Abstract
The Schrodinger’s cat thought experiment highlights the counterintuitive concept of entanglement in macroscopically distinguishable systems. The hallmark of entanglement is the detection of strong correlations between systems, most starkly demonstrated by the violation of a Bell inequality. No violation of a Bell inequality has been observed for a system entangled with a superposition of coherent states, known as a cat state. Here we use the Clauser–Horne–Shimony–Holt formulation of a Bell test to characterize entanglement between an artificial atom and a cat state, or a Bellcat. Using superconducting circuits with highfidelity measurements and realtime feedback, we detect correlations that surpass the classical maximum of the Bell inequality. We investigate the influence of decoherence with states up to 16 photons in size and characterize the system by introducing joint Wigner tomography. Such techniques demonstrate that information stored in superpositions of coherent states can be extracted efficiently, a crucial requirement for quantum computing with resonators.
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Introduction
Quantum information processing necessitates the creation and detection of complex entangled states. Many physical implementations aim to encode quantum information into large registers of entangled twolevel systems, or qubits. Although originally proposed to investigate local hidden variable theory^{1}, a Bell inequality can be used to benchmark the ability to entangle and extract information from an entangled twoqubit system^{2}. Using the Clauser–Horne–Shimony–Holt (CHSH) variant^{3} of the Bell test, this violation has been demonstrated with photons^{4,5}, atoms^{6,7}, solidstate spins^{8} and artificial atoms in superconducting circuits^{9,10}. However, quantum computation necessitates the entanglement of large numbers of qubits. To perform tasks such as quantum error correction, a physical implementation must be capable of highfidelity multiqubit entanglement, as well as the efficient detection multiqubit observables. For these larger, more distinguishable states, creating and preserving entanglement becomes increasingly difficult due to the rapid onset of decoherence^{11}. Alternative encoding schemes that use coherent state superpositions, known as cat states^{12}, take advantage of a cavity resonators much larger Hilbert space, as compared with that of a twolevel system. This architecture allows redundant qubit encodings that can simplify the operations needed to initialize, manipulate and measure the encoded information^{13,14,15}. For such a system to be viable as a quantum computing platform, efficient measurement of such encoded qubit observables must be possible. Using a circuit quantum electrodynamics architecture^{16}, we show efficient, highfidelity measurements of an encoded cat state qubit and demonstrate this technology by detecting a violation of the CHSH Bell inequality between the encoded cat state qubit and a superconducting transmon qubit^{17}. Furthermore, by the use of coherent states in this composite system, we can investigate the effects of decoherence by continuously varying the size of prepared entangled states^{18,19}, something unachievable with discrete systems. These techniques provide an important set of analytical tools for quantum systems composed of entangled qubits and resonators^{14,19,20,21,22,23}, and demonstrate that one can exploit coherent state superpositions in resonators without sacrificing measurement efficiency.
A resonator state can be completely described by direct measurements in the continuous variable basis with the cavity state Wigner function^{24}. We extend this concept to express an entangled qubit–cavity state in what we call the joint Wigner representation. We construct this representation by performing a sequence of two quantum nondemolition measurements (Fig. 1), where a qubit state measurement is correlated with a subsequent cavity state measurement. However, complete cavity state tomography need not be required, and in fact many fewer measurements could be used to characterize a state when operating in a smaller, encoded subspace. By choosing an encoding scheme where states of a quantum bit are mapped onto a superposition of coherent states and , we can condense the joint Wigner representation down to just 16 correlations, equivalent to a twoqubit measurement set. Using direct fidelity estimation (DFE)^{25,26} and CHSH Bell witnesses^{27,28} within this logical basis, we investigate this systems susceptibility to decoherence by continuously increasing the cat state amplitude β. We measure a range in which correlations surpass the Bell inequality threshold and observe its reduction due to decoherence, benchmarking the efficiency of our encoding and detection schemes with cat state qubits.
