Multi-particle entanglement of hot trapped ions
Klaus Mølmer∗ and Anders Sørensen†
Institute of Physics and Astronomy, University of Aarhus,
DK-8000 Århus C, Denmark
arXiv:quant-ph/9810040v2 26 Jan 1999
We propose an efficient method to produce multi-particle entangled states of ions in an ion trap
for which a wide range of interesting effects and applications have been suggested. Our preparation
scheme exploits the collective vibrational motion of the ions, but it works in such a way that this
motion need not be fully controlled in the experiment. The ions may, e.g., be in thermal motion
and exchange mechanical energy with a surrounding heat bath without detrimental effects on the
internal state preparation. Our scheme does not require access to the individual ions in the trap.
PACS:32.80.Pj,03.65 Bz,75.10 Jm
the Jz operator, Jz |J, M i = h̄M |J, M i, M = −J, ..., J,
where h̄ is Planck’s constant. Jz ”counts” the number
of excited ions, so that a state, invariant under permutation of the ions, with a certain number Ne of excited
ions is described as |J = N/2, M = Ne − N/2i, and
the state (1) is a superposition of the states |N/2, −N/2i
and |N/2, N/2i. An increase of the quantum number M
by unity corresponds to a single ionic excitation, and it
is described bypthe angular momentum ladder operator
J+ |J, M i = h̄ (J + M + 1)(J − M )|J, M + 1i, where
the normalization factor accounts precisely for the combinatorial aspects of this excitation of the system. As we
shall detail below, it is possible to apply classical laser irradiation which cannot be absorbed by a single ion, but
which requires two ions to undergo the |gi ↔ |ei transition, simultaneously. When this field is applied to all ions
in the trap, the effective Hamiltonian describing the evolution of the state of the ions in the spin representation
can be written,
We describe in this Letter a method to construct states
of the form
1
|Ψi = √ [eiφg |gg...gi + eiφe |ee...ei],
(1)
2
where |gg...ei and |ee...ei (N terms) are product states
describing N ions which are all in the (same) internal
state g or e. The unique correlation between different
particles in such an entangled state is a significant nonclassical feature of quantum mechanics. Entangled states
with two particles have been employed to test Bell’s inequality and to rule out local realistic descriptions of Nature [1]. States like (1) with three particles, the so-called
GHZ states [2], and with more particles allow a closer
scrutiny of the role of quantum correlations, and there
is a large current interest in carrying out experiments
on such states. It has also been shown that the states
(1) are ideal for spectroscopic investigations [3], and the
study of entangled states of many particles is in line with
the rapidly growing interest in quantum encoding and
quantum information processing [4]. Together with an
analysis of their applicability within high precision frequency measurements, a method to produce such states
in an ion trap was proposed in [3]. That method requires
experimental control of two collective vibrational modes
of the ions, which are involved in stepwise manipulations
of the internal states of the ions.
In order to describe our more straightforward proposal,
we shall show that the desired state (1) can be obtained
by application of one simple interaction Hamiltonian. We
shall then show that this Hamiltonian is realized in an ion
trap by simply illuminating all the ions with laser beams
with two different colours, and we shall present numerical results that display the performance of our proposal
under quite relaxed assumptions for the functioning of
the ion trap.
We begin with all ions in the ground state |gi, and we
assume that our external perturbation interacts in the
same way with all ions, so that the joint internal state remains symmetric with respect to interchange of the ions.
In such a situation, it is convenient to apply the spin representation of atomic states, where a collective state of
all ions is represented by |J, M i which is an eigenstate of
H=
χ
χ 2
2
(J+ + J−
+ J+ J− + J− J+ ) = 4 Jx2
h̄
h̄
(2)
where J− is the hermitian conjugate of J+ and where
Jx = (J+ + J− )/2 denotes the x-component of the effective spin.
Time evolution is accommodated by application of the
operator U (t) = exp(−i4χJx2 t/h̄2 ) to the initial state
|N/2, −N/2i and the population becomes distributed on
all |J, M i states, with M differing from −N/2 by an even
number. If N is even, population is transferred all the
way to the state |N/2, N/2i, which is the second component of (1). Moreover, at the instant t = π/(8χ), the
state is precisely of the form (1), with φg = −π/4 and
φe = π/4 + N π/2. We show in Fig. 1. the time evolution
of the populations of the two extremal states for different
(even) numbers of ions. At t = π/(8χ) and at later odd
multiples of this value, both populations equal one half.
