Quantum Process Tomography on vibrational states of atoms in an optical lattice

Quantum Process Tomography on vibrational states of atoms in an optical lattice
Quantum Process Tomography on vibrational states of atoms in an Optical Lattice
S. H. Myrskog, J. K. Fox, M. W. Mitchell and A. M. Steinberg
Dept. of Physics, University of Toronto, 60 St. George St. Toronto,Ont., Canada, M5S 1A7
Quantum process tomography is used to fully characterize the evolution of the quantum vibrational state of atoms. Rubidium atoms are trapped in a shallow optical lattice supporting only two
vibrational states, which we charcterize by reconstructing the 2x2 density matrix. Repeating this
process for a complete set of inputs allows us to completely characterize both the system’s intrinsic
decoherence and resonant coupling.
arXiv:quant-ph/0312210 v1 29 Dec 2003
PACS numbers: 39.25.+k,03.65.Wj,03.67.Lx
In recent years, there have been remarkable advances
in directly controlling and observing the dynamics of individual quantum systems in a variety of domains. This
degree of control of microscopic systems is one of the
technological advances underlying the myriad proposals for realistic quantum-information processing systems.
For instance, a number of quantum computing proposals rely on atoms trapped in an optical lattice [1, 2], a
system in which a great deal of work has investigated
coherent centre-of-mass motion[3, 4, 5], full characterisation of spin states[6], loading of individual atoms into
lattice sites [7, 8, 9], and coherent interactions between
atoms[10, 11, 12]. Here we demonstrate a technique for
completely reconstructing the quantum state of motion
of an atom trapped in a lattice well. By performing
this density-matrix reconstruction for a complete set of
input states, we are able to completely characterize the
quantum evolution of the system (the “superoperator”),
including decoherence.
The development of a quantum computer will rely on
the success of error-correction to reduce errors to an acceptable level[13, 14, 15, 16]. A quantum computer is
extremely vulnerable to errors and decoherence. Experimental characterization of both the decoherence and
the operations will be required in order to implement
quantum error correction [17, 18, 19, 20]. An arbitrary
operation may be characterized using quantum process
tomography[21, 22], or QPT. The result of QPT is the
superoperator, a positive, linear map from density matrices to density matrices, which governs the evolution of
the density matrix for the operation. Unlike a propagator, the superoperator allows for non-unitary evolution
of the system, thoroughly characterizing decoherence, relaxation and loss in a system. From the superoperator
one can determine the types of errors which occur and
develop procedures to reduce or eliminate errors, without requiring a priori assumptions about the underlying
physical mechanism causing the errors[23].
QPT has recently been demonstrated using spins in a
NMR system[24], the polarization of single photons[25]
and a singlet state filter for photon pairs[26]. QPT is
performed by preparing a complete set of input density
matrices, subjecting each to the operation being tested,
and measuring the resultant output density matrices.
Due to the linearity of quantum mechanics, QPT of a
process on an N dimensional system can be achieved
by sending in N 2 linearly independent density matrices (alternatively, it has recently been shown that one
can use a single state in a larger Hilbert space as the
input[25, 27].)
In this experiment we perform quantum process tomography using the motional states of atoms trapped in
the potential wells of a 1-D optical lattice. We examine
processes which are independent of the electronic state
of the atom and are only dependent upon the motional
states of the atom. The measurements are insensitive to
the long-range degrees of freedom and effectively trace
over the quasimomentum in the Bloch state picture (or
equivalently the well index in the Wannier state picture).
We use a shallow 1-D lattice which only supports 2 bound
bands, which we label as ground (|0i) and excited (|1i).
The lattice is vertically oriented, causing all atoms in
higher energy, classically unbound states to quickly fall
out of the lattice and become spatially separated from
atoms which remain in bound states of the lattice (the
Landau-Zener tunneling rates from the 3 lowest energy
bands are 3·10−7 , 14.5 and 1150 per second in increasing
order). A typical sequence, from state preparation in
the lattice to measurement, lasts 20 ms.
