Download Ramsey fringe measurement of decoherence in a the Cooper pair box

Ramsey fringe measurement of decoherence in a
novel superconducting quantum bit based on
the Cooper pair box
D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier,
C. Urbina, D. Esteve and M.H. Devoret
Quantronics Group
Service de Physique de l’Etat Condensé, CEA-Saclay
F-91191 Gif-sur-Yvette cedex, France
April 12, 2002
Abstract
We have designed and operated a novel superconducting tunnel junction circuit which behaves as a controllable atom, with a ground and first
excited state forming an effective spin 1/2. These two states are the symmetric and antisymmetric combinations of two charge configurations of
a Cooper pair box. Any spin orientation corresponding to an arbritary
superposition of these two states can be prepared by applying NMR-like
microwave pulses to the gate of the box. The spin state is readout by
measuring the voltage response to a probe current pulse applied to the
bridge formed by the two junctions of the box and an additional large,
shunting junction. In a sample whose transition period is 60 ps, coherent
superpositions decay with a 0.5 µs time constant, as measured by a Ramsey fringes experiment. This result is a step towards the realization of a
solid-state quantum information processor.
Among all the practical implementations of quantum bits for quantum information processing [1], those involving solid state electrical circuits are particularly attractive since they could benefit from the parallel fabrication techniques
of microelectronics. However, unlike the electric dipoles of isolated atoms or
ions, the state variables of a circuit like voltages and currents usually undergo
rapid quantum decoherence [2] because, in general, they are coupled to an environment with a large number of uncontrolled degrees of freedom [3]. Tunnel
junction circuits [4, 5, 6, 7, 8] have displayed so far coherence quality factors Qϕ
of several hundreds [9]. Here, the coherence quality factor for a pair of quantum
levels |ai and |bi is defined by Qϕ = πν ab Tϕ where ν ab is the transition frequency
and Tϕ is the inverse of decoherence rate of quantum superpositions of these
two states. A particularly advanced circuit is the so-called single Cooper pair
box [10, 11] in which coherent oscillations on the nanosecond time scale have
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been recently observed [5]. In this article, we report an experiment showing that
coherence quality factors of more than 104 can experimentally be achieved in a
new circuit built around the Cooper pair box, for which two orthogonal access
ports are used for preparing and measuring the quantum state [12, 13].
The basic Cooper pair box consists of a low capacitance superconducting
electrode, the “island”, connected to a superconducting reservoir by a Josephson
tunnel junction with capacitance Cj and Josephson energy EJ . The junction is
biased by a voltage source U in series with a gate capacitance Cg . In addition
to EJ , the box has a second energy scale which is the Cooper pair Coulomb
energy ECP = (2e)2 /2 (Cg + Cj ). The temperature T and the superconducting
gap ∆ satisfy kB T ¿ ∆/ ln N and ECP ¿ ∆, where N is the total number of
paired electrons in the island. The number of excess electrons is then even [14,
15] and the system has discrete quantum states which are in general quantum
superpositions of several charge states with different number N̂ of excess Cooper
pairs in the island. The Hamiltonian of the box is
³
´2
Ĥ = ECP N̂ − Ng − EJ cos θ̂
(1)
where Ng = Cg U/2e is the dimensionless gate charge and θ̂ the phase operator
conjugate to the Cooper pair number N̂ [11].
In our experiment EJ ' ECP and neither N̂ or θ̂ are good quantum numbers. We can restrict ourselves to the Hilbert space spanned by the ground
|0i and first excited energy eigenstate |1i since the system is sufficiently nonharmonic. This Hilbert space corresponds to an effective spin one-half s whose
Zeeman energy hν 01 goes to the minimal value EJ when Ng = 1/2. The spin
up (sz = 1/2, state |0i) and down (sz = −1/2, state |1i) states are there approximately
(|N = 0i ± |N = 1i)/2. Both states have the same average charge
D E
N̂ = 1/2, and consequently the system is immune to first order fluctuations
of the gate charge. With appropriate NMR-like microwave pulses on the gate
u(t) = Uµw (t) sin 2πνt, where ν ' ν 01 , any superposition |Ψi = α |0i + β |1i can
be prepared [16].
