Ultrafast gates for single atomic qubits.

Abstract

We demonstrate single-qubit operations on a trapped atom hyperfine qubit using a single ultrafast pulse from a mode-locked laser. We shape the pulse from the laser and perform a π rotation of the qubit in less than 50 ps with a population transfer exceeding 99% and negligible effects from spontaneous emission or ac Stark shifts. The gate time is significantly shorter than the period of atomic motion in the trap (Ω(Rabi)/ν(trap)>10(4)), demonstrating that this interaction takes place deep within the strong excitation regime.

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arXiv:1005.4144v1 [physics.atom-ph] 22 May 2010
Ultrafast Gates for Single Atomic Qubits
W. C. Campbell,∗J. Mizrahi, Q. Quraishi, C. Senko, D. Hayes,
D. Hucul, D. N. Matsukevich, P. Maunz, and C. Monroe
Joint Quantum Institute, University of Maryland Department
of Physics and National Institute of Standards and Technology,
College Park, Maryland 20742 USA
(Dated: May 25, 2010)
Abstract
We demonstrate single qubit operations on a trapped atom hyperfine qubit using a single ultrafast
pulse from a mode-locked laser. We shape the pulse from the laser and perform a π rotation of
the qubit in less than 50 ps with a population transfer exceeding 99% and negligible effects from
spontaneous emission or ac Stark shifts. The gate time is significantly shorter than the period
of atomic motion in the trap (ΩRabi/νtrap> 104), demonstrating that this interaction takes place
deep within the strong excitation regime.
PACS numbers: 03.67.Lx, 37.10.Vz, 37.25.+k
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Quantum information processing requires the ability to perform operations in an amount
of time shorter than the coherence time of the qubit. The ratio between coherence time
and gate duration can be increased by improving coherence times or developing faster gates.
One experimental extreme is found in atomic systems such as trapped ions, which can have
coherence times of many minutes [1, 2]. However, the precise spectral resolution needed to
address narrow transitions and available excitation power have typically limited gate times to
around a microsecond. In many condensed matter systems, on the other hand, single qubit
operations must be performed on picosecond timescales [3–5] owing to the short coherence
time of the qubit (T∗
2< 10 ns in [3–5]). Here we realize ultrafast operations in an atomic
system with very slow decoherence rates.
Aside from increasing the clock speed to decoherence rate ratio, there are also specific
applications that would benefit immediately from fast single qubit gates, such as noise-
reduction techniques [6], entanglement of material qubits with photon time-bin qubits [7],
and probabilistic gates with repetition rates limited by qubit rotations [8]. In particular, the
ability to apply fast spin-dependent momentum kicks to trapped ions in the strong excitation
regime [9] is a critical ingredient for fast sideband cooling [10] and fast entanglement of
multiple atomic ion qubits [11, 12]. Such entangling gates can be performed faster than a trap
oscillation period, in contrast to motional gates using spectroscopically resolved sidebands.
We previously reported the implementation of an optical frequency comb to perform qubit
operations [13]. In the low excitation regime, each pulse alters the atomic state by a tiny
amount and the effect of the pulse train is manifested through careful coherent accumulation
of spectral density into narrow comb teeth, resulting in gate times of order 100 µs. In this
Letter, we realize the strong excitation regime (ΩRabi ≫ νtrap) [9] and demonstrate fast
control of an atomic qubit by driving stimulated Raman transitions with a single pulse from
a picosecond mode-locked laser. In order to implement a fast π-pulse, we perform simple
pulse shaping in the form of a beam splitter and delay line [14] to fully rotate the qubit in less
than 50 ps with population transfer exceeding 99%. Since the atomic qubit is well isolated
from its environment, the time required to perform such a fast gate is only a small fraction
(< 10−8) of the measured coherence time of our qubit. By setting the delay to zero in a
counter-propagating geometry, the transition probability becomes sensitive to the atomic
motion, which is the first step toward implementing fast cooling [10] and fast entangling
gates between multiple ions [11, 12].
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10 20 30 40 50
Delay Time [ps]
060
Amplitude [arb.]
(a)
(b)
2S1/2
2P1/2
2P3/2
∆FS = 100 THz
∆D1 = 33 THz
= 12.6 GHz
|1>
|0>
νHF
355 nm
variable
delay
vacuum
chamber
mode-locked
355 nm laser
pulse
picker
(c)
σ+
σ -
bs
FIG. 1. Experimental schematic showing (a) the energy level diagram of171Yb+with relevant
states (not to scale), (b) the envelope of the electric field autocorrelation from the mode-locked
laser and (c) the pulsed laser beam path with optional (dashed line) pulse-shaping beam path.
