Preparation and detection of magnetic quantum phases in optical
  superlattices
Preparation and detection of magnetic quantum phases in optical superlattices
A. M. Rey1, V. Gritsev2, I. Bloch 3, E. Demler1,2 and M.D. Lukin1,2
1 Institute for Theoretical Atomic, Molecular and Optical Physics,
Harvard-Smithsonian Center of Astrophysics, Cambridge, MA, 02138.
2 Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA and
3 Johannes Gutenberg-Universität, Institut für Physik, Staudingerweg 7,55099 Mainz, Germany
We describe a novel approach to prepare, detect and characterize magnetic quantum phases in ultra-cold
spinor atoms loaded in optical superlattices. Our technique makes use of singlet-triplet spin manipulations in an
array of isolated double well potentials in analogy to recently demonstrated quantum control in semiconductor
quantum dots. We also discuss the many-body singlet-triplet spin dynamics arising from coherent coupling
between nearest neighbor double wells and derive an effective description for such system. We use it to study
the generation of complex magnetic states by adiabatic and non-equilibrium dynamics.
PACS numbers: 05.50.+q 03.67.Mn 05.30.Fk 05.30.Jp
Recent advances in the manipulations of ultra-cold atoms
in optical lattices have opened new possibilities for exploring
complex many-body systems [1]. A particular topic of con-
tinuous interest is the study of quantum magnetism in spin
systems [2, 3, 4]. By loading spinor atoms in optical lattices it
is now possible to ”simulate” exotic spin models in controlled
environments and to explore novel spin orders and phases.
In this Letter we describe a new approach for prepara-
tion and probing of many-body magnetic quantum states that
makes use of coherent manipulation of singlet-triplet pairs of
ultra-cold atoms loaded in deep period-two optical superlat-
tices. Our approach makes use of a spin dependent energy
offset between the double-well minima to completely control
and measure the spin state of two-atom pairs, in a way anal-
ogous to the recently demonstrated manipulations of coupled
electrons in semiconductor double-dots [5]. As an example,
we show how this technique allows one to detect and analyze
anti-ferromagnetic spin states in optical lattices. We further
study the many-body dynamics that emerge when tunneling
between nearest neighbor double wells is allowed. As two
specific examples, we show how a set of singlet atomic states
can be evolved into singlet-triplet cluster-type states and into a
maximally entangled superposition of two anti-ferromagnetic
states. Finally, we discuss the use of our projection technique
to probe the density of spin defects (kinks) in magnetic states
prepared via equilibrium and non-equilibrium dynamics.
The key idea of this work is illustrated by considering a pair
of ultra-cold atoms with two relevant internal states, which
we identify with spin up and down σ =↑, ↓ in an isolated
double well (DW) potential as shown in Fig.1. By dynam-
ically changing the optical lattice parameters, it is possible
to completely control this system and measure it in an arbi-
trary two-spin basis. For concreteness, we first focus on the
fermionic case. The physics of this system is governed by
three sets of energy scales: i) the on-site interaction energy
U = U↑↓ between the atoms, ii) the tunneling energy of the
σ species: Jσ, and iii) the energy difference between the two
DW minima, 2∆σ for each of the two species. The σ index
in J and ∆ is due to the fact that the lattice that the ↑ and
↓ atoms feel can be engineered to be different by choosing
laser beams of appropriate polarizations, frequencies, phases
and intensities. In the following we assume that the atoms are
strongly interacting, U ≫ Jσ, and that effective vibrational
energy of each well, ~ω0, is the largest energy scale in the
system ~ω0 ≫ U,∆σ, Jσ , i.e deep wells.
Singlet |s〉 and triplet |t〉 states form the natural basis for
the two-atom system. The relative energies of these states
can be manipulated by controlling the energy bias ∆σ be-
tween the two wells. In the unbiased case (U ≫ 2∆σ) only
states with one atom per site (1, 1) are populated, as the large
atomic repulsion energetically suppresses double occupancy
(here, labels (m,n) indicate the integer number of atoms in
the left and right sites of the DW). For weak tunneling and
spin independent lattices (J↑ = J↓ = J , ∆↑ = ∆↓ = ∆)
the states (1, 1)|s〉 and (1, 1)|t〉 are nearly degenerated. The
small energy splitting between them is ∼ 4J2/U , with the
singlet being the low energy state (Fig. 1a). As ∆ is increased
the relative energy of doubly occupied states (0, 2) decreases.
