Stochastic cooling of trapped atoms is considered for a laser-beam configuration with beam waists equal or smaller than the extent of the atomic cloud. It is shown, that various effects appear due to this transverse confinement, among them heating of transverse kinetic energy. Analytical results of the cooling in dependence on size and location of the laser beam are presented for the case of a non-degenerate vapour.
A cooling scheme for trapped atoms is proposed, which combines cavity-enhanced scattering and electromagnetically induced transparency. The cooling dynamics exploits a three-photon resonance, which combines laser and cavity excitations. It is shown that relatively fast ground-state cooling can be achieved in the Lamb-Dicke regime and for large cooperativity. Efficient ground-state cooling is found for parameters of ongoing experiments.
We report a demonstration and quantitative characterization of one-dimensional cavity cooling of a single trapped 88Sr+ ion in the resolved sideband regime. We measure the spectrum of cavity transitions, the rates of cavity heating and cooling, and the steady-state cooling limit. The cavity cooling dynamics and cooling limit of 22.5(3) motional quanta, limited by the moderate coupling between the ion and the cavity, are consistent with a simple model [Phys. Rev. A 64, 033405] without any free parameters, validating the rate equation model for cavity cooling.
Ground state cooling of a nanomechanical resonator coupled to a superconducting flux qubit is discussed. We show that by inducing quantum interference to cancel detrimental carrier excitations, ground state cooling becomes possible in the weak-confinement or non-resolved regime. The qubit is modelled as a three-level system in lambda configuration, and the driving fluxes are applied such that the qubit absorption spectrum exhibits electromagnetically induced transparency, thereby cancelling the unwanted carrier excitation. As our interference-based scheme allows to apply strong cooling fields,…
We study laser cooling of two ions that are trapped in a harmonic potential and interact by Coulomb repulsion. Sideband cooling in the Lamb-Dicke regime is shown to work analogously to sideband cooling of a single ion. Outside the Lamb-Dicke regime, the incommensurable frequencies of the two vibrational modes result in a quasi-continuous energy spectrum that significantly alters the cooling dynamics. The cooling time decreases nonlinearly with the linewidth of the cooling transition, and the effect of trapping states which may slow down the cooling is considerably reduced. We show that cooling…
We sympathetically cool a trapped 112Cd+ ion by directly Doppler-cooling a 114Cd+ ion in the same trap. This is the first demonstration of optically addressing a single trapped ion being sympathetically cooled by a different species ion. Notably, the experiment uses a single laser source, and does not require strong focusing. This paves the way toward reducing decoherence in an ion trap quantum computer based on Cd+ isotopes.
A novel method of ground state laser cooling of trapped atoms utilizes the absorption profile of a three (or multi-) level system which is tailored by a quantum interference. With cooling rates comparable to conventional sideband cooling, lower final temperatures may be achieved. The method was experimentally implemented to cool a single Ca$^+$ ion to its vibrational ground state. Since a broad band of vibrational frequencies can be cooled simultaneously, the technique will be particularly useful for the cooling of larger ion strings, thereby being of great practical importance for initializin…
We present a robust and fast laser cooling scheme suitable for trapped atoms and ions. Based on quantum interference, generated by a special laser configuration, it is able to rapidly cool the system such that the final phonon occupation vanishes to zeroth order in the Lamb-Dicke parameter in contrast to existing cooling schemes. Furthermore, it is robust under conditions of fluctuating laser intensity and frequency, thus making it a viable candidate for experimental applications.
A laser cooling scheme for trapped ions is presented which is based on the fast dynamical Stark shift gate, described in [Jonathan etal, PRA 62, 042307]. Since this cooling method does not contain an off resonant carrier transition, low final temperatures are achieved even in traveling wave light field. The proposed method may operate in either pulsed or continuous mode and is also suitable for ion traps using microwave addressing in strong magnetic field gradients.
Based on a real-time measurement of the motion of a single ion in a Paul trap, we demonstrate its electro-mechanical cooling below the Doppler limit by homodyne feedback control (cold damping). The feedback cooling results are well described by a model based on a quantum mechanical Master Equation.
