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Nature Communications
 
Nat Commun. 2010 August; 1(5): 1–7.
PMCID: PMC2982164

Tunable quantum beam splitters for coherent manipulation of a solid-state tripartite qubit system

Abstract

Coherent control of quantum states is at the heart of implementing solid-state quantum processors and testing quantum mechanics at the macroscopic level. Despite significant progress made in recent years in controlling single- and bi-partite quantum systems, coherent control of quantum wave function in multipartite systems involving artificial solid-state 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 two-level systems (TLSs). The avoided crossings in the system's energy-level 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.

As one of three major forms of superconducting qubits1,2,3, a flux-biased superconducting phase qubit4,5 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 I0, 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 |0An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg and the first excited state |1An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg are usually chosen as the computational basis states of the phase qubit. The energy difference between |1An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg and |0An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg, ω10, 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 |LAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg and |RAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg 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 |gAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg being the ground state and |eAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg the excited state. When the energy difference between |eAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg and |gAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg, ħωTLS=EeEg, is close to [variant Planck's over 2pi]ω10 (ħ[equivalent]h/2π, where h is Planck's constant), coupling between the phase qubit and the TLS becomes significant, which could result in increased decoherence4,5. On the other hand, one can use strong qubit–TLS coupling to demonstrate coherent macroscopic quantum phenomena and/or quantum information processing6,7,8. For instance, recently, a tetrapartite system formed by two qubits, one cavity and one TLS, has been studied5. 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.

Figure 1
Qubit circuit and experimental procedure.

In our experiments, we use two TLSs near 16.5 GHz to form a hybrid tripartite9,10,11 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 π-pulse, 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 An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m1.jpg as marked in Figure 1d, are involved in the dynamics of the system. Notice that these three basis states form a generalized W state10,11,12, An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m2.jpg 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 science13. The system's effective Hamiltonian can be written as

An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m3.jpg
1

where Δ12) is the coupling strength between the qubit and TLS1 (TLS2). ωTLS1 (ωTLS2) is the resonant frequency of TLS1 (TLS2). ω10(t)=ω10,dcsΦ(t), with ω10,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=|dω10(Φ)/dΦ| being the diabatic energy-level slope of state |1g1g2An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg and Φ(t) being the time-dependent 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(−2π2/ν)), where ħν[equivalent]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 states14. 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 qubits15,16,17,18,19,20,21,22. However, in these experiments the avoided crossings of the single-qubit 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 |0g1g2An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg. A resonant microwave π-pulse is applied to coherently transfer the qubit to |1g1g2An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg. A triangular flux bias, Φ(t), with variable width T and amplitude ΦLZS

An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m4.jpg
2

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 |1An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg, that is, the system in the state |1g1g2An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg.

Figure 2
LZS interference in a phase qubit coupled to two TLSs.

Figure 2a shows the measured population of |1An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg as a function of T and ΦLZS. On the top part of the plot, the amplitude is so small that the state could not reach the first avoided crossing M1. Therefore, no LZ transition could occur and only a trivial monotonic behaviour is observed. When the amplitude is large enough to reach M1, 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 P1 by considering the action of the unitary operations on the initially prepared state. Neglecting relaxation and dephasing, we find

An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m5.jpg
3

where PLZi (i=1,2) is the LZ transition probability at the ith avoided crossing Mi, and θI and θII are the phases accumulated in regions I and II, respectively (Fig. 2b). The phase jump An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m6.jpg at the ith avoided crossing is due to the Stokes phase16,22 θSi, which depends on the adiabaticity parameter ηii2/ν in the form θSi=π/4+ηi(ln ηi−1)+arg Γ(1−i), where Γ is the Gamma function. In the adiabatic limit θS→0, while in the sudden limit θS=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 ν1 and ν2 from 2πΔi2/νi=1 (i=1, 2). From the spectroscopy data, we have Δ1/2π=10 MHz and Δ2/2π=32 MHz; thus, ν1/2π=3.94×10−3 GHz ns−1 and ν2/2π=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 M1 (M2) can be viewed as wave function splitters with controllable transmission coefficients set by the sweeping rate ν. ν1 and ν2 thereby define three regions in the T−ΦLZS parameter plane that contain all main features of the measured interference patterns:

