a r X i v : c o n d  m a t / 0 6 0 7 1 8 0 v 2 [ c o n d  m a t . m e s  h a l l ] 1 M a r 2 0 0 7
Controllable Coupling between Flux Qubitand Nanomechanical Resonator by Magnetic Field
Fei Xue, Y. D. Wang, and C. P. Sun
Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100080, China
H. Okamoto, H. Yamaguchi, and K. Semba
NTT Basic Research Laboratories, NTT Corporation, Atsugishi, Kanagawa 2430198, Japan
(Dated: June 26, 2018)We propose an active mechanism for coupling the quantized mode of a nanomechanical resonatorto the persistent current in the loop of superconducting Josephson junction (or phase slip) ﬂux qubit.This coupling is independently controlled by an external coupling magnetic ﬁeld. The whole systemforms a novel solidstate cavity QED architecture in strong coupling limit. This architecture can beused to demonstrate quantum optics phenomena and coherently manipulate the qubit for quantuminformation processing. The coupling mechanism is applicable for more generalized situations wherethe superconducting Josephson junction system is a multilevel system. We also address the practicalissues concerning experimental realization.
PACS numbers: 74.78.Fk, 85.85.+j, 03.65.Lx
I. INTRODUCTION
In recent years, great advances in improving the coherence of superconducting qubit have made it a promising candidate for the physical realization of quantum information processing. Single qubit Rabi oscillation andRamsey fringe have been observed and two qubits entanglement is also achieved. Meanwhile, as an artiﬁcialtwolevel atom, superconducting qubit is adjustable (e.g.by ﬂux, bias voltage, etc.) and scalable. These featuresare favorable for quantum state engineering. A numberof protocols are proposed to engineer the superconducting qubit to form a quantum network. Among them, avery intriguing and successful example is the circuit QEDarchitecture [1]. By coupling the Cooper pair box (chargequbit) to the quantized ﬁeld of a coplanar superconducting transmission line, a macroscopic solidstate analog of cavity QED is realized on chip. Most recently, vacuumRabi oscillations are observed in a coupling system of 3JJ ﬂux qubit and LC circuit [2]. Quantum optical phenomena in traditional cavity QED can be demonstratedin this solid state composite system. Furthermore, dueto its special structure, it oﬀers a number of advantages,such as strong coupling and easy controllability. Thus,some protocols that cannot be realized previously in theoptical cavity QED now becomes possible [3, 4].
The circuit QED experiments motivate us to investigate the possibility of substituting other quantum solidstate devices for the transmission line. It is much desirable to couple Josephson junction qubit to a device withlow energy consuming and small size. If the strong coupling and easy controllability can also be achieved, we getanother favorable cavity QED structure. A possible candidate of this solid state device is the nanomechanical resonator (NAMR). Nanomechanical resonators of GHz oscillation have already been observed. It is supposed thatthe nanomechanical resonator enter the quantum regimeat the attainable temperature of the dilution refrigerator.The schemes of coupling Josephson charge qubit or phasequbit to NAMR have already been proposed. Based onthese coupling mechanisms, several quantum state engineering protocols are put forward [5, 6, 7, 8, 9]. However,
due to the diﬃculty to reach quantum regime of NAMR,those protocols have not been implemented experimentally yet. On the other hand, the coupling mechanism of NAMR and ﬂux qubit is also an attractive problem sincethe ﬂux qubit is supposed to have longer coherence timeas it is less aﬀected by the charge ﬂuctuation in the structure. To our best knowledge, this has not been studied indetail previously. Here, we present a novel mechanism of coupling NAMR to ﬂux qubit. As we present below, thecoupling strength between the NAMR and the ﬂux qubitcan be adjusted conveniently by a coupling magnetic ﬁeldand turned on and oﬀ within the coherence time of ﬂuxqubit. Since the coupling magnetic ﬁeld is independent of the single qubit operation, it is possible to make it strongenough even for GHz oscillation. Therefore our proposalis in principle possible to approach the “strong couplingregime” of cavity QED at attainable temperature of dilution refrigerator. This coupling system acts as an analogof cavity QED system with more ﬂexibility. We expectthat it enables various applications to quantum information processing and quantum state engineering.This paper is organized as follows: in Sec.II, we brieﬂyreview the setup of 3 Josephsonjunction ﬂux qubit andNAMR as well as their experimental progresses. Thenwe get into the coupling mechanism for ﬂux qubit andNAMR. This coupling mechanism can be equally applied to rfSQUID ﬂux qubit and phase slip ﬂux qubit. InSec.III, the spectrum of the coupling system is presentedin the “weak coupling” and the “strong coupling” limitsrespectively. The readout and quantum nondemolitionmeasurement for the ﬂux qubit is studied in Sec.IV. Wealso consider the application of this coupling mechanismin quantum computation in Sec.V and generalize the coupling system to beyond spinboson model in Sec.VI. The
2possible problems on experimental realization and theirsolutions are given in Sec. VII. In Sec.VIII, some discus
sions and remarks are included.
II. COUPLING NANOMECHANICALRESONATOR AND FLUX QUBIT BY ATRANSVERSE MAGNETIC FIELDA. The 3junction ﬂux qubit
Josephson charge qubit system has been used to couple with NAMR. Here, we study another superconducting qubit system – 3 Josephson junction (3JJ) ﬂux qubit[10, 11, 12, 13, 14]. In contrast with charge qubits, the
ﬂux qubit is far less sensitive to charge ﬂuctuations. Estimations show that the ﬂux qubit have a relatively highquality factor [11, 13]. The conﬁguration of ﬂux qubit
consists of a superconducting loop with three Josephson junctions, and the Josephson coupling energy is muchlarger than the charging energy for each junction. Thequantum state of this system is mainly determined bythe phase degree of freedom. The Josephson energy of the three Josephson junction loop reads
U
(
ϕ
1
,ϕ
2
) =
−
E
J
cos
ϕ
1
−
E
J
cos
ϕ
2
−
αE
J
cos(2
πf
−
ϕ
1
−
ϕ
2
)
,
(1)where the constraint of ﬂuxoid quantization has alreadybeen taken into account. Here,
E
J
is the Josephsoncoupling energy of two identical junction and
ϕ
1
,
ϕ
2
are phase diﬀerences across the two junctions respectively. The Josephson energy of the third junction is
αE
J
,
f
= Φ
f
/
Φ
0
with Φ
f
the external ﬂux applied in theloop and Φ
0
=
h/
2
e
the ﬂux quantum. In the vicinityof
f
= 0
.
5, if
α >
0
.
5, a doublewell potential is formedwithin each 2
π
×
2
π
cell in the phase plane and the twolowest stable classical states have persistent circulatingcurrents
I
p
= 2
eE
J
1
−
(1
/
2
α
)
2
/
with opposite directions. Therefore, the ﬂux qubit is also called persistentcurrent qubit. Within the qubit subspace spanned by
{
0
,

