Quantum state reconstruction based on weak continuous measurement has the advantage of being fast, accurate, and almost non-perturbative. In this work we present a pedagogical review of the protocol proposed by Silberfarb et al., PRL 95 030402 (2005), whereby an ensemble of identically prepared systems is collectively probed and controlled in a time-dependent manner so as to create an informationally complete continuous measurement record. The measurement history is then inverted to determine the state at the initial time through a maximum-likelihood estimate. The general formalism is applied to the case of reconstruction of the quantum state encoded in the magnetic sublevels of a large-spin alkali atom, 133Cs. We detail two different protocols for control. Using magnetic interactions and a quadratic ac-Stark shift, we can reconstruct a chosen hyperfine manifold F, e.g., the 7-dimensional F=3 manifold in the electronic-ground state of Cs. We review the procedure as implemented in experiments (Smith et al., PRL 97 180403 (2006)). We extend the protocol to the more ambitious case of reconstruction of states in the full 16-dimensional electronic-ground subspace (F=3 \oplus F=4), controlled by microwaves and radio-frequency magnetic fields. We give detailed derivations of all physical interactions, approximations, numerical methods, and fitting procedures, tailored to the realistic experimental setting. For the case of light-shift and magnetic control, reconstruction fidelities of \sim 0.95 have been achieved, limited primarily by inhomogeneities in the light shift. For the case of microwave/RF-control we simulate fidelity >0.97, limited primarily by signal-to-noise.
Full quantum state tomography is used to characterize the state of an ensemble based qubit implemented through two hyperfine levels in Pr3+ ions, doped into a Y2SiO5 crystal. We experimentally verify that single-qubit rotation errors due to inhomogeneities of the ensemble can be suppressed using the Roos-Moelmer dark state scheme. Fidelities above >90%, presumably limited by excited state decoherence, were achieved. Although not explicitly taken care of in the Roos-Moelmer scheme, it appears that also decoherence due to inhomogeneous broadening on the hyperfine transition is largely suppressed…
Collective effects in the interaction of light with ensembles of identical scatterers play an important role in many fields of physics. However, often the term ``identical'' is not accurate due to the presence of hyperfine fields which induce inhomogeneous transition shifts and splittings. Here we develop a formalism based on the Green function method to model the linear response of such inhomogeneous ensembles in one-dimensional waveguides. We obtain a compact formula for the collective spectrum, which exhibits deviations from the uniform frequency shift and broadening expected of two level s…
We demonstrate quantum control of a large spin-angular momentum associated with the F=3 hyperfine ground state of 133Cs. A combination of time dependent magnetic fields and a static tensor light shift is used to implement near-optimal controls and map a fiducial state to a broad range of target states, with yields in the range 0.8-0.9. Squeezed states are produced also by an adiabatic scheme that is more robust against errors. Universal control facilitates the encoding and manipulation of qubits and qudits in atomic ground states, and may lead to improvement of some precision measurements.
We study the nonlinear dynamics of an ensemble of cold trapped atoms with a hyperfine transition magnetically coupled to a resonant microwave cavity mode. Despite the minute single atom coupling one obtains strong coupling between collective hyperfine qubits and microwave photons enabling coherent transfer of an excitation between the long lived atomic qubit state and the mode. Evidence of strong coupling can be obtained from the cavity transmission spectrum even at finite thermal photon number. The system makes it possible to study further prominent collective phenomena such as superradiant d…
This dissertation studies spin squeezing, entanglement and decoherence in large ensembles of cold, trapped alkali atoms with hyperfine spin f interacting with optical fields. Restricting the state of each atom to a qutrit embedded in the 2f+1 dimensional hyperfine spin enables us to efficiently model the coherent and dissipative dynamics of the ensemble. This formalism also allows us to explore the effects of local control on the internal hyperfine spins of the atoms. State preparation using such control increases the entangling power of the atom-light interface for f>1/2. Subsequent control o…
We study the ability to control d-dimensional quantum systems (qudits) encoded in the hyperfine spin of alkali-metal atoms through the application of radio- and microwave-frequency magnetic fields in the presence of inhomogeneities in amplitude and detuning. Such a capability is essential to the design of robust pulses that mitigate the effects of experimental uncertainty and also for application to tomographic addressing of particular members of an extended ensemble. We study the problem of preparing an arbitrary state in the Hilbert space from an initial fiducial state. We prove that inhomog…
We show that nuclear spin subsystems can be completely controlled via microwave irradiation of resolved anisotropic hyperfine interactions with a nearby electron spin. Such indirect addressing of the nuclear spins via coupling to an electron allows us to create nuclear spin gates whose operational time is significantly faster than conventional direct addressing methods. We experimentally demonstrate the feasibility of this method on a solid-state ensemble system consisting of one electron and one nuclear spin.
