Studies of ultracold gases in optical lattices1 link many disciplines. They allow testing fundamental quantum many-body concepts of condensed-matter physics in well controllable atomic systems 1 , e.g., strongly correlated phases, quantum information processing. Standard methods to observe quantum properties of Bose-Einstein condensates (BEC) are based on matter-wave interference between atoms released from traps 2,3,4,5,6 , destroying the system. Here we propose a new, nondestructive in atom numbers, method based on optical measurements, proving that atomic quantum statistics can be mapped on transmission spectra of high-Q cavities, where atoms create a quantum refractive index. This can be extremely useful for studying phase transitions 7 , e.g. between Mott insulator and superfluid states, since various phases show qualitatively distinct light scattering. Joining the paradigms of cavity quantum electrodynamics (QED) and ultracold gases will enable conceptually new investigations of both light and matter at ultimate quantum levels. We predict effects accessible in experiments, which only recently became possible 8 . All-optical methods to characterize atomic quantum statistics were proposed for homogeneous BEC 9,10,11,12,13 and some modified spectral properties induced by BEC's were attributed to collective emission 9,10 , recoil shifts 12 or local field effects 14 . We show a completely different phenomenon directly reflecting atom quantum statistics due to statedependent dispersion. More precisely, the dispersion shift of a cavity mode depends on the atom number. If the atom number in some lattice region fluctuates from realization to realization, the modes get a fluctuating frequency shift. Thus, in the cavity transmission-spectrum, resonances appear at different frequencies directly reflecting the atom number distribution function. Such a measurement allows then to calculate atomic statistical quantities, e.g., mean value and variance reflected by spectral characteristics such as the central frequency and width.Different phases of a degenerate gas possess similar mean-field densities but different quantum amplitudes. This leads to a superposition of different transmission spectra, which e.g. for a superfluid state (SF) consist of numerous peaks reflecting the discreteness of the matter-field. Analogous discrete spectra reversing the role of atoms and light, thus reflecting the photon structure of electromagnetic fields, were obtained in cavity QED with Rydberg atoms 15 and solid-state superconducting circuits 16 . A quantum phase transition towards a Mott insulator state (MI) is characterized by a reduc- tion of the number of peaks towards a single resonance, because atom number fluctuations are significantly suppressed 17,18 . As our detection scheme is based on nonresonant dispersive interaction independent of a particular level structure, it can be also applied to molecules 19,20 .We consider the quantized motion of N two-level atoms in a deep periodic optical lattice with M sites formed by far off-reso...
We study an ultracold gas of neutral atoms subject to the periodic optical potential generated by a high-Q cavity mode. In the limit of very low temperatures, cavity field and atomic dynamics require a quantum description. Starting from a cavity QED single atom Hamiltonian we use different routes to derive approximative multiparticle Hamiltonians in Bose-Hubbard form with rescaled or even dynamical parameters. In the limit of large enough cavity damping the different models agree. Compared to free space optical lattices, quantum uncertainties of the potential and the possibility of atom-field entanglement lead to modified phase transition characteristics, the appearance of new phases or even quantum superpositions of different phases. Using a corresponding effective master equation, which can be numerically solved for few particles, we can study time evolution including dissipation. As an example we exhibit the microscopic processes behind the transition dynamics from a Mott insulator like state to a self-ordered superradiant state of the atoms, which appears as steady state for transverse atomic pumping.
We study a generalized cold atom Bose Hubbard model, where the periodic optical potential is formed by a cavity field with quantum properties. On the one hand the common coupling of all atoms to the same mode introduces cavity mediated long range atom-atom interactions and on the other hand atomic backaction on the field introduces atom-field entanglement. This modifies the properties of the associated quantum phase transitions and allows for new correlated atom-field states including superposition of different atomic quantum phases. After deriving an approximative Hamiltonian including the new long range interaction terms we exhibit central physical phenomena at generic configurations of few atoms in few wells. We find strong modifications of population fluctuations and next-nearest neighbor correlations near the phase transition point.PACS numbers: 03.75. Fi, 05.30.Jp, 32.80.Pj, 42.50.Vk Laser fields can nowadays be routinely used to create tailored optical potentials for ultracold neutral atoms [1]. Loading an atomic BEC into such a periodic standing light pattern allows to experimentally implement systems, which are well described by a Bose Hubbard Hamiltonian with externally controllable parameters [2,3,4]. In some recent spectacular experiments the predicted Mott-insulator to superfluid quantum phase transition has been observed [5]. As the light fields are normally intense and strongly detuned from any atomic transition, their properties can be safely approximated by prescribed classical fields independent of the atomic state. However, this is invalid if they are confined within an optical resonator. For a sufficient atom number N and atomfield coupling g the fields become dynamical quantities depending on the atoms. In addition in a high-Q cavity the quantum properties of the field get important and the atoms move in quantized potentials [6,7]. Ultimately this allows different states of the lattice field (e.g. different photon numbers) to be quantum correlated with different quantum phases of the atoms. As a striking example the atoms could be in a superposition of a Mott insulator and a superfluid state connected with a different cavity field amplitudes. Even in the classical field limit of high photon numbers all atoms see the same field state and thus we get new long range atom-atom interactions. Interestingly the idea of implementing such combination of cavity QED and a BEC has experimentally made such rapid progress recently, that its success can be expected very soon [8].In this work we discuss the basic physical properties of such a generalized model of ultracold atoms in a periodic potential generated by a quantized field mode. In a first step we derive an approximate Hamiltonian analogous to the Bose Hubbard Hamiltonian including a quantized potential. Its basic physical implications are then exhibited in two cases: (a) a strongly damped cavity, where the field dynamics can be adiabatically eliminated, which leads to a rescaling of the coupling parameters and new long range atom-atom coupling...
