Abstract
Coherent control of quantum states is at the heart of implementing solidstate quantum processors and testing quantum mechanics at the macroscopic level. Despite significant progress made in recent years in controlling single and bipartite quantum systems, coherent control of quantum wave function in multipartite systems involving artificial solidstate qubits has been hampered due to the relatively short decoherence time and lack of precise control methods. Here we report the creation and coherent manipulation of quantum states in a tripartite quantum system, which is formed by a superconducting qubit coupled to two microscopic twolevel systems (TLSs). The avoided crossings in the system's energylevel spectrum due to the qubit–TLS interaction act as tunable quantum beam splitters of wave functions. Our result shows that the Landau–Zener–Stückelberg interference has great potential in precise control of the quantum states in the tripartite system.
Introduction
As one of three major forms of superconducting qubits^{1,2,3}, a fluxbiased superconducting phase qubit^{4,5} consists of a superconducting loop with inductance L interrupted by a Josephson junction (Fig. 1a). The superconducting phase difference ϕ across the junction serves as the quantum variable of coordinate. When biased close to the critical current I_{0}, the qubit can be thought of as a tunable artificial atom with discrete energy levels that exist in a potential energy landscape determined by the circuit design parameters and bias (Fig. 1b). The ground state 0 and the first excited state 1 are usually chosen as the computational basis states of the phase qubit. The energy difference between 1 and 0, ω_{10}, decreases with flux bias. A TLS is phenomenologically understood to be an atom or a small group of atoms tunnelling between two lattice configurations inside the Josephson tunnel barrier, with different wave functions L and R corresponding to different critical currents (Fig. 1c). Under the interaction picture of the qubit–TLS system, the state of the TLS can be expressed in terms of the eigenenergy, with g being the ground state and e the excited state. When the energy difference between e and g, ħω_{TLS}=E_{e}−E_{g}, is close to ℏω_{10} (ħ≡h/2π, where h is Planck's constant), coupling between the phase qubit and the TLS becomes significant, which could result in increased decoherence^{4,5}. On the other hand, one can use strong qubit–TLS coupling to demonstrate coherent macroscopic quantum phenomena and/or quantum information processing^{6,7,8}. For instance, recently, a tetrapartite system formed by two qubits, one cavity and one TLS, has been studied^{5}. However, although multipartite spectral property and vacuum Rabi oscillation have been observed, coherent manipulation of the quantum states of the whole system has not yet been demonstrated.
In our experiments, we use two TLSs near 16.5 GHz to form a hybrid tripartite^{9,10,11} phase qubit–TLS system and demonstrate Landau–Zener–Stückelberg (LZS) interference in such a tripartite system. The avoided crossings due to the qubit–TLS interaction act as tunable quantum beam splitters of wave functions, with which we could precisely control the quantum states of the system.
Results
Experimental results of LZS interference
Figure 1d shows the measured spectroscopy of a phase qubit. The spectroscopy data clearly show two avoided crossings resulting from qubit–TLS coupling. As, after application of the πpulse, the system has absorbed exactly one microwave photon and the subsequent steps of state manipulation are accomplished in the absence of the microwave, conservation of energy guarantees that one and only one of the qubit, TLS1 and TLS2, can be coherently transferred to its excited state. Thus, only as marked in Figure 1d, are involved in the dynamics of the system. Notice that these three basis states form a generalized W state^{10,11,12}, which preserves entanglement between the remaining bipartite system even when one of the qubits is lost and has been recognized as an important resource in quantum information science^{13}. The system's effective Hamiltonian can be written as
where Δ_{1} (Δ_{2}) is the coupling strength between the qubit and TLS1 (TLS2). ω_{TLS1} (ω_{TLS2}) is the resonant frequency of TLS1 (TLS2). ω_{10}(t)=ω_{10,dc}−sΦ(t), with ω_{10,dc} being the initial energy detuning controlled by the dc flux bias line (that is, the second platform holds in the dc flux bias line), s=dω_{10}(Φ)/dΦ being the diabatic energylevel slope of state 1g_{1}g_{2} and Φ(t) being the timedependent flux bias (Fig. 1a).