We study the collective Raman cooling of a polarized trapped Fermi gas in the Festina Lente regime, when the heating effects associated with photon reabsorptions are suppressed. We predict that by adjusting the spontaneous Raman emission rates and using appropriately designed anharmonic traps, temperatures of the order of 2.7% of the Fermi temperature can be achieved in 3D.
Measurement-based cooling is a method by which a quantum system, initially in a thermal state, can be prepared in its ground state through some sort of measurement. This is done by making a measurement that heralds the system being in the desired state. Here we demonstrate the application of a measurement-based cooling technique to a trapped atomic ion. The ion is pre-cooled by Doppler laser cooling to a thermal state with a mean excitation of $\bar n \approx 18$ and the measurement-based cooling technique selects those occasions when the ion happens to be in the motional ground state. The fid…
We develop a theory of quantum feedback cooling of a single ion trapped in front of a mirror. By monitoring the motional sidebands of the light emitted into the mirror mode we infer the position of the ion, and act back with an appropriate force to cool the ion. We derive a feedback master equation along the lines of the quantum feedback theory developed by Wiseman and Milburn, which provides us with cooling times and final temperatures as a function of feedback gain and various system parameters.
Conventional theoretical studies on the ground-state laser cooling of a trapped ion have mostly focused on the weak sideband coupling (WSC) regime, where the cooling rate is inverse proportional to the linewidth of the excited state. In a recent work~[New J. Phys. 23, 023018 (2021)], we proposed a theoretical framework to study the ground state cooling of a trapped ion in the strong sideband coupling (SSC) regime, under the assumption of a vanishing carrier transition. Here we extend this analysis to more general situations with nonvanishing carrier transitions, where we show that by properly …
We derive an equation for the cooling dynamics of the quantum motion of an atom trapped by an external potential inside an optical resonator. This equation has broad validity and allows us to identify novel regimes where the motion can be efficiently cooled to the potential ground state. Our result shows that the motion is critically affected by quantum correlations induced by the mechanical coupling with the resonator, which may lead to selective suppression of certain transitions for the appropriate parameters regimes, thereby increasing the cooling efficiency.
Cooling the trapped atoms toward their motional ground states is key to applications of quantum simulation and quantum computation. By utilizing nonreciprocal couplings between constituent atoms, we present an intriguing dark-state cooling scheme in $Λ$-type three-level structure, which is shown superior than the conventional electromagnetically-induced-transparency cooling in a single atom. The effective nonreciprocal couplings can be facilitated either by an atom-waveguide interface or a free-space photonic quantum link. By tailoring system parameters allowed in dark-state cooling, we identi…
Resolved sideband cooling provides a crucial step in subrecoil cooling the trapped atoms toward their motional ground state, which is essential in atom-based quantum technologies. Here we present an enhanced dark-state sideband cooling in trapped atoms utilizing photon-mediated dipole-dipole interactions among them. By placing the atoms at the magic interparticle distances, we manifest an outperformed cooling behavior in the target atom, which surpasses the limit that a single atom permits. We further investigate various atomic configurations in a multiatom setup with a laser detuning and diff…
We demonstrate broadband laser cooling of atomic ions in an rf trap using ultrafast pulses from a modelocked laser. The temperature of a single ion is measured by observing the size of a time-averaged image of the ion in the known harmonic trap potential. While the lowest observed temperature was only about 1 K, this method efficiently cools very hot atoms and can sufficiently localize trapped atoms to produce near diffraction-limited atomic images.
We theoretically propose and experimentally demonstrate a parallel-electromagnetically-induced transparency (parallel-EIT) cooling technique for ion crystals in the Paul trap. It has less stringent requirements on the cooling resonance condition than the standard electromagnetically-induced transparency (EIT) cooling, thus allowing, in principle, to simultaneously cool the motional mode spectrum with an arbitrary range. A proof-of-principle validation for this cooling scheme is experimentally demonstrated with up to 4 trapped 40Ca+ ions. We observe simultaneous near-ground-state cooling for al…
Resonance fluorescence of a single trapped ion is spectrally analyzed using a heterodyne technique. Motional sidebands due to the oscillation of the ion in the harmonic trap potential are observed in the fluorescence spectrum. From the width of the sidebands the cooling rate is obtained and found to be in agreement with the theoretical prediction.
