Quantum computing with an inhomogeneously broadened ensemble of ions: Suppression of errors from detuning variations by specially adapted pulses and coherent population trapping
Ingela Roos, Klaus Molmer
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6 pages, 6 figures. Revised, extended version. More detailed discussion of frequency tolerance
The proposal for quantum computing with rare-earth-ion qubits in inorganic crystals makes use of the inhomogeneous broadening of optical transitions in the ions to associate individual qubits with ions responding to radiation in selected frequency channels. We show that a class of Gaussian composite pulses and complex sech pulses provide accurate qubit pi-rotations, which are at the same time channel selective on a 5 MHz frequency scale and tolerant to 0.5 MHz deviations of the transition frequency of ions within a single channel. Rotations in qubit space of arbitrary angles and phases are produced by sequences of pi-pulses between the excited state of the ions and coherent superpositions of the qubit states.
We derive composite pulse sequences that achieve high-fidelity excitation of two-state systems in an optically dense, inhomogeneously broadened ensemble. The composite pulses are resistant to distortions due to the back-action of the medium they propagate in and are able to create high-fidelity inversion to optical depths $αz>10$. They function well with smooth pulse shapes used for coherent control of optical atomic transitions in quantum computation and communication. They are an intermediary solution between single $π$-pulse excitation schemes and adiabatic passage schemes, being far more e…
We study collective effects in an inhomogeneously broadened ensemble of two-level emitters coupled to an optical cavity with narrow linewidth. Using second order mean field theory we find that the emitters within a few times the cavity linewidth exhibit synchronous behaviour and undergo collective Rabi oscillations. Under proper conditions, the synchronized oscillations give rise to a modulated intracavity field which can excite emitters detuned by many linewidths from the cavity resonance. To study the synchronization in further detail, we simplify the model and consider two ensembles and stu…
The creation of single collective excitations in atomic ensembles via spontaneous Raman emission plays a key role in several quantum communication protocols, starting with the seminal DLCZ protocol [L.-M.Duan, M.D. Lukin, J.I. Cirac, and P. Zoller, Nature {\bf 414}, 413 (2001).] This process is usually analyzed theoretically under the assumptions that the write laser pulse inducing the Raman transition is far off-resonance, and that the atomic ensemble is only homogeneously broadened. Here we study the impact of near-resonance excitation for inhomogeneously broadened ensembles on the collectiv…
We address the propagation of a single photon pulse with two polarization components, i.e., a polarization qubit, in an inhomogeneously broadened "phaseonium" Λ-type three-level medium. We combine some of the non-trivial propagation effects characteristic for this kind of coherently prepared systems and the controlled reversible inhomogeneous broadening technique to propose several quantum information processing applications, such as a protocol for polarization qubit filtering and sieving as well as a tunable polarization beam splitter. Moreover, we show that, by imposing a spatial variation o…
We show experimentally and describe theoretically how a conventional magnetic resonance Hahn echo sequence can lead to a self-stimulated pulse echo train when an inhomogeneously broadened spin ensemble is coupled to a resonator. Effective strong coupling between the subsystems assures that the first Hahn echo can act as a refocussing pulse on the spins, leading to self-stimulated secondary echoes. Within the framework of mean field theory, we show that this process can continue multiple times leading to a train of echoes. We introduce an analytical model that explains the shape of the first ec…
We present a method for performing quantum state reconstruction on qubits and qubit registers in the presence of decoherence and inhomogeneous broadening. The method assumes only rudimentary single qubit rotations as well as knowledge of decoherence and loss mechanisms. We show that full state reconstruction is possible even in the case where single qubit rotations may only be performed imperfectly. Furthermore we show that for ensemble quantum computing proposals, quantum state reconstruction is possible even if the ensemble experiences inhomogeneous broadening and if only imperfect qubit man…
Here we propose a solid-state quantum memory that does not require spectral holeburning, instead using strong rephasing pulses like traditional photon echo techniques. The memory uses external broadening fields to reduce the optical depth and so switch off the collective atom-light interaction when desired. The proposed memory should allow operation with reasonable efficiency in a much broader range of material systems, for instance Er3+ doped crystals which have a transition at 1.5 um. We present analytic theory supported by numerical calculations and initial experiments.
