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Control of Rydberg atoms to perform Grover’s search
algorithm
C. RANGAN, J. AHN, D. N. HUTCHINSON and P. H.
BUCKSBAU M
FOCUS Center, Physics Department, University of Michigan, Ann
Arbor, MI 48109 -1120, USA
(Received 15 February 2002; revision received 27 May 2002)
Abstract. It is shown that Grover’s search algorithm can be implemented on
a Rydberg atom data register using a short terahertz half-cycle pulse. Using
optimal control theory, a shaped terahertz pulse is designed that can perform
the search algorithm better than an unshaped half-cycle pulse. Starting from an
initial wave packet, it is shown that it is possible to use the search algorithm to
synthesize single-energy eigenstates.
1. Introduction
In this paper, we show that Grover’s search algorithm [1] can be performed on
a Rydberg atom data register using a broadband, terahertz half-cycle pulse (HCP).
The search algorithm is a quantum mechanical method of searching an N-state
database to ®nd a single bit of information stored in one of the states. A general
understanding of the search algorithm can be obtained from reference [2]. The
database consists of quantum states that are otherwise indistinguishable except for
the relative phase di erences between them. The algorithm transforms this phase
information into amplitude information, which can then be measured easily. The
algorithm has been implemented in several physical systems [3±9]; here we present
an implementation in a Rydberg atom.
We present three related, but di erent, results. Using an impulse model of the
HCP, it is shown that the phase retrieval performed by the HCP is closely related
to the inversion-about-the-average operation central to the search algorithm.
Using optimal control theory, shaped terahertz pulses are designed that can
perform the search algorithm better than unshaped HCPs. Finally, it is shown
that a broadband terahertz pulse can be used to drive a Rydberg wave packet
population into a single eigenstate.
The motivation for this research is the implementation of Grover’s search
algorithm [1] in a Rydberg atom. Figure 1 shows a schematic of this process. In
our implementation of the quantum search algorithm, the data register is a
Rydberg wave packet, i.e. a superposition of several eigenstates. In the experi-
mental system, the 24pthrough 29pstates of Cesium are used. Each eigenstate acts
as a bit, and the information is stored in the phases of these eigenstates. A binary
encoding is used: if the phase of an eigenstate relative to a reference state is 0, then
the bit value is binary 0; if the phase of an eigenstate is ºrelative to a reference,
Journal of Modern Optics ISS N 0950±0340 print/ISSN 1362±3044 online #2002 Tayl or & Francis Ltd
http://www.tandf.co.uk/journals
DOI: 10.1080/0950034021 000011365
journal of modern optics, 2002,
vol. 49, no. 14/15, 2339± 2347
then the bit value is binary 1. The search space is restricted to those wave packets
one of whose constituent orbitals has its phase reversed. Thus an initial wave
packet with the constituent orbitals |24pito |29piof Cesium, with the phase of the
|26piorbital reversed with respect to the others is represented by the column
vector …1;1;¡1;1;1;1†T=
N
p, where Nˆ6. The data register is loaded by exciting
the Rydberg atom using a shaped, optical pulse [10]. A schematic of this process is
shown in ®gure 1(a).
This phase information is converted to amplitude information by a terahertz,
broadband, half-cycle pulse (HCP) [11], shown in ®gure 1(b). The half-cycle
pulse reveals the marked bit by redistributing the Rydberg population so that
a signi®cant fraction of the electron’s probability density lies in the orbital that
was initially 1808out of phase with respect to the other orbitals. This amplitude
information is read out by state-selective ®eld ionization of the Rydberg
atoms. This is a well-established experimental methodÐthe ®eld ionization signal
provides a temporal map of the electronic probability distribution, indicated
2340 C. Rangan et al.
Figure 1. Schematic of the phase retrieval process in a Rydberg atom. (a) Read-in of
data by excitation of a Rydberg wave packet. (b) Electric ®eld of the terahertz half-
cycle pulse that performs the search algorithm. (c) Measurement of decoded
information by state-selective ®eld ionization.
schematically in ®gure 1(c). Thus the half-cycle pulse performs the search by
converting the phase information (which state has its phase reversed) into
amplitude information (which state has the largest amplitude), which can then
be measured.