In our experiment, coherent quantum control of multiple qubits is realized with LZ transition. When the system is swept through the avoided crossing, the asymptotic probability of transmission is exp(−2π(Δ^{2}/ν)), where ħν≡dE/dt denotes the rate of the energy spacing change for noninteracting levels, and 2ħΔ is the minimum energy gap. It ranges from 0 to 1, depending on the ratio of Δ and ν. The avoided crossing serves as a beam splitter that splits the initial state into a coherent superposition of two states^{14}. These two states evolve independently in time, while a relative phase is accumulated, causing interference after sweeping back and forth through the avoided crossing. Such LZS interference has been observed recently in superconducting qubits^{15,16,17,18,19,20,21,22}. However, in these experiments the avoided crossings of the singlequbit energy spectrum are used, and microwaves, whose phase is difficult to control, are applied to drive the system through the avoided crossing consecutively to manipulate the qubit state. Here we use a triangular bias waveform with width shorter than the qubit's decoherence time to coherently control the quantum state of the tripartite system. The use of a triangular waveform, with a time resolution of 0.1 ns, ensures precise control of the flux bias sweep at a constant rate and thus the quantum state. The qubit is initially prepared in 0g_{1}g_{2}. A resonant microwave πpulse is applied to coherently transfer the qubit to 1g_{1}g_{2}. A triangular flux bias, Φ(t), with variable width T and amplitude Φ_{LZS}
is then applied immediately to the phase qubit to induce LZ transitions (Fig. 2d). This is followed by a short readout pulse (about 5 ns) to determine the probability of finding the qubit in the state 1, that is, the system in the state 1g_{1}g_{2}.
Figure 2a shows the measured population of 1 as a function of T and Φ_{LZS}. On the top part of the plot, the amplitude is so small that the state could not reach the first avoided crossing M_{1}. Therefore, no LZ transition could occur and only a trivial monotonic behaviour is observed. When the amplitude is large enough to reach M_{1}, the emerging interference pattern can be qualitatively divided into three regions with remarkably different fringe patterns.
Quantitative comparison with the model
To quantitatively model the data, we calculate the probability to return to the initial state P_{1} by considering the action of the unitary operations on the initially prepared state. Neglecting relaxation and dephasing, we find
where P_{LZi} (i=1,2) is the LZ transition probability at the ith avoided crossing M_{i}, and θ_{I} and θ_{II} are the phases accumulated in regions I and II, respectively (Fig. 2b). The phase jump at the ith avoided crossing is due to the Stokes phase^{16,22}θ_{Si}, which depends on the adiabaticity parameter η_{i}=Δ_{i}^{2}/ν in the form θ_{Si}=π/4+η_{i}(ln η_{i}−1)+arg Γ(1−iη_{i}), where Γ is the Gamma function. In the adiabatic limit θ_{S}→0, while in the sudden limit θ_{S}=π/4. In order to give a clear physical picture, hereafter we adopt the terminology of optics to discuss the phenomenon and its mechanism. First of all we define two characteristic sweeping rates of ν_{1} and ν_{2} from 2πΔ_{i}^{2}/ν_{i}=1 (i=1, 2). From the spectroscopy data, we have Δ_{1}/2π=10 MHz and Δ_{2}/2π=32 MHz; thus, ν_{1}/2π=3.94×10^{−3} GHz ns^{−1} and ν_{2}/2π=4.04×10^{−2} GHz ns^{−1}, respectively. These lines of constant sweeping rate characteristic to the system are marked as oblique dotted lines in Figure 2a. The avoided crossings M_{1} (M_{2}) can be viewed as wave function splitters with controllable transmission coefficients set by the sweeping rate ν. ν_{1} and ν_{2} thereby define three regions in the T−Φ_{LZS} parameter plane that contain all main features of the measured interference patterns:
(I) νν_{1} and νν_{2}: M_{1} acts as a beam splitter and M_{2} acts as a total reflection mirror, that is, P_{LZ1}1/2 and P_{LZ2}0. In this case, equation (3) can be simplified as
Apparently, only path 1 and path 2 contribute to the interference. The phase accumulated in region I can be expressed as
where ω_{i}(t) (i=1, 2) denotes the energy frequency corresponding to path i (i=1, 2). It is easy to find that P_{1} is maximized (constructive interference) in the condition
from which we can obtain the analytical expression for the positions of constructive interference fringes
where δ_{1}=ω_{10,dc}−ω_{TLS1}, δ_{2}=ω_{10,dc}−ω_{TLS2}, and δ_{12}=ω_{TLS1}−ω_{TLS2}.
In Figure 2b we show the calculated constructive interference strips, which agree well with the experimental results. Especially, in the limit of sΦ_{LZS}>>δ_{2}, δ_{12}, equation (7) can be simplified as
Intuitively, this result is straightforward to understand, as in the largeamplitude limit the accumulated phase θ_{1} is two times the area of a rectangle with length T/2 and width ω_{TLS1}−ω_{TLS2}.