In large ensembles of identical atoms or spins, the interaction with a mode of the electromagnetic radiation field concentrates in a single superradiant degree of freedom with a collectively enhanced coupling. Given a controllable inhomogeneous broadening, such ensembles may be used for multi-mode storage of quantum states of the radiation field with applications in quantum communication networks and quantum computers. In this paper we analyze how the width and shape of the inhomogeneous broadening influence the collective enhancement and the dynamics of the cavity-ensemble system with focus o…
We propose a multi-mode quantum memory protocol able to store the quantum state of the field in a microwave resonator into an ensemble of electronic spins. The stored information is protected against inhomogeneous broadening of the spin ensemble by spin-echo techniques resulting in memory times orders of magnitude longer than previously achieved. By calculating the evolution of the first and second moments of the spin-cavity system variables for realistic experimental parameters, we show that a memory based on NV center spins in diamond can store a qubit encoded on the |0> and |1> Fock states …
We propose and analyze a scheme for "on demand" creation of coherent superposition of meta-stable states in a tripod-structured atom using frequency-chirped laser pulses. Negligible excitation of the atoms during the creation of the superposition states is a priority in our consideration. The underlying physics of the scheme is explained using the formalism of adiabatic states. By numerically solving master equation for the density matrix operator, we analyze the influence of the spontaneous decay and transverse relaxation on the efficiency of the creation of superposition states. We show that…
Theoretical studies of superradiant lasing on optical clock transitions predict a superb frequency accuracy and precision closely tied to the bare atomic linewidth. Such a superradiant laser is also robust against cavity fluctuations when the spectral width of the lasing mode is much larger than that of the atomic medium. Recent predictions suggest that this unique feature persists even for a hot and thus strongly broadened ensemble, provided the effective atom number is large enough. Here we use a second-order cumulant expansion approach to study the power, linewidth and lineshifts of such a …
We study, theoretically and experimentally, electromagnetically induced transparency (EIT) in two different solid-state systems. Unlike many implementations in homogeneously broadened media, these systems exhibit inhomogeneous broadening of their optical and spin transitions typical of solid-state materials. We observe EIT lineshapes typical of atomic gases, including a crossover into the regime of Autler-Townes splitting, but with the substitution of the inhomogeneous widths for the homogeneous values. We obtain quantitative agreement between experiment and theory for the width of the transpa…
We extend our earlier investigations [Opt. Commun. {\bf 179}, 97 (2000)] on the enhancement of magneto-optical rotation (MOR) to include inhomogeneous broadening. We introduce a control field that counter-propagates with respect to the probe field. We derive analytical results for the susceptibilities corresponding to the two circular polarization components of the probe field. From the analytical results we identify and numerically demonstrate the region of parameters where significantly large magneto-optical rotation (MOR) can be obtained. From the numerical results we isolate the significan…
We propose a method to achieve significant optical signal delays exploiting the effect of Autler-Townes splitting in an inhomogeneously broadened quantum dot medium. The absorption and slow-down effects are compared for three schemes i.e. $Ξ$, V and $Λ$, corresponding to different excitation configurations. Qualitative differences of the V-scheme compared to the $Ξ$- and $Λ$-scheme are found, which show that features of Autler-Townes splitting are only revealed in the V-scheme. The underlying physical mechanisms causing this discrepancy are analyzed and discussed. Finally we compare field prop…
Bad cavity lasers are experiencing renewed interest in the context of active optical frequency standards, due to their enhanced robustness against fluctuations of the laser cavity. The gain medium would consist of narrow-linewidth atoms, either trapped inside the cavity or intersecting the cavity mode dynamically. A series of effects like the atoms finite velocity distribution, atomic interactions, or interactions of realistic multilevel atoms with auxiliary or stray fields can lead to an inhomogeneous broadening of the atomic gain profile. This causes the emergence of instable regimes of lase…
Ensembles of quantum mechanical spins offer a promising platform for quantum memories, but proper functionality requires accurate control of unavoidable system imperfections. We present an efficient control scheme for a spin ensemble strongly coupled to a single-mode cavity based on a set of Volterra equations relying solely on weak classical control pulses. The viability of our approach is demonstrated in terms of explicit storage and readout sequences that will serve as a starting point towards the realization of more demanding full quantum mechanical optimal control schemes.