Results
Creating the Bellcat state
This experiment utilizes a circuit quantum electrodynamics architecture^{16,17} consisting of two waveguide cavities coupled to a single transmon qubit^{22,29}. One longlived cavity (relaxation time τ_{s}=55 μms) is used for quantum information storage, while the other cavity, with fast field decay (relaxation time τ_{r}=30 ns), is used to realize repeated measurements. A transmon qubit (relaxation and decoherence times T_{1}, T_{2}≈10 μs) is coupled to both cavity modes and mediates entanglement and measurement of the storage cavity state. All modes have transition frequencies between 5–8 GHz and are offresonantly coupled. The storage cavity and qubit mode are well described by the dispersive Hamiltonian:
where a is the storage cavity ladder operator, is the excited state qubit projector, ω_{s} and ω_{q} are the storage cavity and qubit transition frequencies, and χ is the dispersive interaction strength between the two modes (1.4 MHz). This interaction creates a shift in the transition frequency of one mode dependent on the other’s excitation number, resulting in qubit–cavity entanglement^{30}. As described in Fig. 1, the system is first prepared in a product state , where are the ground and excited states of the qubit, and is a coherent state of the cavity mode. Under the dispersive interaction, we allow the system to evolve for a time , creating the entangled state:
which we call a Bellcat state^{22,29,30}, mirroring the form of a twoqubit Bell state (for example, ).
Correlating sequential highfidelity measurements of the qubit and cavity allows state tomography of this composite system. We use a Josephson bifurcation amplifier^{31} in a doublepumped configuration in combination^{32,33} with a dispersive readout to perform repeated quantum nondemolition measurements with qubit detection fidelity of 98.0% at a minimum of 800 ns between measurements. With this sequence of two measurements, we characterize the efficacy of our entangling scheme and efficiency of measuring qubit–cavity observables with joint Wigner tomography, DFE and a CHSH inequality. The results of these tests illustrate our ability to recast the state encoded in the cavity as one that has a small, simple set of observables that directly mirrors that of the physical qubit.
Joint Wigner tomography
The first measurement detects the qubit along one of its basis vectors {X, Y, Z}. This value is recorded and the qubit is reset to using realtime feedback. The displaced photon number parity observable P_{α} of the cavity is subsequently mapped onto the qubit using Ramsey interferometry^{24} before a second qubit state detection. The cavity observable , where D_{α} is the displacement operator and P the photon number parity operator, is detected with 95.5% fidelity. The Wigner function is constructed from an ensemble of such measurements with different displacement amplitudes α. The correlations between the qubit and cavity states make up what we refer to as the joint Wigner functions:
where σ_{i} is an observable in the qubit Pauli set {I, X, Y, Z}. These four distributions are a complete representation of the combined qubit–cavity quantum state (Fig. 2). While other representations exist for similar systems^{34,35,36,37}, W_{i}(α) is directly measured with this detection scheme and does not require a density matrix reconstruction. By an overlap integral (Supplementary Note 4), we determine the fidelity to a target state , where are the joint Wigner functions of the ideal state, and W_{i}(α) are the measured joint Wigner functions (normalized), yielding a state fidelity for a displacement amplitude . This amplitude was chosen to ensure orthogonality between logical states with minimal tradeoff due to photon loss. Furthermore, the efficiency of our detection scheme can be quantified by the visibility of the unnormalized joint Wigner measurements . Visibility is primarily limited by measurement fidelity and qubit decoherence between detection events (Supplementary Note 5). The parameters and represent critical benchmarks for creating and retrieving information from entangled states.