1
}
(

0
and

1
denote clockwise and counterclockwise circulating states respectively), the Hamiltonian of the qubit system reads as
H
f
=
ω
f
σ
z
+ ∆
σ
x
= Ω˜
σ
z
,
(2)with
ω
f
=
I
p
Φ
0
(
f
−
0
.
5) is the energy spacing of thetwo classical stable states and ∆ the tunneling splitting between the two states, Ω =
ω
2
f
+ ∆
2
and ˜
σ
z
=cos
θσ
z
+ sin
θσ
x
, tan
θ
= ∆
/ω
f
. The oﬀset of
f
from0
.
5 determines the level splitting of the two states andthe barrier for quantum tunneling between the statesstrongly depends on the value of
α
. If the third junctionis replaced by a dc SQUID, both
f
and
α
are tunable inexperiments by the applied ﬂux or the microwave current[10, 11].
B. The nanomechanical resonator
The ﬂexural modes of thin beams can be described bythe socalled EulerBernoulli equations [15]. In our proposal only the fundamental ﬂexural mode of the NAMRis taken into account. All the other modes have amuch smaller coupling to ﬂux qubit and can be neglected [16, 17]. In this case, the NAMR is modeled as a
harmonic oscillator with a highQ mode of frequency
ω
b
.The Hamiltonian without dissipation reads [6, 7]
H
=
p
2
z
2
m
+ 12
mω
2
b
z
2
.
(3)In pursuing the quantum behavior of macro scale objectthe nano scale mechanical resonator plays an importantrole. At suﬃcient low temperature the zeropoint ﬂuctuation of nano mechanical resonator will be comparableto its thermal Brownian motion. The detection of zeropoint ﬂuctuation of the nano mechanical resonator cangive a direct test of the Heisenberg’s uncertainty principle. With a sensitivity up to 10 times the amplitudeof the zeropoint ﬂuctuation, LaHaye
et al.
have experimentally detected the vibrations of a 20MHz mechanicalbeam of tens micrometers size [18]. For a 20MHz mechanical resonator its temperature must be cooled below1 mK to suppress the thermal ﬂuctuation. For a GHzmechanical resonator a temperature of 50 mK is suﬃcient to eﬀectively freeze out its thermal ﬂuctuation andlet it enter quantum regime. This temperature is alreadyattainable in the dilution refrigerator.The lithographic technology for NAMR is rather mature. The important advantages of NAMR are the potentially higher quality factor and frequency comparable tosuperconducting qubit. Ever since the early demonstration of a radio frequency mechanical resonator at Caltech[19], great advances have been made. The attainable frequencies for the fundamental ﬂexural modes can reach590 MHz for the doublyclamped SiC mechanical resonator of the size 1
×
0
.
05
×
0
.
05
µ
m [20] and 1 GHzoscillation frequency has also been measured [21]. It isargued that quantized displacements of the mechanicalresonator were observed despite of some opposite opinions [22]. For a 1
µ
m beam a quality factor
Q
of 1700has been observed at a frequency of 110 MHz [23]. In acarefully designed antenna shape, Gaidarzhy
et al.
haveachieved
Q
= 11000for 21 MHz oscillation at the temperature of 60 mK and
Q
= 150 for 1
.
49 GHz oscillation atthe temperature of 1K with a comparatively large doubleclamped beam [24]. The signiﬁcantly small size (
∼
µ
m)of the NAMR is also favorable for incorporating it in thesuperconducting qubit circuit.
C. The composite system with tunable coupling
To achieve a “strong” interaction, the coupling dynamical variable usually should be the dominant one in thedynamics of the composite system. For the Josephson
3
yzx
f
0
FIG. 1: (Color on line) A 3Josephsonjunction ﬂux qubitloopis located in the xy plane and a NAMR is integrated in theloop(indicated by a green line). The zdirection oscillationof the NAMR couples to the current in the ﬂux qubit loopby a transverse magnetic ﬁeld
B
0
