Efficient tomography of microwave photonic cluster states
Stanford University（斯坦福大学）；Department of Applied Physics, Graduate School of Engineering, The University of Tokyo（东京大学工学研究科应用物理系）；RIKEN Center for Quantum Computing (RQC)（理化学研究所量子计算中心）；Research Center for Advanced Science and Technology (RCAST), The University of Tokyo（东京大学先端科学研究中心）；NTT Computer and Data Science Laboratories（NTT计算机与数据科学实验室）；PRESTO, Japan Science and Technology Agency（日本科学技术振兴机构 PRESTO 项目）；Institute for Physics of Intelligence, The University of Tokyo（东京大学智能物理学研究所）
Entanglement among a large number of qubits is a crucial resource for many quantum algorithms. Such many-body states have been efficiently generated by entangling a chain of itinerant photonic qubits in the optical or microwave domain. However, it has remained challenging to fully characterize the generated many-body states by experimentally reconstructing their exponentially large density matrices. Here, we develop an efficient tomography method based on the matrix-product-operator formalism and demonstrate it on a cluster state of up to 35 microwave photonic qubits by reconstructing its $2^{35} \times 2^{35}$ density matrix. The full characterization enables us to detect the performance degradation of our photon source which occurs only when generating a large cluster state. This tomography method is generally applicable to various physical realizations of entangled qubits and provides an efficient benchmarking method for guiding the development of high-fidelity sources of entangled photons.
Quantum tomography is an essential method of the photonic technology toolbox and is routinely used for evaluation of experimentally prepared states of light and characterization of devices transforming such states. The tomography procedure consists of many different sequentially performed measurements. We present considerable tomography speedup by optimally arranging the individual constituent measurements, which is equivalent to solving an instance of the traveling salesman problem. As an example, we obtain solutions for photonic systems of up to five qubits, and conclude that already for sys…
We present a novel method to realize a multi-target-qubit controlled phase gate with one microwave photonic qubit simultaneously controlling $n-1$ target microwave photonic qubits. This gate is implemented with $n$ microwave cavities coupled to a superconducting flux qutrit. Each cavity hosts a microwave photonic qubit, whose two logic states are represented by the vacuum state and the single photon state of a single cavity mode, respectively. During the gate operation, the qutrit remains in the ground state and thus decoherence from the qutrit is greatly suppressed. This proposal requires onl…
We present the experimental quantum tomography of 7- and 8-dimensional quantum systems based on projective measurements in the mutually unbiased basis (MUB-QT). One of the advantages of MUB-QT is that it requires projections from a minimal number of bases to be performed. In our scheme, the higher dimensional quantum systems are encoded using the propagation modes of single photons, and we take advantage of the capabilities of amplitude- and phase-modulation of programmable spatial light modulators to implement the MUB-QT.
Stimulated emission tomography is a powerful and successful technique to both improve the resolution and experimentally simplify the task of determining the modal properties of biphotons. In the present manuscript we provide a theoretical description of SET valid for any quadratic coupling regime between a non-linear medium and pump fields generating photons by pairs. We use our results to obtain not only information about the associated modal function modulus but also its phase, for any mode, and we discuss the specific case of time-frequency variables as well as the quantities and limitation…
Manipulation of qudits in optical tables is a difficult and nonscalable task. The use of integrated optical circuits opens new possibilities for the generation, manipulation, and characterization of high dimensional states besides the ease of transmission of these states through an optical fiber. In this work we propose photonic circuits to perform minimum quantum state tomography of path qudits and show how to determine all the constituents parameters of these circuits (beam splitters and phase shifters). Our strategies were based on the symmetries of the involved POVMs (positive operator-val…
In this work, we propose three-dimensional photonic circuit designs that guarantee a considerable reduction in the complexity of circuits for the purpose of performing quantum state tomography of N-dimensional path qudits. The POVM (Positive Operator-Valued Measure) chosen in this work ensures that, for odd dimensions, such process is minimal. Our proposal consists of organizing the waveguides that form the circuit as a square array formed by N vertical sectors composed of N waveguides each, arranged in the vertical direction. Based on the symmetry of the chosen POVM, the interferometer acting…
We propose to generate photon-pair blockade in a Josephson-photonics circuit that consists of a dc voltage-biased Josephson junction in series with a superconducting charge qubit and two nondegenerate microwave resonators. The two-level charge qubit is utilized to create anticorrelations of the charge transport; that is, the simultaneous Cooper pair tunneling events are inhibited. When the Josephson frequency matches the sum of resonance frequencies of the charge qubit and the two resonators, we demonstrate that the two resonators can release their energies in the form of antibunched pairs of …
As one of the most fundamental functionalities of microwave photonics, microwave frequency mixing plays an essential role in modern radars and wireless communication systems. However, the commonly utilized intensity modulation in the systems often leads to inadequate spurious-free dynamic range (SFDR) for many sought-after applications. Quantum microwave photonics technique offers a promising solution for improving SFDR in terms of higher-order harmonic distortion. In this paper, we demonstrate two types of quantum microwave photonic mixers based on the configuration of the intensity modulator…