(I) νAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m45.jpgν1 and νAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m45.jpgν2: M1 acts as a beam splitter and M2 acts as a total reflection mirror, that is, PLZ1An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m45.jpg1/2 and PLZ2An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m45.jpg0. In this case, equation (3) can be simplified as

An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m7.jpg
4

Apparently, only path 1 and path 2 contribute to the interference. The phase accumulated in region I can be expressed as

An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m8.jpg
5

where ωi(t) (i=1, 2) denotes the energy frequency corresponding to path i (i=1, 2). It is easy to find that P1 is maximized (constructive interference) in the condition

An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m9.jpg
6

from which we can obtain the analytical expression for the positions of constructive interference fringes

An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m10.jpg
7

where δ1=ω10,dcωTLS1, δ2=ω10,dcωTLS2, and δ12=ωTLS1ωTLS2.

In Figure 2b we show the calculated constructive interference strips, which agree well with the experimental results. Especially, in the limit of sΦLZS>>δ2, δ12, equation (7) can be simplified as

An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m11.jpg
8

Intuitively, this result is straightforward to understand, as in the large-amplitude limit the accumulated phase θ1 is two times the area of a rectangle with length T/2 and width ωTLS1ωTLS2.

(II) νAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m45.jpgν2 and ν>>ν1: M1 acts as a total transmission mirror and M2 acts as a beam splitter, that is, PLZ1An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m45.jpg1 and PLZ2An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m45.jpg1/2. In this case, equation (3) can be simplified as

An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m12.jpg
8

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:

An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m13.jpg
9

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 sΦLZS>>δ2, equation (10) has the simple form,

An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m14.jpg
11

which is also readily understood because in the large-amplitude limit the accumulated phase θII is two times the area of a triangle with base length T/2 and height sΦLZS.

(III) ν1<ν<ν2: This region is more interesting and complex. Here, M1 acts as a beam splitter, while M2 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 PLZ corresponding to the transmission coefficient of M1 and M2 for comparison with the experimental data. We find that for certain sweeping rates, LZ transition probability resulting from M2 is quite low. Therefore, M2 can be treated as a total reflection mirror, while M1 is still acting as a good beam splitter. The interference fringes generated by M2 thus disappear (the fringes tend to fade out) and the interference fringes generated by M1 dominate, displayed as a chain of 'hot spots' marked by the circles in Figure 2a.

When both M1 and M2 can be treated as beam splitters, all three paths (1, 2, and 3) contribute to the interference. According to equation (3), P1 is maximized in the condition

An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m15.jpg
12

It is noted that under this condition the term An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m16.jpg in equation (3) equals 2. 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:

An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m17.jpg
13

where Γ[ρ] includes the effects of energy relaxation. Figure 3a shows the calculated population of |1An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg as a function of T and ΦLZS. Figure 3b shows the extracted data for different T and ΦLZS 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 ΦLZS=10mΦ0 (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 M1 is always greater because Δ1 is much smaller than Δ2. 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 M2 is a minimum. M2 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.

Figure 3
Numerically simulated LZS interference pattern and control of a generalized W state in a phase qubit coupled to two TLSs.

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, |ψAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg=α|1g1g2An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg+β|0e1g2An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg+γ|0g1e2An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg, evolving in the sub-space spanned by the three product states during the operation of sweeping flux bias. In order to quantify the generalized W state, we define An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m18.jpg where σ=α, β, γ. In Figure 3d, w is plotted as a function of T and ΦLZS. 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 environment-induced decoherence and thus can be used as a hybrid qubit. The excellent agreement between our data and theory over the entire T−ΦLZS 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 solid-state qubits and/or hybrid qubits6,8.

Methods

Experimental detail

Figure 1a shows the principal circuitry of the measurement. The flux bias and microwave are fed through the on-chip 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 |0An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg and |1An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg in the left well. A microwave π-pulse is applied to rotate the qubit from |0An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg to |1An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg. 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 |1An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg or remain in the left well if it was in |0An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg. The flux bias is then lowered to the third platform, where the double-well 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 dc-SQUID magnetometer inductively coupled to the qubit. By mapping the states |0An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg and |1An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg into the left and right wells, respectively, the probability of finding the qubit in state |1An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg is obtained. We obtained T1An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m45.jpg70 ns from energy relaxation measurement (Supplementary Fig. S1a), TRAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m45.jpg80 ns from Rabi oscillation (Supplementary Fig. S1b), T2*An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m45.jpg60 ns from Ramsey interference fringe (Supplementary Figs S1c and S1d) and T2An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m45.jpg137 ns from spin-echo (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 as23,24

