Time-Resolved Measurement of a Charge Qubit

Time-Resolved Measurement of a Charge Qubit Georg M. Reuther, David Zueco, Peter Hänggi, and Sigmund Kohler arXiv:0806.2786v2 [cond-mat.mes-hall] 27 Oct 2008 Institut für Physik, Universität Augsburg, Universitätsstraße 1, D-86135 Augsburg, Germany (Dated: October 27, 2008) We propose a scheme for monitoring coherent quantum dynamics with good time-resolution and low backaction, which relies on the response of the considered quantum system to high-frequency ac driving. An approximate analytical solution of the corresponding quantum master equation reveals that the phase of an outgoing signal, which can directly be measured in an experiment with lock-in technique, is proportional to the expectation value of a particular system observable. This result is corroborated by the numerical solution of the master equation for a charge qubit realized with a Cooper-pair box, where we focus on monitoring coherent oscillations. PACS numbers: 42.50.Dv, 03.65.Yz, 03.67.Lx, 85.25.Cp An indispensable requirement for a quantum computer is the readout of its state after performing gate operations. For that purpose it is sufficient to distinguish between two possible logical states. At the same time, it is desirable to demonstrate the coherence of time evolution explicitly. For solid-state qubits, this has been accomplished by Rabi-type experiments [1, 2]. In general the qubit state measurement is destructive, so that an interference pattern emerges only after a number of experimental runs. Certainly, it would be preferable to observe signatures of coherent dynamics already in a single run. Both the charge and the flux degree of freedom of superconducting qubits can be measured by coupling the qubit to a low-frequency “tank” circuit that is excited resonantly [3, 4]. In doing so one makes use of the fact that the resonance frequency of the slow oscillator depends on the qubit state which, in turn, influences the phase of the oscillator response [3, 4, 5]. The drawback of this scheme, however, is that the coherent qubit dynamics is considerably faster than the driving. Thus, one can only observe the time-average of the qubit state, but not time-resolve its dynamics. Measuring the qubit by driving it at resonance is possible as well [6]. This however induces Rabi oscillations, making the qubit dynamics differ significantly from the undriven case [7, 8]. Here, by contrast, we propose to probe the qubit by a weak high-frequency driving that directly acts upon the qubit without a tank circuit being present. We find that the resulting outgoing signal possesses sidebands which are related to a phase shift and demonstrate that the latter is related to a qubit observable. Validating this relation numerically for a Cooper-pair box, we show that the underlying measurement scheme principally enables monitoring the coherent qubit dynamics experimentally in a single run with good time-resolution and fidelity, whereas the backaction on the qubit, induced by the driving, stays at a tolerable level. Dissipative quantum circuit.—Although later on we focus on the dynamics of a superconducting charge qubit as sketched in Fig. 1, our measurement scheme is rather generic and can be applied to any open quantum system. We employ the system-bath Hamiltonian [9, 10, 11] H = H0 + X  p2 (qk − λk Ck Q)2  k , + 2Lk 2Ck (1) k where H0 denotes the system Hamiltonian and Q is a system operator. The bath is modelled by LC circuits with charges qk and conjugate momenta pk , where Ck and Lk are effective capacitances and inductances, respectively, and λk are the corresponding coupling constants. It is Pconvenient to introduce the spectral density I(ω) = π2 k λ2k (Ck /Lk )1/2 δ(ω − ωk ) which we assume to be ohmic, i.e. I(ω) = ωZ0 with an effective impedance Z0 [11, 12, 13]. By standard techniques we obtain the Bloch-Redfield master equation for the reduced system density operator ρ in the weak-coupling limit [14], 1 Z0 i ρ̇ = − [H0 , ρ] − [Q, [Q̂, ρ]] − i [Q, [Q̇, ρ]+ ], ~ ~ ~ (2) with the anticommutator [A, B]+ = AB + BA and the operators Q̇ = i[H0 , Q]/~ and Z ∞ Z 1 ∞ Q̂ = dω S(ω) cos(ωτ )Q̃(−τ ). (3) dτ π 0 0 Here, S(ω) = I(ω) coth(~ω/2kB T ) is the Fourier transform of the symmetrically ordered equilibrium correlation function 12 h[ξin (τ ), ξin (0)]+ ieq at temperature P T with regard to the collective bath coordinate ξin = k λk qk . The notation X̃(t) is a shorthand for the Heisenberg operator U0† (t)XU0 (t), where U0 is the system propagator. CPB Vgate transmission line LC chain: ξin ξout eff. Ohmic resistor Z0 φext FIG. 1: (color online) Cooper pair box (CPB) coupled to a transmission line with ohmic effective impedance Z0 . 