Vanadium in silicon carbide (SiC) is emerging as an important candidate system for quantum technology due to its optical transitions in the telecom wavelength range. However, several key characteristics of this defect family including their spin relaxation lifetime (T1), charge state dynamics, and level structure are not fully understood. In this work, we determine the T1 of an ensemble of vanadium defects, demonstrating that it can be greatly enhanced at low temperature. We observe a large spin contrast exceeding 90% and long spin-relaxation times of up to 25s at 100mK, and of order 1s at 1.3…
Quantum state tomography is a fundamental tool in quantum information processing. It allows us to estimate the state of a quantum system by measuring different observables on many identically prepared copies of the system. This is, in general, a very time-consuming task that requires a large number of measurements. There are, however, systems in which the data acquisition can be done more efficiently. In fact, an ensemble of quantum systems can be prepared and manipulated by external fields while being continuously and collectively probed, producing enough information to estimate its state. Th…
We report on the experimental implementation of a polarimeter based on a scheme known to be optimal for obtaining the polarization vector of ensembles of spin-1/2 quantum systems, and the alignment procedure for this polarimeter is discussed. We also show how to use this polarimeter to estimate the polarization state for identically prepared ensembles of single photons and photon pairs and extend the method to obtain the density matrix for generic multi-photon states. State reconstruction and performance of the polarimeter is illustrated by actual measurements on identically prepared ensembles…
Dynamic nuclear polarisation (DNP) refers to a class of techniques used to increase the signal in nuclear magnetic resonance measurements by transferring spin polarisation from ensembles of highly polarised electrons to target nuclear analytes. These techniques, however, require the application of strong magnetic fields to maximise electron spin polarisation, limiting pathways for electron-nuclear (hyperfine) spin coupling and transfer. In this work we show that, for systems of electronic spin $S\geq1$ possessing an intrinsic zero-field splitting, a separate class of stronger hyperfine interac…
We have numerically simulated quantum tomography of single-qubit and two-qubit quantum gates with qubits represented by mesoscopic ensembles containing random numbers of atoms. Such ensembles of strongly interacting atoms in the regime of Rydberg blockade are known as Rydberg superatoms. The Stimulated Raman Adiabatic Passage (STIRAP) in the regime of Rydberg blockade is used for deterministic Rydberg excitation in the ensemble, required for storage of quantum information in the collective state of the atomic ensemble and implementation of two-qubit gates. The optimized shapes of the STIRAP pu…
The one-dimensional homonuclear periodic array of nuclear spins I = 1/2, owing to hyperfine interaction of nuclear spins with electronic magnetic moments in antiferromagnetic structure, is considered. The neighbor nuclear spins in such array are opposite oriented and have resonant frequencies determined by hyperfine interaction constant, applied magnetic field value and interaction with the left and right nuclear neighbor spins. The resonant frequencies difference of nuclear spins, when the neighbor spins have different and the same states, is used to control the spin dynamics by means of sele…
We present a proposal and a feasibility study for the creation and quantum state tomography of a single polariton state of an atomic ensemble. The collective non-classical and non-Gaussian state of the ensemble is generated by detection of a single forward scattered photon. The state is subsequently characterized by atomic state tomography performed using strong dispersive light-atoms interaction followed by a homodyne measurement on the transmitted light. The proposal is backed by preliminary experimental results showing projection noise limited sensitivity and a simulation demonstrating the …
Subnatural-linewidth single-photon source is a potential candidate for exploring the time degree of freedom in photonic quantum information science. This type of single-photon source has been demonstrated to be generated and reshaped in atomic ensembles without any external cavity or filter, and is typically characterized through photon-counting technology. However, the full complex temporal mode function(TMF) of the photon source is not able to be revealed from direct photon counting measurement. Here, for the first time, we demonstrate the complete temporal mode of the subnatural-linewidth s…
We present the first complete optimization of quantum tomography, for states, POVMs, and various classes of transformations, for arbitrary prior ensemble and arbitrary representation, giving corresponding feasible experimental schemes.