We have studied a general technique for laser cooling a cloud of polarized trapped atoms down to the Doppler temperature. A one-dimensional optical molasses using polarized light cools the axial motional degree of freedom of the atoms in the trap. Cooling of the radial degrees of freedom can be modelled by reabsorption of scattered photons in the optically dense cloud. We present experimental results for a cloud of chromium atoms in a magnetic trap. A simple model based on rate equations shows quantitative agreement with the experimental results. This scheme allows us to readily prepare a dens…
We theoretically analyze the cooling dynamics of an atom which is tightly trapped inside a high-finesse optical resonator. Cooling is achieved by suitably tailored scattering processes, in which the atomic dipole transition either scatters a cavity photon into the electromagnetic field external to the resonator, or performs a stimulated emission into the cavity mode, which then dissipates via the cavity mirrors. We identify the parameter regimes in which the atom center-of-mass motion can be cooled into the ground state of the external trap. We predict, in particular, that for high cooperativi…
We analyse the effects of atom-atom collisions on collective laser cooling scheme. We derive a quantum Master equation which describes the laser cooling in presence of atom-atom collisions in the weak-condensation regime. Using such equation, we perform Monte Carlo simulations of the population dynamics in one and three dimensions. We observe that the ground-state laser-induced condensation is maintained in the presence of collisions. Laser cooling causes a transition from a Bose-Einstein distribution describing collisionally induced equilibrium,to a distribution with an effective zero tempera…
This paper analyses the cooling of a single particle in a harmonic trap with red-detuned laser light with fewer approximations than previously done in the literature. We avoid the adiabatic elimination of the excited atomic state but are still interested in Lamb-Dicke parameters $η\ll 1$. Our results show that the Rabi frequency of the cooling laser can be chosen higher than previously assumed, thereby increasing the effective cooling rate but {\em not} affecting the final outcome of the cooling process. Since laser cooling is already a well established experimental technique, the main aim of …
We demonstrate sympathetic cooling of a 43Ca+ trapped-ion "memory" qubit by a 40Ca+ "coolant" ion near the ground state of both axial motional modes, whilst maintaining coherence of the qubit. This is an essential ingredient in trapped-ion quantum computers. The isotope shifts are sufficient to suppress decoherence and phase shifts of the memory qubit due to the cooling light which illuminates both ions. We measure the qubit coherence during 10 cycles of sideband cooling, finding a coherence loss of 3.3% per cooling cycle. The natural limit of the method is O(0.01%) infidelity per cooling cycl…
We discuss the dynamical behavior of the entanglement between the internal and the external degrees of freedom of a trapped atom in electromagnetically-induced transparency (EIT) laser cooling. It is shown that essential features of the intricate entanglement dynamics observed in full numerical simulations of the underlying quantum master equation can be understood in terms of a two-state model on the basis of Landau-Zener splittings in the atom-laser field Hamiltonian. An extension of this model to an effective non-Hermitian Hamiltonian is constructed which describes the decay of entanglement…
We investigate theoretically the mechanical effects of light on atoms trapped by an external potential, whose dipole transition couples to the mode of an optical resonator and is driven by a laser. We derive an analytical expression for the quantum center-of-mass dynamics, which is valid in presence of a tight external potential. This equation has broad validity and allows for a transparent interpretation of the individual scattering processes leading to cooling. We show that the dynamics are a competition of the mechanical effects of the cavity and of the laser photons, which may mutually int…
We propose a scheme that allows to laser cool trapped atoms to the ground state of a one-dimensional confining potential. The scheme is based on the creation of a dark state by designing the laser profile, so that the hottest atoms are coherently pumped to another internal level, and then repumped back. The scheme works beyond the Lamb-Dicke limit. We present results of a full quantum treatment for a one-dimensional model.