2. Phase retrieval and the Grover’s search algorithm
The initial experiment and calculations presented in reference [11] show that
the terahertz pulse ampli®es the phase-¯ipped orbital of a Rydberg wave packet
data register. Calculations using both an impulse model and a ®tted model of the
HCP (full solution to the time-dependent Schro
Èdinger equation) agree very well
with the experimental results. This indicates that the impulse approximation,
when the width of such a half-cycle pulse is much shorter than the Kepler orbital
period of the wave packet, is valid for this system. Therefore, we use the impulse
model to understand the how the HCP performs the database search.
The action of the half-cycle pulse on the initial wave packet can be written (in
atomic units) as
jª…T†i ˆ exp ¡i…T
0
E…t†dt
³ ´jª…0†i …1†
In the impulse limit, when the pulse width is much smaller than the Kepler orbital
period, the electron does not evolve while the pulse is on. The pulse transfers a
momentum Qto the electron, equal to the integrated area of the electric ®eld as a
function of time:
jª…T†i ˆ exp…iQz†jª…0†i …2†
The matrix elements of the impulse operator in the energy basis can be written as
Mn0l0nl ˆ hn0;l0;mˆ0jexp…iQz†jn;l;mˆ0i …3†
For the database states (24pto 29pof Cs),
Mn0l0nl ˆ hn0pjexp…iQz†jnpi …4†
As this operator connects states of the same parity, only the even powers of Qz
contribute to the matrix elements, and the matrix elements are all real. This is not
a requirement for the search algorithm, but an extra symmetry imposed by the
experimental implementation. These matrix elements can be calculated analyti-
cally [11]. Examining the matrix elements as a function of Q, the total momentum
transferred by the HCP [12], we see that the diagonal matrix elements (that
connect a database state to itself) and the o -diagonal matrix elements (that
connect a database state to its neighbors in the database) both oscillate as a
function of Q. For a range of Q-values around Q
0
, the diagonal elements of the
M-matrix have the opposite sign from the o -diagonal elements. At QˆQ0, the
magnitudes of the o -diagonal elements are roughly equal, so we can write the
action of the matrix on the initial state as
Control of Rydberg atoms to perform Grover’s search algorithm 2341
¡a b b b b b
b¡abbbb
b b ¡a b b b
b b b ¡a b b
b b b b ¡a b
bbbbb¡a
0
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
A
1
1
¡1
1
1
1
0
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
A
1
N
pˆ
¡a‡N¡3†b
¡a‡ …N¡3†b
‡a‡ …N¡1†b
¡a‡ …N¡3†b
¡a‡ …N¡3†b
¡a‡ …N¡3†b
0
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
A
1
N
p…5†
If one of the states is marked, i.e. has its phase reversed with respect to those of the
others, the multimode interference conditions are appropriate for constructive
interference to the marked state, and destructive interference to the others. That is,
the population in the marked bit is ampli®ed. The form of this matrix is identical
to that obtained through the inversion about the average procedure in Grover’s
search algorithm. This operator produces a contrast in the probability density of
the ®nal states of the database of …a‡ …N¡1†b†2=Nfor the marked bit versus
…a¡ …N¡3†b†2=Nfor the unmarked bits; where, Nis the number of bits in the
data register.
3. Optimal phase retrieval via shaped terahertz pulses
In the physical system, phase retrieval is not performed with the e ciency
expected from the matrix Min equation (5). A major reason for this is that the far
o -diagonal matrix elements are, in general, smaller than the near o -diagonal
elements. The ampli®cation of population in the marked bit occurs by `di usion’
(to borrow Grover’s term) of population from the nearby bits with a relative phase
between them. Since the overlap between states that are far apart in the register is
small, the far o -diagonal terms in the matrix are also small. If the marked bit is
one of the outer states in the register, its phase retrieval is not as e cient as
expected from the matrix.