in ydirection. Anothertunable magnetic ﬂux Φ
f
penetrates this loop tunes the freeHamiltonian of the 3JJ system.
phase qubit [25, 26], the phase degree of freedom domi
nates the dynamics and the bias current coupled with thephase is modiﬁed by the dilatational motion of the piezoelectric dilatational resonator [27]. While for the Josephson charge qubit, the coupling mechanism is that theresonator displacement modiﬁes the eﬀective bias chargeof a Cooperpair box [7, 8, 17, 28]. These previous inves
tigations enlighten us to consider the coupling betweenpersistent current in superconducting ﬂux qubit loop andthe motion of nano mechanical resonator.Since the Josephson coupling energy of each junctionin the ﬂux qubit is much larger than that in the chargequbit, the persistent current in the loop could be abouthundreds of nano ampere [12] in contrast with the criticalcurrent of the charge qubit (usually about 20
∼
50 nA).The magnitude of this persistent current naturally leadsus to consider the magnetomotive displacement actuation and sensing technique [15, 19]. It is well known when
a current passes through a beam with conducting material, the perpendicular arrangeed of an external magneticﬁeld and the direction of the current generates a Lorentzforce in the plane of the beam. This is just the actuationpart of the magnetomotive technique. Meanwhile, the resulted displacement of the beam under the Lorentz forcegenerates an electromotive force, or voltage, which servesas measurement. Thus, if the doublyclamped nano beamcoated with superconducting material is incorporated inthe superconducting qubit loop, the persistent current induces a Lorentz force with opposite directions for clockwise and counterclockwise current. The oscillation of theNAMR is modulated by these Lorentz force. In this way,the quantized harmonic oscillation mode of the beam iscoupled to the quantum state of the ﬂux qubit system.And this is just the coupling mechanism considered inour paper.Our proposal is illustrated in Fig.1. A 3JJ system isfabricated on the xy plane. The external applied magnetic ﬂux Φ
f
is enclosed in the loop modulated by thecontrol lines (the lines are not plotted). The 4JJ version of ﬂux qubit system can also be used here to allowthe modulation of the eﬀective Josephson energy of thethird junction and hence the tunneling amplitude of thetwo current states. One side of the loop (indicated bythick (green) rod) is suspended from the substrate andclamped at both ends. This can be fabricated with adoublyclamped nanomechanical beam coated with superconductor or with the superconductor itself as the mechanical resonator. A magnetic ﬁeld
B
0
is applied in the
y
direction. As we discussed above, the circulating suppercurrent under the magnetic ﬁeld generates a Lorentzforce in the zdirection. The magnitude of the force is
B
0
I
p
L
, with
L
the eﬀective length of the resonator along
x
direction (
L
=
ξL
0
and
L
0
is the actual length of theresonator,
ξ
a factor depending on the oscillation mode[29], for the fundamental oscillation mode of a doublyclamped beam
ξ
≈
0
.
8). This force results a forced termin the Hamiltonian, which reads
H
fb
=
Fz
=
B
0
I
p
Lz
.With the twolevel approximation of 3JJ loop and thesinglemode boson approximation, the coupling is writtenas
H
fb
=
g
a
+
a
†
σ
z
,
(4)for
z
∼
a
+
a
†
. Here,
g
=
B
0
(
t
)
I
p
Lδ
z
(5)and
δ
z
=
/
2
mω
b
is the amplitude of zero point motionin zdirection of the NAMR, with
m
the eﬀective massof the resonator,
ω
b
the frequency of the fundamentalﬂexural mode;
a
(
a
†
) is the creation (annihilation) operator of the mode of the ﬂexural motion in zdirection.
σ
z
=