Our experimental apparatus is shown in Fig. 1(c). A171Yb+ion is trapped in a linear
radio frequency Paul trap. The mF= 0 “clock states” of the two2S1/2hyperfine levels act
as the basis states of our qubit separated in frequency by νHF= 12.642815 GHz. Doppler
cooling, state preparation, and state detection are accomplished on the2P1/2↔2S1/2“D1
line” at 370 nm as described in Ref. [15]. The mode-locked laser for Raman transitions is a
frequency-tripled yttrium vanadate laser operating at a repetition rate of frep.= 121 MHz
with an average power of 4 W at 355 nm [16]. Single pulses are extracted by a Pockels cell
pulse picker [17] and focused onto the ion.
Fig. 1(a) shows the relevant energy levels of171Yb+. The2S1/2qubit states |0? and |1?
are coupled through excited2P states via far-detuned light that is polarized to drive either
σ+or σ−transitions, as Raman transitions from π light are forbidden by selection rules.
The σ±Raman transition (two-photon) Rabi frequency is [18, 19]
Ω0,1= ±g2
6
?
1
∆D1
+
1
∆FS− ∆D1
?
, (1)
where g is the resonant one-photon Rabi frequency of the2P3/2(F = 2,mF= 2) ↔2S1/2(F =
1,mF = 1) “D2 line” cycling transition, ∆D1is the detuning of the light above the2P1/2
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state, ∆FS− ∆D1is the detuning of the light below the2P3/2state, and ∆FS≈ 100 THz is
the fine structure splitting. The ac Stark shifts of the qubit states from σ±light are [18, 19]
δ0=g2
12∆D1
∆FS− ∆D1
δ1=g2
12∆D1+ νHF
where we have neglected the hyperfine splitting of the2P3/2state.
?
?
1
−
2
?
(2)
1
−
2
∆FS− ∆D1− νHF
?
, (3)
Eqs. 1-3 show that if the Raman laser is tuned between the D1 and D2 lines (0 < ∆D1<
∆FS) the stimulated Raman transition amplitudes due to the2P1/2and2P3/2couplings add
constructively while the ac Stark shift contributions interfere destructively. The Stark shifts
of the qubit states each cross zero near an optimal wavelength of ∆opt.
D1= ∆FS/3, which for
Yb+corresponds to a Raman laser wavelength of λopt.= 355 nm. The differential ac Stark
shift of the qubit states does not exactly cancel, but has a local minimum near λopt., reaching
a value of δ0−δ1≈ Ω0,1×3νHF/2∆FS, corresponding to 2×10−4Ω0,1for Yb+. We measure
a differential Stark shift of 1.1(5)×10−4Ω0,1through microwave Ramsey spectroscopy with
linearly polarized 355 nm light.
The spontaneous emission rate can be estimated by calculating the excited state popu-
lations during the Raman transition to give [18, 19]
Γspon= γg2
6
?
1
∆2
D1
+
2
(∆FS− ∆D1)2
?
, (4)
where γ ∼ 2π × 20 MHz is the spontaneous emission rate from the2P states [20]. The
probability of a spontaneous emission event during a π-pulse near λopt.can be estimated as
Pspon,π≈ 3γ/2∆FS. The local minima in Γsponand Pspon,πlie close to λopt.for Yb+at 349
nm and 352 nm, respectively. At 355 nm we estimate Pspon,π< 10−5.
In order to operate in this low Stark shift, low spontaneous emission regime, sufficient
optical power at λopt.must be provided to make Ω0,1appreciable. For Yb+, frequency tripled
mode-locked Nd:YAG and vanadate lasers are available that provide many Watts at a center
wavelength of 355 nm [21].
The two-photon Rabi frequency in Eq. 1 is time-dependent due to the shape of the pulse.
As long as the pulse bandwidth is small compared to the single photon detunings ∆, the
adiabatic elimination of the excited states remains valid and we can treat the qubit as a
two-level system with a time-dependent coupling:
Heff/h = −νHF
2
ˆ σz−Ω0,1(t)
2
ˆ σx. (5)
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Numerical solutions to the Schr¨ odinger equation with this Hamiltonian can be obtained
for a general time-dependent Ω(t). However, there is an analytic solution due to Rosen and
Zener for a coupling with a hyperbolic secant time-dependence [22], which is the electric field
envelope expected from picosecond mode-locked pulses [23]. Fig. 1(b) shows the envelope
of an electric field autocorrelation of a single pulse as measured with a scanning Mach-
Zender interferometer and a fast photodiode, which is consistent with the autocorrelation of
a hyperbolic secant with Tpulse= 14.8 ps having a linear pulse chirp of 8 × 10−3ps−2. Eq.