This result can be understood from the rotation properties of angular momenta:
The initial state can P
be expanded on eigenstates of Jx :
|N/2, −N/2i = M cM |N/2, M ix, and from the properties of the Wigner rotation functions [5], it follows
1
P
N/2−M
|N/2, M ix . Inthat |N/2, N/2i =
M cM (−1)
serting the values for φg and φe mentioned above, we
P
can therefore write the state (1) as M cM √12 (e−iπ/4 +
and we assume η1 = η2 = η. Numerical solutions of
the Schrödinger equation with the Hamiltonian given by
Eqs.(3,4), are readily obtained within the basis of 2N
internal states and a sufficiently large set of vibrational
states {|f1 f2 ...fN i|ni}, fi = e or g. First, however, we
discuss a perturbative analysis pointing to appropriate
choices of parameters which will effectively yield the spin
Hamiltonian (2).
(−1)M eiπ/4 )|N/2, M ix . The net factors multiplying the
initial amplitudes cM are unity for M even and −i for
M odd. The action of U (t) in the Jx basis amounts to a
multiplication of each amplitude cM by exp(−4iχM 2 t),
and for t = π/(8χ) this factor just attains the desired
value of unity for M even and −i for M odd.
Note that the state (1) is not only a multi-particle entangled state; it is a superposition of two “mesoscopically distinguishable” states. In Quantum Optics, such
states have been studied experimentally for single quantum systems, such as a harmonically trapped ion [6] and
a single mode of the quantized radiation field [7], excited
into superpositions of states of “mesoscopic” separation.
Mathematically, our analysis above resembles the one for
production of such superpositions by propagation of an
optical field through an amplitude dispersive medium [8].
We now turn to a “microscopic” analysis of a collection
of N two-level ions in a linear ion trap. The choice of a
linear trap eliminates the so-called micro-motion for ions
on the axis, and by choosing the transverse confinement
much stronger than the longitudinal one, we can neglect
coupling to transverse vibrations. Due to the Coulomb
repulsion among the ions, the various longitudinal vibrational modes have different frequencies and by choosing
the laser detuning close to the center-of-mass vibrational
frequency we may exclude other modes from our analysis
[9].
The quantized energy of the total system is composed
of the center-of-mass vibrational energy of the ion string
and the internal electronic energy of the ions. The vibrational energy is represented by a harmonic oscillator
with ladder operators a and a† and frequency ν, and the
internal energy is described by Pauli matrices σzi and the
energy difference h̄ωeg ,
H0 = h̄ν(a† a + 1/2) +
h̄ωeg X
σzi .
2
i
The energy levels of the ion string constitute a ladder of equidistant vibrational levels added to the internal
ionic energies. This ladder structure has been extensively
used in laser manipulation of ions in traps, where laser
frequencies differing from the atomic transition frequency
by a multiple of the trap frequency may be used to control the internal and external dynamics [6], and excitation
of collective vibrational modes for several ions has been
proposed as a means for implementing quantum logic in
an ion trap quantum computer [11].