Emitters strongly coupled to a photonic crystal provide a powerful platform for realizing novel quantum light-matter interactions. Here we study the optical properties of a three-level artificial atomic chain coupled to a one-dimensional superconducting microwave photonic crystal. A sharp minimum-energy dip appears in the transmission spectrum of a weak input field, which reveals rich behavior of the long-range interactions arising from localized bound states. We find that the dip frequency scales linearly with both the number of the artificial atoms and the characteristic strength of the long…
Quantum microwave photonics aims at generating, routing, and manipulating propagating quantum microwave fields in the spirit of optical photonics. To this end, the strong nonlinearities of superconducting quantum circuits can be used to either improve or move beyond the implementation of concepts from the optical domain. In this context, the design of a well-controlled broadband environment for the superconducting quantum circuits is a central task. In this work, we place a superconducting transmon qubit in one arm of an on-chip Mach-Zehnder interferometer composed of two superconducting micro…
Quantum state reconstruction involves measurement devices that are usually described by idealized models, but not known in full detail in experiments. For weak propagating microwaves, the detection process requires linear amplifiers which obscure the signal with random noise. Here, we introduce a theory which nevertheless allows one to use these devices for measuring all quadrature moments of propagating quantum microwaves based on cross-correlations from a dual-path amplification setup. Simultaneously, the detector noise properties are determined, allowing for tomography. We demonstrate the f…
While measuring the orbital angular momentum state of bright light beams can be performed using imaging techniques, a full characterization at the single-photon level is challenging. For applications to quantum optics and quantum information science, such characterization is an essential capability. Here, we present a setup to perform the quantum state tomography of photonic qubits encoded in this degree of freedom. The method is based on a projective technique using spatial mode projection via fork holograms and single-mode fibers inserted into an interferometer. The alignment and calibration…
The protocol of measurement the probability density of a rotated quadrature of microwave mode in an arbitrary quantum state is presented and numerically investigated. The protocol is adapted to the measuring procedure used in the CQED experiments. The method is based on measurement of integrated probability of detection of atoms - probes in the top working state from atoms flying via the cavity. Connection of random variables and is found, when is a qudrature phase. Connection of the tomogram and the tomogram is found. The protocol prescribes the rule of choice of problem parameters, such as i…
We study a pair of anharmonic microwave cavities that is connected by an optical fiber. The photonic spectral density characterizes the evolution of the coupled cavities after the system has been prepared in a Fock or N00N state. We evaluate the photonic spectral density within the recursive projection method and find that the anharmonicity has no substantial effect on the spectral properties. In all cases the photonic spectral density has a Gaussian envelope for large photon numbers.
In this article, we investigate the efficiency of photonic state tomography in the presence of fiber attenuation. The theoretical formalism of the photon loss is provided by implementing methods from the theory of open quantum systems. The quantum state is reconstructed from photon counts obtained for symmetric informationally complete POVMs. The number of photons that reach the detectors is numerically modeled by the binomial distribution, which describes the loss of light caused by the medium. This approach allows us to study the quality of state tomography versus the length of the fiber. In…
Experimental verification of Threshold Quantum State Tomography on a fully-reconfigurable photonic integrated circuit
Sapienza Università di Roma（罗马第一大学）；Università di Pavia（帕维亚大学）；European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*, Fondazione Bruno Kessler)（核物理及相关领域欧洲理论研究中心（ECT*，布鲁诺·凯斯勒基金会））；Istituto di Fotonica e Nanotecnologie, Consiglio Nazionale delle Ricerche (IFN-CNR)（意大利国家研究委员会光子学与纳米技术研究所）；Politecnico di Milano（米兰理工大学）
If one seeks to test quantum theory against many alternatives in a landscape of possible physical theories, then it is crucial to be able to analyze experimental data in a theory-agnostic way. This can be achieved using the framework of Generalized Probabilistic Theories (GPTs). Here, we implement GPT tomography on a three-level system corresponding to a single photon shared among three modes. This scheme achieves a GPT characterization of each of the preparations and measurements implemented in the experiment without requiring any prior characterization of either. Assuming that the sets of re…
Microwave photonics is a remarkably powerful system for quantum simulation and technologies, but its integration in superconducting circuits, superior in many aspects, is constrained by the long wavelengths and impedance mismatches in this platform. We introduce a solution to these difficulties via compact networks of high-kinetic inductance microstrip waveguides and coupling wires with strongly reduced phase velocities. We demonstrate broadband capabilities for superconducting microwave photonics in terms of routing, emulation and generalized linear and nonlinear networks.
