequency squeezing), (2) two-mode squeezed optical vacuum (i.e., high-frequency squeezing) transferred to both atomic inputs, and (3) two-mode squeezed optical vacuum transferred to a single atomic input. Crucially, our analysis considers incomplete quantum state transfer (QST) between the optical and atomic modes, and the effects of depleting the initially prepared atomic source. Unsurprisingly, incomplete QST degrades the sensitivity in all three schemes. We show that by measuring the transmitted photons and using information recycling [Phys. Rev. Lett. 110, 053002 (2013)], the degrading effects of incomplete QST on the sensitivity can be substantially reduced. In particular, information recycling allows scheme (2) to operate at the Heisenberg limit irrespective of the QST efficiency, even when depletion is significant. Although we concentrate on Bose-condensed atomic systems, our scheme is equally applicable to ultracold thermal vapors.
As an ``input'' state of a linear (Mach-Zehnder or Ramsey) interferometer, the phase-squeezed state proposed by Berry and Wiseman exhibits the best sensitivity approaching to the Heisenberg limit [Phys. Rev. Lett. 85, 5098 (2000)]. Similar with the Berry and Wiseman's state, we find that two kinds of phase-squeezed states can be generated dynamically with atomic Bose-Einstein condensates confined in a symmetric double-well potential, which shows squeezing along spin operator ${\mathrm{S\ifmmode \hat{}\else \^{}\fi{}}}_{y}$ and antisqueezing along ${\mathrm{S\ifmmode \hat{}\else \^{}\fi{}}}_{z}$, leading to subshot-noise phase estimation.
In recent years, atom interferometry has become established as an indispensable tool in both fundamental and applied physics. With present state-of-the-art devices based on thermal atoms reaching limits imposed by the momentum spread of the initial atomic wavepacket, it seems natural to ask whether colder sources such as Bose-Einstein condensates may prove beneficial in advancing the precision of interferometric measurements. The thesis at hand aims to inform this question, specifically by examining the role played by atomic interactions in interferometers based on Bose-condensed atoms. Interactions can have both advantageous and deleterious consequences in the context of atom interferometry. They provide a means to control the momentum width of the condensate, and facilitate the generation of nonclassical squeezed states which may enhance the phase sensitivity beyond the shot noise limit. Conversely, the condensate self-interaction causes mean-field shifts, multimode excitations and phase diffusion which can erode both the precision and the accuracy of an interferometric measurement. The question of when and in which systems the detrimental effects of interactions outweigh the advantages of using Bose-Einstein condensates is an important one, and warrants investigation. This thesis presents experimental studies into the role of interactions in both internal- and external-state atom interferometers. As a foundation for these investigations, we describe the design and construction of an apparatus for creating Bose-Einstein condensates of the two stable rubidium isotopes in an optical trap. By sympathetic cooling with a rubidium-87 reservoir, we are able to produce condensates of rubidium-85 in which the interactions may be adjusted by means of a magnetic Feshbach resonance. The tunability afforded by the Feshbach resonance is used to study inelastic losses in ultracold rubidium-85 clouds, as well as the effect of interactions on condensate stability and on the ground state of dual-species mixtures. In particular, we offer new experimental data on the dynamics of collapsing condensates with attractive interactions, over which some controversy has existed since the first experiments more than a decade ago. Good agreement is found between the measured collapse times and a simple mean-field model. Proceeding to interferometry, we present results from Ramsey interferometers operating on the clock transition of rubidium-87 Bose-Einstein condensates. In free-space operation with Raman beamsplitters, we demonstrate projection-noise-limited performance, an important prerequisite for the realisation of squeezing-enhanced sensitivity. Using large condensates of up to 10⁶ atoms and microwave coupling, we study the effect of interactions on the Ramsey fringe contrast. The dominant source of decoherence is found to be spatial dynamics driven by the difference in interparticle interaction strengths, which are analysed using the spin-echo technique and numerical simulations of the Gross-Pitaevskii equation. Finally, we turn our attention to external-state interferometry, implementing a Mach-Zehnder gravimeter using Bragg transitions in a freely falling rubidium-87 condensate. Large-momentum-transfer beamsplitters composed of higher-order Bragg diffraction and Bloch oscillations are used to increase the accumulated phase and thus the sensitivity of the interferometer. The role of interactions in this system is examined, and we canvass methods for achieving further increases in sensitivity.
We analyze methods designed to go beyond the standard quantum limit for a class of atomic interferometers, where the quantity of interest is the difference of phase shifts obtained by two independent atomic ensembles. An example is given by an atomic Sagnac interferometer, where for two ensembles propagating in opposite directions in the interferometer this phase difference encodes the angular velocity of the experimental setup. We discuss methods of separately or jointly squeezing observables of the two atomic ensembles, and compare in detail the advantages and drawbacks of such schemes. In particular, we show that the method of joint squeezing may improve the variance by up to a factor of 2. We take into account fluctuations of the number of atoms in both the preparation and the measurement stage, and obtain bounds on the difference between the numbers of atoms in the two ensembles, as well as on the detection efficiency, which have to be fulfilled in order to surpass the standard quantum limit. Under realistic conditions, the performance of both schemes can be improved significantly by reading out the phase difference via a quantum nondemolition measurement. Finally, we discuss a scheme using macroscopically entangled ensembles.