We demonstrate a new method to directly manipulate the state of individual
two-level systems (TLS) in phase qubits. It allows one to characterize the
coherence properties of TLS using standard microwave pulse sequences, while the
qubit is used only for state readout. We apply this method to measure the
temperature dependence of TLS coherence for the first time. The energy
relaxation time $T_1$ is found to decrease quadratically with temperature for
the two TLS studied in this work, while their dephasing time measured in Ramsey
and spin-echo experiments is found to be $T_1$ limited at all temperatures.Comment: 4 pages, 5 figure

Bushev, P.

Cole, J. H.

Lisenfeld, J.

Lukashenko, A.

Müller, C.

Shnirman, A.

Ustinov, A. V.

English

arXiv

J. Lisenfeld

C. Müller

J. H. Cole

P. Bushev

A. Lukashenko

A. Shnirman

A. V. Ustinov

Crossref

Physical Review Letters

Measuring the Temperature Dependence of Individual Two-Level Systems by Direct Coherent Control

We demonstrate a new method to directly manipulate the state of individual two-level systems (TLSs) in phase qubits. It allows one to characterize the coherence properties of TLSs using standard microwave pulse sequences, while the qubit is used only for state readout. We apply this method to measure the temperature dependence of TLS coherence for the first time. The energy relaxation time T(1) is found to decrease quadratically with temperature for the two TLSs studied in this work, while their dephasing time measured in Ramsey and spin-echo experiments is found to be T(1) limited at all temperatures

Lisenfeld, J

Muller, C

Cole, J

Bushev, P

Lukashenko, A

Shnirman, A

Ustinov, A

RMIT Research Repository

Measuring the temperature dependence of individual two-level systems by direct coherent control

We demonstrate a new method to directly manipulate the state of individual two-level systems (TLSs) in phase qubits. It allows one to characterize the coherence properties of TLSs using standard microwave pulse sequences, while the qubit is used only for state readout. We apply this method to measure the temperature dependence of TLS coherence for the first time. The energy relaxation time T1 is found to decrease quadratically with temperature for the two TLSs studied in this work, while their dephasing time measured in Ramsey and spin-echo experiments is found to be T1 limited at all temperatures

Mueller, C.

