Abstract
Thorough control of quantum measurement is key to the development of quantum information technologies. Many measurements are destructive, removing more information from the system than they obtain. Quantum nondemolition (QND) measurements allow repeated measurements that give the same eigenvalue^{1}. They could be used for several quantum information processing tasks such as error correction^{2}, preparation by measurement^{3} and oneway quantum computing^{4}. Achieving QND measurements of photons is especially challenging because the detector must be completely transparent to the photons while still acquiring information about them^{5,6}. Recent progress in manipulating microwave photons in superconducting circuits^{7,8,9} has increased demand for a QND detector that operates in the gigahertz frequency range. Here we demonstrate a QND detection scheme that measures the number of photons inside a highqualityfactor microwave cavity on a chip. This scheme maps a photon number, n, onto a qubit state in a singleshot by means of qubitâ€“photon logic gates. We verify the operation of the device for n=0 and 1 by analysing the average correlations of repeated measurements, and show that it is 90% QND. It differs from previously reported detectors^{5,8,9,10,11} because its sensitivity is strongly selective to chosen photon number states. This scheme could be used to monitor the state of a photonbased memory in a quantum computer.
Main
Several teams have engineered detectors that are sensitive to single microwave photons by strongly coupling atoms (or qubits) to highqualityfactor (highQ) cavities. This architecture, known as cavity quantum electrodynamics (cavity QED), can be used in various ways to detect photons. One destructive method measures quantum Rabi oscillations of an atom or qubit resonantly coupled to the cavity^{8,9,10}. The oscillation frequency is proportional to , where n is the number of photons in the cavity, so this method essentially measures the timedomain swap frequency.
Another method uses a dispersive interaction to map the photon number in the cavity onto the phase difference of a superposition of atomic states . Each photon number n corresponds to a different phase Ï†, so repeated Ramsey experiments^{5} can be used to estimate the phase and extract n. This method is QND, because it does not exchange energy between the atom and photon. However, as the phase cannot be measured in a single operation, it does not extract full information about a particular Fock state nã€‰ in a single interrogation. Nonetheless, using Rydberg atoms in cavity QED, remarkable experiments have shown quantum jumps of light and the collapse of the photon number by measurement^{5,12}.
Here we report a new method that implements a set of programmable controlledNOT (CNOT) operations between an nphoton Fock state and a qubit, asking the question â€˜are there exactly n photons in the cavity?â€™ A single interrogation consists of applying one such CNOT operation and readingout the resulting qubit state. To do this we use a quasidispersive qubitâ€“photon interaction that causes the qubit transition frequency to depend strongly on the number of photons in the cavity. Consequently, frequency control of a pulse implements a conditional Ï€ rotation on the qubitâ€”the qubit state is inverted if and only if there are n photons in the storage cavity. To ensure that this is QND, the qubit and storage cavity are adiabatically decoupled before carrying out a measurement of the qubit state.
To realize this method we extend circuitbased cavity QED (ref. 13) by coupling a single transmon qubit^{14,15} simultaneously to two cavities. In this scheme, the transmon qubit is used to interrogate the state of one cavity; then the transmon is measured with the other cavity. This separation of functions allows one cavity to be optimized for coherent storage of photons (highQ) and the other for fast qubit readout (lowQ). Related work by Leek et al. ^{16} realized a single transmon coupled to two modes of a single cavity, where the two modes were engineered to have very different quality factors. A schematic of the twocavity device is shown in Fig. 1a. The cavities are realized as Nb coplanar waveguide resonators with Î»/2 resonances at Ï‰_{s}/2Ï€=5.07 GHz and Ï‰_{m}/2Ï€=6.65 GHz, respectively. The cavities are engineered, by design of the capacitors C_{s} and C_{m}, to have very different decay rates (Îº_{s}/2Ï€=50 kHz and Îº_{m}/2Ï€=20 MHz) so that the qubit state can be measured several times per photon lifetime in the storage cavity. By having these cavities at different frequencies, the fast decay of the readout cavity does not adversely affect photons in the storage cavity. A transmon qubit is endcoupled to the two cavities, with finger capacitors controlling the individual coupling strengths (g_{s}/2Ï€=70 MHz and g_{m}/2Ï€=83 MHz). The usual shunt capacitor between the transmon islands is replaced with capacitors to the ground planes to reduce direct coupling between the cavities. Furthermore, a flux bias line^{17} allows fast, local control of the magnetic field near the transmon. This facilitates manipulations of the detunings Î”_{s}=Ï‰_{g,e}âˆ’Ï‰_{s} and Î”_{m}=Ï‰_{g,e}âˆ’Ï‰_{m} between the transmon and cavities, where we use the convention of labelling the transmon states from lowest to highest energy as (g,e,f,h,â€¦).