D E
For readout, instead of measuring the charge N̂ which requires moving
away from Ng = 1/2 [5, 17], we entangle s with a new degree of freedom more
advantageous to measure. For this purpose, the single junction of the basic
Cooper pair box has been split into two nominally identical junctions inserted
into a superconducting loop (see Fig. 1). The new degree of freedom is the
phase difference δ̂ across the two junctions, and the Josephson energy EJ in
Eq. (1) becomes EJ cos(δ̂/2) [18]. The information about s is transferred onto
ˆ (2e/h̄)∂ Ĥ/∂ δ̂ ' i0 sz sin(δ̂/2) where
δ̂ through the supercurrent D
in the
¯ ¯ loop
E I=
D ¯ ¯ E
¯ ¯
¯ ˆ¯
i0 = eEJ /h̄. The currents 0 ¯I ¯ 0 and 1 ¯Iˆ¯ 1 in the two states grow with
D E
opposite signs when δ = δ̂ moves away from zero.
Our readout strategy is to exploit the two corresponding evolutions of δ
generated by its entanglement with s. It has been implemented by inserting in
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the loop a large Josephson junction biased with a current Ib [19]. This junction
has a Josephson energy EJ0 which is about 20 times larger than EJ . Its phase
difference is denoted by γ̂. We have designed the loop dimensions to be as small
as possible, thus making the loop inductance much smaller than the effective inductance of all junctions. This condition imposes the constraint δ̂ = φ+γ̂, where
φ = 2eΦ/h̄, Φ being the externally imposed flux through the loop. By placing a
large capacitance C in parallel with the large junction so that (2e)2 /2C ¿ EJ0 ,
the phase γ̂, and consequently δ̂, can be treated in first approximation as classical variables. Under these conditions, δ ' φ + arcsin(Ib /I0 ), neglecting terms
of order i0 /2I0 ' 0.01 where I0 = 2eEJ0 /h̄. Just as the system is immune to
charge noise at Ng = 1/2, it is immune to flux and current noise at φ = 0 and
Ib = 0, where Iˆ = 0. The preparation of the quantum state and its manipulation are therefore performed at this working point in charge-flux-current space.
Readout is then achieved by displacing the system adiabatically along the Ib
axis to make the loop supercurrent non-zero.
In practice, a trapezoidal pulse of current with amplitude slightly below I0
is applied to the large junction (see Fig. 2). Depending on whether the effective
spin s is up or down, a value of order i0 /2 will be subtracted or added to Ib in the
large junction. The junction is extremely sensitive to the total current running
through it when a value close to γ = π/2 is reached at the top of the pulse. A
small excess current induces the switching of the junction from the zero-voltage
state to the finite voltage state. With a precise adjustment of the amplitude and
duration of the Ib (t) pulse, the large junction switches to the voltage state with
a large probability if sz = −1/2 and with a small probability if sz = +1/2 [12].
For the parameters given above, the efficiency of this projective measurement
should be η = p1 − p0 = 0.95, where p1 and p0 are the switching probabilities
in the excited and ground states, respectively, for optimum readout conditions.
The readout is also designed so as to minimize the relaxation of s using a
Wheatstone-bridge-like symmetry. The large ratios EJ 0 /EJ and C/Cj provide
further protection from the environment.
It is worthwhile to develop analogies between our experiment on an electrical
circuit and experiments on atoms. Our circuit can be viewed as an artificial
two-level atom, which we have nicknamed “quantronium”. The gate voltage
and the magnetic flux through the loop, the two bias parameters used to tune
the transition frequency, play a role similar to that of the static electric and
magnetic fields for atoms. Like in many atomic physics experiments, transitions
are induced by microwave pulses. Finally, the readout scheme can be compared
with a Stern and Gerlach experiment: s is the analog of the spin of the Ag atom
while the applied bias current is the analog of the magnetic field gradient. The
loop current response is to be compared with the transverse acceleration of the
Ag atom, the phase γ being the analog of the transverse position of the atom.