0
0
−
1
1

is the Pauli matrix deﬁned in thebasis of
{
0
,

1
}
. We see that this interaction
H
fb
actually couples the two systems. Together with the freeHamiltonian of ﬂux qubit and NAMR, the Hamiltonianof whole system reads
H
=
ω
b
a
†
a
+
ω
f
σ
z
+ ∆
σ
x
+
g
a
+
a
†
σ
z
.
(6)An important advantage of this coupling mechanismis the convenient controllability. As seen from Eq.(5),the coupling constant is directly dependent on the applied coupling magnetic ﬁeld
B
0
. Thus, both the magnitude and sign of the coupling constant can be modiﬁed.What’s more important is that the control parameter
B
0
in coupling coeﬃcient Eq.(5) is independent from the parameters of free Hamiltonian, such as bias voltage andexternal magnetic ﬂux Φ
f
. This means the free Hamiltonian and the interaction Hamiltonian can be manipulatedindependently. This full controllability is a rather favorable feature for quantum state engineering and quantuminformation processing protocols. It is in contrast withthe coupling of charge qubit and NAMR, where the coupling strength is controlled by the bias voltage which isalso the crucial parameter to determine the energy spacing of the charge qubit. For example, for bangbang
4cooling of NAMR by charge qubit [8], the bias voltageshould be set to certain value to induce desirable damping. Therefore the onandoﬀ of the interaction betweenthe qubit and NAMR can only approximately controlledby detuning and this can result in harmful reheating of the NAMR. But in our present coupling mechanism, boththe coupling coeﬃcient and the energy spacing are independent. Thus, this “bangbang” cooling protocol shouldbe implemented more reliable by the ﬂux qubit and theNAMR with above coupling mechanism.To estimate the coupling strength, we use the following parameters in refs. [7, 30, 31]:
I
p
= 660 nA,
L
0
= 3
.
9
µ
m,
ω
b
= 100 MHz,
δ
z
= 2
.
6
×
10
−
13
m,
Q
= 2
×
10
4
and assume the applied magnetic ﬁeld to be
B
0
= 5 mT.Then we have
g
≈
4
.
01 MHz. Hence we see that the“strong coupling” regime for cavity QED is potentiallyrealizable in our scheme. This regime requires the periodof the Rabi oscillation 1
/g
, is much shorter than boththe decoherence time 1
/γ
of the twolevel system andthe average lifetime 1
/κ
=
Q/ω
b
of the “photon” in the“cavity”[32]. For this composite system, the decoherencetime for ﬂux qubit is 1
−
10
µ
s and the cavity lifetime isabout 200
µ
s, while the Rabi oscillation time 0
.
016
µ
s ismuch shorter than the two lifetime scale. For GHz oscillation, the quality factor is rather low[24] (1
.
49 GHz with
Q
= 150). This corresponding to a much shorter cavitylifetime (about 0
.
1
µ
s. However, the coupling strengthcan be increased by larger coupling magnetic ﬁeld. Forexample, if we take
B
0
= 50 mT, the Rabi oscillation period 1
/g
≈
0
.
016
µ
s which is still short enough to reach“strong coupling regime”. Therefore, this protocol mightbe promising in dilution refrigerator (several tens of millikelvin).
D. Phase slip ﬂux qubit and NAMR
Most recently, a new type of ﬂux qubit – phase slipﬂux qubit is proposed based on coherent quantum phaseslip [33, 34]. Phase slip ﬂux qubit is formed by a
highresistance superconducting thin wire instead of theJosephson junctions. The computational basis are alsothe two opposed persistent current states. Our couplingscheme can be equally applicable to this type of ﬂuxqubit. There are two important advantages to consider
yzx
f
0
FIG. 2: (Color on line) A superconducting thin wire in theloop acts as the center of phase slip ﬂux qubit and a NAMR.
phase slip ﬂux qubit. First, since the superconductingthin wire can be fabricated by a suspended carbon nanotube, it acts as the phase slip center and the NAMR simultaneously. The circuit conﬁguration is simpliﬁed (seeFig. 2). Secondly, if one use this qubit, one can free fromany ﬂuctuator due to imperfection or twolevel systemshidden in the dielectric layer of Josephson junction.
III. ENERGY SPECTRUM OF COUPLINGSYSTEM
The Larmor frequency of superconducting qubit isabout the order of 10 GHz, while the frequency of NAMRonly reaches several hundred MHz with quality factor10
4
at present stage. Thus, the composite system of ﬂuxqubit and NAMR is in large detuning regime of cavityQED, i.e., the following condition is satisﬁed
g