(II) νν_{2} and ν>>ν_{1}: M_{1} acts as a total transmission mirror and M_{2} acts as a beam splitter, that is, P_{LZ1}1 and P_{LZ2}1/2. In this case, equation (3) can be simplified as
Only path 2 and path 3 contribute to the interference. Using the same method in dealing with region I, we obtain the analytical formula governing the positions of constructive interference fringes:
As shown in Figure 2c, the positions of the constructive interference fringes obtained from equation (10) agree with experimental results very well. Similarly, in the limit sΦ_{LZS}>>δ_{2}, equation (10) has the simple form,
which is also readily understood because in the largeamplitude limit the accumulated phase θ_{II} is two times the area of a triangle with base length T/2 and height sΦ_{LZS}.
(III) ν_{1}<ν<ν_{2}: This region is more interesting and complex. Here, M_{1} acts as a beam splitter, while M_{2} can act either as a beam splitter or as a total reflection mirror. This effect cannot be described by the asymptotic LZ formula because in this region LZS interference occurs only in a relatively small range around the avoided crossings. As the analytical solution is extremely complicated and does not provide clear intuition about the underlying physics, we use a numerically calculated LZ transition probability P_{LZ} corresponding to the transmission coefficient of M_{1} and M_{2} for comparison with the experimental data. We find that for certain sweeping rates, LZ transition probability resulting from M_{2} is quite low. Therefore, M_{2} can be treated as a total reflection mirror, while M_{1} is still acting as a good beam splitter. The interference fringes generated by M_{2} thus disappear (the fringes tend to fade out) and the interference fringes generated by M_{1} dominate, displayed as a chain of 'hot spots' marked by the circles in Figure 2a.
When both M_{1} and M_{2} can be treated as beam splitters, all three paths (1, 2, and 3) contribute to the interference. According to equation (3), P_{1} is maximized in the condition
It is noted that under this condition the term in equation (3) equals 2nπ. Considering different weights in each path, it is more convenient to obtain a theoretical prediction from a numerical simulation. Here we utilize the Bloch equation to describe the time evolution of the density operator of the tripartite system:
where Γ[ρ] includes the effects of energy relaxation. Figure 3a shows the calculated population of 1 as a function of T and Φ_{LZS}. Figure 3b shows the extracted data for different T and Φ_{LZS} values. The agreement between the theoretical and experimental results is remarkable. In order to better understand the origin of the 'hot spots', we also plot the probabilities of LZ transition as a function of the pulse width at fixed amplitude Φ_{LZS}=10mΦ_{0} (Fig. 3c). Notice that both LZ transition probabilities oscillate with T, which are quite different from the general asymptotic LZ transition probabilities. The transition probability at M_{1} is always greater because Δ_{1} is much smaller than Δ_{2}. The three oblique dotted lines in Figure 3a represent lines of constant sweeping rate. The 'hot spots' are located on these lines, where the transition probability of M_{2} is a minimum. M_{2} thereby acts as a total reflection mirror, resulting in the 'hot spots' in transition probability. This feature further confirms that the avoided crossings play the role of quantum mechanical wave function splitters, analogous to continuously tunable beam splitters in optical experiments. The transmission coefficient of the wave function splitters (the avoided crossings) in our experiment can be varied in situ from zero (total reflection) to unity (total transmission) or any value in between by adjusting the duration and amplitude of the single triangular bias waveform used to sweep through the avoided crossings.
Precise control of the quantum states in the tripartite system
We emphasize that the method of using LZS interference for the precise quantum state manipulation described above is performed within the decoherence time of the tripartite system, which is about 140 ns. Through coherent LZ transition, we can thus achieve a high degree of control over the quantum state of the qubit–TLS tripartite system. For example, one may take advantage of LZS to control the generalized W state, ψ=α1g_{1}g_{2}+β0e_{1}g_{2}+γ0g_{1}e_{2}, evolving in the subspace spanned by the three product states during the operation of sweeping flux bias. In order to quantify the generalized W state, we define where σ=α, β, γ. In Figure 3d, w is plotted as a function of T and Φ_{LZS}. Note that with precise control of the flux bias sweep, the states with w=1, which are generalized W states with equal probability in each of the three basis product states, are obtained, demonstrating the effectiveness of this new method. It should be pointed out that when one of the three qubits is lost, the remaining two qubits are maximally entangled.
Discussion
Our tripartite system includes a macroscopic object, which is relatively easy to control and read out, coupled to microscopic degrees of freedom that are less prone to environmentinduced decoherence and thus can be used as a hybrid qubit. The excellent agreement between our data and theory over the entire T−Φ_{LZS} parameter plane indicates strongly that the states created are consistent with the generalized W states. The coherent generation and manipulation of generalized W states reported here demonstrate an effective new technique for the precise control of multipartite quantum states in solidstate qubits and/or hybrid qubits^{6,8}.