We propose a new dark-state cooling method of trapped ion systems in the Lamb-Dicke limit. With application of microwave dressing the ion, we can obtain two electromagnetically induced transparency structures. The heating effects caused by the carrier and the blue sideband transition vanish due to the EIT effects and the final mean phonon numbers can be much less than the recoil limit. Our scheme is robust to fluctuations of microwave power and laser intensities which provides a broad cooling bandwidth to cool motional modes of a linear ion chain. Moreover, it is more suitable to cool four-lev…
Trapped ion in the Lamb-Dicke regime with the Lamb-Dicke parameter $η\ll1$ can be cooled down to its motional ground state using sideband cooling. Standard sideband cooling works in the weak sideband coupling limit, where the sideband coupling strength is small compared to the natural linewidth $γ$ of the internal excited state, with a cooling rate much less than $γ$. Here we consider cooling schemes in the strong sideband coupling regime, where the sideband coupling strength is comparable or even greater than $γ$. We derive analytic expressions for the cooling rate and the average occupation …
We systematically investigate the multi-stability behaviour and cooling of both librational and translational modes of an optically levitated nonspherical nanoparticle. By expanding the trapping potential to the fourth order of both the translational and librational freedom degrees, we deduce the nonlinearity of them and their nonlinear coupling. Through stability analysis, we find that the system presents multi-stability when either the librational or the translational drive is red-detuned. The system will be stabilized if and only if these two drives are both blue-detuned. In the steady stat…
The collective Raman cooling of trapped one- and two-component Fermi gases is considered. We obtain the quantum master equation that describes the laser cooling in the festina lente regime, for which the heating due to photon reabsorption can be neglected. For the two-component case the collisional processes are described within the formalism of quantum Boltzmann master equation. The inhibition of the spontaneous emission can be overcome by properly adjusting the spontaneous Raman rate during the cooling. Our numerical results based in Monte Carlo simulations of the corresponding rate equation…
We study the laser cooling of one atom in an harmonic trap beyond the Lamb-Dicke regime. By using sequences of laser pulses of different detunings we show that the atom can be confined into just one state of the trap, either the ground state or an excited state of the harmonic potential. The last can be achieved because under certain conditions an excited state becomes a dark state. We study the problem in one and two dimensions. For the latter case a new cooling mechanism is possible, based on the destructive interference between the effects of laser fields in different directions, which allo…
We discuss continuous observation of the momentum of a single atom by employing the high velocity sensitivity of the index of refraction in a driven $Λ$-system based on electromagnetically induced transparency (EIT). In the ideal limit of unit collection efficiency this provides a quantum limited measurement with minimal backaction on the atomic motion. A feedback loop, which drives the atom with a force proportional to measured signal, provides a cooling mechanism for the atomic motion. We derive the master equation which describes the feedback cooling and show that in the Lamb-Dicke limit th…
We report preparation in the ground state of collective modes of motion of two trapped 9Be+ ions. This is a crucial step towards realizing quantum logic gates which can entangle the ions' internal electronic states. We find that heating of the modes of relative ion motion is substantially suppressed relative to that of the center-of-mass modes, suggesting the importance of these modes in future experiments.
We study cooling of the collective vibrational motion of two 138Ba+ ions confined in an electrodynamic trap and irradiated with laser light close to the resonances S_1/2-P_1/2 (493 nm) and P_1/2-D_3/2 (650 nm). The motional state of the ions is monitored by a spatially resolving photo multiplier. Depending on detuning and intensity of the cooling lasers, macroscopically different motional states corresponding to different ion temperatures are observed. We also derive the ions' temperature from detailed analytical calculations of laser cooling taking into account the Zeeman structure of the ene…
It is shown that a two-qubit phase gate and SWAP operation between ground states of cold trapped ions can be realised in one step by simultaneously applying two laser fields. Cooling during gate operations is possible without perturbing the computation and the scheme does not require a second ion species for sympathetic cooling. On the contrary, the cooling lasers even stabilise the desired time evolution of the system. This affords gate operation times of nearly the same order of magnitude as the inverse coupling constant of the ions to a common vibrational mode.