Therefore, states (1, 1)|s〉 and (0, 2)|s〉 will hybridize. When
2∆ & U the atomic repulsion is overwhelmed and conse-
quently the (0, 2)|s〉 becomes the ground state. At the same
time, Pauli exclusion results in a large energy splitting ~ω0
between doubly occupied singlet and triplet states as the lat-
ter must have an antisymmetric orbital wave function. Hence,
(1, 1)|t〉 does not hybridize with its doubly occupied counter-
part, and its relative energy becomes large as compared to the
singlet state. Thus the energy difference between singlet and
triplet states can be controlled using ∆.
Further control is provided by changing Jσ and ∆σ in spin
dependent lattices (see Fig.1b). Specifically, let us now con-
sider the regime 2∆σ ≪ U in which only (1, 1) subspace is
populated. Within this manifold we define [6]
|s〉 = ŝ†|0〉 ≡
(| ↑↓〉 − | ↓↑〉), (1)
|tz〉 = t̂†z|0〉 ≡
(| ↑↓〉+ | ↓↑〉), (2)
http://arxiv.org/abs/0704.1413v3
|tx〉 = t̂†x|0〉 ≡
(| ↑↑〉 − | ↓↓〉), (3)
|ty〉 = t̂†y|0〉 ≡
(| ↑↑〉+ | ↓↓〉) (4)
Here t̂†α and ŝ are operators that create triplet and sin-
glet states from the vacuum |0〉 (state with no atoms).
They satisfy bosonic commutation relations and the constrain
α=x,y,z t̂
αt̂α) + ŝ
†ŝ = 1, due to the physical restriction
that the state in a double well is either a singlet or a triplet. In
the rest of the letter we will omit the label (1, 1) for the singly
occupied states.
When ∆σ depends on spin, i.e Υ ≡ ∆↑ − ∆↓ 6= 0, the
|tz〉 component mixes with |s〉(see Fig.1c). Note that on the
other hand |tx,y〉 remain decoupled from |tz〉 and |s〉 . As a
result the states |s〉 and |tz〉 form an effective two-level system
whose dynamics is driven by the Hamiltonian:
ĤJ1 = −ζ(ŝ†ŝ− t̂†z t̂z)−ΥS̃z + const, (5)
Here ζ ≡ 2J↑J↓/Ũ , is the exchange coupling energy (with
Ũ ≡ U
2−(∆↑+∆↓)2
) and S̃z = ŝ†t̂z + t̂
z ŝ. If Υ = 0, ex-
change dominates and |s〉 and |tz〉 becomes the ground and
first excited states respectively. However if Υ ≫ ζ, exchange
can be neglected and the ground state becomes either | ↑↓〉 or
| ↓↑〉 depending on the sign of Υ.
0.0 0.2 0.4 0.6 0.8 1.0
-U -U
-4J  /U2
Y=     -
(a)(0,2)s + (2,0)s
(1,1)s
(1,1)t
(0,2)s
(1,1)s,(1,1)t
FIG. 1: (color online) a) Energy levels of fermionic atoms in a spin
independent double well as ∆/U is varied: While in the regime
2∆ ≪ U , (1, 1)|s〉 is the lowest energy state, when 2∆ & U ,
(0, 2)|s〉 becomes the state with lowest energy. b) In spin depen-
dent potentials the two species feel different lattice parameters c)
Restricted to the (1, 1) subspace Υ acts as an effective magnetic field
gradient and couples the |s〉 and |tz〉 states .
These considerations indicate that it is possible to perform
arbitrary coherent manipulations and robust measurement of
atom pair spin states. The former can be accomplished by
combining time-dependant control over ζ,Υ to obtain effec-
tive rotations on the spin-1/2 Bloch sphere within |s〉 − |tz〉
state. In the parameter regime of interest, ζ,Υ, can be var-
ied independently in experiments. In addition, by applying
pulsed (uniform) magnetic fields it is possible to rotate the
basis, thereby changing the relative population of the |tx,y,z〉
states. Atom pair spin states can be probed by adiabatically
increasing ∆ until it becomes larger than U/2, in which case
atoms in the |s〉 will adiabatically follow to (0, 2)|s〉 while the
atoms in |tα〉 will remain in (1,1) state (Fig. 1a). A subse-
quent measurement of the number of doubly occupied wells
will reveal the number of singlets in the initial state. Such
a measurement can be achieved by efficiently converting the
doubly occupied wells into molecules via photoassociation or
using other techniques such as microwave spectroscopy and
spin changing collisions [7]. Alternatively, one can continue
adiabatically tilting the DW until it merges to one well. In
such a way the |s〉 will be projected to the (0, 2)|s〉, while the
triplets will map to (0, 2)|tα〉. As (0, 2)|tα〉 has one of the
atoms in the first vibrational state of the well, by measuring
the population in excited bands one can detect the number of
initial |tα〉 states. Hence the spin-triplet blockade [5] allows
to effectively control and measure atom pairs.