An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m19.jpg
14

In the two-level approximation the effective Hamiltonian of the qubit is An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m20.jpg here the flux bias (Φ) dependent energy-level spacing of the qubit, ħω10=E1E0, can be obtained numerically by solving the eigenvalues problem associated with the full Hamiltonian of the phase qubit25. The Hamiltonian of the ith TLS can be written as An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m21.jpg where ħωTLSi is the energy-level spacing of the ith TLS. The interaction Hamiltonian between the qubit and the ith TLS is An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m22.jpg where Δi is the coupling strength between the qubit and the ith TLS and An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m23.jpg 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 |0An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg and |1An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg (|giAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg and |eiAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg) 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 |0g1g2An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg. When the couplings between the qubit and TLSs are off, we use a π-pulse to pump the qubit to |1An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg (thus the system is in |1g1g2An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg). 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 π-pulse 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, |1g1g2An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg, |0e1g2An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg, and |1g1e2An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg, 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 method16,22 to obtain the probability of finding the system in |1g1g2An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg at the end of the triangular pulse. We use |aAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg=[1,0,0]T, |bAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg=[0,1,0]T and |cAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg=[0,0,1]T to denote the instantaneous eigenstates of the time-dependent 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 |aAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg=|1g1g2An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg. At the crossing times t=t1 and t=t2, the incoming and outgoing states are connected by the transfer matrix:

An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m24.jpg
15

and

An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m25.jpg
16

respectively. Here sin2(θi/2)=PLZi (i=1, 2) is the LZ transition probability at the ith avoided crossing. An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m26.jpg where θSi is the Stokes phase16,22, the value of which depends on the adiabaticity parameter ηii2/υ in the form of θSi=π/4+ηi (ln ηi−1)+arg Γ(1−iηi), where Γ is the Gamma function. In the adiabatic limit θS→0, and in the sudden limit θS=π/4. At crossing times t=t3 and t=t4, we have An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m27.jpg respectively. The outgoing state at t=ti and the incoming state at t=ti+1 (i=0, 1, 2, 3, 4) are thus connected by the propagator

An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m28.jpg
17

where ωi(t) is the energy-level spacing frequency of |iAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg (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

An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m29.jpg
18

The probability of finding the system remaining at the initial state is P1=|An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m47.jpg1g1g2|Û|1g1g2An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg|2. Its concrete form is equation (3), in which An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m30.jpg and An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m31.jpg 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: An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m32.jpg

path 2: An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m33.jpg

path 3: An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m34.jpg

Denoting ωi(t) as the energy-level spacing frequency corresponding to path i (i=1, 2, 3), then θI and θII have the forms An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m35.jpg and An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m36.jpg 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:

An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m37.jpg
19

where

An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m38.jpg

where ω10(t)=ω10,dcνt, ν[equivalent]2sΦLZS/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 |0gAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg and dephasing process phenomenologically. In a concrete expression, equation (19) can be written as (for ease of discussion, we relabel |1gAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg and |0eAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg as |aAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg and |bAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg, respectively)

An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m39.jpg
20

with ρba=ρab*. Here Γα (α=a, b) is the relaxation rate from state |αAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg to the ground state |0gAn external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg. The decoherence rate An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m40.jpg 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 Mi (i=1, 2), that is, the LZ tunneling probability PLZ, 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 ħωTLS1 and ħωTLS2. The Hamiltonian in the basis of |1g1g2An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg, |0e1g2An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg, |0g1e2An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m46.jpg 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 An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m48.jpg respectively)

An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m41.jpg
21

where the diagonal elements ρii are the populations, off-diagonal elements ρij(ij) describe coherence, and An external file that holds a picture, illustration, etc.
Object name is ncomms1050-m42.jpg are the rates of decoherence. The remaining three elements' equations are determined by ρij*=ρji. The numerically simulated LZS interference pattern is shown in Figure 3a, which agrees with the experimental results excellently.

Author contributions

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.

Additional information

How to cite this article: Sun, G. et al. Tunable quantum beam splitters for coherent manipulation of a solid-state tripartite qubit system. Nat. Commun. 1:51 doi: 10.1038/ncomms1050 (2010).