This e ect was seen in the published experiment [11] where the 26pand 27p
states were retrieved with higher ®delity than the 25pand 28pstates. This problem
has been addressed in a recent theoretical work [13], where, by using a shaped
terahertz pulse designed by optimal control theory, it is shown that the phase
retrieval can be performed successfully for all but the two outermost database
orbitals. The challenge here is to design a pulse that does more than perform a
unitary transformation; it should perform an algorithm. That is, the same terahertz
pulse should optimally take several initial eigenstates to the appropriate ®nal state.
Explicitly,
…1¡1 1 1 1 1 †T! … 010000†
…1 1 ¡1 1 1 1 †T! …001000†
…111¡1 1 1 †T! …000100†
…1111¡1 1 †T! …000010†
…6†
2342 C. Rangan et al.
We use the standard formalism of optimal control theory [14±16], where the
optimal ®eld E…t†;04t4Tis calculated by optimizing an unconstrained
functional
·
JJ ˆ hª…T†jPkjª…T†i ¡ …T
0
dtl…t†jE…t†j2¡2<eh_
¶¶…t†jª…t†i ¡ …T
0
dt2<eh¶…t†jiHjª…t†i
…7†
Here, P
k
is the operator that must be optimized at time T(the ¯ipped bit
population). The parameter l(t) imposes a cost on the total energy of the pulse,
and makes the ®eld go smoothly to zero at the temporal end points. The third and
fourth terms are the constraints imposed by the Schro
Èdinger equation. To ®nd the
electric ®eld that simultaneously optimizes several unitary operations (i.e. start
from several initial states, and go to the appropriate ®nal states), we rede®ne the
optimal control problem. We consider an initial state that is a product state of
independent wave packets with singly ¯ipped orbitals.
jª…0†i ˆ jª1
25pi « jª2
26pi « jª3
27pi « jª4
28pi
where jª26pi ˆ … 11¡1 1 1 1 †T1
N
p…8†
The terahertz pulse acts simultaneously, but independently on all these wave
packets. The desired ®nal state is also a product state of independent wave packets
with the ¯ipped bit correctly decoded.
jª…T†i ˆ j25pi « j26pi « j27pi « j28pi …9†
At every time step, the updated THz ®eld is found by using a modi®ed version of
the algorithm used to optimize a single target state [13]. Using this new method,
we ®nd the terahertz pulse that detects any ¯ipped orbital of the N-bit data
register.
We have chosen a ®xed pulse length Tof roughly 8ps. The THz ®eld that
optimizes the population in any marked state is shown in ®gure 2(a). The pulse has
a strong initial lobe similar to the positive lobe of a half-cycle pulse. This pulse
clearly retrieves the marked bit information much better than an unshaped HCP.
The evolution of the wave packet as a function of time while the optimal pulse is
on, seen in ®gure 2(b) and ®gure 2(c) shows that the HCP-like lobe does the phase
retrieval, and the rest of the pulse performs the ampli®cation of the population in
the marked bit. This corresponds well with our impulse-model understanding of
the search algorithm.
The optimal shaped terahertz pulse has some interesting characteristics. No-
tably, the strong peaks in the spectrum and the Husimi distribution (seen in ®gure
3(a) and ®gure 3(b) respectively) do not correspond to resonances between the
energy levels of the selected state basis. The optimal terahertz pulse does not drive
the system to any particular resonant condition. Instead, it alters the phases of the
constituent orbitals of the wave packet so that they interfere to produce the desired
probability distribution. Another interesting feature of this optimal pulse is that
the peak ®eld of roughly 1 kV cm
1
lasts for roughly 0.5ps. For a wave packet
centred at nˆ26, this ®eld, which is beyond the ®eld ionizatio n limit, lasts for
more than half the Kepler period …¹ 2ºn3†. Yet, 99% of the population remains in
Control of Rydberg atoms to perform Grover’s search algorithm 2343
2344 C. Rangan et al.
Figure 2. Shaped terahertz pulse that optimally performs the phase retrieval. (a)
Electric ®eld of the pulse as a function of time. (b) Target state population during
the pulse as a function of time. (c) Temporal evolution of the Rydberg wave packet
during the shaped terahertz pulse.