The presence or absence of the voltage pulse corresponds to the impact of the
Ag atom in the upper or lower spot of the “screen”.
The actual sample on which measurements have been performed is shown
in Fig. 1b. Tunnel junctions have been fabricated using the standard technique
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of Al evaporation through a shadow-mask obtained by e-beam lithography [20].
With the external microwave circuit capacitor C = 1 pF, the plasma frequency
of the large junction with I0 = 0.77 µA is ω p /2π ' 8 GHz. The sample and last
filtering stage were anchored to the mixing chamber of a dilution refrigerator
with 15 mK base temperature. The switching of the large junction to the voltage
state is detected by measuring the voltage across it with a room temperature
preamplifier followed by a discriminator with a threshold voltage Vth well above
the noise level (Fig. 2). By repeating the experiment, we can determine the
2
2
switching probability, and hence, the occupation probabilities |α| and |β| .
We have first tested the readout part of the circuit by measuring at thermal
equilibrium, for a current pulse duration of τ r = 100 ns, the switching probability p as a function of the pulse height Ip . We have found that the discrimination
between the currents corresponding to the |0i and |1i states had an efficiency
of η = 0.6, lower than the expected η = 0.95. Measurements of the switching
probability as a function of temperature and repetition rate indicate that the
discrepancy between the theoretical and experimental readout efficiency could
be due to an incomplete thermalization of our last filtering stage in the bias
current line.
We have then performed spectroscopic measurements of ν 01 by applying to
the gate a weak continuous microwave irradiation suppressed just before the
readout current pulse. The variations of the switching probability as a function
of the irradiation frequency display a resonance whose center frequency evolves
as a function of the dc gate voltage and flux as the Hamiltonian (1) predicts,
reaching ν 01 ' 16.5 GHz at the optimal working point (see Fig. 3). A small
discrepancy between theory and experiment can be observed in the variations of
the center frequency with flux, but we attribute it to a residual flux penetration
in the small junctions not taken into account in the model. We have used these
spectroscopic data to determine precisely the relevant circuit parameters and
found i0 = 18.1 nA and EJ /ECP = 1.27. At the optimal working point, the
linewidth was found to be minimal with a 0.8 MHz full width at half-maximum.
When varying the delay between the end of the irradiation and the measurement
pulse, the peak height decays with a time constant T1 = 1.8 µs (see Fig. 4).
Supposing that the energy relaxation of the system is only due to the bias
circuitry, a calculation along the lines of Ref. [21] predicts that T1 ∼ 10 µs for
a crude discrete element model. This result shows that no detrimental sources
of dissipation have been seriously overlooked in our circuit design.
We have then addressed the fidelity of controlled rotations of s around an axis
x perpendicular to the quantization axis z. Prior to readout, a single pulse at the
transition frequency with variable amplitude Uµw and duration τ was applied.
The resulting change in switching probability is an oscillatory function of the
product Uµw τ (see Fig. 5), in agreement with the theory of Rabi oscillations
[22, 16]. It provides direct evidence that the resonance indeed corresponds to
an effective spin rather than to a spurious harmonic oscillator resonance in the
circuit. The proportionality ratio between the Rabi period and Uµw τ was used
to calibrate microwave pulses for the application of controlled rotations of s.
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The main result of this paper, the coherence time of s during free evolution,
was obtained by performing the classic Ramsey fringes experiment [23] on which
atomic clocks are based. One applies on the gate two phase coherent microwave
pulses corresponding each to a π/2 rotation around x [24] and separated by a
delay ∆t during which the spin precesses freely around z. For a given detuning
of the pulse center frequency, we have observed decaying oscillations of the
switching probability as a function of ∆t (see Fig. 6), which correspond to
the “beating” of the spin precession with the external microwave field. The
oscillation period agrees exactly with the inverse of the detuning, allowing a
measurement of the transition frequency with an accuracy of 6 × 10−6 [25]. The
envelope of the oscillations yields the decoherence time Tϕ ' 0.5 µs. Given the
transition period 1/ν 01 ' 60 ps, this means that s can perform on average 8000
coherent free precession turns.