We begin by cooling and trapping Rubidium-85 atoms
in a standard vapour cell MOT to a temperature of 7 µK
with an rms radius of approximately 1 mm. During the
optical molasses stage we turn on an optical lattice in
a vertical orientation. The optical lattice is created by
interfering two laser beams which are detuned 30 GHz below the Rb D2 resonance at 780.03 nm. Each beam travels through an acousto-optic modulator, each of which
is driven by a function generator, providing control of
the relative phase between the lattice beams. By modulating the phase of one of the lasers the lattice may be
displaced by up to a lattice spacing within one microsecond. The beams are superposed after the acousto-optic
modulators, and co-propagate to the vacuum chamber
with orthogonal polarizations, to reduce phase fluctuations between the beams which would impress noise on
the lattice. The beams are separated on a polarizing
beam splitter near the MOT. The polarization of one
beam is then rotated such that the beams have parallel polarizations in the MOT. The beams have an angle of 50 degrees between them, creating an optical lat-
2
tice with a lattice constant of L = 0.93 microns. The
depth of the lattice is controlled by the intensity and detuning of the beams, and is chosen to be 18 Er (where
Er = h2 /8L2 m = h·690 Hz is the effective recoil energy of
the lattice) at which depth it contains two bound states.
The energy separation between the states is h̄ · 2π · 5.0
kHz. The scattering rate from the lattice beams is on
the order of 4 Hz, which is insignificant on the timescale
of the experiment.
The population in each band of the lattice is
determined by adiabatically decreasing the lattice
potential[28, 29]. In order to satisfy the adiabatic criterion we must decrease the potential slower than h ·108 /s2 ,
whereas the fastest decrease we use is h · 4.1 · 106/s2 with
non-adiabatic effects appearing if the turn-off is faster
than h · 1.4 · 107 /s2 . As the depth of the wells decreases
the energy bands gradually get closer to the top of the
potential. Once an energy band becomes classically unbound, then all the atoms in that band accelerate downwards due to gravity. Since each state becomes unbound
at a different time, each band becomes mapped into a different location in space. The spatial distribution can be
recorded on a CCD camera by fluorescence imaging. Figure 1a shows a sample spatial distribution after ramping
down the potential over a time of 45 milliseconds. Alternatively, the lattice depth may be quickly, but still
adiabatically, lowered to a depth of 9 Er . At this depth
the ground state has a Landau-Zener-limited lifetime of
250 milliseconds while the excited state has a lifetime of
0.5 milliseconds. The excited state atoms quickly escape
the lattice while ground state atoms remain trapped. After holding the depth constant for some time (typically
about 20 ms), the relative populations can be determined
by fluorescence imaging, as shown in Figure 1b. The
beams creating the lattice have a Gaussian shape, with
rms width of 3 mm, causing the lattice to be shallower
as we move farther away from the center. The spatial
variation of the lattice depth causes atoms at the edge
of the lattice to tunnel out sooner than atoms near the
center, resulting in the curved clouds seen in Figure 1a.
To reduce broadening effects due to inhomogeneous well
depths we integrate the flourescence signal only over the
central 600 µm of the cloud where the potential varies
slowly.
A sample of ground state atoms is prepared by filtering out the excited state atoms from the lattice. This
is accomplished by reducing the well depth to a depth of
approximately 9 recoil energies, at which point only the
ground state is bound. The potential depth is held at
this value for 3 ms, long enough for almost all the excited
state atoms to escape. The well depth is then adiabatically increased back to the original depth, preparing a
sample of atoms with up to 95% occupation of the ground
state.
To prepare a variety of initial states, we make use of
our ability to displace the lattice, and of the atoms’ free
evolution. Displacement of the lattice is equivalent to a
spatial translation of the atom cloud in the lattice’s ref-
erence frame, constituting a coherent coupling between
the energy eigenstates. In addition to coupling the two
bound states, this induces some transitions to unbound
states, which are lost from the lattice. Spatial translations change the coefficients of the states as described by
the following equations.