Detection and diagnostics of many-body spin phases such
as antiferromagnetic (AF) states is an example of direct ap-
plication of the singlet-triplet manipulation and measurement
technique. The procedure to measure the AF state population
is the following; after inhibiting tunneling between the various
DWs, one can abruptly increase Υ, such that the initial state
is projected into the new eigenstates | ↑↓〉 and | ↓↑〉 at time
τ = τ0. For τ > τ0 Υ can then be adiabatically decreased to
zero, in which case the | ↑↓〉 pairs will be adiabatically con-
verted into |s〉 and | ↓↑〉 pairs to |tz〉. Finally, the singlet pop-
ulation can be measured using the spin blockade. As a result,
a measure of the doubly occupied sites (or excited bands pop-
ulation) will detect the number of | ↑↓〉 pairs and thus probe
antiferromagnetic states of the type | ↑↓↑↓ ...〉.
These ideas can be directly generalized to perform mea-
surements of the more complex magnetic states that can be
represented as products of two atom pairs. For example, a
pulse of RF magnetic field can be used to orient all spins,
thus providing the ability to detect |AF 〉 states aligned along
an arbitrary direction. Moreover, one can determine the rela-
tive phase between singlet and triplet pairs in |AF 〉 states of
the form
|s〉 + eiφ|tz〉 by performing Ramsey-type spec-
troscopy. After letting the system evolve freely (with Υ = 0)
so that the |s〉 and |tz〉 components accumulate an additional
relative phase due to exchange, a read-out pulse (controlled
by pulsing Υ) will map the accumulated phase onto popula-
tion of singlet and triplet pairs. To know φ is important as it
determines the direction of the anti-ferromagnetic order. Fur-
thermore, by combining the blockade with noise correlation
measurements [8] it is possible to obtain further information
about the magnetic phases. While the blockade probes local
correlation in the DWs, noise measurements probe non-local
spin-spin correlations and thus can reveal long range order.
Before proceeding we note that similar ideas to that out-
lined above can be used for bosonic atoms if initially no |tx,y〉
states are populated. The latter can be done by detuning the
|tx,y〉 states by means of an external magnetic field. In the
bosonic case the doubly occupied tz states will be the ones
that have the lowest energy. They will be separated by an
energy ~ω0 from the doubly occupied singlets as the latter
are the ones that have antisymmetric orbital wave function in
bosons. Consequently, the role of |s〉 in fermions will be re-
placed by |tz〉 in bosons. The read-out procedure would then
be identical to that described above, while the coherent dy-
namics will be given by the Hamiltonian Eq.(5) apart from
the sign change ζ → −ζ.
Up to now our analysis has ignored tunneling between dif-
ferent DWs, but in practice this tunneling can be controlled by
tuning the lattice potential. How will singlet and triplet pairs
evolve due to this coupling? We will now discuss the many-
body dynamics that emerges when nearest neighbor DW tun-
neling is allowed, i.e. tσ > 0. When atoms can hop between
DWs, the behavior of the system will depend on the dimen-
sionality. For simplicity we will restrict our analysis to a 1D
array ofN double-wells, where tσ corresponds to hopping en-
ergy of σ-type atoms between the right site of the jth −DW
and the left site of the (j + 1)th −DW .
In the regime Jσ, tσ,∆σ ≪ U , multiply occupied wells
are energetically suppressed and the effective Hamiltonian is
given by Ĥeff = ĤJ+Ĥt . Here the first term corresponds to
the sum over N independent HJj Hamiltonians (see Eq.(5)),
ĤJ =
j , each of which acts on its respective j
DW. On the other hand Ĥt is non-local as it couples different
DWs and quartic as it consists of terms with four singlet-triplet
operators [9]. The coupled DWs system is in general complex
and the quantum spin dynamics can be studied only numeri-
cally. However, there are specific parameter regimes where an
exact solution can be found. For this discussion we will set
∆σ = 0. If t↑/t↓ → 0, and at time τ = 0, no |tx〉, |ty〉 triplet
states are populated, their population will remain always zero.