Supplementary Material

Supplementary Figures:

Supplementary Figures S1-S3

Acknowledgments

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. DMR-0325551. 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.

References

  • Makhlin Y., Schön G. & Shnirman A. Quantum-state engineering with Josephson-junction devices. Rev. Mod. Phys. 73, 357–400 (2001).
  • You J. & Nori F. Superconducting circuits and quantum informations. Phys. Today 58, 42–47 (2005).
  • Clarke J. & Wilhelm F. K. Superconducting quantum bits. Nature 453, 1031–1042 (2008). [PubMed]
  • Martinis J. M. Superconducting phase qubits. Quantum Inf. Process. 8, 81–103 (2009).
  • Simmonds R. W. et al. . Coherent interactions between phase qubits, cavities, and TLS defects. Quantum Inf. Process. 8, 117–131 (2009).
  • Zagoskin A. M., Ashhab S., Johansson J. R. & Nori F. Quantum two-level systems in Josephson junctions as naturally formed qubits. Phys. Rev. Lett. 97, 077001 (2006). [PubMed]
  • Wei L. F., Johansson J. R., Cen L. X., Ashhab S. & Nori F. Controllable coherent population transfers in superconducting qubits for quantum computing. Phys. Rev. Lett. 100, 113601 (2008). [PubMed]
  • Neeley M. et al. . Process tomography of quantum memory in a Josephson-phase qubit coupled to a two-level state. Nature Phys. 4, 523–526 (2009).
  • Xu H. et al. . Spectroscopy of three-particle entanglement in a macroscopic superconducting circuit. Phys. Rev. Lett. 94, 027003 (2005). [PubMed]
  • Dür W., Vidal G. & Cirac J. I Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62, 062314 (2000).
  • Roos C. F. et al. . Control and measurement of three-qubit entangled states. Science 304, 1478–1480 (2004). [PubMed]
  • Häffner H. et al. . Scalable multiparticle entanglement of trapped ions. Nature 438, 643–646 (2005). [PubMed]
  • Bennett C. H. & DiVincenzo D. P. Quantum information and computation. Nature 404, 247–255 (2000). [PubMed]
  • Petta J. R., Lu H. & Gossard A. C. A coherent beam splitter for electronic spin states. Science 327, 669–672 (2010). [PubMed]
  • Oliver W. D. et al. . Mach-Zehnder interferometry in a strongly driven superconducting qubit. Science 310, 1653–1657 (2005). [PubMed]
  • Sillanpää M., Lehtinen T., Paila A., Makhlin Y. & Hakonen P. Continuous-time monitoring of Landau-Zener interference in a cooper-pair box. Phys. Rev. Lett. 96, 187002 (2006). [PubMed]
  • Wilson C. M. et al. . Coherence times of dressed states of a super conducting qubit under extreme driving. Phys. Rev. Lett. 98, 257003 (2007). [PubMed]
  • Izmalkov A. et al. . Consistency of ground state and spectroscopic measurements on flux qubits. Phys. Rev. Lett. 101, 017003 (2008). [PubMed]
  • Rudner M. S. et al. . Quantum phase tomography of a strongly driven qubit. Phys. Rev. Lett. 101, 190502 (2008). [PubMed]
  • Sun G. et al. . Population inversion induced by Landau–Zener transition in a strongly driven rf superconducting quantum interference device. Appl. Phys. Lett. 94, 102502 (2009).
  • LaHaye M. D., Suh J., Echternach P. M., Schwab K. C. & Roukes M. L. Nanomechanical measurements of a superconducting qubit. Nature 459, 960–964 (2009). [PubMed]
  • Shevchenko S., Ashhab S. & Nori F. Landau-Zener-Stückelberg interferometry. Phys. Rep. 492, 1–30 (2010).
  • Ku L. -C. & Yu C. C. Decoherence of a Josephson qubit due to coupling to two-level systems. Phys. Rev. B 72, 024526 (2005).
  • Lupascu A., Bertet P., Driessen E. F. C., Harmans C. J. P. M. & Mooij J. E. One- and two-photon spectroscopy of a flux qubit coupled to a microscopic defect. Phys. Rev. B 80, 172506 (2009).
  • Han S., Rouse R. & Lukens J. E. Generation of a population inversion between quantum states of a macroscopic variable. Phys. Rev. Lett. 76, 3404–3407 (1996). [PubMed]

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