In all our time domain experiments, the oscillation period of the switching
probability closely agrees with theory, meaning a precise control of the preparation of s and of its evolution. However, the amplitude of the oscillations is
smaller than expected by a factor of three to four. This loss of contrast is likely
to be due to a relaxation of the level population during the measurement itself.
In principle the current pulse, whose rise time is 50 ns, is sufficiently adiabatic
not to induce transitions directly between the two levels. Nevertheless, it is possible that the readout or even the preparation pulses excite resonances in the
bias circuitry which in turn could induce transitions in our two-level manifold.
Experiments using better shaped readout pulses and a bias circuitry with better
controlled high-frequency impedance are needed to clarify this point.
In order to understand what limits the coherence time of the circuit, we
have performed measurements of the linewidth ∆ν 01 of the resonant peak as
a function of U and Φ. The linewidth varies linearly when departing from the
optimal point (Ng = 1/2, φ = 0, Ib = 0), the proportionality coefficients being
∂∆ν 01 /∂Ng ' 250 MHz and ∂∆ν 01 /∂(φ/2π) ' 430 MHz (see Fig. 7). These
values can be translated into RMS deviations ∆Ng = 0.004 and ∆(φ/2π) =
0.002 of the transition frequency during the time needed to record the resonance.
The residual linewidth at the optimal working point is well-explained by the
second order contribution of these noises and is not therefore limited by any
fundamental factor or unknown noise sources. The amplitude of the charge
noise is in agreement with measurements of 1/f charge noise [26], and its effect
could be minimized by increasing the EJ /EC ratio. By contrast, the amplitude
of the flux noise is unusually large [27], and we think that implementation of
magnetic shielding and better Al layer configuration will significantly reduce it.
An improvement of Qϕ by an order of magnitude seems thus possible [28].
In conclusion, we have designed and operated a superconducting tunnel junction circuit which behaves as a tunable two-level atom that can be decoupled
from its environment. When the readout is off, the coherence of this “quantronium” atom is of sufficient quality (Qϕ = 2.5 × 104 ) that an arbitrary quantum
evolution can be programmed with a series of NMR-like microwaves pulses. Coupling several of these circuits can be achieved using on-chip capacitors. Since
we could tune and address them individually, we could produce entangled states
5
and probe their quantum correlations. These fundamental physics experiments
would then lead to the realization of solid state quantum logic gates, an important step towards the practical implementation of quantum processors.
Acknowledgements: The indispensable technical work of Pief Orfila is gratefully acknowledged. This work has greatly benefited from direct inputs from
J. M. Martinis and Y. Nakamura. The authors acknowledge discussions with
P. Delsing, G. Falci, D. Haviland, H. Mooij, R. Schoelkopf, G. Schön and G.
Wendin. This work is partly supported by the European Union through contract
IST-10673 SQUBIT.
References
[1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum
Information (Cambridge University Press, Cambridge, 2000).
[2] W. H. Zurek, Physics Today 44, 36 (1991); W. H. Zurek and J. P. Paz, in
Coherent atomic matter waves, edited by R. Kaiser, C. Westbrook and F.
David, (Springer-Verlag Heidelberg 2000) [quant-ph/0010011].
[3] Y. Makhlin, G. Schön, A. Shnirman, Rev. Mod. Phys. 73, 357 (2001).
[4] J. M. Martinis, M. H. Devoret, and J. Clarke, Phys. Rev. B 35, 46824698 (1987); M. H. Devoret, D. Esteve , C. Urbina, J. M. Martinis , A. N.
Cleland, and J. Clarke, in Quantum Tunneling in Condensed Media edited
by Y. Kagan and A.J. Leggett (Elsevier Science Publishers, 1992).
[5] Y. Nakamura, Yu. A. Pashkin and J. S. Tsai, Nature 398, 786-788,
(1999); Phys. Rev. Lett. 87, 246601 (2001); ibid. to be published [condmat/0111402].