Electromagnetically induced transparency (EIT) is a phenomenon that can provide strong and robust interfacing between optical signals and quantum coherence of electronic spins. In its archetypical form, mainly explored with atomic media, it uses a (near-)homogeneous ensemble of three-level systems, in which two low-energy spin-1/2 levels are coupled to a common optically excited state. We investigate the implementation of EIT with c-axis divacancy color centers in silicon carbide. While this material has attractive properties for quantum device technologies with near-IR optics, implementing EI…
Electromagnetically induced transparency (EIT) has mainly been modelled for three-level systems. In particular, a considerable interest has been dedicated to the Lambda-configuration, with two ground states and one excited state. However, in the alkali-metal atoms, which are commonly used, hyperfine interaction in the excited state introduces several levels which simultaneously participate in the scattering process. When the Doppler broadening is comparable with the hyperfine splitting in the upper state, the three-level Lambda model does not reproduce the experimental results. Here we theoret…
A very promising recent trend in applied quantum physics is to combine the advantageous features of different quantum systems into what is called "hybrid quantum technology". One of the key elements in this new field will have to be a quantum memory enabling to store quanta over extended periods of time. Systems that may fulfill the demands of such applications are comb-shaped spin ensembles coupled to a cavity. Due to the decoherence induced by the inhomogeneous ensemble broadening, the storage time of these quantum memories is, however, still rather limited. Here we demonstrate how to overco…
We study the dynamics of a spin ensemble strongly coupled to a single-mode resonator driven by external pulses. When the mean frequency of the spin ensemble is in resonance with the cavity mode, damped Rabi oscillations are found between the spin ensemble and the cavity mode which we describe very accurately, including the dephasing effect of the inhomogeneous spin broadening. We demonstrate that a precise knowledge of this broadening is crucial both for a qualitative and a quantitative understanding of the temporal spin-cavity dynamics. On this basis we show that coherent oscillations between…
Specific features of nonlinear interference processes at quantum transitions in near- and fully-resonant optically-dense Doppler-broadened medium are studied. The feasibility of overcoming of the fundamental limitation on a velocity-interval of resonantly coupled molecules imposed by the Doppler effect is shown based on quantum coherence. This increases the efficiency of nonlinear-optical processes in atomic and molecular gases that possess the most narrow and strongest resonances. The possibility of all-optical switching of the medium to opaque or, alternatively, to absolutely transparent, or…
We describe how stimulated Raman adiabatic passage (STIRAP) can be applied to create spectral holes in an inhomogeneously broadened system. Due to the robustness of STIRAP, our proposal guarantees high flexibility and accuracy and, at variance with traditional spectral hole burning techniques, it may require substantially less time resources since it does not rely upon the spontaneous decay of an intermediate excited state. We investigate the effects on the scheme of dephasing and dissipation as well as of unintentional driving of undesired transitions due to a finite splitting of the initial …
Quantum coherence control is presented for wavelength convertible quantum memory in a double-lambda-type solid ensemble whose spin states are inhomogeneously broadened. Unlike typical atomic media whose spin decay is homogeneous, a spin inhomogeneously broadened solid ensemble requires a counter-intuitive access in the quantum coherence control to avoid spontaneous emission-caused quantum noises. Such quantum coherence control in a solid ensemble results in a near perfect retrieval efficiency and is applicable to ultralong photon storage up to the spin phase relaxation time. Here, the basic ph…
Pulse propagation is considered in an inhomogeneously broadened medium of three-level atoms in a V-configuration, dressed by a counter-propagating pump pulse. A significant signal slowdown is demonstrated in this of the three frequency windows of a reduced absorption and a steep normal dispersion, which is due to a cross-over resonance. Particular properties of the group index in the vicinity of such a resonance are demonstrated in the case of closely spaced upper levels.