We propose to use a collective excitation blockade mechanism to identify errors that occur due to disturbances of single atoms in ensemble quantum registers where qubits are stored in the collective population of different internal atomic states. A simple error correction procedure and a simple decoherence-free encoding of ensemble qubits in the hyperfine states of alkali atoms are presented.
We report the first experimental realization of single-qubit manipulation for single spinwaves stored in an atomic ensemble quantum memory. In order to have high-fidelity gate operations, we make use of stimulated Raman transition and controlled Lamor precession jointly. We characterize the gate performances with quantum state tomography and quantum process tomography, both of which imply that high-fidelity operations have been achieved. Our work complements the experimental toolbox of atomic-ensemble quantum memories by adding the capability of single-qubit manipulation, thus may have importa…
Atomic ensembles have many potential applications in quantum information science. Owing to collective enhancement, working with ensembles at high densities increases the overall efficiency of quantum operations, but at the same time also increases the collision rate and leads to faster decoherence. Here we report on experiments with optically trapped 87Rb atoms demonstrating a 20-fold increase of the coherence time when a dynamical decoupling sequence with more than 200 pi-pulses is applied. We perform quantum process tomography and demonstrate that using the decoupling scheme a dense ensemble…
We study quantum tomography from a continuous measurement record obtained by measuring expectation values of a set of Hermitian operators obtained from unitary evolution of an initial observable. For this purpose, we consider the application of a random unitary, diagonal in a fixed basis at each time step and quantify the information gain in tomography using Fisher information of the measurement record and the Shannon entropy associated with the eigenvalues of covariance matrix of the estimation. Surprisingly, very high fidelity of reconstruction is obtained using random unitaries diagonal in …
We measure the hyperfine structure of $nP_{1/2}$ Rydberg states using mm-wave spectroscopy on an ensemble of laser-cooled $^{85}$Rb atoms. Systematic uncertainties in our measurement from the Zeeman splittings induced by stray magnetic fields and dipole-dipole interactions between two Rydberg atoms are factored in with the obtained statistical uncertainty. Our final measurement of the $nP_{1/2}$ hyperfine coupling constant is $A_{\text{hfs}}=1.443(31)$~GHz. This measurement is useful for studies of long-range Rydberg molecules, Rydberg electrometry, and quantum simulation with dipole-dipole in…
In contrast to the standard quantum state tomography, the direct tomography seeks the direct access to the complex values of the wave function at particular positions (i.e., the expansion coefficient in a fixed basis). Originally put forward as a special case of weak measurement, it can be extended to arbitrary measurement setup. We generalize the idea of "quantum metrology," where a real-valued phase is estimated, to the estimation of complex-valued phase, and apply it for the direct tomography of the wave function. It turns out that the reformulation can help us easily find the optimal measu…
We characterize correlations in polarization and time of photon pairs generated from a cold cloud of Rb-87 atoms via a four-wave mixing process in a cascade level scheme. Quantum state tomography reveals entangled polarization states of high purity for each of the decay paths through two different intermediate hyperfine levels. When allowing both decay paths, we observe quantum beats in time-resolved correlation measurements.
Neutral atom systems are an appealing platform for the development and testing of quantum control and measurement techniques. This dissertation presents experimental investigations of control and measurement tools using as a testbed the 16-dimensional hyperfine manifold associated with the electronic ground state of $^{133}$Cs atoms. On the control side, we present an experimental realization of a protocol to implement robust unitary transformations in the presence of static and dynamic perturbations. We also present an experimental realization of inhomogeneous quantum control. Specifically, w…
I propose an iterative expectation maximization algorithm for reconstructing a quantum optical ensemble from a set of balanced homodyne measurements performed on an optical state. The algorithm applies directly to the acquired data, bypassing the intermediate step of calculating marginal distributions. The advantages of the new method are made manifest by comparing it with the traditional inverse Radon transformation technique.