∆x
|0i =⇒ c00 |0i + c10 |1i + loss
|1i =⇒ −c∗10 |0i + c11 |1i + loss
(1)
The coefficients c00 , c10 and c11 are determined by displacing the lattice with different initial state populations.
A typical displacement used during the measurement is
116 nm. For this displacement we measure c00 ,c10 and
c11 to be 0.86(2), 0.50(2) and 0.53(10) respectively, close
to the theoretical values of 0.87,0.45 and 0.63. During a time period ∆t of free evolution in the lattice, the
ground and excited states acquire a relative phase shift
of ω∆t, where ω is the oscillation frequency in the lattice.
Using a combination of displacement and time delay we
can prepare superposition states with arbitrary relative
phase.
State tomography[30] is performed by projecting the
unknown state onto a set of known states. We use a set of
non-orthogonal states {Φ1 , ..., Φ4 } = {|0i , |1i , |θx i , |θy i}
where[31] |θx i = cos θ |0i + sin θ |1i and |θy i = cos θ |0i −
i sin θ |1i . We project onto states of the form cos θ |0i +
sin θ |1i by spatially displacing the trapping potential before separating the resulting energy eigenstates. We
choose our displacements to be small in order to have
negligible coupling to the higher energy, unbound states.
We use a displacement of L/8=116 nm, generating states
with an experimental value of θ ≈ 0.5 radians. The
state |θx i can be changed into state |θy i by a time delay of a quarter period after displacement. State tomography is performed by measuring the projections
mi = hΦi | ρ |Φi i for all states |Φi i. The resulting measurements {m1 , ..., m4 } are used to reconstruct the density matrix.
ρ=
m1
(m3−i∗m4)−m1 cos2 θ−m2 sin2 θ
2 sin θ cos θ
(m3+i∗m4)−m1 cos2 θ−m2 sin2 θ
2 sin θ cos θ
m2
(2)
Process tomography is first performed on the free evolution of a quantum state in the lattice for one period.
This allows us to completely characterize decoherence intrinsic to the lattice. To perform process tomography we
prepare a complete set of density matrices as input states.
The four linearly independent density matrices we used
correspond to a ground state, prepared by filtering out
the excited state population; a coherent state with real
coherence prepared by displacing a ground state; a coherent state with imaginary coherence prepared by adding
a quarter-period time delay after displacement; and a
mixed state. The mixed state can be prepared by either
skipping the filtering step, or by preparing a coherent
state and waiting 3 ms for it to decohere (see discussion
below).
!
3
QPT proceeds as follows: one of the input density matrices is prepared, and characterized with state tomography as outlined above; the same state is again prepared
and allowed to freely evolve in the lattice for 200µs (one
oscillation period), and then state tomography is performed on the resulting state. Figure 2 shows the projection of each input density matrix onto the projection
states, before and after the ‘operation’.
The super-operator E resulting from QPT can be expressed in a number of ways. One common form is the
operator sum representation,
X
b†
bi ρin A
(3)
A
ρout = E (ρin ) =
i
i
bi are operational elements, often called Kraus
where A
P b† b
Ai Ai = I .
operators[32], subject to the constraint
i
The Choi matrix[33, 34] provides a straightforward procedure to obtain experimental Kraus operators. The
Choi matrix is defined as
X
C=
|ii hj| ⊗ E(|ii hj|)
(4)
i,j
where |ii hj| is an outer product of basis states and
E(|ii hj|) is the super-operator acting on the matrix given
by the outer product |ii hj|. Then
X
Ci,j ρi,j
E (ρ) =
i,j
where Ci,j = E(|ii hj|) is the i,jth 2×2 submatrix of C and
ρi,j is the i,jth element of the density matrix. The eigenvalues and eigenvectors of C can then be used to deterbi = √κi kbi
mine the canonical Kraus operators, given by A
where κi is the ith eigenvalue and kbi is the corresponding
eigenvector written in matrix form. The matrix |ii hj| is
not necessarily a density matrix, but can be written as
a linear combination of the measured input density matrices. Using a maximum-likelihood technique, we find
the Choi matrix which best predicts the measured output states given the measured input states. This search
is limited to physical, i.e., completely positive, Choi matrices. The resulting Choi matrix is shown in Figure 3.