We propose a scheme to couple short single photon pulses to superconducting qubits. An optical photon is first absorbed into an inhomogeneously broadened rare-earth doped crystal using controlled reversible inhomogeneous broadening. The optical excitation is then mapped into a spin state using a series of $π$-pulses and subsequently transferred to a superconducting qubit via a microwave cavity. To overcome the intrinsic and engineered inhomogeneous broadening of the optical and spin transitions in rare earth doped crystals, we make use of a special transfer protocol using staggered $π$-pulses.…
We demonstrate experimentally a quantum memory scheme for the storage of weak coherent light pulses in an inhomogeneously broadened optical transition in a Pr^{3+}: YSO crystal at 2.1 K. Precise optical pumping using a frequency stable (about 1kHz linewidth) laser is employed to create a highly controllable Atomic Frequency Comb (AFC) structure. We report single photon storage and retrieval efficiencies of 25%, based on coherent photon echo type re-emission in the forward direction. The coherence property of the quantum memory is proved through interference between a super Gaussian pulse and t…
In this paper initial experiments towards constructing simple quantum gates in a solid state material are presented. Instead of using specially tailored materials, the aim is to select a subset of randomly distributed ions in the material, which have the interaction necessary to control each other and therefore can be used to do quantum logic operations. The experimental results demonstrate that part of an inhomogeneously broadened absorption line can be selected as a qubit and that a subset of ions in the material can control the resonance frequency of other ions. This opens the way for the c…
Pulses applied to an inhomogeneously broadened set of harmonic oscillators, previously prepared in squeezed states, can lead to a recovery of coherence, manifesting itself as echoes, similar to those exhibited by an ensemble of spins when excited by properly designed electromagnetic pulses. Such echoes, of classical or quantum nature, are expected to arise in the squeezing of linear systems of various sorts and, in particular, light and vibrational modes.
Inhomogeneously broadened spin ensembles play an important role in present-day implementation of hybrid quantum processing architectures. When coupled to a resonator such an ensemble may serve as a multi-mode quantum memory for the resonator field, and by employing spin-refocusing techniques the quantum memory time can be extended to the coherence time of individual spins in the ensemble. In the present paper we investigate such a memory protocol capable of storing an unknown resonator-field state, and we examine separately the various constituents of the protocol: the storage and read-out par…
We investigate the performance of inhomogeneously broadened spin ensembles as quantum memories under continuous dynamical decoupling. The role of the continuous driving field is two-fold: first, it decouples individual spins from magnetic noise; second and more important, it suppresses and reshapes the spectral inhomogeneity of spin ensembles. We show that a continuous driving field, which itself may also be inhomogeneous over the ensemble, can enhance the decay of the tails of the inhomogeneous broadening distribution considerably. This fact enables a spin ensemble based quantum memory to exp…
Continuous spectral hole burning and spin-level population oscillations are studied in an inhomogeneously broadened diamond-based spin ensemble composed of substitutional nitrogen and nitrogen-vacancy centres created through neutron irradiation and annealing. The burnt spectral features highlight a detuning-dependent homogeneous hole linewidth that is up to three orders of magnitude narrower than the total inhomogeneous ensemble linewidth. Continuous population oscillations are observed to quickly decay beyond a pump and probe detuning of 5 Hz, and are numerically modelled using a five-level s…
Photon-echo based quantum memories use inhomogeneously broadened, optically thick ensembles of absorbers to store a weak optical signal and employ various protocols to rephase the atomic coherences for information retrieval. We study the application of two consecutive, frequency-chirped control pulses for coherence rephasing in an ensemble with a 'natural' inhomogeneous broadening. Although propagation effects distort the two control pulses differently, chirped pulses that drive adiabatic passage can rephase atomic coherences in an optically thick storage medium. Combined with spatial phase mi…