Figure 3. (a) Fourier transform and (b) Husimi distribution of the shaped terahertz pulse.
the selected state basis. This feature is an example of interferometric stabilization
[17], seen in other atomic systems.
4. Creating states outside the database
According to Grover, `The quantum search algorithm can be looked at as a
technique for synthesizing a particular kind of superpositionÐone whose ampli-
tude is concentrated in a single basis state’ [18]. This is exactly what is done in the
general algorithm. The question is: Can we use the HCP to drive amplitude into
eigenstates other than those within the database? These outside states are initially
unpopulated. For example, if the initial wave packet has population in the 24pto
29p`database’ states, we would like to use the HCP to produce a 27s-eigenstate. In
the Cesium spectrum, the s-states are spectrally separated from the database states.
The graph of the matrix elements between the database states and the target state
as a function of Qin ®gure 4(a) shows that in order to produce an s-state, all the
Control of Rydberg atoms to perform Grover’s search algorithm 2345
Figure 4. (a) Matrix elements of the HCP interaction between the database states and
the desired non-database states as a function of the impulse Q. (b) Experimental
(circles) and calculated (lines) results of the desired non-database state population as
a function of Q. The solid line corresponds to the impulse model calculation, and
the dashed line corresponds to a full calculation by solving the time-dependent
Schro
Èdinger equation.
higher p-states have to be in phase, but out of phase with the all the lower p-
states[12]. An impulse interacting with a wave packet that has this phase structure
creates the constructive interference condition for that s-state. The phase structure
of the initial wave packet that produces the j27sistate (that is in between the j26pi
and j27pistates) is …1;1;1;¡1;¡1;¡1†=
N
p. The experimental spectra
of the decoded wave packet as a function of the peak ®eld of the HCP show that s-
states appear at a HCP strength of Qˆ0:0024 a.u. as seen in ®gure 4(b). Previous
experiments have shown that an HCP can interact with a Rydberg eigenstate to
produce a wave packet [19]. This experiment shows that the inverse is also
achievableÐa properly programmed wave packet can be driven towards an
eigenstate by a broadband HCP interaction.
5. Scaling
The advantage of this method is that the search is completed by a single query
(a single HCP) by exploiting the massive parallelism obtained by having many
identical Rydberg atoms for the measurement. The extension of the Rydberg atom
implementation of the search algorithm to larger registers is limited. The scaling of
the peak HCP with change in n, the principal quantum number of the marked bit,
is suggested by the form of the impulse operator exp(iQz). The average radial
extent of a Rydberg state scales as n
2
. For marked bits with di erent n-values, the
Qof the HCP that retrieves them can be expected to scale as n¡2[12]. Experi-
mentally, it is possible to load a larger register at a higher central n-value, but the
®eld-ionization signal loses resolution in that regime. This limits the size of the
Rydberg register. The Rydberg system has limited scaling capability, but other-
wise contains all of the features needed to execute the algorithm [20].
6. Conclusions
We have shown that Grover’s search algorithm can be performed optimally on
a Rydberg atom data register. Optimal control theory can be extended to design
broadband terahertz pulses for the control of an atom to execute an algorithm. A
half-cycle pulse can be used to drive a wave packet towards a single energy
eigenstate. These results provide the motivation for terahertz pulse shaping and
control.
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Control of Rydberg atoms to perform Grover’s search algorithm 2347
