Quantum rifling: protecting a qubit from measurement back-action
Quantum mechanics postulates that measuring the qubit’s wave function results in its collapse, with the recorded discrete outcome designating the particular eigenstate the qubit collapsed into. We show this picture breaks down when the qubit is strongly driven during measurement. More specifically, for a fast evolving qubit the measurement returns the time-averaged expectation value of the measurement operator, erasing information about the initial state of the qubit, while completely suppressing the measurement back-action. We call this regime “quantum rifling”, as the fast spinning of the Bloch vector protects it from deflection into either of its two eigenstates. We study this phenomenon with two superconducting qubits coupled to the same probe field and demonstrate that quantum rifling allows us to measure either one of the two qubits on demand while protecting the state of the other from measurement back-action. Our results allow for the implementation of selective read out multiplexing of several qubits, contributing to efficient scaling up of quantum processors for future quantum technologies.
The Stern-Gerlach experiment, originally conducted to demonstrate quantization in atomic-scale systemsGerlach and Stern (1922), is the prototypical example of a quantum measurement with a linear detector: an electron (or qubit) flying through a magnetic field is deflected from its straight path either up or down, with probabilities dependent on the qubit’s initial state (see Figure 1a). The measurement projects the state of the qubit onto either of its two eigenstates, resulting in two possible values for the spin: .
This picture becomes more interesting when the qubit is externally driven with Rabi frequency during the measurement, leading to competition between the state evolution and the measurement projection. Such a scenario has been thoroughly studied both theoretically Milburn (1988); Blais et al. (2004); Gambetta et al. (2008) and experimentally Hatridge et al. (2013); Ficheux et al. (2018) in the strong measurement regime , where is the measurement rate at which information is extracted from the qubit. This regime is commonly described by the Quantum Zeno effectItano et al. (1990): a strong quantum measurement freezes the qubit’s state, with occasional transitions occurring as sudden quantum jumpsSchulman (1998); Streed et al. (2006); Vijay et al. (2011); Slichter et al. (2016) with rate .
The regime of strong driving , referred to as the sub-Zeno limit, has attracted attention in the context of continuous weak measurementsStace and Barrett (2004). When the probe’s bandwidth exceeds the Rabi frequency, , signatures of coherent Rabi oscillations appear in the detector signal Goan and Milburn (2001); Korotkov and Averin (2001); Korotkov (2001); Gurvitz et al. (2003). It has been shown that the back-action introduced by the measurement imposes a fundamental limit on the detection of oscillations and can be used to determine the quantum efficiency of the detectorKorotkov and Averin (2001); Jordan and Büttiker (2005); Jordan and Korotkov (2006), or even to test the Leggett-Garg inequalityJordan et al. (2006); Palacios-Laloy et al. (2010). The opposite limit , where the Rabi frequency exceeds the bandwidth, is suitably described by the average Hamiltonian theory Maricq (1982). This regime however, has not yet been investigated in the context of continuous qubit measurement neither theoretically nor experimentally.
In this Letter, we study the measurement of a continuously driven qubit in the regime where the Rabi frequency dominates all the other relevant parameters: . First, we show that when the probe’s bandwidth is not sufficient to follow the qubit’s state, the probe signal reveals only the expectation value of the time-averaged measurement operator , leading to the erasure of any information contained in the probe about the state of the qubit and thus canceling the measurement back-action on the qubit. In the language of the Stern-Gerlach experiment, the fast rotation of the spin allows the electron to fly through the measurement apparatus in a straight line without experiencing a force. Thus we call this effect quantum rifling, in analogy to the rifling of bullets, which stabilizes the trajectory of the projectile (see Figure 1b). We then investigate the driving threshold to achieve rifling by measuring the Rabi decay rate of a probed qubit for different probe field amplitudes. Finally, using tomographic reconstruction of the qubit’s state, we demonstrate read out multiplexing of two qubits coupled to the same measurement apparatus: quantum rifling is used to suppress the measurement back-action on one of the qubits on demand, while still extracting full information about the state of another qubit.
We use a typical circuit quantum electrodynamics system comprising of two superconducting transmon qubitsKoch et al. (2007) coupled dispersively to a microwave resonatorBlais et al. (2004). The driven stationary microwave mode passing through the resonator acts as our measurement probe: its interaction with the qubit leads to a dispersive shift of the resonator frequency depending on the state of the qubit. Rifling of the qubit state is achieved by applying a resonant Rabi drive to a charge line coupled directly to the qubit. For weak Rabi driving the resonator transmission measurement returns two peaks weighted by the corresponding populations of the ground and excited states of the qubit (see Figure 1c). When the driving strength reaches a threshold, the transmission spectrum yields a single peak in the middle analogous to a straight line for a spin in the Stern-Gerlach apparatus.