University of Queensland eSpace

Measuring the Temperature Dependence of Individual Two-Level Systems
by Direct Coherent Control
J. Lisenfeld,1,4 C. Mu¨ller,2,4 J. H. Cole,3,4 P. Bushev,1,4 A. Lukashenko,1,4 A. Shnirman,2,4 and A.V. Ustinov1,4,*
1Physikalisches Institut, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany
2Institut fu¨r Theorie der Kondensierten Materie, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany
3Institut fu¨r Theoretische Festko¨rperphysik, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany
4DFG-Center for Functional Nanostructures (CFN), D-76128 Karlsruhe, Germany
(Received 2 August 2010; published 3 December 2010)
We demonstrate a new method to directly manipulate the state of individual two-level systems (TLSs)
in phase qubits. It allows one to characterize the coherence properties of TLSs using standard microwave
pulse sequences, while the qubit is used only for state readout. We apply this method to measure the
temperature dependence of TLS coherence for the first time. The energy relaxation time T1 is found to
decrease quadratically with temperature for the two TLSs studied in this work, while their dephasing time
measured in Ramsey and spin-echo experiments is found to be T1 limited at all temperatures.
DOI: 10.1103/PhysRevLett.105.230504 PACS numbers: 03.67.Lx, 03.65.Yz, 74.50.+r, 85.25.Am
In the early 1970s, measurements of the thermal prop-
erties of amorphous materials [1] led to the development of
a phenomenological model to explain their specific heat
and thermal conductivity at low temperature. Anderson,
Halperin, and Varma [2], as well as Phillips [3], suggested
the presence of an ensemble of two-level systems in the
amorphous material, originating from quantum tunneling
of individual atoms or a small group of atoms between two
metastable lattice positions.
The tunneling model was intensively tested experimen-
tally by ensemble measurements performed on samples
having a large two-level system (TLS) density, such as
glasses. As an example, the life time of thermally exited
states could be measured from the heat release of a sample
after a rapid cool-down. Also, quantum coherent measure-
ments of the decoherence times of (near-) resonantly
excited subsets of TLSs were performed by monitoring
the response to acoustic [4] or electric echo pulses.
Interpretation of these experiments inherently requires a
statistical analysis of an inhomogeneous ensemble of
TLSs, which is characterized by a distribution of dipole
orientations and strengths as well as a spread of local strain
fields. Since the microscopic nature of TLSs remains an
actively debated topic, it is crucial to gain a deeper under-
standing by observing the properties of individual TLSs
and hereby raise the veil imposed by averaging.
Experiments on individual TLSs became possible with
the advent of superconducting quantum bit (qubit) circuits
[5]. Individual TLSs can couple strongly to the qubit via
their electric dipole moment when they are located in the
dielectric of the thin (  2 nm) tunnel barrier of the
Josephson junctions forming the qubit. This coupling
manifests itself as an avoided level crossing in the qubit
spectrum at bias values for which a certain TLS energy
splitting E matches the energy difference between the
two qubit states [6]. The coupling strength between a TLS
and the qubit follows directly from the magnitude of the
avoided level crossing S as indicated in Fig. 1(b).
Time-resolved experiments on phase qubits have dem-
onstrated that an individual TLS can be manipulated using
the qubit as a tool to both fully control and read out its
state, and their possible use as a quantummemory has been
demonstrated [7]. Recently [8], we showed that there exists
an effective qubit mediated coupling between TLSs and an
externally applied electromagnetic ac field.
In this work, we demonstrate that this coupling allows
one to directly control the quantum state of individual
TLSs by coherent single-pulse resonant driving. Since
the qubit always remains detuned during TLS operation
and merely acts as a detector to measure its resulting state,
we can apply standard microwave pulse sequences at the
TLS frequency in order to characterize its coherence prop-
erties. This supersedes the need for coherent qubit-TLS
population exchange involving decoherence-limited ex-
cited qubit states, which would cause errors in the TLS
x
x
x
x
microwave pulses
TLS
flux
bias
SQUID
(a) (b)
7.7
7.8
7.9
8
µ  w
 fr
eq
ue
nc
y 
(G
Hz
)
flux bias (arb.units)  
 