Methods
Experimental detail
Figure 1a shows the principal circuitry of the measurement. The flux bias and microwave are fed through the onchip thin film flux lines coupled inductively to the qubit. The slowly varying flux bias is used to prepare the initial state of the qubit and to read out the qubit state after coherent state manipulation. In the first platform of the flux bias, the potential is tilted quite asymmetrically to ensure that the qubit is initialized in the left well. Then we increase the flux bias to the second platform until there are only a few energy levels, including the computational basis states 0 and 1 in the left well. A microwave πpulse is applied to rotate the qubit from 0 to 1. This is followed by a triangular waveform with adjustable width and amplitude applied to the fast flux bias line, which results in LZ transition. A short readout pulse of flux bias is then used to adiabatically reduce the well's depth so that the qubit will tunnel to the right well if it was in 1 or remain in the left well if it was in 0. The flux bias is then lowered to the third platform, where the doublewell potential is symmetric, to freeze the final state in one of the wells. The state in the left or right well corresponds to clockwise or counterclockwise current in the loop, which can be distinguished by the dcSQUID magnetometer inductively coupled to the qubit. By mapping the states 0 and 1 into the left and right wells, respectively, the probability of finding the qubit in state 1 is obtained. We obtained T_{1}70 ns from energy relaxation measurement (Supplementary Fig. S1a), T_{R}80 ns from Rabi oscillation (Supplementary Fig. S1b), T_{2}*60 ns from Ramsey interference fringe (Supplementary Figs S1c and S1d) and T_{2}137 ns from spinecho (Supplementary Fig. S1e) in the region free of qubit–TLS coupling.
Hamiltonian in our tripartite system
For the coupled qubit–TLS system, the Hamiltonian can be written as^{23,24}
In the twolevel approximation the effective Hamiltonian of the qubit is here the flux bias (Φ) dependent energylevel spacing of the qubit, ħω_{10}=E_{1}−E_{0}, can be obtained numerically by solving the eigenvalues problem associated with the full Hamiltonian of the phase qubit^{25}. The Hamiltonian of the ith TLS can be written as where ħω_{TLSi} is the energylevel spacing of the ith TLS. The interaction Hamiltonian between the qubit and the ith TLS is where Δ_{i} is the coupling strength between the qubit and the ith TLS and are the Pauli operators acting on the states of the qubit (the ith TLS). By adjusting the flux bias, the qubit and TLSs can be tuned into and out of resonance, effectively turning on and off the couplings. Below 0 and 1 (g_{i} and e_{i}) are used to denote the ground state and excited state of the qubit (the ith TLS). In our experiment the initial state is prepared in the system's ground state 0g_{1}g_{2}. When the couplings between the qubit and TLSs are off, we use a πpulse to pump the qubit to 1 (thus the system is in 1g_{1}g_{2}). We then sweep the flux bias through the avoided crossing(s) to turn on the coupling(s) between the qubit and the TLS(s). Since after the application of the πpulse the system has absorbed exactly one microwave photon and the subsequent steps of state manipulation are accomplished in the absence of the microwave, conservation of energy guarantees that one and only one of the qubit, TLS1 and TLS2, can be coherently transferred to its excited state. Therefore, states with only one of the three subsystems in excited state, 1g_{1}g_{2}, 0e_{1}g_{2}, and 1g_{1}e_{2}, are relevant in discussing the subsequent coherent dynamics of the system. In the subspace spanned by these three basis states, the Hamiltonian (14) can be written explicitly as Hamiltonian (1) in the main text.