We show that optically active coupled quantum dots embedded in a superconducting microwave cavity can be used to realize a fast quantum interface between photonic and transmon qubits. Single photon absorption by a coupled quantum dot results in generation of a large electric dipole, which in turn ensures efficient coupling to the microwave cavity. Using cavity parameters achieved in prior experiments, we estimate that bi-directional microwave-optics conversion in nanosecond timescales with efficiencies approaching unity is experimentally feasible with current technology. We also outline a prot…
The exponential growth of photonic quantum technologies is driving the demand of tools for measuring the quality of their information carriers. One of the most prominent is Stimulated Emission Tomography (SET), which uses classical coherent fields to measure the Joint Spectral Amplitude (JSA) of photon pairs with high speed and resolution. While the modulus of the JSA can be directly addressed from a single intensity measurement, the retrieval of the Joint Spectral Phase (JSP) is far more challenging and received minor attentions. However, a wide class of spontaneous sources of technological r…
We perform state tomography of an itinerant squeezed state of the microwave field prepared by a Josephson parametric amplifier (JPA). We use a second JPA as a pre-amplifier to improve the quantum efficiency of the field quadrature measurement (QM) from 2% to 36 +/- 4%. Without correcting for the detection inefficiency we observe a minimum quadrature variance which is 69 +/- 8% of the variance of the vacuum. We reconstruct the state's density matrix by a maximum likelihood method and infer that the squeezed state has a minimum variance less than 40% of the vacuum, with uncertainty mostly caused…
In this letter we introduce a novel approach to Hamiltonian tomography of non-interacting tight-binding photonic lattices. To begin with, we prove that the matrix element of the low-energy effective Hamiltonian between sites $i$ and $j$ may be obtained directly from $S_{ij}(ω)$, the (suitably normalized) two-port measurement between sites $i$ and $j$ at frequency $ω$. This general result enables complete characterization of both on-site energies and tunneling matrix elements in arbitrary lattice networks by spectroscopy, and suggests that coupling between lattice sites is actually a topologica…
Continuous-variable (CV) photonic states are of increasing interest in quantum information science, bolstered by features such as deterministic resource state generation and error correction via bosonic codes. Data-efficient characterization methods will prove critical in the fine-tuning and maturation of such CV quantum technology. Although Bayesian inference offers appealing properties -- including uncertainty quantification and optimality in mean-squared error -- Bayesian methods have yet to be demonstrated for the tomography of arbitrary CV states. Here we introduce a complete Bayesian qua…
Microwave photonic graph states provide a promising approach for robust quantum communication between remote superconducting chips using microwave photons. Recently, Besse et al. [Nat. Commun. 11, 4887 (2020)] demonstrated that 1D graph states can be generated using two transmon qubits. In this paper, we propose to use transmon qubits combined with other microwave devices to construct more complex graph states. Specifically, we consider 2D lattice and tree-like graph states. We compare the performance using fixed- versus tunable-frequency transmon qubits and also for different photonic qubit e…
Single-acquisition tomography of photonic qubits with structured media
University of Ottawa（渥太华大学）；Università degli Studi di Napoli Federico II（那不勒斯费德里科二世大学）；National Research Council of Canada（加拿大国家研究委员会）；McGill University（麦吉尔大学）；Chapman University（查普曼大学）
With the rapid development of microwave photonics, which has expanded to numerous applications of commercial importance, eliminating the emerging bottlenecks becomes of vital importance. For example, as the main branch of microwave photonics, radio-over-fiber technology provides high bandwidth, low-loss, and long-distance propagation capability, facilitating wide applications ranging from telecommunication to wireless networks. With ultrashort pulses as the optical carrier, huge capacity is further endowed. However, the wide bandwidth of ultrashort pulses results in the severe vulnerability of…
Generation of frequency-bin-encoded dual-rail cluster states via time-frequency multiplexing of microwave photonic qubits
RIKEN Center for Quantum Computing (RQC)（理化学研究所量子计算中心）；Department of Applied Physics, Graduate School of Engineering, The University of Tokyo（东京大学大学院工学系研究科应用物理系）；Department of Applied Physics, Stanford University（斯坦福大学应用物理系）