Microfabricated ion-trap devices offer a promising pathway towards scalable quantum computing. Research efforts have begun to focus on the engineering challenges associated with developing large-scale ion-trap arrays and networks. However, increasing the size of the array and integrating on-chip electronics can drastically increase the power dissipation within the ion-trap chips. This leads to an increase in the operating temperature of the ion-trap and limits the device performance. Therefore, effective thermal management is an essential consideration for any large-scale architecture. Present…
We present a method of treating the interaction of a single three-level ion with two laser beams. The idea is to apply a unitary transformation such that the exact transformed Hamiltonian has one of the three levels decoupled for all values of the detunings. When one takes into account damping, the evolution of the system is governed by a master equation usually obtained via adiabatic approximation under the assumption of far-detuned lasers. To go around the drawbacks of this technique, we use the same unitary transformation to get an effective master equation.
Resolved sideband cooling is a standard technique for cooling trapped ions below the Doppler limit to near their motional ground state. Yet, the most common methods for sideband cooling implicitly rely on low Doppler-cooled temperatures and tightly confined ions, and they cannot be optimized for different experimental conditions. Here we introduce a framework which calculates the fastest possible pulsed sideband cooling sequence for a given number of pulses and set of experimental parameters, and we verify its improvement compared to traditional methods using a trapped $^{171}$Yb$^+$ ion. Afte…
The efficiency of cavity sideband cooling of trapped molecules is theoretically investigated for the case where the IR transition between two rovibrational states is used as a cycling transition. The molecules are assumed to be trapped either by a radio-frequency or optical trapping potential, depending on whether they are charged or neutral, and confined inside a high-finesse optical resonator which enhances radiative emission into the cavity mode. Using realistic experimental parameters and COS as a representative molecular example, we show that in this setup cooling to the trap ground state…
Two techniques are described that simplify the experimental requirements for measuring and manipulating quantum information stored in trapped ions. The first is a new technique using electron shelving to measure the populations of the Zeeman sublevels of the ground state, in an ion for which no cycling transition exists from any of these sublevels. The second technique is laser cooling to the vibrational ground state, without the need for a trap operating in the Lamb-Dicke limit. This requires sideband cooling in a sub-recoil regime. We present a thorough analysis of sideband cooling on one or…
Trapped ions are a promising candidate for large scale quantum computation. Several systems have been built in both academic and industrial settings to implement modestly-sized quantum algorithms. Efficient cooling of the motional degrees of freedom is a key requirement for high-fidelity quantum operations using trapped ions. Here, we present a technique whereby individual ions are used to cool individual motional modes in parallel, reducing the time required to bring an ion chain to its motional ground state. We demonstrate this technique experimentally and develop a model to understand the e…
Ground state laser cooling of a single trapped ion is achieved using a technique which tailors the absorption profile for the cooling laser by exploiting electromagnetically induced transparency in the Zeeman structure of a dipole transition. This new method is robust, easy to implement and proves particularly useful for cooling several motional degrees of freedom simultaneously, which is of great practical importance for the implementation of quantum logic schemes with trapped ions.
Roton excitations constitute a key feature of dipolar gases, connecting these gases with superfluid helium. We show that the density dependence of the roton minimum results in a spatial roton confinement, particularly relevant in pancake dipolar condensates with large aspect ratios. We show that roton confinement plays a crucial role in the dynamics after roton instability, and that arresting the instability may create a trapped roton gas revealed by confined density modulations. We discuss the local susceptibility against density perturbations, which we illustrate for the case of vortices. Ro…
This book chapter gives an introduction to, and an overview of, methods for cooling trapped ions. The main addressees are researchers entering the field. It is not intended as a comprehensive survey and historical account of the extensive literature on this topic. We present the physical ideas behind several cooling schemes, outline their mathematical description, and point to relevant literature useful for a more in-depth study of this topic.