The outcome statistics of an informationally complete quantum measurement for a system in a given state can be used to evaluate the ensemble expectation of any linear operator in the same state, by averaging a function of the outcomes that depends on the specific operator. Here we introduce two novel data-processing strategies, non-linear in the frequencies, which lead to faster convergence to theoretical expectations.
It is known that mutually unbiased bases, whenever they exist, are optimal in an information theoretic sense for the determination of unknown state of a quantum ensemble. These bases may not exist in most dimensions and some suboptimal choices have to be made. The present paper deals with estimates of the information loss in suboptimal choice of bases. The information is calculated directly in terms of transition probabilities. I give estimates for the information content of measurement in some approximate MUBs proposed recently.
The spin-coherent-state positive-operator-valued-measure (POVM) is a fundamental measurement in quantum science, with applications including tomography, metrology, teleportation, benchmarking, and measurement of Husimi phase space probabilities. We prove that this POVM is achieved by collectively measuring the spin projection of an ensemble of qubits weakly and isotropically. We apply this in the context of optimal tomography of pure qubits. We show numerically that through a sequence of weak measurements of random directions of the collective spin component, sampled discretely or in a continu…
We study the coupling between collective variables of atomic spin and light polarization in an ensemble of cold 87Rb probed with polarized light. The effects of multiple hyperfine levels manifest themselves as a rank-2 tensor polarizability, whose irreducible components can be selected by means of probe detuning. The D1 and D2 lines of Rb are explored and we identify different detunings which lead to Hamiltonians with different symmetries for rotations. As possible applications of these Hamiltonians, we describe schemes for spin squeezing, quantum cloning, quantum memory, and measuring atom nu…
We find quantum signatures of classical chaos in various metrics of information gain in quantum tomography. We employ a quantum state estimator based on weak collective measurements of an ensemble of identically prepared systems. The tomographic measurement record consists of a sequence of expectation values of a Hermitian operator that evolves under repeated application of the Floquet map of the quantum kicked top. We find an increase in information gain and hence higher fidelities in the reconstruction algorithm when the chaoticity parameter map increases. The results are well predicted by r…
We demonstrate coherent microwave control of rotational and hyperfine states of trapped, ultracold, and chemically stable $^{23}$Na$^{40}$K molecules. Starting with all molecules in the absolute rovibrational and hyperfine ground state, we study rotational transitions in combined magnetic and electric fields and explain the rich hyperfine structure. Following the transfer of the entire molecular ensemble into a single hyperfine level of the first rotationally excited state, $J{=}1$, we observe collisional lifetimes of more than $3\, \rm s$, comparable to those in the rovibrational ground state…
The spins of gaseous alkali atoms are commonly assumed to oscillate at a constant hyperfine frequency, which for many years has been used to define the Second. Indeed, under standard experimental conditions, the spins oscillate independently, only weakly perturbed and slowly decaying due to random spin-spin collisions. Here we consider a different, unexplored regime of very dense gas, where collisions, more frequent than the hyperfine frequency, dominate the dynamics. Counter-intuitively, we find that the hyperfine oscillations become significantly longer-lived, and their frequency becomes dep…
We report on the implementation of quantum state tomography for an ensemble of Eu$^{3+}$ dopant ions in a \YSO crystal. The tomography was applied to a qubit based on one of the ion's optical transitions. The qubit was manipulated using optical pulses and measurements were made by observing the optical free induction in a phase sensitive manner. Fidelities of $>90$% for the combined preparation and measurement process were achieved. Interactions between the ions due to the change in the ions' permanent electric dipole moment when excited optically were also measured. In light of these results,…
We demonstrate the generation of heralded Fock-basis qubits $(a|0\rangle + b|1\rangle)$ from transient collective spin excitations in a hot atomic vapor. The collective spin excitation is heralded by Raman-scattered photons in a four-wave mixing process seeded by a weak coherent optical excitation. The amplitude and phase of the seed field allow arbitrary control over the qubit coefficients. The state of the ensemble is read out optically and then characterized using balanced homodyne tomography. This work represents a step towards quantum engineering of collective atomic spin states.