A photonic transistor that can switch or amplify an optical signal with a single gate photon requires strong non-linear interaction at the single-photon level. Circuit quantum electrodynamics provides great flexibility to generate such an interaction, and thus could serve as an effective platform to realize a high-performance single-photon transistor. Here we demonstrate such a photonic transistor in the microwave regime. Our device consists of two microwave cavities dispersively coupled to a superconducting qubit. A single gate photon imprints a phase shift on the qubit state through one cavi…
Characterizing Photonic Partial Distinguishability Through Gram Matrix Tomography
Center for Quantum Enabled-Computing, Center for Theoretical Physics of the Polish Academy of Sciences（波兰科学院理论物理中心量子使能计算中心）；International Iberian Nanotechnology Laboratory (INL)（伊比利亚纳米技术国际实验室）；Centre for Quantum Information and Communication, École polytechnique de Bruxelles, CP 165/59, Université libre de Bruxelles（布鲁塞尔自由大学布鲁塞尔理工学院量子信息与通信中心）；Instituto de Física, Universidade Federal Fluminense（弗鲁米嫩塞联邦大学物理研究所）
In the past 20 years, impressive progress has been made both experimentally and theoretically in superconducting quantum circuits, which provide a platform for manipulating microwave photons. This emerging field of superconducting quantum microwave circuits has been driven by many new interesting phenomena in microwave photonics and quantum information processing. For instance, the interaction between superconducting quantum circuits and single microwave photons can reach the regimes of strong, ultra-strong, and even deep-strong coupling. Many higher-order effects, unusual and less familiar in…
We propose a method for transferring quantum entangled states of two photonic cat-state qubits (cqubits) from two microwave cavities to the other two microwave cavities. This proposal is realized by using four microwave cavities coupled to a superconducting flux qutrit. Because of using four cavities with different frequencies, the inter-cavity crosstalk is significantly reduced. Since only one coupler qutrit is used, the circuit resources is minimized. The entanglement transfer is completed with a single-step operation only, thus this proposal is quite simple. The third energy level of the co…
Quantum information processing with hybrid protocols making use of discrete- and continuous-variable currently attracts of great interest because of its promising applications in scalable quantum computer and distant quantum network. By inducing a giant cross-Kerr nonlinearity between two cavities, we propose a general protocol for hybrid quantum gate and quantum entanglement with high fidelity between a stationary, discrete photonic qubit and a flying photonic state. Interestingly, our protocol can be used to conduct a controlled-Z quantum gate between a stationary microwave photon stored in …
Light-sound interactions have long been exploited in various acousto-optic devices based on bulk crystalline materials. Conventionally these devices operate in megahertz frequency range where the acoustic wavelength is much longer than the optical wavelength and a long interaction length is required to attain significant coupling. With nanoscale transducers, acoustic waves with sub-optical wavelengths can now be excited to induce strong acousto-optic coupling in nanophotonic devices. Here we demonstrate microwave frequency surface acoustic wave transducers co-integrated with nanophotonic reson…
Continuous variable entanglement between two modes of a radiation field is usually studied at optical frequencies. As an important step towards the observation of entanglement between propagating microwave photons we demonstrate the experimental state reconstruction of two field modes in the microwave domain. In particular, we generate two-mode correlated states with a Josephson parametric amplifier and detect all four quadrature components simultaneously in a two-channel heterodyne setup using amplitude detectors. Analyzing two-dimensional phase space histograms for all possible pairs of quad…
We study different architectures for a photonic crystal in the microwave regime based on superconducting transmission lines interrupted by Josephson junctions, both in one and two dimensions. A study of the scattering properties of a single junction in the line shows that the junction behaves as a perfect mirror when the photon frequency matches the Josephson plasma frequency. We generalize our calculations to periodic arrangements of junctions, demonstrating that they can be used for tunable band engineering, forming what we call a quantum circuit crystal. Two applications are discussed in de…
In 2015 experiments were performed with superconducting microwave photonic crystals emulating artificial graphene B. Dietz, T. Klaus, M. Miski-Oglu, and A. Richter, Phys. Rev. B 91, 035411 (2015)]. The associated density of states comprises two Dirac points with adjacent bands including van Hove singularities, thus exhibiting the characteristic features originating from the extraordinary electronic band structure of graphene. They are separated by a narrow region of particularly high resonance density corresponding to a nearly flatband in the band structure, which is reminiscent of that of a h…