2 In order to relate the quantum dynamics of the central circuit to the response via the transmission line, we employ the input-output formalism [15] which is an established tool in quantum optics and has also been used in quantum circuit theory [5]. It starts from the Heisenberg equation of motion for the environmental mode k which reads q̈k + ωk2 qk = ωk2 Q, where ωk = (Lk Ck )−1/2 denotes the angular frequency of mode k. Owing to its linearity, this equation of motion can be solved formally. Inserting the obtained solution into the Heisenberg equations of motion for the system operators, one arrives at the so-called quantum Langevin equation [16, 17]. For an ohmic environment, the latter possesses the inhomogeneity ξin (t) − Z0 Q̇(t), where the noise operator ξin is fully determined by the correlation functions given above. Alternatively, one can write the quantum Langevin equation in terms of the outgoing fluctuations [15]. The result differs only by the sign of the dissipative term, so that the inhomogeneity now reads ξout (t) + Z0 Q̇(t). The difference between these two equations relates the input and the output fluctuations via ξout − ξin = −2Z0 Q̇ = −  2iZ0  H0 , Q , ~ (4) which is a cornerstone of the input-output formalism [15] and holds for any weakly coupled  ξin . We used that for weak dissipation, Q̇ ≈ i H0 , Q /~ is essentially bathindependent. Response to high-frequency driving.—We next probe the system by driving it via the transmission line with an ac signal A cos(Ωt) that also couples to the system operator Q. Then, the Hamiltonian acquires an additional term: H0 → H0 + QA cos(Ωt), and the master equation (2) changes accordingly. For the input ξin , this corresponds to one coherently excited incoming mode such that hξin (t)i = A cos(Ωt), while the r.h.s. of the inputoutput relation (4) remains unchanged. Because the driving must not significantly alter the system dynamics, we assume that the amplitude A is sufficiently small, so that the driving can be treated perturbatively. This yields the ansatz ρ(t) = ρ0 (t) + ρ1 (t), where ρ0 (t) is the unperturbed state. To lowest order in A, ρ1 obeys ρ̇1 = L0 ρ1 − ~i A[Q, ρ0 ] cos(Ωt), where L0 denotes the superoperator on the r.h.s. of Eq. (2). This linear inhomogeneous equation of motion can be solved formally in terms of a convolution between the propagator of the undriven system and the inhomogeneity. If the driving frequency Ω is much larger than all relevant system frequencies, one may separate time scales to obtain A [Q, ρ0 (t)] sin(Ωt) , ρ(t) = ρ0 (t) − i ~Ω (5) which identifies eA/~Ω as the necessarily small perturbation parameter. Together with the input-output relation (4) this solution allows us to compute the response of the system. In an experiment it is possible to employ a lock-in technique with the incoming signal providing the reference oscillator. This singles out the high-frequency components of the outgoing signal, which correspond to the second term of the density operator (5) and read hξout (t)i = A cos(Ωt)+ 2AZ0 h[[H0 , Q], Q]i0 sin(Ωt), (6) ~2 Ω where h. . .i0 = tr[ρ0 (t) . . .] refers to the undriven dynamics. Writing next hξout (t)i = A′ cos[Ωt − φ0hf (t)], we find the central expression φ0hf (t) = 2Z0 h[[H0 , Q], Q]i0 , ~2 Ω (7) which relates a small phase shift φ0hf (t) between the input and the output signal to a hermitian system observable describing the unperturbed low-frequency system dynamics. This means that the time-resolved evolution of the open quantum system can be monitored in a single run by continuously measuring the phase shift φ0hf (t) with appropriate experimental techniques. In this connection, Eq. (7) represents the basis for our proposed measurement scheme. Below we will explore its validity and limitations for a specific system by comparing the phase of the output hξout (t)i with the expectation value h[[H0 , Q], Q]i0 , both computed from the numerical solution of the master equation (2) in the presence of ac driving. Monitoring coherent qubit dynamics.