An efficient multi-mode quantum memory is a crucial resource for long-distance quantum communication based on quantum repeaters. We propose a quantum memory based on spectral shaping of an inhomogeneously broadened optical transition into an atomic frequency comb (AFC). The spectral width of the AFC allows efficient storage of multiple temporal modes, without the need to increase the absorption depth of the storage material, in contrast to previously known quantum memories. Efficient readout is possible thanks to rephasing of the atomic dipoles due to the AFC structure. Long-time storage and o…
We show that dissipative quantum state preparation processes can be protected against qubit dephasing by interlacing the state preparation control with dynamical decoupling (DD) control consisting of a sequence of short $π$-pulses. The inhomogeneous broadening can be suppressed to second order of the pulse interval, and the protection efficiency is nearly independent of the pulse sequence but determined by the average interval between pulses. The DD protection is numerically tested and found to be efficient against inhomogeneous dephasing on two exemplary dissipative state preparation schemes …
We present a light-storage experiment in a praseodymium-doped crystal where the light is mapped onto an inhomogeneously broadened optical transition shaped into an atomic frequency comb. After absorption of the light the optical excitation is converted into a spin-wave excitation by a control pulse. A second control pulse reads the memory (on-demand) by reconverting the spin-wave excitation to an optical one, where the comb structure causes a photon-echo type rephasing of the dipole moments and directional retrieval of the light. This combination of photon echo and spin-wave storage allows us …
Quantum plasmonic systems suffer from significant decoherence due to the intrinsically large dissipative and radiative dampings. Based on our quantum simulations via a quantum tensor network algorithm, we numerically demonstrate the mitigation of this restrictive drawback by hybridizing a plasmonic nanocavity with an emitter ensemble with inhomogeneously-broadened transition frequencies. By burning two narrow spectral holes in the spectral density of the emitter ensemble, the coherent time of Rabi oscillation for the hybrid system is increased tenfold. With the suppressed decoherence, we move …
Quantum sensing with solid-state systems finds broad applications in diverse areas ranging from material and biomedical sciences to fundamental physics. Several solid-state spin sensors have been developed, facilitating the ultra-sensitive detection of physical quantities such as magnetic and electric fields and temperature. Exploiting collective behaviour of non-interacting spins holds the promise of pushing the detection limit to even lower levels, while to date, those levels are scarcely reached due to the broadened linewidth and inefficient readout of solid-state spin ensembles. Here, we e…
It has been shown that an inhomogeneously broadened optical transition shaped into an atomic frequency comb can store a large number of temporal modes of the electromagnetic field at the single photon level without the need to increase the optical depth of the storage material. The readout of light modes is made efficient thanks to the rephasing of the optical-wavelength coherence similarly to photon echo-type techniques and the re-emission time is given by the comb structure. For on-demand readout and long storage times, two control fields are used to transfer back and forth the optical coher…
An overview of current status and prospects of the development of quantum computer hardware based on inorganic crystals doped with rare-earth ions is presented. Major parts of the experimental work in this area has been done in two places, Canberra, Australia and Lund, Sweden, and the present description follows more closely the Lund work. Techniques will be described that include optimal filtering of the initially inhomogeneously broadened profile down to well separated and narrow ensembles, as well as the use of advanced pulse-shaping in order to achieve robust arbitrary single-qubit operati…
We study the effects of dispersion in carrier waves on the properties of soliton self--induced transparency (SIT) in two level media. We found substantial impact of dispersion effects on typical SIT soliton features. For example, the degree of SIT pulse velocity slowing down (acceleration) is determined by the ratio of the incoming pulse frequency over atomic transition frequency - $x=ω/ω_0$. Specifically, an immediate pulse stopping is predicted for absorbing media when pulse duration time exceeds some critical value. In the sharp line limit stopping may emerge only for frequency ratio above …