Here, we make use of the idea we put forward in [12]
to apply laser fields with two different frequencies so that
the two-photon process, exciting any pair of ions in the
trap |ggi → |eei is resonant, i.e., ω1 + ω2 = 2ωeg , but
neither of the frequencies are resonant with single excitations of the ions. By choosing ω1 = ωeg − δ and
ω2 = ωeg + δ, where δ is close to, but not resonant with,
the center-of-mass vibration frequency we may limit our
perturbative analysis to involve only the intermediate
states with one excited ion and a vibrational quantum
number raised or lowered by unity. Note that the lack of
degeneracy of vibrational modes is crucial because it prevents a number of modes to interfere and obstruct the collective transitions in the multi-particle system. We show
in Fig. 2 the effective transition paths between states
|ggi|ni and |eei|ni and between |egi|ni and |gei|ni mediated by the interaction with the laser fields within any
pair of ions. Although they serve as intermediate states
in the process, the intermediate states are only virtually
i.e., negligibly, populated. The interaction strengths of
the different one-photon transitions in Fig. 2 depend on
the value √
of the vibrational quantum number n and in the
limit of η n + 1 <<√1, terms between states n and n + 1
whereas terms between n − 1
are proportional to n + 1 √
and n are proportional to n. When we add the contributions from the four different transition paths shown
in either panel of Fig. 2, a “miracle” occurs: The paths
(virtually) involving
√ intermediate states with n+1 vibration quanta yield ( √n + 1Ωη)2 /(δ − ν) and the ones with
n − 1 quanta yield ( nΩη)2 /(ν − δ). The signs of the denominators cause a destructive interference which exactly
removes the n-dependence from the total effective Rabi
frequency of the two-photon transition. This implies that
the internal state evolves in a manner independent of the
external state, which may be in a mixed (e.g., thermal)
state and which may even evolve with time.
(3)
The ions interact with two laser fields with frequencies
ωj (j = 1, 2). We assume that both fields have the same
intensity and we also assume that the ion-laser interaction strength characterized by the Rabi frequency Ω is
the same for all ions. Pauli matrices σ±i describe internal atomic transitions and the interaction Hamiltonian
can be written
X h̄Ω
†
(σ+i ei(ηj (a+a )−ωj t) + h.c.).
(4)
Hint =
2
i,j
(h.c. in (4) denotes the hermitian conjugate.) The Dicke
parameters ηj are products of the projection of the kvector of the j’th laser field along the direction of the
string of ions and the rms excursion of the ionic centerof-mass along this direction [10]. In practical realizations
the k-vectors can be nearly identical for the two fields,
When the summation over all ions is carried out, the
couplings lead to the spin Hamiltonian (2) with
2
χ=
η 2 Ω2 ν
,
2(ν 2 − δ 2 )
mal motion. Our analysis adds the surprising possibility of coherently controlling the mutual internal state of
several particles by explicit use of the uncontrolled incoherently excited degrees of freedom. We believe that
much wider possibilities for quantum state control become feasible with the introduction of the crucial elements of our method: resonance conditions allowing only
pairwise transitions, virtual excitations of un-controlled
degrees of freedom, and partially destructive interference
between transition paths. We have recently applied these
ideas to quantum computation in an ion trap [12] and
we are currently investigating their possible use in other
physical systems.
(5)
and to the extent that our perturbative treatment is
valid, we therefore expect to realize the results, presented
in Fig. 1. In Fig. 3. we show numerical simulations of the
dynamics governed by the correct Hamiltonian (3,4) for 4
ions, i.e. beyond perturbation theory and to all orders in
the Dicke parameter η. We assume a distribution on vibrational levels with quantum numbers ranging between
0 and 40, and we assume that the center-of-mass motion
exchanges energy with a surrounding thermal
p reservoir
represented
by
relaxation
operators
c
=
Γ(nth + 1)a
1
√
and c2 = Γnth a† . Our calculation for a reservoir with
a mean excitation of nth = 5 vibrational quanta at the
trap frequency and a coupling strength Γ = 0.0001 may
represent a heating mechanism towards higher temperatures counteracted by laser cooling on one ion in the
string, reserved for this purpose. The action of the reservoir is described by quantum jumps and a non-Hermitian
Hamiltonian in a Monte Carlo wave function simulation
[13]. Part (a) of the figure represents the result of a single Monte Carlo realisation with quantum jumps in the
vibrational states at instants of time, indicated by arrows
in the figure. Part (b) is an average over 10 such realisations. The full lines present the populations and coherences of the internal state of the ions, obtained by tracing
out the vibrational degrees of freedom. For comparison
we also show the result of a calculation with the corresponding spin Hamiltonian (2) (dashed curve). With
the realistic parameters chosen, the state at νt ∼ 1500 is
very close to the desired state in (1). Small oscillatory
deviations from the smooth results of Fig. 1. are seen.
The magnitude of the oscillations is given by the physical
parameters, so that in the limit of small η and Ω, they
can be made arbitrarily small [12].