The incorporation of multiplexing techniques used in Microwave Photonics to Quantum Key Distribution (QKD) systems bring important advantages enabling the simultaneous and parallel delivery of multiple keys between a central station and different end-users in the context of multipoint access and metropolitan networks, or by providing higher key distribution rates in point to point links by suitably linking the parallel distributed keys. It also allows the coexistence of classical information and quantum key distribution channels over a single optical fibre infrastructure. Here we show, for the…
Electrically actuated optomechanical resonators provide a route to quantum-coherent, bidirectional conversion of microwave and optical photons. Such devices could enable optical interconnection of quantum computers based on qubits operating at microwave frequencies. Here we present a novel platform for microwave-to-optical conversion comprising a photonic crystal cavity made of single-crystal, piezoelectric gallium phosphide integrated on pre-fabricated niobium circuits on an intrinsic silicon substrate. The devices exploit spatially extended, sideband-resolved mechanical breathing modes at $\…
By harnessing quantum superposition and entanglement, remarkable progress has sprouted over the past three decades from different areas of research in communication computation and simulation. To further improve the processing ability of microwave pho-tonics, here, we have demonstrated a quantum microwave photonic processing system using a low jitter superconducting nanowire single photon detector (SNSPD) and a time-correlated single-photon counting (TCSPC) module. This method uniquely combines extreme optical sensitivity, down to a single-photon level (below -100 dBm), and wide processing ban…
We experimentally study the phase space distribution (PSD) of a mechanical resonator that is simultaneously coupled to two electromagnetic cavities. The first one, operating in the microwave band, is employed for inducing either cooling or self-excited oscillation, whereas the second one, operating in the optical band, is used for displacement detection. A tomography technique is employed for extracting the PSD from the signal reflected by the optical cavity. Measurements of PSD are performed in steady state near the threshold of self-excited oscillation while sweeping the microwave cavity det…
Practical quantum networks require low-loss and noise-resilient optical interconnects as well as non-Gaussian resources for entanglement distillation and distributed quantum computation. The latter could be provided by superconducting circuits but - despite growing efforts and rapid progress - existing solutions to interface the microwave and optical domains lack either scalability or efficiency, and in most cases the conversion noise is not known. In this work we utilize the unique opportunities of silicon photonics, cavity optomechanics and superconducting circuits to demonstrate a fully int…
Superconducting circuits offer a scalable platform for the construction of large-scale quantum networks where information can be encoded in multiple temporal modes of propagating microwaves. Characterization of such microwave signals with a method extendable to an arbitrary number of temporal modes with a single detector and demonstration of their phase-robust nature are of great interest. Here we show the on-demand generation and Wigner tomography of a microwave time-bin qubit with superconducting circuit quantum electrodynamics architecture. We perform the tomography with a single heterodyne…
Continuous-Variable Quantum State Tomography Enabled by Quantum Mirrors
Universidad de Concepción（康塞普西翁大学）；ANID - Millennium Science Iniciative Program - Millennium Institute for Research in Optics（智利国家研究与发展局-千年科学倡议计划-光学研究千年研究所）；Universidad de Chile（智利大学）
We report strong coupling between an ensemble of N impurity (P1) centres in diamond and microwave photons using a unique double post re-entrant cavity. The cavity is designed so that the magnetic component of the cavity field is spatially separated from the electric component and focused into the small volume in which the diamond sample is mounted. The novelty of the structure simultaneously allows high magnetic filling factor (38.4\%) and low frequencies necessary to interact, at low magnetic field, with transitions in diamond such as those in NV$^{-}$ and P1 centres. Coupling strength (or no…
When connecting a voltage-biased Josephson junction in series to several microwave cavities, a Cooper-pair current across the junction gives rise to a continuous emission of strongly correlated photons into the cavity modes. Tuning the bias voltage to the resonance where a single Cooper pair provides the energy to create an additional photon in each of the cavities, we demonstrate the entangling nature of these creation processes by simple witnesses in terms of experimentally accessible observables. To characterize the entanglement properties of the such created quantum states of light to the …