Figure 2a shows the transmission spectroscopy of the resonator when Qubit 1 is continuously driven on resonance. The resonator probe amplitude is adjusted to yield a mean photon number of . By changing the power of the qubit drive we are able to identify different characteristic regimes of the measurement. For very low drive power the qubit remains in its ground state and only a single transmission peak is visible at . As the Rabi drive becomes sufficient to excite the qubit, a second peak appears at in the cavity spectrum. The two peaks reach equal height when the qubit is saturated by the drive and the populations of the qubit in its ground and excited state becomes equal.
When increases further the two cavity peaks split, and the outer diverging peaks vanish with increasing drive. The inner peaks converge to form a single peak at the average frequency , which we identify as the onset of the quantum rifling regime. When the Rabi drive becomes comparable to the anharmonicity of the transmon, the central cavity peak splits again as we populate higher levels, setting an upper bound on the drive for rifling (see Supplementary information Sec. II.).
The system can be described by the Jaynes-Cummings Hamiltonian transformed into a doubly rotating frame at both the qubit and probe drive frequencies respectively:
where , and are creation and annihilation operators of resonator excitation modes, are Pauli matrices acting on the qubit, is the probe amplitude, and we assume resonant driving with the qubit’s ground to excited state transition frequency, . Taking into account decoherence and losses for such a systemIthier et al. (2005), the full time-evolution is described by the master equation
where , MHz is the cavity decay rate and , are the qubit relaxation and pure dephasing times, respectively.
We compute both numericallyJohansson et al. (2013) and analytically the steady-state of the resonator probe amplitude by solving Eq. (2) (as well as its extension to a three level transmon) in the low photon number limit , plotting the result in Figure 2b-c. The additional splitting of the central resonance peak around 100 MHz, where the Rabi drive frequency is comparable to the qubit anharmonicity, is well accounted for by the multi-level model.
When truncated to the single cavity photon subspace, the diagonalization of the Hamiltonian (1) for leads to four eigenstates in the dressed-state pictureGolter et al. (2014); Laucht et al. (2016), each a superposition of both atom and resonator states. The unnormalized expressions are:
where are the ground- and excited states of the qubit and are resonator states with photons. The corresponding eigenenergies in the rotating frame at can be found as and .
The four cavity transmission peaks correspond to the four transitions between eigenstates with differing photon parity (see Figure 2b dashed lines). In the limit of , yielding two degenerate transitions at frequencies . The splitting of the peaks is proportional to , which becomes visible when their splitting exceeds the linewidth MHz. In the opposite limit of strong driving , thus . The matrix elements corresponding to the outer transitions at vanish for large , making the outer peaks disappear. The inner transitions converge in energy, merging into the central cavity peak once their separation is too small to still resolve two peaks , leading to the condition for the quantum rifling regime MHz. The emergence of a single peak can also be understood by observing that in the strong driving limit the dressed eigenstates are equal superposition of qubit ground and excited states, leading to a vanishing dispersive shift of the cavity and inability to obtain any information about the qubit. For larger cavity probe powers, the evolution of the transmission peaks remains qualitatively the same, with less pronounced side peaks and the critical drive amplitude shifting to higher drive frequencies (see Supplementary information Sec III. for details).
We have also verified that the effect of two resonator transmission peaks merging into one can also be induced by fast incoherent qubit dynamics (see Supplementary information Sec. IV.). As for a qubit interacting with a heat bath at different temperatures such that in the steady state, simulations show that the position in frequency of the central peak shifts proportionally to the asymmetry in population between the ground and excited state (see Supplementary information Sec. II. B1).
The decay rates of Rabi oscillations with and without the simultaneous presence of a resonator probe tone are presented in Figure 3, varying the Rabi frequency for each specific probe power.
For a weak Rabi drive , there is a clear degradation of the Rabi coherence time caused by the measurement: the probe extracts information from the qubit leading to its dephasing. For strong Rabi drive , the qubit coherence times are comparable to the standard Rabi decay time measured without applying a cavity pulse simultaneously with the Rabi drive. Driving the cavity with more photons leads to a stronger suppression of the coherence time for small Rabi frequencies, consistent with the measurement rate being proportional to the cavity’s photon number population. The threshold Rabi drive frequency at which converges to the standard Rabi coherence time of the qubit is, however, independent of the cavity tone strength for low photon population (). This is in agreement with the observed threshold drive required for the emergence of a single cavity peak in continuous wave spectroscopy (see Figure 2 and Supplementary information Sec. III.).