The pairwise interaction with ions does not produce
a coherent coupling of the two components in (1) if N
is odd. In this case, however, one can verify within the
spin model, that it suffices to apply a linear coupling
H1 = 4ξJx for the duration t = π/(8ξ) in addition to
application of our quadratic term (2). Since Jx and Jx2
commute, the linear Hamiltonian may actually be applied before, after or simultaneously with the quadratic
one. This additional operation is easy to implement in
the ion trap, where one only needs to drive the ions on
the atomic resonance frequency ωeg .
In summary, we have presented a method to prepare a
multi-particle entangled state of trapped ions. Our suggestion aims at experimental application, and it is relevant both for experiments with limited control of the
vibrational motion, where the ions heat up for different
technical reasons, or where there is not access to individual ions. The possibility to exercise control over some
quantum degrees of freedom of a system while others are
uncontrolled is common place in coherent optical excitation experiments, where the internal states of atoms in
a gas may be controlled regardless of the external ther-
∗
†
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
3
Email: moelmer@ifa.au.dk
Email: anderss@ifa.au.dk
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Physics World 11, 33-57, March 1998. Special issue on
Quantum Information.
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B. Yurke and D. Stoler, Phys. Rev. Lett. 57, 13 (1986).
Numerical tests including an additional vibrational mode
prove that it is sufficient to deal with the near-resonant
center-of-mass mode of the ions.
D. M. Meekhof et al, Phys. Rev. Lett. 76, 1796 (1996).
This Letter shows how the two-level formalism also describes two stable states coupled by Raman transitions.
The k-vector in the definition of the Dicke parameter is
then a k-vector difference, and η can be varied more or
less freely in an experiment.
J. I Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995).
Anders Sørensen and Klaus Mølmer, Quantum computation with ions in thermal motion, to appear in Phys. Rev.
Lett., quant-ph/9810039.
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68, 580 (1992).
1
1
(a)
0.8
0.8
0.6
0.6
0.4
0.2
0.4
0
0.2
-0.2
0
0
0
0.2
0.4
χt
0.6
0.8
1000
1
2000
νt
3000
1
FIG. 1. Time evolution of the population of the joint ionic
ground state |gg...gi = |N/2, −N/2i (curves starting from
the value of unity at t = 0), and the joint ionic excited state
|ee...ei = |N/2, N/2i. Results are presented for different values of the number of ions: N=2 (solid curves), N=4 (dashed
curves) and N=8 (dotted curves). At t = π/(8χ) the states
are in a 50-50 superposition and the state (1) is obtained.
4000
(b)
0.8
0.6
0.4
0.2
0
-0.2
|een>
ω
1
|egn+1>
|egn>
|egn-1>
ω
ω
ω
1
ω
|gen+1>
|gen>
|gen-1>
2
2
2
ω
|egn>
ω
ω
|ggn>
|een+1>
|een>
|een-1>
2
1
0
1000
2000
νt
3000
4000
FIG. 3. Results of numerical integration of the Schrödinger
equation for 4 ions. The physical parameters are δ = 0.9ν,
Ω = 0.1ν, η = 0.1, Γ = 0.0001ν and nth = 5. (a) Single
Monte Carlo realisation with quantum jumps in the vibrational states at instants of time, indicated by arrows in the
figure. (b) Average over 10 realisations. The first full curve
(counted from above at νt < 1000) is the population of the
joint ground state, the second one is the imaginary part of
the internal density matrix element between the joint ground
and the joint excited state, the third one is the population of
the joint excited state, and the fourth one is the real part of
the off-diagonal density matrix element. The dashed curves
are equivalent to the N = 4 curves of Fig. 1, obtained by
η 2 Ω2 ν
application of the spin Hamiltonian (2) with χ = 2(ν
2 −δ 2 ) .
|gen>
1
|ggn+1>
|ggn>
|ggn-1>
FIG. 2. Level scheme for a pair of ions sharing an oscillator
degree of freedom. Left: by application of laser light with frequencies ωeg ± δ, where δ is somewhat smaller than the vibrational frequency ν, we identify four transition paths between
the states |ggi|ni and |eei|ni, which interfere as described in
the text. Right: four similar transition paths are identified
between states |egi|ni and |gei|ni, yielding the same effective
coupling among these states as between the states in the left
panel.
4