Consequently, in this limit, the relevant Hilbert space reduces
to that of an effective spin one-half system with |s〉 and |tz〉
representing the effective ±1/2 states, which we denote as
| ⇑〉 and | ⇓〉. Ĥt couples such effective spin states. In the
restricted Hilbert space Ĥeff maps exactly to an Ising chain
in a magnetic field:
Ĥeff = ∓ζ
σ̂zj − λz
σ̂xj σ̂
j+1 (6)
where σ̂α are the usual Pauli matrices which act of the effec-
tive | ⇑〉 and | ⇓〉 spins. In terms of singlet-triplet operators
they are given by σ̂zj = (ŝ
j ŝj − t̂
zj t̂zj), σ̂
j = ŝ
j t̂zj + t̂
zj ŝj
and σ̂yj = (ŝ
j t̂zj − t̂
zj ŝj)/i. Here λz =
and the up-
per and lower signs are for fermions and bosons respectively.
For fermions in the lowest vibrational level the onsite interac-
tion energy between the same type of atoms U↑↑, U↓↓ → ∞
due to the Pauli exclusion principle.
The 1D quantum Ising model exhibits a second order quan-
tum phase transition at the critical value |g| ≡ |λz/ζ| = 1.
For fermions (upper sign) when g ≪ 1 the ground state cor-
responds to all effective spins pointing up, i.e |G〉 = | ⇑ . . . ⇑
〉 = Πj |s〉j . On the other hand when g ≫ 1, there are two
degenerate ground states which are, in the effective spin ba-
sis, macroscopic superpositions of oppositely polarized states
along x. In terms of the original fermionic spin states this su-
perposition correspond to the states |AF±〉 = 1√
(| ↑↓ . . . ↑↓
〉 ± | ↓↑ . . . ↓↑〉). Therefore, by adiabatic passage one could
start with |G〉 and convert it into AF state(s). Due to vanish-
ing energy gap at the quantum critical point g = 1, adiabatic-
ity is difficult to maintain as N → ∞ [11, 12, 13, 14]. In
that respect, our projection scheme is useful to test adiabatic
following. It can be done either by measuring the number of
| ↑↓〉 pairs in the final state or by adiabatically ramping down
g back to zero and measuring the number of singlets/triplet
pairs. The remaining number of triplets will determine the
number of excitations created in the process.
We now turn to non-adiabatic dynamics. We will discuss
the situation where initially the system is prepared in a prod-
uct of singlet states (λz = 0 ground state ) and then one lets it
evolve for τ > 0 with a fixed |λz | > 0. Generically the cou-
pling between DWs results in oscillations between singlet and
triplet pairs with additional decay on a slower time scale. We
present two important special cases involving such dynamics:
i) Singlet-triplet cluster state generation: If the value of λz
is set to be |λz | ≫ ζ, then the Hamiltonian reduces to a pure
Ising Hamiltonian and thus at particular times, τc, given by
λzτc/~ = π/4 mod π/2 the evolving state becomes a d = 1
cluster state |C〉 in the effective spin basis [15]. Up to single
spin rotations |C〉 = 1
j=1(| ⇑〉j σ̂zj+1 + | ⇓〉j). Cluster
states are of interest for the realization of one-way quantum
computation proposals where starting from the state |C〉 com-
putation can be done via measurements only. Preparation of
cluster states encoded in the logical ⇑,⇓ qubits may have sig-
nificant practical advantages since the ⇑,⇓ states have zero
net spin along the quantization axis and hence are not affected
by global magnetic field fluctuations. Additionally, the use
of such singlet-triplet states for encoding might allow for the
generation of decoherence free subspaces insensitive to col-
lective and local errors [16] and for alternative schemes for
measured-based quantum computation [17].
ii) Non-equilibrium generation and probing of AF corre-
lations: The second situation is when the value of λz is set
to the critical value, |λz| = ζ (or g = 1). We will first
focus on the fermionic system λz > 0. To discuss it, we
remind that the dynamics driven by Ĥeff is exactly solv-
able as Ĥeff can be mapped via the Jordan Wigner trans-
formation into a quadratic Hamiltonian of fermionic opera-
tors which can be diagonalized by a canonical transforma-
tion [10, 14]. Using such transformation it is possible to
show that at specific times, the shortest of them we denote
by τm ≈ ~N+14ζ , long range AF correlations build up and for
small atom number the state approaches |AF+〉. To quan-
tify the resulting state in Fig. 2(inset) we plot the fidelity,
defined as F1(τm) = |〈AF+|ψ(τm)〉g=1|2, as a function of
N . The figure shows that while an almost perfect |AF+〉 is
dynamically generated for small N , its fidelity exponentially
degrades with increasing atom number.