Direct fidelity estimation
The number of measurement settings required to perform cavity state tomography can be resource intensive. Restricting to an encoded qubit subspace, only four values of the cavity Wigner function W(α) are required to reconstruct the state, known as a DFE^{25,26}. For large cat states , the encoded state observables map to cavity observables as:
where {I_{c}, X_{c}, Y_{c}, Z_{c}} form the Pauli set for the encoded qubit state in the cavity (Supplementary Note 8). Cuts in the joint Wigner function (Fig. 3) show these observables and their correlations to the qubit as a function of cat state size. As the superposition state is made larger, interference fringe oscillations increase, while fringe amplitude decreases due to photon loss. For a state with , we estimate a direct fidelity putting a fidelity bound on the target state with no corrections for visibility. This estimate is related to the benchmarks reported above and far surpasses the 50% threshold for a classically correlated state. This indicates both highfidelity state preparation and measurement, and demonstrates that strong correlations are directly detectable using joint Wigner tomography.
Bell inequality measurements
To place a stricter bound on observed entanglement, we perform a Bell test on the measured state. Although proposed to investigate local hidden variable theory, the Bell test here serves to benchmark the performance of a system that creates and measures entangled states^{8,9,10}. Bell tests using homodyne measurements have been proposed^{38,39}; however, here we choose the CHSH Bell test which states that the sum of four classical correlations will be bounded such that:
where, in this experiment, A and B are two qubit observables and A_{c} and B_{c} are two cavity observables. We perform two Bell tests (Fig. 4) with correlations taken shot by shot with no post selection or compensation for detector inefficiencies. In the first, we take observables X(θ)=Xcos(θ/2)+Zsin(θ/2), Z(θ)=Zcos(θ/2)−Xsin(θ/2), X_{c}, Z_{c} and sweep both qubit detector angle θ (Supplementary Note 11) and cat state amplitude β. We observe a Bell signal with a maximal value at for β=1. We witness a Bell signal surpassing bounded values up to cat states of size photons^{19,29}.
Measurements along Z_{c} require assumptions on the symmetry of the prepared state (Supplementary Note 8); we can instead employ an alternative Bell test. Using a scheme similar to ref. 28, and choosing observables , where α is a displacement amplitude corresponding to a rotation of the encoded cavity state detector (Supplementary Note 11), we observe a maximal value for β=1. A lower Bell signal is observed in the second test due to its greater sensitivity to photon loss, yet in both tests two regimes are evident. For small cat state amplitudes, the initial Bell signal is limited by the nonorthogonality of the coherent state superpositions (Supplementary Note 7), while for large displacements the system’s sensitivity to photon loss results in a reduction of the Bell signal. Larger, more distinguishable states quickly devolve into a classical mixture due to the onset of decoherence, corresponding to the resolution of Schrödinger’s thought experiment. However, for intermediate cat state sizes, we observe Bell signals surpassing classical predictions larger than statistical uncertainties in both tests.
Discussion
In this letter, we have demonstrated the efficient detection of an artificial atom and a cat state in a cavity mode. We determine the entangled state using sequential detection with highfidelity state measurement and realtime feedback on the quantum state. We benchmark the capabilities of this detection scheme with DFE and Bell test witnesses, which both reveal nonclassical correlations of our system. Besides characterizing the high degree of entanglement in our Bellcat, the tests detailed above also demonstrate that simple encoding techniques allow for the efficient extraction of information from states stored in a cavity, illustrating the viability of measuring redundantly encoded states in multilevel systems^{13,14}. Furthermore, this implementation provides a resource for quantum state tomography and quantum process tomography of continuous variable systems and creates a platform for measurementbased quantum computation and quantum error correction using superconducting cavity resonators^{15}. Finally, these features can extend to multicavity systems^{27}, which will require entanglement detection between continuous variable degrees of freedom and entanglement distribution of complex oscillator states.
Methods
Measurement setup
Experiments are performed in a cryogenfree dilution refrigerator at a base temperature of ∼10 mK. Our output signal amplification chain consists of two stages. A Josephson bifurcation amplifier^{31} operating in a doublepumping configuration^{32,33} serves as the first stage, which is followed by a high electron mobility transistor amplifier.
Fabrication techniques of the transmon qubit and the design of storage and readout resonators follow the methods described in ref. 29. The refrigerator wiring (Fig. 5), including the filters and attenuators used, are similar to that of ref. 22, but with the addition of a feedback system, the details of which are discussed in a following section.