To achieve high photonnumber selectivity of the CNOT operations, there must be a large separation between the numberdependent qubit transition frequencies. To obtain this, we use small detunings (Î”_{s}/g_{s}<10) between the qubit and storage cavity. Figure 2 shows spectroscopy in this quasidispersive regime as a function of flux bias when the storage cavity is populated with a coherent state . A numerical energylevel calculation is overlaid, showing the positions of various transitions. We define as the photonnumberdependent transition frequency n,gã€‰â†’n,eã€‰. Other transitions, such as 2,gã€‰â†’0,hã€‰, are allowed because of the small detuning. Fortunately, we also see that the separation between and grows rapidly to order âˆ¼2 g_{s}=140 MHz as the qubit approaches the storage cavity.
To test the photon meter, we generate single photons in the storage cavity with an adiabatic protocol. Our method uses the avoided crossing between the 0,eã€‰ and 1,gã€‰ levels to convert a qubit excitation into a photon. The preparation of a photon begins with the qubit detuned below the storage cavity (Î”_{s}â‰ƒâˆ’3g_{s}), where we apply a Ï€pulse to create the state 0,eã€‰. We then adiabatically tune the qubit frequency through the avoided crossing with the storage cavity, leaving the system in the state 1,gã€‰. The sweep rate is limited by Landauâ€“Zener transitions that keep the system in 0,eã€‰. The preparation protocol changes the qubit frequency by 600 MHz in 50 ns, giving a spurious transition probability less than 0.1% (calculated with a multilevel numerical simulation). This protocol actually allows for the creation of arbitrary superpositions of 0,gã€‰ and 1,gã€‰ by changing the rotation angle of the initial pulse. For example, if we use a Ï€/2pulse, after the sweep the system ends up in the state , where Ï† is determined by the rotation axis of the Ï€/2pulse. One could also use a resonant swap scheme, which has been successfully used to create Fock states^{9} up to n=15ã€‰. The method used here has the advantage of being very robust to timing errors.
After the photon is prepared in the storage cavity, the qubit frequency is adjusted such that Î”_{s}/g_{s}â‰ƒ5. At this detuning, the separation between and is âˆ¼65 MHz. In Fig. 3a, we show pulsed spectroscopy at this detuning for several rotation angles of the initial preparation pulse. We observe wellresolved dips in the reflected phase of a pulsed signal sent at the measurement cavity frequency. The locations of these dips correspond to the qubit transition frequencies for n=0 () and n=1 (), and the relative heights match expectations from the different preparationpulse rotations (for example, a Ï€/2pulse results in equalheight signals).
To show selective driving of these transitions, we carry out Rabi experiments at and for the cases where we prepare 0,gã€‰ and 1,gã€‰. In each experiment we ensemble average measurements of the resulting qubit state after further decoupling the qubit from the storage cavity. For the 0,gã€‰ case (Fig. 3b) there is a largeamplitude oscillation when the drive is at (red, R_{0}(Î¸)) and almost no oscillation when the drive is at (blue, R_{1}(Î¸)). When we prepare 1,gã€‰ the situation is reversed (Fig. 3c); however, in this case the residual oscillation of R_{0}(Î¸) (red) is substantial because of small errors in the preparation of 1,gã€‰ associated with the initial rotation of the qubit and, more importantly, the âˆ¼10% probability of energy decay during the subsequent adiabatic sweep through the cavity.
The responses R_{i}(Î¸) are a result of driving and the far offresonant drive of , where . The crosstalk is seen in the small residual oscillation of R_{1}(Î¸) in Fig. 3b. In the Supplementary Information, we derive a method for extracting a selectivity and preparation fidelity from these data, giving a selectivity â‰¥95% for both interrogations and a preparation fidelity of ã€ˆn=1Ïˆã€‰^{2}â‰ˆ88%. These numbers were confirmed by doing equivalent experiments over a range of preparationpulse rotation angles between 0 and 2Ï€ (not shown).