Unitary operation in our tripartite system
We use the transfer matrix method^{16,22} to obtain the probability of finding the system in 1g_{1}g_{2} at the end of the triangular pulse. We use a=[1,0,0]^{T}, b=[0,1,0]^{T} and c=[0,0,1]^{T} to denote the instantaneous eigenstates of the timedependent Hamiltonian (14), as shown in Supplementary Figure S2. It is noted that at the initial flux bias point, which is far from the avoided crossings, the system is in a=1g_{1}g_{2}. At the crossing times t=t_{1} and t=t_{2}, the incoming and outgoing states are connected by the transfer matrix:
and
respectively. Here sin^{2}(θ_{i}/2)=P_{LZi} (i=1, 2) is the LZ transition probability at the ith avoided crossing. where θ_{Si} is the Stokes phase^{16,22}, the value of which depends on the adiabaticity parameter η_{i}=Δ_{i}^{2}/υ in the form of θ_{Si}=π/4+η_{i} (ln η_{i}−1)+arg Γ(1−iη_{i}), where Γ is the Gamma function. In the adiabatic limit θ_{S}→0, and in the sudden limit θ_{S}=π/4. At crossing times t=t_{3} and t=t_{4}, we have respectively. The outgoing state at t=t_{i} and the incoming state at t=t_{i+1} (i=0, 1, 2, 3, 4) are thus connected by the propagator
where ω_{i}(t) is the energylevel spacing frequency of i (i=a, b, c) at time t. The net effect of a triangular pulse is to cause the state vector to evolve according to the unitary transformation
The probability of finding the system remaining at the initial state is P_{1}=1g_{1}g_{2}Û1g_{1}g_{2}^{2}. Its concrete form is equation (3), in which and are the relative phases accumulated in regions I and II, respectively, as shown in Supplementary Fig. S2. The LZS in our experiment can be viewed as interferences among the three paths, which are labelled 1, 2 and 3, starting from the same initial state:
path 1:
path 2:
path 3:
Denoting ω_{i}(t) as the energylevel spacing frequency corresponding to path i (i=1, 2, 3), then θ_{I} and θ_{II} have the forms and respectively.
Numerical simulation of LZS interference in the bipartite qubit–TLS system
For the bipartite qubit–TLS system discussed here, the qubit is coupled only to a single TLS. The quantum dynamics of the system, including the effects of dissipation, is described by the Bloch equation of the time evolution of the density operator:
where
where ω_{10}(t)=ω_{10,dc}−νt, ν≡2sΦ_{LZS}/T is the energy sweeping rate and Δ is the qubit–TLS coupling strength. The second term, Γ[ρ], describes the relaxation process to the ground state 0g and dephasing process phenomenologically. In a concrete expression, equation (19) can be written as (for ease of discussion, we relabel 1g and 0e as a and b, respectively)
with ρ_{ba}=ρ_{ab}*. Here Γ_{α} (α=a, b) is the relaxation rate from state α to the ground state 0g. The decoherence rate includes contributions from both relaxation and dephasing. Supplementary Figures S3a and S3b give the numerically simulated LZS interference pattern for the qubit coupled with the first TLS and second TLS, respectively. To calculate the transmission coefficient of M_{i} (i=1, 2), that is, the LZ tunneling probability P_{LZ}, as shown in Figure 3c, we cannot directly use the asymptotic LZ formula, which is based on sweeping the system across the avoided crossing from negative to positive infinities. In contrast, in our experiment the LZS occurs near the avoided crossings. Therefore, our numerical results are obtained by solving the Bloch equations directly.
Numerical simulation of LZS interference in the tripartite qubit–TLS system
For the tripartite qubit–TLS system discussed below, the qubit is coupled resonantly to two TLSs (TLS1 and TLS2) with different excited state energies ħω_{TLS1} and ħω_{TLS2}. The Hamiltonian in the basis of 1g_{1}g_{2}, 0e_{1}g_{2}, 0g_{1}e_{2} is Hamiltonian (1) in the main text. The Bloch equations that govern the evolution of the density operator can be written as (for simplicity, we relabel respectively)
where the diagonal elements ρ_{ii} are the populations, offdiagonal elements ρ_{ij}(i ≠ j) describe coherence, and are the rates of decoherence. The remaining three elements' equations are determined by ρ_{ij}*=ρ_{ji}. The numerically simulated LZS interference pattern is shown in Figure 3a, which agrees with the experimental results excellently.
Additional information
How to cite this article: Sun, G. et al. Tunable quantum beam splitters for coherent manipulation of a solidstate tripartite qubit system. Nat. Commun. 1:51 doi: 10.1038/ncomms1050 (2010).
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Acknowledgements
This work is partially supported by NCET, NSFC (10704034,10725415), 973 Program (2006CB601006), the State Key Program for Basic Research of China (2006CB921801) and NSF Grant No. DMR0325551. We thank Northrop Grumman ES in Baltimore, MD, for technical and foundry support and thank R. Lewis, A. Pesetski, E. Folk and J. Talvacchio for technical assistance. We thank B. Ruzicka for editing the paper.
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G.S. and S.H. conceived the experiments; G.S. carried out the measurements with the help of B.M. and analysed the data with the help of X.W., Y.Y., J.C., P.W. and S.H.; X.W. performed the numerical calculations; G.S., Y.Y. and S.H. wrote the paper.
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Sun, G., Wen, X., Mao, B. et al. Tunable quantum beam splitters for coherent manipulation of a solidstate tripartite qubit system . Nat Commun 1, 51 (2010). https://doi.org/10.1038/ncomms1050
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DOI: https://doi.org/10.1038/ncomms1050
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