However, the fidelity is a very strict probe, as it drops to
zero when a single spin is flipped. As N increases the sys-
tem ends at τm in a quantum superposition of states like
| · · · ⇒⇐⇐⇐⇐⇐⇒⇒⇒⇒⇒⇒⇐ . . . 〉 with finite do-
mains of ”effective spins” pointing along ±x, separated by
kinks where the polarization of the spins change its orien-
tation (we used the convention | ↑↓〉 ≡ | ⇒〉). Conse-
quently, one gets more realistic information about the AF
order of the state, by measuring the average size of the do-
mains or the average density of kinks, the latter defined as
ν ≡ 1
j(1 − 〈ψ(τ)|σ̂xj σ̂xj+1|ψ(τ)〉).
Our read-out technique can be used to detect the kink-
density as for an arbitrary fixed g energy conservation imposes
a relation between ν and the triplet-z density, Nt:
ν(τ, g) =
− Nt(τ, g)
. (7)
A simple analytical expression for Nt(τ, g) can be ob-
tained by using the Jordan Wigner transformation [10]:
Nt(τ, g) =
sin2(2πk/N) sin2(2ωkτ)
where ~ωk =
g2 + 1 + 2g cos(2πk/N) are quasi-particle frequencies of
Ĥeff . The fact that it remains always below 0.2 (see Fig.
2) confirms the idea that regardless of the reduced fidelity at
large N , the state does retain AF correlations. We point out
that |AF+〉 states are only generated at g = 1, a feature that
illustrates the special character of the critical dynamics.
0 20 40 60 80 100
0.025
0.075
0.125
0 20 40 60 80 100N
FIG. 2: Using the the Jordan-Wigner transformation [10] we cal-
culated the density of kinks vs N at τ = τm and the fidelity
|〈ψ(τm)|AF
+〉|2 vsN (inset). Our projection technique can be used
to measure ν(τ ) as it is directly related to the triplet density, Nt(τ )
(see Eq.(7)).
Let us now discuss the bosonic case. If λz > 0, the
fermionic results apply for bosons by simply interchanging
the role of |s〉 ↔ |tz〉. On the other hand if λz < 0, not only
one has to interchange |s〉 ↔ |tz〉 but additionally, the adia-
batic and non-equilibrium dynamics will generate, instead of
|AF±〉 states, 1√
(| ⇒⇐ · · · ⇒⇐〉 ± | ⇐⇒ · · · ⇐⇒〉) i.e
macroscopic superpositions of AF states along the x-direction
in the effective spin basis. With these modifications, the re-
sults derived for fermions hold for bosons[21].
Before concluding we briefly mention that spin dependent
superlattices of the form
j=1,2
(Aj +Bjσz) cos
2[kz/j + θj ] (8)
can be experimentally realized by superimposing two inde-
pendent lattices, generated by elliptically polarized light, one
with twice the periodicity of the other [18, 19, 20]. Com-
plete control over the DW parameters is achieved by control-
ling the phases (which determine ∆), intensities (which deter-
mine U ,J and t) and polarization of the laser beams (which
allow for spin dependent control). For example lattice con-
figurations with t↑ ≪ t↓ can be achieved by setting the laser
parameters such that B1 = 0 and A2 = B2 ≫ 1.
In summary we have described a technique to prepare, de-
tect and manipulate spin configurations in ultra-cold atomic
systems loaded in spin dependent period-two superlattices.
By studying the many-body dynamics that arises when tun-
neling between DWs is allowed, we discussed how to dynam-
ically generate singlet-triplet cluster states and AF cat states,
which are of interest for quantum information science, and
how to probe AF correlations in far from equilibrium dynam-
ics. Even though in this Letter we restrict our analysis to 1D
systems the ideas developed here can be extended to higher
dimensions and more general kinds of interactions.
We acknowledge useful discussions with G. Morigi. This
work was supported by ITAMP, NSF (Career Program),
Harvard-MIT CUA, AFOSR, Swiss NF, the Sloan Founda-
tion, and the David and Lucille Packard Foundation.
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[21] In this case a different sign in the definition of kink density
σ̂xj σ̂
j+1 → −σ̂
j+1 is required.