From the Choi matrix we find that the populations are
preserved to within experimental uncertainty while the
coherences decay by 36 percent.
The same data may be visually displayed using a Bloch
sphere representation[25, 35], which has the advantage of
showing how any state on the surface of the Bloch sphere
evolves into a new state. Figure 4a shows the initial,
undisturbed Bloch sphere before evolution in the lattice,
and Figure 4b shows the Bloch sphere after one oscillation. The sphere becomes prolate, by contracting toward
the z-axis by 36%, as expected. The Kraus operators are
determined from the Choi matrix. The most significant
b1 = 0.90Ib + R
b1 and
Kraus operators are found to be A
b
b
b
A2 = −0.41b
σz + R2 where I is the identity matrix, σ
bz
b1 , R
b2 are small remainders
is the z Pauli matrix and R
b†R
b
with magnitudes bound by T r[R
i i ] ≤ 0.03. The other
two Kraus operators are insignificant on the scale of our
experimental resolution, also satisfying a similar bound.
Kraus operators of the form Ib and σ
bz are consistent with
pure dephasing, as expected for from either inter-well
tunneling or inhomogeneous broadening.
The coherence time is thus found to be 555 µs (2.78
periods); this is shorter than the coherence time of 2 ms
expected based on the width of the excited band (the
inter-well tunneling rate) and on the variation in well
depth across the finite Gaussian lattice beams. The
number of oscillations is however consistent with oscillations observed in other work[4, 36, 37]. It is believed
that the observed decoherence is caused by small-scale inhomogeneity, such as fringes, in the lattice beams. The
lattice beams have been spatially filtered before travelling to the MOT. Unlike earlier experiments, we use a
two-level system, and anharmonicity is therefore not a
factor in the observed decoherence.
An example of an operation necessary for quantum
information processing is a single-qubit rotation. To
demonstrate the applicability of process tomography to
characterizing such operations we attempt to perform
a rotation of the Bloch sphere by a method equivalent
to a Rabi oscillation. We sinusoidally drive the displacement of the lattice at the trap frequency, thereby
coupling neighbouring states coherently. Process tomography is performed after a single period of this drive.
The lattice displacement is driven by applying a sinusoidal phase modulation with an amplitude of π/9
radians to one of the lattice beams.
The displacement is kept small to ensure that coupling is predominantly between neighbouring states.
We test both
a sine drive, ∆x(t) = xm sin(ω0 t), and a cosine drive,
∆x(t) = xm (cos(ω0 t) − 1), where xm = 26 nm and ω0
is the oscillation frequency in the lattice and the pulse
lasts from t = 0 to t = 2π/ω0 . We again find the Choi
matrix from a maximum-likelihood model, but find the
Bloch sphere representation to be the most intuitive. Figures 4c and 4d show the resulting Bloch spheres, which
have rotations of 35.5 degrees and 36.4 degrees about
the y-axis and x-axis respectively. The Bloch spheres
are rotated 90 degrees out of phase with one another as
expected for driving fields 90 degrees out of phase. The
resulting shape is close to a sphere, but the radius has
decreased in all dimensions. In particular, the length of
the semi-minor axis’ for the sinusoidal drive is 0.69 (while
it should be noted that in the absence of the coherent
drive, it decayed to a value of 0.64). A simulation using
a truncated harmonic-oscillator model predicts a rotation
of 35.0 degrees about the y-axis.