—A particular case of a quantum circuit which recently attracted much interest is a Cooper pair box (CPB) which is sketched in Fig. 1 and described by the Hamiltonian H0CPB = 4EC (N̂ − Ng )2 − ∞ EJ X (|N +1ihN | + h.c.), 2 N =−∞ (8) where N is the number of excess Cooper pairs in the box, P so that the charge operator reads Q = 2eN̂ = 2e N N |N ihN |. The charging energy EC is determined by various capacitances, while the scaled gate voltage Ng and the effective Josephson energy EJ are controllable. If the charging energy is sufficiently large and Ng ≈ 1/2, only the two charge states |0i ≡ |↓i and |1i ≡ |↑i matter and form a qubit [1, 4, 11] described by the Hamiltonian 1 1 H0qb = − Eel σz − EJ σx , 2 2 (9) where the Pauli matrices σi are defined in the qubit subspace and Eel = 4EC (1 − 2Ng ). The qubit energy split2 ting reads ~ωqb = (Eel + EJ2 )1/2 . Moreover, Qqb = eσz while by virtue of relation (7) the phase of the output signal is linked to the qubit observable σx according to φ0hf (t) = − 4e2 Z0 EJ hσx (t)i0 . ~2 Ω (10) 3 1 (a) 0.01 0 −0.01 190 0.5 1◦ 190.5 191 0◦ 0 −0.5 ◦  φhf0 (t) hQ̇(t)i −1 0 10 ◦  φout (t) 20 30 −1 ◦ 40 t [h̄/EJ ] (b) 10 2∆Ω 0 2ωqb 10−2 hQ̇(ω)i 2 hQ̇qb (ω)i 10−4 2 ωqb 2 Ω 5 10 20 50 ω [EJ /h̄] FIG. 2: (color online) Decaying qubit oscillations with initial state |↑i in a weakly probed CPB with 6 states for α = Z0 e2 /~ = 0.08, A = 0.1EJ /e, EC = 5.25EJ and Ng = 0.45, so that Eel = 2.1EJ and ωqb = 2.3EJ /~. (a) Time evolution of the measured difference signal hQ̇i ∝ hξout i−hξin i (in units of 2eEJ /~) of the full CPB and its lock-in amplified phase φout (frequency window ∆Ω = 5EJ /~), compared to the estimated phase φ0hf ∝ hσx i0 in the qubit approximation. The inset resolves the underlying small rapid oscillations with frequency Ω = 15EJ /~ in the long-time limit. (b) Power spectrum of hQ̇i for the full CPB Hamiltonian (solid) and for the two-level approximation (dashed). This means that the high-frequency component of hQ̇i, which is manifest in the phase of the outgoing signal (4), contains information about the low-frequency qubit dynamics in terms of the unperturbed hσx i0 . We now turn to the question how relation (10) allows one to retrieve information about the coherent qubit dynamics in an experiment. Figure 2(a) shows the time evolution of the expectation value hQ̇(t)i for the initial state |↑i ≡ |1i, obtained via numerical integration of the master equation (2) with the full Cooper-pair box Hamiltonian (8) in the presence of the ac driving which in principle may excite higher states. The driving, due to its rather small amplitude, is barely noticeable on the scale chosen for the main figure, but only on a refined scale for long times; see inset of Fig. 2(a). This already insinuates that the backaction on the dynamics is weak. In the cor- responding power spectrum of hQ̇i depicted in Fig. 2(b), the driving is nevertheless reflected in sideband peaks at the frequencies Ω and Ω ± ωqb . In the time domain these peaks correspond to a signal cos[Ωt − φout (t)]. Moreover, non-qubit CPB states leads to additional peaks at higher frequencies, while their influence at frequencies ω . Ω is minor. Experimentally, the phase φout (t) can be retrieved by lock-in amplification of the output signal, which we mimic numerically in the following way [18]: We only consider the spectrum of ξout in a window Ω ± ∆Ω around the driving frequency and shift it by −Ω. The inverse Fourier transformation to the time domain provides φout (t) which is expected to agree with φ0hf (t) and, according to Eq. (10), to reflect the unperturbed time evolution of hσx i0 with respect to the qubit. Although the condition of high-frequency probing, Ω ≫ ωqb , is not strictly fulfilled and despite the presence of higher charge states, the lock-in amplified phase φout (t) and the predicted phase φ0hf (t) are barely distinguishable for an appropriate choice of parameters as is shown in Fig. 2(a). In order to quantify this agreement, we introduce the measurement fidelity F = (φout , hσx i0 ), where (f, g) = R R R dt f g/( dt f 2 dt g 2 )1/2 with time integration over the decay