[6] Caspar H. van der Wal, A. C. J. ter Haar, F. K. Wilhelm, R.N. Schouten,
C. J. P. M. Harmans, T. P. Orlando, Seth Lloyd and E. Mooij, Science
290, 773 (2000).
[7] Siyuan-Han, R. Rouse, J. E. Lukens, Phys. Rev. Lett. 84, 1300 (2000).
[8] Siyuan-Han, Yang-Yu, Xi-Chu, Shih-I-Chu, Zhen-Wang, Science. 293, 1457
(2001).
[9] J. M. Martinis, Sae woo Nan, J. Aumentado, and C. Urbina (unpublished)
have recently obtained Qϕ ’s of the order of 500 for a current biased Josephson junction.
[10] M. Büttiker, Phys. Rev. B 36, 3548 (1987).
[11] V. Bouchiat, D. Vion, P. Joyez, D. Esteve, M. H. Devoret, Phys. Scr. T76,
165-170 (1998).
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[12] A. Cottet, D. Vion, P. Joyez, A. Aassime, D. Esteve, and M. H. Devoret,
to be published in Physica C.
[13] Another two-port design has been proposed by A. B. Zorin (condmat/0112351; to be published in Physica C)
[14] M. T. Tuominen, J. M. Hergenrother, T. S. Tighe, and M. Tinkham, Phys.
Rev. Lett. 69, 1997 (1992).
[15] P. Lafarge, P. Joyez, D. Esteve, C. Urbina, and M.H. Devoret, Nature 365,
422 (1993).
[16] A. Abragam, The principles of nuclear magnetism (Oxford University
Press, 1961).
[17] A. Aassime, G. Johansson, G. Wendin, R. J. Schoelkopf, and P. Delsing,
Phys. Rev. Lett. 86, 3376 (2001).
[18] D. V. Averin and K. K. Likharev, in Mesoscopic Phenomena in Solids,
edited by B. L. Altshuler, P. A. Lee, and R. A. Webb (Elsevier, Amsterdam,
1991).
[19] A different Cooper pair box readout scheme using a large Josephson junction is discussed by F. W. J. Hekking, O. Buisson, F. Balestro, and M. G.
Vergniory, in Electronic Correlations: from Meso- to Nanophysics, T. Martin, G. Montambaux and J. Trân Thanh Vân, eds. (EDPSciences, 2001),
p. 515.
[20] G. J. Dolan and J. H. Dunsmuir, Physica (Amsterdam) 152B, 7 (1988).
[21] A. Cottet, A. H. Steinbach, P. Joyez, D. Vion, H. Pothier, D. Esteve, and
M. E. Huber, in Macroscopic quantum coherence and quantum computing,
edited by D. V. Averin, B. Ruggiero, and P. Silvestrini, Kluwer Academic,
Plenum publishers, New-York, 2001, p. 111-125.
[22] I. I. Rabi, Phys. Rev. 51, 652 (1937).
[23] N. F. Ramsey, Phys. Rev. 78, 695 (1950).
[24] In practice, the rotation axis does not need to be x, but the rotation angle
of the two pulses is always adjusted so as to bring a spin initially along z
into a plane perpendicular to z.
[25] At fixed ∆t, the switching probability displays a decaying oscillation as a
function of detuning, the maximum corresponding to zero detuning.
[26] H. Wolf, F.-J. Ahlers, J. Niemeyer, H. Scherer, Th. Weimann, A. B. Zorin,
V. A. Krupenin, S. V. Lotkhov, D. E. Presnov, IEEE Trans. on Instrum.
and Measurement 46, 303 (1997).
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[27] F. C. Wellstood, C. Urbina, and J. Clarke, Appl. Phys. Lett. 50, 772-774
(1987).
[28] Critical current noise [B. Savo, F. C. Wellstood, and J. Clarke, Appl. Phys.
Lett. 50, 1758-1760 (1987)] seems to be of a lesser concern since none of
our results forces us to invoke it.