Ω
−
ω
b
 ≪
1
.
(7)However, the superconducting ﬂux qubit and the NAMRis nonresonant, i.e. Ω
≫
ω
b
. This is in contrast withYale’s circuit QED experiment where the Cooper pairbox is resonant with the 1D transmission line [35]. In thefollowing, we discuss the energy spectrum of our modelin two diﬀerent regimes:
g
≪
ω
b
(denoted as “weak coupling”) and
g
≈
ω
b
(denoted as “strong coupling”). Inour proposal, the two regimes can be reached by varyingthe applied coupling magnetic ﬁeld
B
0
. And the energyspectrum are qualitatively diﬀerent from each other. Itis notable that Ref.[36] has proved that the dispersivemeasurement back action can be enhanced or reduced bycavity damping respectively in the two regimes.
A. “Weak coupling” and sideband spectrum
For the parameters
B
0
= 5 mT,
g
= 4
.
01 MHz, both
g/
Ω and
g/ω
b
are much smaller than 1. In this case, theenergy spectrum can be calculated by Floquet approachor by Fr¨olich transformation. After performing a unitarytransformation on the srcinal Hamiltonian (6), we getthe eﬀective Hamiltonian
H
eﬀ1
≈
ω
b
a
†
a
+ Ω˜
σ
z
+
ig
2
sin2
θω
b
a
2
−
a
†
2
˜
σ
y
+
g
2
sin
θ
Ω
a
+
a
†
2
(cos
θ
˜
σ
x
+ sin
θ
˜
σ
z
)
.
(8)The spectrum are
nω
b
+
m
Ω plus some small oﬀdiagonaltransition terms that are of order
O
(
g/ω
) or
O
(
g/
Ω).As shown in Fig.3, the energy levels of the two subsystems are weakly perturbed by the coupling due to thelarge detuning. By applying microwave pulse to inducethe transition between those levels, the blue sideband(

00
→ 
11
) and red sideband transition (

01
→ 
10
)can be observedin addition to the main zerophotontransition

00
→
10
[37]. This atomic physics phenomenon