If Ï€pulses are used in the interrogation step, measurement results of the average qubit state directly correlate with the probability of being in the states n=0ã€‰ or n=1ã€‰. Details of the scaling needed to do this transformation when the selectivity is <100% are presented in the Supplementary Information. These are the desired CNOT operations of the photon meter. If we now insert a variable delay before interrogating, we find that P_{0} (P_{1}), the probability of being in n=0ã€‰ (n=1ã€‰), decays exponentially towards 1 (0), as shown by the red (orange) trace in Fig. 4b. The decay constant of T_{1}â‰ƒ3.11Â±0.02 Î¼s agrees with the linewidth of the storage cavity, 1/Îº_{s}=1/(2Ï€ 50 kHz)=3.2Â±0.1 Î¼s, measured in a separate, lowpower reflection experiment.
Strong QND measurements are projective, such that if the measurement observable commutes with the Hamiltonian, the system will remain in an eigenstate of both operators between measurements. Consequently, comparing the results of successive interrogations provides a mechanism to test whether a particular protocol causes perturbations on the system beyond the expected projection. Here, we compare only ensemble average results, because the singleshot qubit readout fidelity for the device is âˆ¼55%. This is sufficient to reveal processes that change the photon number, and technical improvements to interrogation speed or qubit readout fidelity should allow for realtime monitoring of the photon state.
The protocol cannot be repeated immediately, though, because the first interrogation may leave the qubit in the excited state. To circumvent this problem, we use the fast decay rate of the measurement cavity to cause the qubit to spontaneously decay into the 50 Î© environment. The â€˜resetâ€™ protocol brings the qubit into resonance with the measurement cavity for a time, Ï„_{reset}=50 ns, which is sufficient to reset the qubit with probability âˆ¼98%. The procedure is described in detail in ref. 18.
After resetting the qubit, we can interrogate a second time. The full protocol for a repeated interrogation sequence is shown in Fig. 4a. The combination of a CNOT_{0} (CNOT_{1}), a qubit measurement and a qubit reset define an interrogation process I_{0} (I_{1}). Data for the four possible combinations of interrogating n=0ã€‰ and n=1ã€‰ are shown in Fig. 4b as a function of delay between the first and second interrogations. The data are ensemble averaged over all results from the first interrogation, so we do not observe projection onto number states. Instead, we again observe exponential decay, where the result of the second measurement is essentially indistinguishable from the first, indicating that the interrogation is highly QND.
Small deviations between first and second interrogations stem from finite photon lifetime in the storage cavity and nonQND processes that cause transitions to other photon numbers (Fig. 4c). A single interrogation process takes âˆ¼550 ns, which accounts for the time gap between the single and repeated interrogation points in Fig. 4b. Recording the second interrogation results for different delays allows us to subtract the effect of photon T_{1} and calculate the transition probabilities for the I_{0} and I_{1} processes^{19}. In principle, I_{0} and I_{1} can cause transitions to photon numbers outside the nâˆˆ{0,1} manifold; however, the absence of statistically significant deviations from P_{0}+P_{1}=1 suggests that any such effects are negligible. Instead, we consider only transitions from n=1ã€‰â†’n=0ã€‰ or n=0ã€‰â†’n=1ã€‰, characterized by the probabilities Î³_{i} and Î´_{i}, respectively, where i references the interrogation process I_{0} or I_{1}. We observe Î³_{0}(Î³_{1})=1(10)Â±3% and Î´_{0}(Î´_{1})=7(3)Â±3%, demonstrating that this protocol is highly QND.
The protocol presented here is a fast and highly QND measurement of single photons, which we believe can be extended to detect higher photon numbers. It should be possible to demonstrate the projective nature of the interrogation and create highly nonclassical states of light by postselection, and eventually with higher fidelity readout it should be possible to observe quantum jumps of light in a circuit.
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Acknowledgements
We thank J.Â Chow and M.Â Devoret for helpful discussions. This work was supported by NSF grants DMR0603369 and PHY0653073. J.M.G. was supported by a CIFAR Junior Fellowship, MITACS, MRI and NSERC. L.F. was partially supported by CNRIstituto diÂ Cibernetica.
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B.R.J. and M.D.R. carried out measurements and data analysis. A.A.H., D.I.S. and L.D. provided further experimental contributions. L.S.B., E.G. and J.M.G. provided theoretical contributions. Devices were fabricated by B.R.J., L.F. and L.D. The experiment was designed by B.R.J., A.A.H., D.I.S., J.M.G., S.M.G. and R.J.S.
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Johnson, B., Reed, M., Houck, A. et al. Quantum nondemolition detection of single microwave photons in a circuit. Nature Phys 6, 663â€“667 (2010). https://doi.org/10.1038/nphys1710
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DOI: https://doi.org/10.1038/nphys1710
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