8
Figure 1: Top: schematic diagram of our quantum coherent circuit with its tuning, preparation and readout blocks. The coherent circuit, nicknamed “quantronium”, consists of a Cooper pair box island (black node) delimited by two small
Josephson junctions (cross symbols) in a superconducting loop. The loop also
includes a third, much larger Josephson junction shunted by a capacitance C.
The Josephson energy of the box and the large junction are EJ and EJ0 . The
Cooper pair number N and the phases δ and γ are the degrees of freedom of the
circuit. A dc voltage U applied to the gate capacitance Cg and a dc current Iφ
applied to a coil producing a flux Φ in the circuit loop tune the quantum energy
levels. Microwave pulses u(t) applied to the gate prepare arbitrary quantum
states of the circuit. The states are readout by applying a current pulse Ib (t) to
the large junction and by monitoring the voltage V (t). Filters in the microwave
and dc lines have not been represented for simplicity. Bottom: scanning electron
micrograph of a quantronium sample.
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Figure 2: Signals involved in quantum state manipulations and measurement
of the “quantronium”. Top: microwave voltage pulses are applied to the gate
for state manipulation. Middle: a readout current pulse Ib (t) with amplitude
Ip is applied to the large junction td after the last microwave pulse. Bottom:
voltage V (t) across the junction. The occurence of a pulse depends on the occupation probabilities of the energy eigenstates. A discriminator with threshold
Vth converts V (t) into a boolean 0/1 output for statistical analysis.
10
Figure 3: Measured center frequency (symbols) of the resonance as a function of
reduced gate charge Ng for reduced flux φ = 0 (right panel) and as a function of
φ for Ng = 0.5 (left panel), at 15 mK. Spectroscopy is performed by measuring
the switching probability p (105 events) when a continuous microwave irradiation of variable frequency is applied to the gate before readout (td < 100 ns).
Continuous line: theoretical best fit (see text). Inset: minimum width lineshape
measured at the optimal working point φ = 0 and Ng = 0.5 (dots). Lorentzian
fit with a FWHM ∆ν 01 = 0.8 MHz and a center frequency ν 01 = 16463.5 MHz
(curve).
11
s w it c h in g p r o b a b i lit y
0 .3 2
0 .2 8
W ON
0 .2 4
0 .2 0
W OFF
0 .1 6
0
1
2
3
4
5
6
7
t im e d e la y ( s )
Figure 4: Decay of the switching probability as a function of the delay time
after a continuous excitation at the center frequency of the line shown in the
inset of Fig. 3 (data points labeled ”µW ON”). A control experiment has been
performed by doing the same measurement as a function of the delay time, but
without excitation (data points labeled ”µW OFF”).
12
Figure 5: Top: Rabi oscillations of the switching probability p (5 × 104 events)
measured just after a resonant microwave pulse of duration τ . Data taken at
15mK for a nominal pulse amplitude Uµw = 10µV (dots). The Rabi frequency
is extracted from an exponentially damped sinusoidal fit (continuous line). Bottom: the measured Rabi frequency (dots) varies linearly with Uµw , as expected.
13
Figure 6: Ramsey fringes of the switching probability p (5×104 events) after two
phase coherent microwave pulses separated by ∆t. Dots: data at 15mK; The
total acquisition time was 5 mn. Continuous line: fit by exponentially damped
sinusoid with time constant Tϕ = 500 ± 50 ns. The oscillation corresponds
to the “beating” of the free evolution of the spin with the external microwave
field. Its period indeed coincides with the inverse of the detuning frequency
(here ν − ν 01 = 20.6 MHz).
14
re s o n a n c e p e a k F W H M (M H z )
60
60
40
40
20
20
0
0 .2
0 .1
0
/ 2
0 .5
0 .6
N g- 1 / 2
Figure 7: Variations of the resonance width with deviations of bias point from
the optimal bias point. The dashed lines indicate fits of the data assuming
gaussian charge and flux fluctuations in bias with RMS deviations 4 × 10−3
Cooper pair and 2 × 10−3 flux quantum.
15