We describe how the transient behavior of a tuned and matched resonator circuit and a ringdown suppression pulse may be integrated into an optimal control theory (OCT) pulse-design algorithm to derive control sequences with limited ringdown that perform a desired quantum operation in the presence of resonator distortions of the ideal waveform. Inclusion of ringdown suppression in numerical pulse optimizations significantly reduces spectrometer deadtime when using high quality factor (high-Q) resonators, leading to increased signal-to-noise ratio (SNR) and sensitivity of inductive measurements.…
We show that quantum frequency conversion (QFC) can overcome the spectral distinguishability common to inhomogeneously broadened solid-state quantum emitters. QFC is implemented by combining single photons from an InAs quantum dot (QD) at 980 nm with a 1550 nm pump laser in a periodically-poled lithium niobate (PPLN) waveguide to generate photons at 600 nm with a signal-to-background ratio exceeding 100:1. Photon correlation and two-photon interference measurements confirm that both the single photon character and wavepacket interference of individual QD states are preserved during frequency c…
We study quantum control techniques, specifically Adiabatic Rapid Passage (ARP) and Gradient Ascent Pulse Engineering (GRAPE), for transferring atoms trapped in an optical lattice between different vibrational states. We compare them with each other and with previously studied coupling schemes in terms of performance. In our study of ARP, we realize control of the vibrational states by tuning the frequency of a spatial modulation through the inhomogeneously broadened vibrational absorption spectrum. We show that due to the presence of multiple crossings, the population transfer depends on the …
The ability to induce, observe and control quantum coherent interactions in room temperature, electrically driven optoelectronic devices is of outmost significance for advancing quantum science and engineering towards practical applications. We demonstrate here a quantum interference phenomena, Ramsey fringes, in an inhomogeneously broadened InAs/InP quantum dot (QD) ensemble in the form of a 1.5 mm long optical amplifier operating at room temperature. Observation of Ramsey fringes in semiconductor QD was previously achieved only at cryogenic temperatures and only in isolated single dot system…
The success of the emerging field of solid-state optical quantum information processing (QIP) critically depends on the access to resonant optical materials. Rare-earth ions (REIs) are suitable candidates for QIP protocols due to their extraordinary photo-physical and magnetic quantum properties such as long optical and spin coherence lifetimes ($T_2$). However, molecules incorporating REIs, despite having advantageous properties such as atomically exact quantum tunability, inherent scalability, and large portability, have not yet been studied for QIP applications. As a first testimony of the …
Superradiance is the archetypical collective phenomenon where radiation is amplified by the coherence of emitters. It plays a prominent role in optics, where it enables the design of lasers with substantially reduced linewidths, quantum mechanics, and is even used to explain cosmological observations like Hawking radiation from black holes. Hybridization of distinct quantum systems allows to engineer new quantum metamaterials pooling the advantages of the individual systems. Superconducting circuits coupled to spin ensembles are promising future building blocks of integrated quantum devices an…
A quantum emitter in a dynamic environment may have its energy levels drift uncontrollably in time with the fluctuating bath. This can result in an emission/absorption spectrum that is spread over a broad range of frequencies and presents a challenging hurdle for various applications. We consider a quantum emitter in an environment that alters the energy levels so that the emission frequency is represented by a Gaussian random distribution around a given mean value with given standard deviation and correlation time. We study the emission spectrum of this system when it is placed under the infl…
Ensembles of electron spins in hybrid microwave systems are powerful and versatile components for future quantum technologies. Quantum memories with high storage capacities are one such example which require long-lived states that can be addressed and manipulated coherently within the inhomogeneously broadened ensemble. This broadening is essential for true multimode memories, but induces a considerable spin dephasing and together with dissipation from a cavity interface poses a constraint on the memory's storage time. In this work we show how to overcome both of these limitations through the …