Qubit–cavity parameters
The twocavity, singlequbit system is well described by the approximate dispersive Hamiltonian:
Where ω_{s}, ω_{r} and ω_{q} are the storage, readout and qubit transition frequencies, a_{s}, a_{r} and b are the associated ladder operators, and K and χ are the modal anharmonicities and dispersive shifts, respectively. Supplementary Table 1 details the Hamiltonian parameters of our system. The resonant frequency of the readout resonator ω_{r}/2π is determined by transmission spectroscopy. The qubit frequency ω_{s}/2π and storage cavity frequencies ω_{q}/2π are found using twotone spectroscopy.
Qubit anharmonicity K_{q} is measured using twotone spectroscopy to observe the 0–2 twophoton transition^{17}. Storage cavity anharmonicity K_{s} is determined by displacing the cavity with a coherent state and observing its time evolution with Wigner tomography. The resulting dynamics are characterized by state reconstruction and K_{s} is observed by the state’s quadratic dependence of phase on photon number. Finally, we predict the readout cavity anharmonicity K_{r} using its approximate dependence on the measured values of K_{q} and the qubitreadout dispersive shift χ_{qr} (ref. 40).
The dispersive shift between the qubit and the readout resonator χ_{qr} is found by taking the difference in frequency between the readout resonance when the qubit is in the ground and excited state. The dispersive shift between the qubit and the storage resonator χ_{qs} is found using two methods: photon numberdependent qubit spectroscopy^{41}, and observing qubit state revival using Ramsey interferometry^{29}. Finally, χ_{rs} is predicted using its approximate relationship between K_{s} and K_{r} (ref. 40).
Lifetimes and thermal populations
The lifetime of the storage cavity is determined by displacing to a coherent state, waiting a variable length of time, and then applying a qubit rotation conditioned on zero photons in the storage cavity. This allows a measurement of the timedependent overlap of the cavity state with its ground state dependent on time. The lifetime of the readout cavity is found from its line width. The thermal population of the qubit is determined from a histogram of one million singleshot measurements of the qubit thermal state, where the signaltonoise ratio provided by the Josephson bifurcation amplifier allows discrimination between and all states not . The thermal population of the storage cavity is found by taking the difference between parity measurements of the thermal and vacuum states of the cavity. A vacuum state is prepared by first performing two parity measurements on the thermal state and then postselecting such that all results give even parity, projecting the thermal state onto . Finally, the known thermal population of the readout cavity is bounded by the dephasing rate Γ_{φ} of the qubit: , where is the readout cavity’s thermal occupation and κ is the readout singlephoton decay rate^{42}. Coherence properties are summarized in Supplementary Table 2.
Measurement fidelities
We define singleshot measurement fidelity as , where P(gg) and P(ee) are the probabilities to get , knowing that we start with . The state is prepared through purification of the qubit thermal state with realtime feedback (see the following section). Given a preparation of , we have a 98.5% chance of measuring again (P(gg)=0.985). Likewise, we find P(ee)=0.975 by preparing and rotating the state to . This gives a singleshot measurement fidelity of F_{q}=98%. We find our cavity parity measurement fidelity by purifying the storage cavity thermal state into then performing one of two kinds of parity measurement (Supplementary Figs 7 and 8; Supplementary Table 3). We report a parity measurement fidelity for n=0 photons as , where P(gE_{1}) and (P(eE_{2})) are the probabilities to measure , given that the parity is even for each of the two measurement settings. We expect F_{c} to decrease with increasing numbers of photons in the cavity due to singlephoton loss during the measurement sequence.
Directly from these readout fidelities, the estimated visibility^{43} for correlated observables . This allows us to predict the maximum Bell violation possible given only measurement inefficiencies . In practice, is directly related to the contrast of the joint Wigner function (Supplementary Note 5), which we measure to be 85%. This discrepancy is due to qubit decoherence, which is studied further in Supplementary Note 2, and puts a more conservative estimate for the maximum Bell violation achievable: .