1, however, shows that the two-photon Rabi frequency is proportional to the square of the
electric field envelope. Nonetheless, sech2and sech are sufficiently similar that numerical
solutions to the Schr¨ odinger equation with Ω(t) ∝ sech2[2t/T] match the analytic solutions
with Ω(t) ∝ sech[πt/T] to within a few percent for all simulations shown here. The Rosen-
Zener transition probability from an initial state |0? to |1? is [22]
P0→1= sin2(πu) sech2[πνHFTpulse] (6)
for a two-photon Rabi frequency of Ω1,0(t) =
u
Tpulsesech[
πt
Tpulse] where u is the pulse area. Eq.
6 indicates that in order to create a high-fidelity π-pulse, the pulse duration must be many
times shorter than the hyperfine period.
Fig. 2(a) shows the measured transition probability as a function of pulse energy (pro-
portional to u in Eq. 6 and monitored with a fast photodiode). Both plots in Fig. 2 were
taken with a single beam path, as represented by the solid line in Fig. 1(c). The maximum
transition probability of 72% corresponds (see Eq. 6) to a pulse duration of Tpulse= 14.8 ps.
We also studied the dynamics of a train of identical pulses (with pulse energy denoted by
the arrow in Fig. 2(a)) at the laser repetition rate, shown in Fig. 2(b). The solid curve is
an analytic solution including free evolution with a fitted Gaussian decay in contrast. The
fitted pulse duration is Tpulse= 14 ps, in good agreement with the fitted pulse width from
Fig. 2(a).
It is clear from Eq. 6 and Fig. 2(a) that for Tpulse= 14.8 ps, even with unlimited pulse
energy the population transfer probability cannot exceed 72% without changing the pulse
shape. In order to execute a full π-pulse, we introduced a simple pulse-shaping element into
the beam path in the form of a beamsplitter and a delay line [14], resulting in two counter-
propagating pulses as shown in Fig. 1(c). By setting the energy of each half of the pulse
to transfer 50% of the population, we are able to perform a fast Ramsey experiment with a
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Pulse Energy [arb.]
0246810 12
Transition Probability
0
0.2
0.4
0.6
0.8
1
(a)
Transition Probability
0
0.2
0.4
0.6
0.8
1
Number of Pulses ( frep. x time)
05 10 1520
(b)
FIG. 2. (a) Qubit transition probability from a single pulse vs. pulse energy. The solid curve
is a fit to Eq. 6 and the peak corresponds to a pulse energy of 12 ± 2 nJ. (b) Qubit transition
probability vs. the number of identical (≈ 8 nJ) pulses at the laser repetition rate. The solid curve
is a fit to the Rosen-Zener solution with a Gaussian contrast decay. The single pulse energy for
(b) is shown with an arrow in (a).
variable free evolution time between two π/2 Raman pulses. A π-pulse then corresponds to
the top of a Ramsey fringe.
The results of the pulse shape delay scan are shown in Fig. 3(a)-(c). We show results
from experiments with three different polarization orientations for the two π/2-pulses. In
(a), both beams drive only σ+transitions. The Ramsey fringes are first maximized for a
delay of 72 ps, corresponding to a net π-pulse after 80 ps. In the frequency domain, the
shaped pulse spectrum in this case is a crude frequency comb with sinusoidal teeth separated
by 14 GHz. The reason this separation is not exactly 12.6 GHz = νHFis the additional ˆ σz
rotation introduced by the Rosen-Zener solution dynamics. In the limit of infinitesimally
short pulses, the comb spacing would converge to νHF. Fig. 3(b) shows the same experiment
with one beam driving σ+transitions and the other driving σ−. The two central peaks each
represent a ∼ 40 ps net π-pulse, which is shorter than the 80 ps pure σ+π-pulse due to
the fact that the Rabi frequency for σ+and σ−transitions have opposite sign and therefore
rotate the Bloch vector about different axes. Fig. 3(c) shows the same experiment with
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Transition Probability
0
1
0.2
0.4
0.6
0.8
1
(a)
σ+ / σ+
0
1
0.2
0.4
0.6
0.8
(b)
σ+ / σ -
0
0.2
0.4
0.6
0.8
(c)
Ramsey Zone Delay [ps]
100
-1000 -50
50
lin ┴ lin
FIG. 3. Ramsey fringes for three different polarization orientations for each half of the shaped
pulse obtained by fitting ensemble histograms of photon counts. The polarization configurations
are (a) pure σ+light, (b) one is σ+and the other σ−, and (c) lin ⊥ lin. The overlap region near
zero delay results in an optical standing wave with a period of 177 nm. The solid curves of (a) and
(c) are calculations based on independent measurements of the optical pulse chirp and envelope
with no free fit parameters. The bold curve also incorporates a thermal average of the intial ion
position at the Doppler cooling limit of ¯ n = 40, washing out the fine standing wave fringes.
orthogonal linear polarizations (“lin ⊥ lin”) where neither beam drives Raman transitions
by itself.