0
0.3
S “TLS 2”
“TLS 1”
Pe
FIG. 1 (color online). (a) Schematic of the phase qubit circuit.
(b) Qubit spectroscopy. Probability to measure the excited qubit
state Pe after a long (300 ns) microwave pulse of small ampli-
tude as a function of the qubit flux bias and microwave fre-
quency. Two avoided level crossings, denoted ‘‘TLS 1’’ and
‘‘TLS 2’’, are observed at 7.735 and 7.947 GHz with magnitudes
S ¼ 23 and 36 MHz, respectively.
PRL 105, 230504 (2010) P HY S I CA L R EV I EW LE T T E R S
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0031-9007=10=105(23)=230504(4) 230504-1  2010 The American Physical Society
manipulation, and enhances the possibility of using TLSs
as the computational qubits [9].
We use a phase qubit [5] which consists of a super-
conducting LC—resonator, realized by shorting a capaci-
tively shunted Josephson junction with an inductor as
shown schematically in Fig. 1(a). The Josephson junction
acts as a nonlinear inductance providing anharmonicity
to the oscillator, which is required to selectively excite
transitions between the two lowest energy eigenstates.
One can distinguish these qubit states by the rate at which
one magnetic flux quantum enters the superconducting
loop during application of a short (2 ns) readout flux pulse
[10]. The final flux state of the qubit is measured by an
inductively coupled dc-SQUID.
The sample used in this work had a qubit energy relaxa-
tion time Tq1  110 ns and a qubit dephasing time Tq2 
95 ns and is otherwise identical to the sample presented in
Ref. [11]. Microwaves are applied to the qubit via an on-
chip planar transmission line being capacitively coupled to
the shunting capacitor. During our experiments, the chip
was maintained at a temperature of 35 mK. In several cool-
downs of the sample studied, we observed between 3 and 4
TLSs coupling to the qubit with a strength larger then
10 MHz, which is about the spectroscopic resolution given
by the linewidth of the resonance peak. Their energy split-
ting, coupling strength as well as coherence times changed
after temperature cycles in which the superconducting
transition temperature of aluminum  1:2 K was ex-
ceeded, but otherwise remained constant during several
months of measurements [12].
In order to use the qubit as a detector measuring the state
of a certain TLS, it is first initialized in the ground state and
biased at a flux at which both systems are sufficiently
detuned such that their coupling can be neglected. A flux
pulse then brings both systems into resonance, which gives
rise to coherent oscillations within the coupled system.
By choosing the flux pulse duration such that exactly half
a period of oscillation occurs, the TLS state is mapped
to the qubit state (under acquisition of a phase factor).
Accordingly such a flux pulse is called an iswap pulse.
The duration of the iswap pulse, Tswap ¼ S1=2, corre-
sponds to half the inverse magnitude of the avoided level
crossing found spectroscopically. A subsequent readout of
the qubit state then reflects the TLS population [7].
Direct manipulation of the TLS is straightforward: a
microwave pulse applied to the qubit at the TLS resonance
frequency TLS ¼ E=h prepares its state via Rabi oscil-
lation in a similar way as if the TLS would directly couple
to the externally applied microwave. Similar to the situ-
ation we described in Ref. [8], this coupling to the driving
field is mediated by a second order Raman process involv-
ing excited states of the qubit.
In Fig. 2, we present data obtained on TLS 1 whose
resonance frequency was TLS1  7:735 GHz and which
was coupled to the qubit with a strength of S1  23 MHz.
The energy relaxation time of the TLS is measured by
applying a resonant microwave pulse of duration equal to
half the Rabi period (a so-called  pulse), which prepares
the TLS in its excited state. By delaying the TLS readout
sequence, which always consists of the iswap pulse fol-
lowed by the qubit readout pulse, we observe exponential
decay of the TLS excited state at a characteristic time
TTLS11 as shown in Fig. 2(a). In Fig. 2(b), we show data
obtained by applying a microwave pulse of varying dura-
tion to observe Rabi oscillations in the time domain, which
decay at a characteristic time TTLS1d . Figure 2(c) shows
0.2
0.5
P e
(a)
0.2
0.5
P e
(b)
0 500 1000 1500
0.2
0.4
P e
∆ t (ns)
(c)
0
200
400
T 1
  
