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Nature Communications

Nat Commun. 2010 August; 1(5): 1–7.

doi: 10.1038/ncomms1050

PMCID: PMC2982164

Guozhu Sun,^{a,}^{1,}^{2,}^{3} Xueda Wen,^{3} Bo Mao,^{2} Jian Chen,^{1,}^{3} Yang Yu,^{b,}^{3} Peiheng Wu,^{1,}^{3} and Siyuan Han^{c,}^{1,}^{2}

Received 2010 March 31; Accepted 2010 July 6.

Copyright © 2010, Nature Publishing Group, a division of Macmillan Publishers Limited. All Rights
Reserved.

This work is licensed under a Creative Commons Attribution-NonCommercial-Share
Alike 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/

This article has been cited by other articles in PMC.

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 *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}*−

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.

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 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, 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

1

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}−*s*Φ(*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*=|d*ω*_{10}(Φ)/dΦ| being the diabatic energy-level slope of state
|1*g*_{1}*g*_{2} 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(−2*π*(Δ^{2}/*ν*)), where
*ħν*d*E*/d*t* denotes the rate of the energy spacing change for
noninteracting levels, and 2*ħ*Δ 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 |0*g*_{1}*g*_{2}. A
resonant microwave *π*-pulse is applied to coherently transfer the qubit to
|1*g*_{1}*g*_{2}. A triangular flux bias, Φ(*t*), with
variable width *T* and amplitude Φ_{LZS}

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 |1, that is, the system in the state
|1*g*_{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.

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

3

where *P*_{LZi} (*i*=1,2) is the LZ transition probability at
the *i*th 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 *i*th avoided crossing is due to the Stokes phase16,22
*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
2*π*Δ_{i}^{2}/*ν*_{i}=1 (*i*=1,
2). From the spectroscopy data, we have Δ_{1}/2*π*=10 MHz and
Δ_{2}/2*π*=32 MHz; thus,
*ν*_{1}/2*π*=3.94×10^{−3} GHz ns^{−1} and
*ν*_{2}/2*π*=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) *ν**ν*_{1} and *ν**ν*_{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

4

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

5

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

6

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

7

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
*s*Φ_{LZS}>>*δ*_{2}, *δ*_{12}, equation (7)
can be simplified as

8

Intuitively, this result is straightforward to understand, as in the large-amplitude
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) *ν**ν*_{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

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:

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 *s*Φ_{LZS}>>*δ*_{2},
equation (10) has the simple form,

11

which is also readily understood because in the large-amplitude limit the accumulated
phase *θ*_{II} is two times the area of a triangle with base length
*T*/2 and height *s*Φ_{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

12

It is noted that under this condition the term in equation (3) equals 2*nπ*. 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:

13

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}=10*m*Φ_{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.

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*=*α*|1*g*_{1}*g*_{2}+*β*|0*e*_{1}*g*_{2}+*γ*|0*g*_{1}*e*_{2},
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 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.

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*−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 solid-state qubits and/or hybrid qubits6,8.

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 |0 and |1 in the left well. A microwave *π*-pulse 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 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 |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 spin-echo (Supplementary Fig. S1e) in the region free of
qubit–TLS coupling.

For the coupled qubit–TLS system, the Hamiltonian can be written as23,24

14

In the two-level approximation the effective Hamiltonian of the qubit is here the flux bias (Φ) dependent
energy-level 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 qubit25. The Hamiltonian of the *i*th TLS can be written as where
*ħω*_{TLSi} is the energy-level spacing of the *i*th TLS. The
interaction Hamiltonian between the qubit and the *i*th TLS is where Δ_{i} is the coupling
strength between the qubit and the *i*th TLS and are the Pauli operators acting on the states of the
qubit (the *i*th 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 *i*th TLS). In our experiment the
initial state is prepared in the system's ground state
|0*g*_{1}*g*_{2}. When the couplings between the qubit and
TLSs are off, we use a *π*-pulse to pump the qubit to |1 (thus the system is in
|1*g*_{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 *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, |1*g*_{1}*g*_{2},
|0*e*_{1}*g*_{2}, and
|1*g*_{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.

We use the transfer matrix method16,22 to obtain the probability of
finding the system in |1*g*_{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 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
|*a*=|1*g*_{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:

15

and

16

respectively. Here
sin^{2}(*θ*_{i}/2)=*P*_{LZi}
(*i*=1, 2) is the LZ transition probability at the *i*th avoided crossing. where *θ*_{Si} is
the Stokes phase16,22, 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−i*η*_{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

17

where *ω*_{i}(*t*) is the energy-level 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

18

The probability of finding the system remaining at the initial state is
*P*_{1}=|1*g*_{1}*g*_{2}|*Û*|1*g*_{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 energy-level spacing frequency
corresponding to path *i* (*i*=1, 2, 3), then *θ*_{I} and
*θ*_{II} have the forms and
respectively.

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:

19

where

where *ω*_{10}(*t*)=*ω*_{10,dc}−*νt*,
*ν*2*s*Φ_{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 |0*g* and dephasing process phenomenologically. In a
concrete expression, equation (19) can be written as (for ease of discussion, we relabel
|1*g* and |0*e* as |*a* and |*b*, respectively)

20

with *ρ*_{ba}=*ρ*_{ab}*. Here
Γ_{α} (*α*=*a*, *b*) is the relaxation rate from state
|*α* to the ground state |0*g*. 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.

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
|1*g*_{1}*g*_{2},
|0*e*_{1}*g*_{2}, |0*g*_{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)

21

where the diagonal elements *ρ*_{ii} are the populations,
off-diagonal 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.

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.

**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).

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.

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