We also numerically simulate such rifled Rabi oscillations for Qubit 1 in the time domain and plot their decay rate in Figure 3b. Note that only a single fit parameter was used to scale the curve with lowest drive strength.
As shown above, strong Rabi driving of a qubit leads to the emergence of a single resonator transmission peak independent of the qubit state. This peak can then experience state-dependent dispersive shifts due to other qubits coupled to the same resonator. Hence rifling allows measurements on other qubits, while keeping the rifled qubit in superposition.
To demonstrate this multiplexing capability, we perform a two-qubit algorithm with Qubit 1 and 2 coupled to the same readout resonator (see Figure 4). First we prepare a full superposition of the four basis states by applying a rotation to both qubits. We then rifle Qubit 1 for 1142 ns, while performing tomography on Qubit 2, followed by tomography on Qubit 1. We measure the density matrix of Qubit 1 in full superposition with 92.8% fidelity (corrected for qubit decoherence) confirming the qubit remained in superposition following the first read out (Figure 4a). Conversely, omitting the rifling pulse (Figure 4b) leads to vanishing non-diagonal terms in the density matrix and therefore to the collapse of coherence of Qubit 1, induced by the read out of Qubit 2.
It is worth noting that following the initial pulse, the qubit can be rotated around two different axes of the Bloch sphere: either around the x-axis, inducing full rotations around the Bloch sphere, or around the y-axis, effectively spin-locking the qubitYan et al. (2013). We report that in both cases the coherence is preserved (see Supplementary information Sec. V.). We also performed rifling of Qubit 2 while extracting information from Qubit 1 with similar results (see Supplementary information Sec. VI.).
Other protocols, such as stroboscopicSuh et al. (2014) or engineered longitudinal couplingTouzard et al. (2019), have been devised previously to suppress measurement back-action on qubits. Contrary to quantum rifling, however, these methods require additional time-dependent pump drives beyond the Rabi drive and thus offer a less practical implementation. Interestingly, the merging of the two cavity peaks to a single peak with increasing qubit modulation is reminiscent of motional averaging of a linewidth of a moleculeLi et al. (2013). In analogy, one could view our experiment as motional narrowing of the resonator, where the qubit acts as noisy environment shifting the resonator’s resonance frequency. Likewise, the absence of measurement dephasing for a fast rotating qubit bears similarities with well-known dynamical decoupling schemesBylander et al. (2011), where noise during successive periods of free evolution interferes destructively at specific moments in time. In contrast, quantum rifling allows controlled decoupling of individual qubits from their measurement apparatus for all times during rifling.
In conclusion, our results reveal an intuitive picture for the regime of a strongly driven, continuously measured qubit, where the Rabi frequency exceeds both the measurement rate and the meter bandwidth. In this regime, the resonator photons are not able to extract information about the qubit’s state, leading to the measurement revealing only the time-averaged population of the qubit and importantly imposing no back-action onto the qubit. We have also demonstrated that a strong Rabi drive can be utilized as a useful experimental knob to turn the measurement back-action between a qubit and its probe on and off, without affecting the ability to measure other qubits probed simultaneously by the same field. This capability allows for many qubits to be connected to an individual detector, thus facilitating scalability in architectures where connecting individual detectors to every qubit might be technically challengingHeinsoo et al. (2018).
The authors thank Gerard Milburn, Tom Stace and Sergey N. Shevchenko for fruitful discussions. This research was supported by the Australian Research Council Centre of Excellence for Engineered Quantum Systems (EQUS, CE170100009), the ARC Future Fellowship FT140100338 and by the Swiss National Science Foundation through the NCCR QSIT.
The sample consists of four 2D transmon qubits pair-wise coupled to four superconducting co-planar waveguide in a circuit-QED arrangement. Only two qubits and one resonator were used for the experiment. The sample substrate is sapphire and the junctions were fabricated using a double-angle aluminum shadow evaporation technique. Qubit 1 has maximum ground to excited state transition frequency of GHz and the relaxation times s s. For Qubit 2, GHz and relaxation times s 3.39 s. Both qubits are operated at their maximum transition frequencies.
Measurement fidelities were calculated using the equation .
The description of the electronics used for signal generation and detection can be found in the Supplementary information Sec I.
D.S. and A.F. designed the experiment. D.S. and A.G.F. performed the experiment. T.J. and M.J. contributed to the experiment. C.M. carried out the theoretical analysis. T.J. and A.G.F. performed the numerical simulations. D.S., A.F., C.M. and A.G.F. co-wrote the manuscript.
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