(ns
)
(d)
0 0.5 1 1.5
0
400
800
T 2
 
 
(ns
)
detuning δ [GHz]
(f) 0
250
500
T d
 
 
(ns
)(e)
Φ r
e
a
do
ut
t
iswap
∆t
π/2 π/2
µw
π
Φ re
a
do
ut
t
iswap
∆t
µw
Φ re
a
do
ut
t
iswap
∆t
µw
-0.5
energy relaxation
Rabi oscillation
Ramsey oscillation
FIG. 2 (color online). Coherent response of TLS 1 to standard microwave pulse sequences as depicted in the insets. (a) Relaxation of
the excited TLS state after its preparation using a microwave  pulse of duration 62 ns. Here, t indicates the time delay between state
preparation and readout. (b) Rabi oscillation observed by driving the TLS resonantly for a duration t. (c) Ramsey oscillation
measured by varying the delay t between two =2 pulses (of duration 31 ns each) which were detuned from the TLS’ resonance
frequency by about 10 MHz. The data shown in panels (a)–(c) were obtained at a detuning  ¼ 502 MHz. Panels (d),(e) and (f) show
the energy relaxation time TTLS11 , the Rabi decay time T
TLS1
d , and the dephasing time T
TLS1
2 as a function of the detuning  between
TLS and qubit.
PRL 105, 230504 (2010) P HY S I CA L R EV I EW LE T T E R S
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Ramsey fringes created by applying two delayed =2
pulses whose frequency was detuned from the exact TLS
resonance by approximately 10 MHz. From the exponen-
tial decay of the Ramsey oscillations, we extract the de-
phasing time TTLS12 .
To check whether the obtained coherence times are
influenced by the coupling to the qubit, we repeated the
measurements described above at different flux bias values.
In Figs. 2(d) and 2(e), we plot the dependence of the
coherence times on the detuning between qubit and TLS,
 ¼ TLS  qubit. We see that the measured TLS coher-
ence times do not depend on the detuning as long as jj is
larger than a minimal value of about 500 MHz. From these
data we conclude that for large detuning, the TLS coher-
ence is not limited by effects arising from the coupling to
the qubit. For smaller detunings the qubit may be excited,
leading to population of higher excited qubit states by
multiphoton processes due to the relatively large power
of the applied microwave pulse. In addition, the time
evolution will be more complicated as the coupling be-
tween the systems is no longer negligibly small [13].
Off-resonant, multiphoton excitation of the qubit to
higher excited states is also the reason why the Rabi
frequency of the TLS can not be increased arbitrarily by
stronger driving. It is interesting to note that while this
problem can in principle be circumvented by increasing the
detuning, this counteracts faster TLS driving because for a
given driving strength the TLS Rabi frequency decreases
with increasing detuning as approximately R / 1= as it
is shown in Fig. 3(a). The theoretical curves in Fig. 3(a)
indicate the Rabi frequencies of the system when driving
resonantly with the TLS. The calculations included 5
higher levels in the qubit [8,14].
We note that the Ramsey frequency is not only deter-
mined by the drive detuning but is additionally modified
by the coupling to the qubit (since the uncoupled states are
no longer eigenstates), which gives rise to its dependence
on the detuning  as we show in Fig. 3(b).
We emphasize that for TLS 1, the average dephasing
time TTLS12  810 ns was very close to twice the average
value of TTLS11  410 ns (indicated by dashed lines in
Figs. 2(d) and 2(f). This is expected from a quantum
system whose decoherence is limited purely by energy
relaxation. Accordingly, we argue that this TLS was not
coupled to any low-frequency noise sources which intro-
duced dephasing on the observed time scale, such as
fluctuating local fields affecting the TLS asymmetry E.
We measured a second TLS found in the sample during
the same cool-down (TLS2). We obtained a similar depen-
dence on the detuning parameter  (data not shown here).
However, TLS 2 had a maximal dephasing time of TTLS22 
580 ns whereas its energy relaxation time was TTLS21 
380 ns. We found that application of a three-pulse echo
sequence corrected for the excess dephasing in this TLS.
We illustrate this in Fig. 4, where we compare the decay of
the signal in a Ramsey protocol consisting of two resonant