I/O control parameters
As shown in Fig. 5, we employ a fieldprogrammable gate array (FPGA) to implement an active feedback scheme. We use an X61000M board from Innovative Integration that contains two 1 GS/s analoguetodigital converters, two 1 GS/s digitaltoanalogue converter channels and digital inputs/outputs all controlled by a Xilinx VIRTEX6 FPGA loaded with custom logic. We synchronize two such boards in a master/slave configuration to have IQ control of both the qubit/storage cavity. IQ control over the readout cavity is performed with a Tektronix AWG, which is triggered by the master board. The readout and reference signals are routed to the analoguetodigital converters on the master board, whereafter the FPGA demodulates the signal and decides whether the qubit is in or . The feedback latency of the FPGA logic (last in, first out LIFO) is 320 ns. Additional delay for active feedback includes cable delay (∼100 ns) and readout pulse length with resonator decay time (320 ns). Thus, in total, the qubit waits τ_{wait} ∼740 ns between the time at which photons first enter the readout resonator and the time at which the feedback pulse resets the qubit.
Implementations of feedback
Feedback is used three times during a single iteration of the experiment. Before the state preparation (Fig. 6), we purify the qubit state to by measuring the qubit and applying a rotation if measured in . We succeed in preparing with a probability of 99%. Second, when performing qubit tomography we reset the qubit to if it is measured to be in . Since we must wait τ_{wait} before feedback can be applied, the cavity state will acquire an additional phase χ_{qs}τ_{wait} if the qubit is in . In this case, in addition to resetting the qubit, the FPGA applies an equivalent phase shift on the subsequent Wigner tomography pulse. This feedback implementation does not close the ‘locality’ loophole for a CHSH Bell test and therefore cannot be used to test local realism.
Quantum measurement back action
The sequential measurement protocol allows us to observe the result of quantum measurement back action of the qubit on the cavity state. We prepare the system in a Bellcat state as in equation (2) and measure along one of the three qubit axes M_{q}∈{X, Y, Z}. For each measurement, we observe one of two possible outcomes of the projected cavity state :
See Fig. 7 for each of these projective measurements on the Bellcat state . The method of using strong projective measurements for the create of cat states has been demonstrated in previous works^{19}. A second example of quantum measurement back action using an entangled Fock state can be found in Supplementary Fig. 9.
Additional information
How to cite this article: Vlastakis, B. et al. Characterizing entanglement of an artificial atom and a cavity cat state with Bell’s inequality. Nat. Commun. 6:8970 doi: 10.1038/ncomms9970 (2015).
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Acknowledgements
We thank R. Heeres, W. Pfaff and A. Narla for discussions. This research was supported by the National Science Foundation (NSF; PHY1309996), the Multidisciplinary University Research Initiatives program through the Air Force Office of Scientific Research (FA95501410052) and the US Army Research Office (W911NF1410011). Facilities use was supported by the Yale Institute for Nanoscience and Quantum Engineering and the NSF (MRSECDMR 1119826).
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B.V. and A.P. performed the experiment and analysed the data. N.O. and Y.L. designed the feedback architecture. L.S. provided further experimental implementation support. K.S. and M.H. provided the Josephson Parametric Converter technology under the supervision of M.H.D. Z.L., M.M. and L.J. provided theoretical support. J.B. and L.F. fabricated the transmon qubit. R.J.S. supervised the project. B.V., A.P., L.F. and R.J.S. wrote the manuscript with feedback from all authors.
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Supplementary Figures 19, Supplementary Tables 13, Supplementary Notes 115 and Supplementary References (PDF 856 kb)
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Vlastakis, B., Petrenko, A., Ofek, N. et al. Characterizing entanglement of an artificial atom and a cavity cat state with Bell’s inequality. Nat Commun 6, 8970 (2015). https://doi.org/10.1038/ncomms9970
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DOI: https://doi.org/10.1038/ncomms9970
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