We have introduced a new technique for determining
the motional quantum state of atoms in an optical lattice,
and applied it to a demonstration of quantum process tomography. We have extracted the “superoperators” fully
characterizing the action of several different operations
on an arbitrary state of atoms in the lattice, specifically,
4
free evolution for one period and two different resonantcoupling protocols. In this way, we have characterized
the intrinsic dephasing of the system over time, and the
effectiveness of single-qubit rotations induced by resonant
modulation of the lattice phase. We plan to extend these
techniques to test the Markovian approximation; to characterize and optimize bang-bang methods[38, 39] for removing inhomogeneous-broadening effects; and to study
the well-depth-dependence of the decoherence, investigating the role of inter-well tunneling, Wannier-Stark
transitions, and Bloch oscillations. The procedure can
be extended to higher-dimensional Hilbert spaces, although the number of measurements required grows exponentially with the dimensionality. Process tomography should prove essential for tailoring error-correction
protocols to the observed behaviour of particular physical
realisations of quantum-information systems[40, 41]; it is
reasonable to expect that such system-by-system tailor-
ing may prove necessary if error thresholds on the order of
10−4 or 10−5 are ever to be reached[13, 14, 15, 16]. More
generally, it is the only method to permit complete characterisation of the evolution of open quantum systems,
and should play a central role in the toolbox for control and study of individual quantum systems, whether
for quantum-information processing or for other applications.
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We would like to thank Daniel Lidar and Samansa
Maneshi for helpful discussions, and we thank Sara
Schneider for assistance with the maximum-likelihood
methods. We would also like to thank the DARPAQuIST program (managed by AFOSR under agreement
No. F49620-01-1-0468), the National Science and Engineering Research Council of Canada, the Canadian Institute for Photonics Inovation, and eMPOWR for financial
support of this project.
Figures:
Figure 1) a) A fluorescence image of the state populations in a lattice obtained by adiabatic decrease of the
lattice potential is shown . A stepwise decrease of the
potential leads to a clearer separation of the states as
shown in b).
Figure 2) Matrix of measured projections. Input density matrices (reading left to right: ground state; mixed
state; superposition with real coherence and superposition with imaginary coherence) are shown along the top
while the post-selected states are listed on the side. The
table on the left shows the projections for the input states
while the table on the right shows the corresponding projections after one period. Note the decreased contrast
in the θx and θy projections for the coherent states. All
populations are unchanged to within experimental error.
Figure 3) The Choi matrix after one oscillation in the
lattice characterizing decoherence. The left graph shows
the real part of the Choi matrix and the right graph shows
the imaginary. The matrix is dominated by real components at the corners. The diagonal corners represent
the mapping of populations into populations. The offdiagonal corners, which map coherences into coherences,
are significantly less than one, showing a loss of coherence. The dotted lines separates the 2 × 2 submatrices
of the Choi matrix.
Figure 4) Bloch sphere representation of process to-
5
mography. a) The initial Bloch sphere representing the
space of pure input states. b) after 1 period of free
evolution the sphere contracts horizontally due to a loss
of coherence. c) B.S. after sine drive showing rotation
about the y-axis. d) B.S. after cosine drive, showing
rotation about the x-axis.
arXiv:quant-ph/0312210 v1 29 Dec 2003
a)
b)
arXiv:quant-ph/0312210 v1 29 Dec 2003
b)
a)
ρ0 ρm ρre
ρim
ρ0 ρm ρre
|x
| y
|0
|1
|x
|0
|1
|
y
ρim
Projection
arXiv:quant-ph/0312210 v1 29 Dec 2003
C11
C12
C11
1
2
1
2
3
3
4
4
1
1
0.5
0.5
0
C21
1
2
Re(C)
3
4
C22
0
2
3
C121
Im(C)
4
C22
arXiv:quant-ph/0312210 v1 29 Dec 2003
y 0.5
0
a)
1
1
b)
-0.5
-1
1
y
0.5
0
-0.5
-1
1
0.5
0.5
z 0
z
-0.5
0
-0.5
-1
-1
-1
-1
-0.5
-0.5
0
0
x
x
0.5
0.5
1
c)
1
y
1
d)
0.5
0
1
y
0.5
0
-0.5
-0.5
-1
1
-1
1
0.5
0.5
z
0
z
0
-0.5
-0.5
-1
-1
-1
-1
-0.5
-0.5
0
x
0
0.5
x
1
0.5
1
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