We describe a method for creating small quantum processors in a crystal stoichiometric in an optically active rare earth ion. The crystal is doped with another rare earth, creating an ensemble of identical clusters of surrounding ions, whose optical and hyperfine frequencies are uniquely determined by their spatial position in the cluster. Ensembles of ions in each unique position around the dopant serve as qubits, with strong local interactions between ions in different qubits. These ensemble qubits can each be used as a quantum memory for light, and we show how the interactions between qubit…
We generate entangled states of an ensemble of 5*10^4 rubidium-87 atoms by optical quantum nondemolition measurement. The resonator-enhanced measurement leaves the atomic ensemble, prepared in a superposition of hyperfine clock levels, in a squeezed spin state. By comparing the resulting reduction of quantum projection noise (up to 8.8(8) dB) with the concomitant reduction of coherence, we demonstrate a clock input state with spectroscopic sensitivity 3.0(8) dB beyond the standard quantum limit.
We derive an expression for a density operator estimated via Bayesian quantum inference in the limit of an infinite number of measurements. This expression is derived under the assumption that the reconstructed system is in a pure state. In this case the estimation corresponds to an averaging over a generalized microcanonical ensemble of pure states satisfying a set of constraints imposed by the measured mean values of the observables under consideration. We show that via the ``purification'' ansatz, statistical mixtures can also be consistently reconstructed via the quantum Bayesian inference…
A quantum gravity-gradiometer consists of two spatially separated ensembles of atoms interrogated by pulses of a common laser beam. Laser pulses cause the probability amplitudes of atomic ground-state hyperfine levels to interfere, producing two motion-sensitive phase shifts which allow the measurement of the average acceleration of each ensemble, and, via simple differencing, of the acceleration gradient. Here I propose entangling the quantum states of the two ensembles prior to the pulse sequence, and show that entanglement encodes their relative acceleration in a single interference phase w…
Magnetometers based on ensembles of nitrogen-vacancy centres are a promising platform for continuously sensing static and low-frequency magnetic fields. Their combination with phase-sensitive (lock-in) detection creates a highly versatile sensor with a sensitivity that is proportional to the derivative of the optical magnetic resonance lock-in spectrum, which is in turn dependant on the lock-in modulation parameters. Here we study the dependence of the lock-in spectral slope on the modulation of the spin-driving microwave field. Given the presence of the intrinsic nitrogen hyperfine spin trans…
We consider two qubits interacting by means of an optical cavity, where each qubit is represented by a single electron spin confined to a quantum dot. It is known that electron spins in III-V semiconductor quantum dots are affected by the decoherence due to the hyperfine interaction with nuclear spins. Here we show that the interaction between two qubits is influenced by the Overhauser field as well. Starting from an unpolarizied nuclear ensemble, we investigate the dependance of the fidelities for two-qubit gates on the Overhauser field. We include the hyperfine interaction perturbatively to …
The need to perform quantum state tomography on ever larger systems has spurred a search for methods that yield good estimates from incomplete data. We study the performance of compressed sensing (CS) and least squares (LS) estimators in a fast protocol based on continuous measurement on an ensemble of cesium atomic spins. Both efficiently reconstruct nearly pure states in the 16-dimensional ground manifold, reaching average fidelities FCS = 0.92 and FLS = 0.88 using similar amounts of incomplete data. Surprisingly, the main advantage of CS in our protocol is an increased robustness to experim…
2-designs -- ensembles of quantum pure states whose 2nd moments equal those of the uniform Haar ensemble -- are optimal solutions for several tasks in quantum information science, especially state and process tomography. We show that Gaussian states cannot form a 2-design for the continuous-variable (quantum optical) Hilbert space L2(R). This is surprising because the affine symplectic group HWSp (the natural symmetry group of Gaussian states) is irreducible on the symmetric subspace of two copies. In finite dimensional Hilbert spaces, irreducibility guarantees that HWSp-covariant ensembles (s…