In order to quantify the population transfer probability, we compared detection his-
tograms [15] from a fast Raman π-pulse to a π-pulse applied with microwaves and measured
a population transfer probability of 99.3%. We also investigated the population transfer
efficiency of the π-pulses by repeating the gate multiple times at the laser repetition rate
and monitoring the transfer probability while increasing the number of π-pulses. Since the
laser repetition period is very close to a half integer number of hyperfine evolution peri-
ods (νHF/frep.≈ 104.5), there is some natural spin-echo-type error cancellation for multiple
pulses for our system.For the σ+/σ+configuration, we measure a population transfer
probability contrast of 91% after 25 π-pulses while for σ+/σ−we measure 77%.
In the overlap region (near zero delay time) in Fig. 3, the two counterpropagating π/2-
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pulses begin to overlap in time at the position of the atom and form an optical standing
wave. For the σ+/σ+and lin ⊥ lin configurations, the two-photon Rabi frequency has 177
nm period spatial interference fringes. In the overlap regions of Fig. 3(a) and (c), Raman
transitions are sensitive to the motion of the ion since momentum kicks of 2¯ hk are being
transferred from the optical field. The pulse’s spectrum, however, is much wider than the
trap frequency (100 GHz compared to 500 kHz), so the pulse simultaneously drives many
motional sidebands (including the carrier transition). For the σ+/σ−configuration (Fig.
3(b)), the overlap region contains linear polarization and does not drive Raman transitions.
The thin solid curves in Fig. 3 show the results of a numerical solution of the Schr¨ odinger
equation vs. delay, which oscillate at optical-frequency delays of less than 1 fs in the overlap
region. In this region, the ion experiences momentum kicks and phase shifts as spin states
spread out in motional phase space, leaving the interferometer open. Since the final spin
phase of different motional states depends sensitively on the ion’s intial position, thermal
averaging tends to wash out the fast spatial variation of the transition probability. The thick
curves show a thermal average over the ion’s initial state, assumed to be a thermal state
at the Doppler cooling limit of ¯ n = 40. The thin and thick curves in Fig. 3 are not fit to
the data as these numerical solutions are fully constrained by the pulse duration and chirp
measurements shown in Fig. 1(b) and 2(a).
Implementation of the fast cooling [10] and fast entangling gates [11, 12] will require
repeated spin-dependent momentum kicks generated through ultrafast interferometry in the
strong excitation regime [9]. In this case, the interferometer will be closed and therefore will
not be sensitive to optical wavelength interference such as the overlap region in Fig. 3(c).
We have demonstrated ultrafast single-qubit gates with a mode-locked laser pulse using
an atomic qubit. For a single trapped ion, the free-evolution of the qubit can be used to
perform ˆ σz rotations, and delaying the pulse arrival time will allow a rotation about an
arbitrary axis in the x-y plane of the Bloch sphere. As such, the fundamental limit on the
gate speed is the hyperfine period (analogous to the Larmor precession time), which would
yield a gate time of ≈ 100 ps. Previous results with these same qubit levels in171Yb+
have demonstrated coherence times in excess of 1000 s [2], so this single-qubit gate can be
performed in a vanishingly small fraction (< 10−13) of the coherence time.
We acknowledge helpful discussions with Michael Biercuk, Ming-Shien Chang, Kihwan
Kim, and Steven Olmschenk. This work is supported by the ARO with funds from the
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DARPA Optical Lattice Emulator (OLE) Program, IARPA under ARO contract, the NSF
Physics at the Information Frontier Program, the IC Postdoctoral Program, and the NSF
Physics Frontier Center at JQI.
∗wes3000@umd.edu
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[20] For Eq. 1-4 we have ignored the small contributions from the3[3/2]o
3/2state transition at
348 nm caused by the2P3/2admixture in that state. We have also neglected the difference
in transition dipole moment between the D1 and D2 lines. Both effects contribute less than
15% corrections to the two-photon Rabi frequency, differential Stark shift, and spontaneous
emission rate at 355 nm.
[21] Despite being slightly outside the fine-structure interval, a laser at 532 nm (= 1064 nm/2)
has a two-photon Rabi frequency for Ba+that is only slightly lower than Yb+at 355 nm with
comparable pulse energy.
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