We calculate the finite-temperature Tan's contact for N SU(2) fermions, characterized by repulsive contact interaction, trapped in a 1D harmonic confinement within a local density approximation on top of a thermodynamic Bethe Ansatz. The Tan's contact for such a system, as in the homogeneous case, displays a minimum at a very low temperature. By means of an exact canonical ensemble calculation for two fermions, we provide an explicit formula for the contact at very low temperatures that reveals that the minimum is due to the mixing of states with different exchange symmetries. In the unitary r…
We present the results of simulations of optical sideband cooling of atomic ions in a trap with a shallow potential well. In such traps, an ion cannot be Doppler cooled near to the Lamb-Dicke regime ($η^2(2n+1) \ll 1$). Outside the Lamb-Dicke regime, the sideband cooling dynamics are altered by the existence of various Fock states where the cooling becomes very slow. A calcium ion trapped in our Penning trap realises such a situation, hence single stage cooling is inefficient to prepare the ion in the motional ground state. For these systems, it is necessary to study the cooling dynamics in de…
The two most widely used ion cooling methods are laser cooling and sympathetic cooling by elastic collisions (ECs). Here we demonstrate another method of cooling ions that is based on resonant charge exchange (RCE) between the trapped ion and the ultracold parent atom. Specifically, trapped Cs$^{+}$ ions are cooled by collisions with co-trapped, ultracold Cs atoms and, separately, by collisions with co-trapped, ultracold Rb atoms. We observe that the cooling of Cs$^{+}$ ions by Cs atoms is more efficient than cooling of Cs$^{+}$ ions by Rb atoms. This signals the presence of a cooling mechanis…
In this tutorial we review physical implementation of quantum computing using a system of cold trapped ions. We discuss systematically all the aspects for making the implementation possible. Firstly, we go through the loading and confining of atomic ions in the linear Paul trap, then we describe the collective vibrational motion of trapped ions. Further, we discuss interactions of the ions with a laser beam. We treat the interactions in the travelling-wave and standing-wave configuration for dipole and quadrupole transitions. We review different types of laser cooling techniques associated wit…
We propose a cooling scheme based on depolarisation of a polarised cloud of trapped atoms. Similar to adiabatic demagnetisation, we suggest to use the coupling between the internal spin reservoir of the cloud and the external kinetic reservoir via dipolar relaxation to reduce the temperature of the cloud. By optical pumping one can cool the spin reservoir and force the cooling process. In case of a trapped gas of dipolar chromium atoms, we show that this cooling technique can be performed continuously and used to approach the critical phase space density for BEC
A mixed system of cooled and trapped, ions and atoms, paves the way for ion assisted cold chemistry and novel many body studies. Due to the different individual trapping mechanisms, trapped atoms are significantly colder than trapped ions, therefore in the combined system, the strong binary ion$-$atom interaction results in heat flow from ions to atoms. Conversely, trapped ions can also get collisionally heated by the cold atoms, making the resulting equilibrium between ions and atoms intriguing. Here we experimentally demonstrate, Rubidium ions (Rb$^+$) cool in contact with magneto-optically …
We investigate the temporal dynamics of Doppler cooling of an initially hot single trapped atom in the weak binding regime using a semiclassical approach. We develop an analytical model for the simplest case of a single vibrational mode for a harmonic trap, and show how this model allows us to estimate the initial energy of the trapped particle by observing the fluorescence rate during the cooling process. The experimental implementation of this temperature measurement provides a way to measure atom heating rates by observing the temperature rise in the absence of cooling. This method is techn…
We report the electromagnetically-induced-transparency (EIT) cooling of a large trapped $^{171}$Yb$^+$ ion chain to the quantum ground state. Unlike conventional EIT cooling, we engage a four-level tripod structure and achieve fast sub-Doppler cooling over all motional modes. We observe simultaneous ground-state cooling across the complete transverse mode spectrum of up to $40$ ions, occupying a bandwidth of over $3$ MHz. The cooling time is observed to be less than $300\,μ$s, independent of the number of ions. Such efficient cooling across the entire spectrum is essential for high-fidelity qu…