Different quantum states of atoms in optical lattices can be nondestructively monitored by offresonant collective light scattering into a cavity. Angle resolved measurements of photon number and variance give information about atom-number fluctuations and pair correlations without single-site access. Observation at angles of diffraction minima provides information on quantum fluctuations insensitive to classical noise. For transverse probing, no photon is scattered into a cavity from a Mott insulator phase, while the photon number is proportional to the atom number for a superfluid. While mean-field approaches describe only the average atomic density, the main goal is to study quantum properties of these gases. They are most prominent in lattices, where one has phase transitions between states of similar density but radically different quantum fluctuations.Standard methods to measure quantum properties are based on matter-wave interference of atoms released from a trap [2] destroying the system. "Bragg spectroscopy" using stimulated matter-wave scattering by laser pulses proved successful [3, 4] but destructive. Alternative less destructive methods observing scattered light were proposed mainly for homogeneous Bose-Einstein condensates (BEC) [5,6,7,8], but not yet implemented.Here we show that specifically for periodic lattices, light scattering can help to overcome experimental difficulties. In contrast to homogeneous BECs, scattering from a lattice allows to determine local and nonlocal correlations without single-atom optical access using the suppression of strong classical scattering at Bragg minima and monitoring much richer angular distributions. This looks extremely useful for studying phase transitions between, e.g., Mott insulator (MI) and superfluid (SF) states, without destruction, since various quantum phases show even qualitatively distinct scattering.Joining two fields, cavity quantum electrodynamics (QED) and ultracold gases, will enable new investigations of both light and matter at ultimate quantum levels, which only recently became experimentally possible [9].Our model is based on nonresonant interaction, not relying on a particle level structure. Thus it also applies to molecular physics, where new quantum phases were obtained [10]. It can be also applied for semiconductors [11], as, e.g., were used for BEC of exciton-polaritons [12].Model. We consider N two-level atoms in an optical lattice with M sites. A region of K ≤ M sites is illuminated by probe light which is scattered into another mode (cf. Fig. 1). Although, each mode could be a freely propagating field, we will consider cavity modes whose geometries (i.e. axis directions or wavelengths) can be varied. A related manybody Hamiltonian is given bywhere a 0 (a 1 ) are the annihilation operators of the probe (scattered) light with the frequencies ω 0,1 , wave vectors k 0,1 , and mode functions u 0,1 (r); Ψ(r) is the atom-field operator. In the effective single-atom Hamiltonian H a1 , p and r are the momentum and position operators of an a...
We study off-resonant collective light scattering from ultracold atoms trapped in an optical lattice. Scattering from different atomic quantum states creates different quantum states of the scattered light, which can be distinguished by measurements of the spatial intensity distribution, quadrature variances, photon statistics, or spectral measurements. In particular, angle-resolved intensity measurements reflect global statistics of atoms (total number of radiating atoms) as well as local statistical quantities (single-site statistics even without an optical access to a single site) and pair correlations between different sites. As a striking example we consider scattering from transversally illuminated atoms into an optical cavity mode. For the Mott insulator state, similar to classical diffraction, the number of photons scattered into a cavity is zero due to destructive interference, while for the superfluid state it is nonzero and proportional to the number of atoms. Moreover, we demonstrate that light scattering into a standing-wave cavity has a nontrivial angle dependence, including the appearance of narrow features at angles, where classical diffraction predicts zero. The measurement procedure corresponds to the quantum non-demolition (QND) measurement of various atomic variables by observing light.
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