The high-fidelity analysis of many-body quantum states of indistinguishable atoms requires the accurate counting of atoms. Here we report the tomographic reconstruction of an atom-number-resolving detector. The tomography is performed with an ultracold rubidium ensemble that is prepared in a coherent spin state by driving a Rabi coupling between the two hyperfine clock levels. The coupling is followed by counting the occupation number in one level. We characterize the fidelity of our detector and show that a negative-valued Wigner function is associated with it. Our results offer an exciting p…
The determination of the density matrix of an ensemble of identically prepared quantum systems by performing a series of measurements, known as quantum tomography, is minimal when the number of outcomes is minimal. The most accurate minimal quantum tomography of qubits, sometimes called a tetrahedron measurement, corresponds to projections over four states which can be represented on the Bloch sphere as the vertices of a regular tetrahedron. We investigate whether it is possible to implement the tetrahedron measurement of double slit qubits of light, using measurements performed on a single pl…
Gate Synthesis in Dephasing-Limited Nitrogen-Vacancy Ensembles
Institut quantique, Université de Sherbrooke（舍布鲁克大学量子研究所）；Département de physique, Université de Sherbrooke（舍布鲁克大学物理系）；Keysight Technologies Canada（是德科技加拿大）；Département de génie électrique et de génie informatique, Université de Sherbrooke（舍布鲁克大学电气与计算机工程系）
We consider how to measure collective spin states of an atomic ensemble based on the recent multi-pass approaches for quantum interface between light and atoms. We find that a scheme with two passages of a light pulse through the atomic ensemble is efficient to implement the homodyne tomography of the spin state. Thereby, we propose to utilize optical pulses as a phase-shifter that rotates the quadrature of the spins. This method substantially simplifies the geometry of experimental schemes.
We numerically study the hyperfine induced nuclear spin dynamics in a system of two coupled quantum dots in zero magnetic field. Each of the electron spins is considered to interact with an individual bath of nuclear spins via homogeneous coupling constants (all coupling coefficients being equal). In order to lower the dimension of the problem, the two baths are approximated by two single long spins. We demonstrate that the hyperfine interaction enables to utilize the nuclear baths for quantum information purposes. In particular, we show that it is possible to swap the nuclear ensembles on tim…
Quantum state reconstruction on a finite number of copies of a quantum system with informationally incomplete measurements does, as a rule, not yield a unique result. We derive a reconstruction scheme where both the likelihood and the von Neumann entropy functionals are maximized in order to systematically select the most-likely estimator with the largest entropy, that is the least-bias estimator, consistent with a given set of measurement data. This is equivalent to the joint consideration of our partial knowledge and ignorance about the ensemble to reconstruct its identity. An interesting st…
Mapping the positions of single electron spins is a highly desired capability for applications such as nanoscale magnetic resonance imaging and quantum network characterization. Here, we demonstrate a method based on rotating an external magnetic field to identify the precise location of single electron spins in the vicinity of a quantum spin sensor. We use a nitrogen-vacancy center in diamond as a quantum sensor and modulate the dipolar coupling to a proximate electron spin in the crystal by varying the magnetic field vector. The modulation of the dipolar coupling contains information on the …
The relaxation of a quantum field stored in a high-$Q$ superconducting cavity is monitored by non-resonant Rydberg atoms. The field, subjected to repetitive quantum non-demolition (QND) photon counting, undergoes jumps between photon number states. We select ensembles of field realizations evolving from a given Fock state and reconstruct the subsequent evolution of their photon number distributions. We realize in this way a tomography of the photon number relaxation process yielding all the jump rates between Fock states. The damping rates of the $n$ photon states ($0\leq n \leq 7$) are found …
Shadow tomography via classical shadows is a state-of-the-art approach for estimating properties of a quantum state. We present a simplified, combinatorial analysis of a recently proposed instantiation of this approach based on the ensemble of unitaries that are both fermionic Gaussian and Clifford. Using this analysis, we derive a corrected expression for the variance of the estimator. We then show how this leads to efficient estimation protocols for the fidelity with a pure fermionic Gaussian state (provably) and for an $X$-like operator of the form ($|\mathbf 0\rangle\langleψ|$ + h.c.) (via…