duration. Thus, the ideal value F = 1 is assumed if φout (t) and hσx (t)i0 are proportional to each other, i.e. if the agreement between the measured phase and the unperturbed expectation value hσx i0 is perfect. Figure 3(a) depicts the fidelity as a function of the driving frequency. As expected, whenever non-qubit CPB states are excited resonantly, we find F ≪ 1, indicating a significant population of these states. Far-off such resonances, the fidelity increases with the driving frequency Ω. A proper frequency lies in the middle between the qubit doublet and the next higher state. In the present case, Ω ≈ 15EJ /~ appears as a good choice. Concerning the driving amplitude, one has to find a compromise, because as A increases, so does the phase contrast of the outgoing signal (6), while the driving perturbs more and more the low-frequency dynamics. For the frequency chosen above, the fidelity is best in the range A = 0.1–1~ωqb/e. This corresponds to eA/~Ω ≈ 10−2 –10−3 , which justifies our perturbative treatment. Measurement quality and backaction.—For any quantum measurement, one has to worry about backaction on the system in terms of decoherence. In our measurement scheme, decoherence plays a particular role, because both the driving and the ohmic environment couple to the CPB via the same mechanism. This is reflected by the fact that the predicted phase (10) is proportional to the dimensionless dissipation strength α = Z0 e2 /~. However, α should not exceed 0.1 in order to preserve a predominantly coherent time evolution, which also means that our measurement is weak, but destructive. The condition α . 0.1 together with the above conditions on the driving amplitude and frequency provides phase shifts φout of the order 1◦ , which is small but still measurable 4 δF = 1−F 100 (a) A = 5 EJ /e A = 0.1 EJ /e A = 0.01 EJ /e 10−2 10−4 D 10−2 ∝ 1/Ω (b) 10−4 10−6 10 20 50 100 200 Ω [EJ /h̄] FIG. 3: (color online) (a) Fidelity defect δF = 1 − F and (b) time-averaged trace distance between the driven and the undriven density operator of the CPB for various driving amplitudes as a function of the driving frequency. All other parameters are the same as in Fig. 2. with present technologies. The additional decoherence due to the driving, by contrast, is not noticeable. This is in agreement with the first-order result tr(ρ2 ) = tr(ρ20 ) which follows from Eq. (5). In order to investigate to what extent the driving affects the quantum state of the CPB, we compute the trace distance D(t) = 21 tr |ρ(t) − ρ0 (t)| [19] between the density operators of the driven system ρ(t) and the undriven reference ρ0 (t). Its time average D quantifies the perturbation due to the driving. Figure 3(b) indicates that D ∝ A/Ω, unless the driving is in resonance with higher levels. This confirms the picture drawn by studying the measurement fidelity F . For practically all parameters used in Fig. 3, we found that the total population of levels outside the qubit doublet is always less than 0.1%. The only exception occurs again in the case of resonances with non-qubit CPB levels. Far from these resonances, the system is faithfully described with only the qubit levels. In our investigations, we have not considered excitations of quasiparticles which are relevant once the driving frequency becomes of the order of the gap frequency of the superconducting material. Thus, for an aluminum CPB our model is valid only for Ω . 100 GHz. Since a typical Josephson energy is of the order of some GHz, a driving frequency Ω ≈ 20EJ /~ is still within this range while it provides already a good measurement quality. Conclusions.—We have proposed a method for the time-resolved monitoring of the dynamics of a quantum system coupled to a dissipative environment. The crucial requirement for this is the possibility to drive the system coherently via one environmental high-frequency mode accompanied by measuring the phase of the outgoing signal via lock-in techniques. By analyzing the highfrequency response, we have found that the phase of the output signal is related to a particular system observable. We have substantiated this relation by computing both quantities numerically for a charge qubit implemented with a Cooper-pair box. 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