2 pulses to the Hahn-echo sequence [15] which adds a
refocussing  pulse in the middle. The T2 time obtained
by the Hahn-echo sequence is close to twice the energy
relaxation time TTLS21 , from which we conclude that the
dephasing mechanism acting on this particular TLS indu-
ces low-frequency energy fluctuations.
With the help of the established direct TLSmanipulation
procedure described so far, it is straightforward to measure
the temperature dependence of TLS coherence times. As
we showed in a previous work [16], the coherence time of
this phase qubit shows a weak temperature dependence up
to a point where the thermal energy kBT approaches the
qubit splitting Ej1i  Ej0i. To prevent the qubit from being
thermally excited, we bias it above the studied TLS’ reso-
nance frequency at a chosen detuning of  ¼ 500 MHz.
In Fig. 5, we show the temperature dependencies of the
TLSs’ T1 and T2 times which were obtained by addressing
the TLSs directly with the described pulse sequences.
We did not observe any indication that the TLSs’ reso-
nance frequency and coupling strength to the qubit varied
with temperature. For both TLSs, we observed an approxi-
mately quadratic decrease of the energy relaxation time
T1 with temperature. This is illustrated by the solid lines in
Fig. 5 which are plots of the equation T1ðTÞ¼T1ð0ÞaT2,
−0.5 0.5 1.5
0
5
10
15
20
25
R
ab
i f
re
q.
 (M
Hz
)
(a)
Detuning δ [GHz]
−0.5 0.5 1.5
9
10
11
R
am
se
y 
fre
q.
 (M
Hz
)
Detuning δ [GHz]
(b)
FIG. 3 (color online). (a) Rabi frequency vs detuning between
TLS 1 and qubit. The two curves correspond to microwave
powers differing by 9 dBm. (b) Ramsey frequency vs detuning
between TLS 1 and qubit. Solid lines are calculated from theory.
Ramsey
Φ re
a
do
ut
t
iswap
∆t
π/2 π/2µw
Hahn Echo
0 1 2 3
0.2
0.4
0.6
P e
time ∆ t  (µs)
Φ r
e
a
do
ut
t
iswap
∆t
π/2 π/2
µw
FIG. 4 (color online). Signal amplitudes obtained from
Ramsey (curve starting at Pe ¼ 0:6) and Hahn-echo (curve
starting at Pe ¼ 0:2) sequences for TLS 2. Solid lines are
exponential fits resulting in a dephasing time of 551 41 ns
(Ramsey) and 743 62 ns (Hahn echo), respectively.
PRL 105, 230504 (2010) P HY S I CA L R EV I EW LE T T E R S
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230504-3
where a is a fitting parameter with the values a ¼ 4:96
0:71 s=K2 for TLS 1 and a ¼ 4:18 0:36 s=K2 for
TLS 2. Such behavior is not expected from a simple model,
e.g., coupling the TLS to a bath of harmonic oscillators
which would give T1ðTÞ / tanhðE=2kBTÞ [17] as shown
by the dotted lines in Fig. 5. We stress that the Tq1 time of
the qubit shows a qualitatively different behavior as com-
pared to the TLSs. This is illustrated by the data shown in
the inset to Fig. 5(a), which was measured at a qubit
splitting of h 8:235 GHz during the same cool-down.
We note that in TLS 1, the dephasing time T2 remains
close to twice its T1 value in the whole temperature range.
This TLS thus appears to remain immune to pure dephas-
ing even at elevated sample temperatures. In contrast, for
TLS 2, which showed T2 < 2T1, the dephasing time is not
well fitted by a parabolic temperature dependence. We
illustrate this by the dashed line in Fig. 5(b), which is a
three-parameter fit resulting in the equation T2ðTÞ ¼
681 ns 3:01sK1:24T1:24. This can also be seen from
the temperature dependence of the low-frequency noise
contribution T’ as shown in the inset of Fig. 5(b). The
Hahn-echo pulse suppressed excess dephasing in this TLS
also at higher temperatures as shown in the same figure.
However, the visibility of the echo signal decreased rapidly
with temperature as the T1 time became comparable to
the duration of the microwave pulse sequence, in which
we used a  pulse of duration 62 ns.
In conclusion, we have demonstrated a new method to
directly control TLSs in phase qubits, and applied it to
measure TLS coherence times using standard pulse se-
quences. Using this technique, we obtained the first data
illustrating the temperature dependence of individual
two-level systems in an amorphous solid. We observed
an approximately quadratic decrease of the energy relaxa-
tion time T1 with temperature in both measured TLSs, of
which one did not show any excess dephasing.
We are grateful to J. Martinis (UCSB) for providing the
qubit used in this work. We acknowledge financial support
from the European projects SOLID and MIDAS and the
U.S. ARO under contract W911NF-09-1-0336.
*alexey.ustinov@kit.edu
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T 1
 
(ns
)
T2*
1
T1
T2
T  2
T 
 
(µs
)
ϕ
  0 0.1 0.2 0.3
(b)
  0 0.1 0.2 0.3
0
200
400
600
800
temperature (K) temperature (K)
(ns
)
(a)
0 0.30
100
T (K)
0 0.30
3
T (K)
FIG. 5 (color online). Temperature dependence of coherence
times in (a) TLS 1 and (b) TLS 2. The value of 2  T1 is indicated
by open circles. Solid lines are parabolas as described in the text,
dotted lines are functions T1ðTÞ / tanhðE=2kBTÞ. Insets:
(a) Temperature dependence of the qubit T1 time.
(b) Extracted pure dephasing time T’ ¼ 1=½T12  ð2T1Þ1.
PRL 105, 230504 (2010) P HY S I CA L R EV I EW LE T T E R S
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230504-4


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Measuring the temperature dependence of individual two-level systems by
  direct coherent control

Measuring the temperature dependence of individual two-level systems by direct coherent control

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