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[To be submitted to Phys. Rev. A]
Effects Of The Bloch-Siegert Oscillation On The Precision Of
Qubit Rotations: Direct Two-Level Vs. Off-Resonant Raman
Excitation
Prabhakar Pradhan, George C. Cardoso and Selim M. Shahriar
Department of Electrical and Computer Engineering
Northwestern University
Evanston, IL 60208
In a direct two-level qubit system, when the Rabi frequency is comparable to the resonance
frequency, the rotating wave approximation is not appropriate. In this case the Rabi
oscillation is accompanied by another oscillation at twice the frequency of the driving field,
the so called Bloch-Siegert oscillation (BSO), which depends on the initial phase of the
field. This oscillation may restrict the precise rotation of a qubit. Here, we show that for an
effectively two-level (lambda) system, the BSO is inherently negligible, implying in a
greater precision for rotation of a lambda qubit when compared to a direct two-level qubit in
a strong driving field. Different aspects of BSO in direct two-level system in a strong
driving field have been discussed in detail.
PACS Number(s): 03.67.-a, 03.67.Hk, 03.67.Lx, 32.80.Qk
2
1 . Introduction
Quantum computation [1-3] has drawn much attention for more research in recent years due to
its potential for exponentially faster computation relative to the classical case. Though the
experimental realization of a quantum computer has remained a big challenge, there are
several proposals for realizing them physically. Any quantum algorithm can be decomposed
into controlled-NOT gate operations and quantum bit rotations. All qubits are composed of
either two- or three-level quantum systems, which in turn are made of electric or magnetic
dipoles or multipoles. Qubit rotations, which are key operations in quantum computation and
communication, are performed by applying external classical electromagnetic fields, make use
of Rabi flopping. So far, successful attempts for quantum bit operations and toy model
quantum computations have been done by nuclear magnetic resonance, ion traps and two
levels systems using Josephson junctions. In most of these qubit systems, the energy levels
generally differ in a range of MHz to GHz. For these low energy level qubits, a fast
operation requires a high Rabi frequency which may be of the order of the resonance frequency.
In a recent short paper [4-6], we have shown that when a two-level system is resonantly
driven by a strong Rabi frequency, an additional oscillation called the Bloch-Siegert Oscillation
(BSO) [7,8] is present at twice the driving frequency of the field. In this paper we have made
an extensive and systematic study of the BSO for a direct two-level qubit system and an
effective two level lambda system. The origin of the BSO for a two-level system is an effective
virtual two-photon transition which occurs at an off-resonance frequency of twice the frequency
of the driving field. The magnitude of the BSO is proportional to the ratio of the Rabi
frequency to the resonance frequency and also depends on the absolute phase of the driving
field. In a strong driving field, the presence of the BSO changes the population of the ground
and excited states compared to the case of a weak driving field when the rotating wave
approximation (RWA) is valid. Furthermore, the BSO’s dependence on the absolute phase of
the field complicates the reliable prediction of the effect of the Rabi transition on the final
populations.
2. Potential trouble with fast quantum-bit operation due to BSO and possible solutions
The BSO correction to the usual Rabi oscillation could be a significant fraction for a low
frequency qubit system with a fast Rabi driving field. For example, in a recent experiment by
Martinis et al [9], the BSO amplitude was of the order of 1% of the usual Rabi oscillation. For
a still stronger driving field, this correction could be as large as 10%. For fault tolerant quantum
computation, the maximum permissible error rate scales as the inverse of the number of qubits
involved in the computation. For example, a quantum computer which is made of 106 qubits
could tolerate well a 10-6 error rate per operation [10], but the introduction of a 1% error
through an imprecise rotation of a qubit is unacceptable for the protocol .
3
A quantum computer operates better in a low frequency transition because of lower
decoherence. One wants as fast an operation as possible [3], so the Rabi frequency is not
necessarily small [1,11,12] and may be comparable to the transition frequency . The flipping
probability of the target qubit depends not only on the amplitude of the field, but also on the
phase of the field at that spatial point, i.e. propagation delay. To avoid this, one can creat a
situation where the Rabi frequency is much less than the transition frequency. However, in this
case many operations are not possible within the limited decoherence time. Another solution is
to keep track of the phase by measuring the phase of the field locally before each operation,
but this is technically difficult and may complicate the quantum computation software. The most
practical solution seems to be to use indirect two-photon Raman excitation for the qubit rotation.
This is the solution that we offer here and we will show later that this process does not suffer
from the BSO effect under the condition that the effective Rabi frequency is close to or even
greater than the effectively two-level transition frequency.
In this paper, we first extend our previous results [4] of a direct two-level qubit system. Then,
we derive a quantum description of the joint state of the field and the atom, showing that for a
two-level system, the BSO is present at twice the driving field frequency. Finally, we consider
an optical lambda system where the energy difference between its two lower energy levels is
much smaller than the optical frequency, just as in the systems regularly used to demonstrate
electromagnetically induced transparency (EIT). By using composite state transition arguments
similarly to those described for a two-level system, we show that for a Raman transition in an
effectively two-level lambda system, the BSO is inherently negligible for the effective twolevel transition.
3. Direct two level system: Hamiltonian and its solution without RWA
We consider an ideal two-level system where a ground state |0> is coupled to a higher
energy state |1>. We also assume that the |0> ↔|1> transition is magnetic dipolar with a
transition frequency ω and that the magnetic field is of the form B=B0 cos(ωt+φ) [13]. In the
dipole approximation, the Hamiltonian can be written as
)  0
g (t )
H =
,
 g (t ) ε 
(1)
where g(t) = -go [exp(iωt+iφ)+c.c.]/2 and ε=ω, corresponding to resonant excitation. The state
vector is written as
C (t )
ξ (t ) =  0  .
 C1 (t ) 
(2)
4
We now perform a rotating wave transformation by operating on |ξ(t)> with the unitary operator
Q given by
0
1

Qˆ = 
.
0 exp(iωt + iφ )
(3)
The Schroedinger equation then takes the form (setting h=1):
~
∂ | ξ (t ) >
~
~
= −iH (t ) | ξ (t ) > ,
∂t
(4)
where the effective Hamiltonian is given by
~
α (t )
 0
,
H = *
0 
α (t )
(5)
with α(t)= -go[1 + exp(-i2ωt-i2φ)]/2 and the rotating frame state vector
~
C 0 (t )
~
ˆ
| ξ (t ) >≡ Q | ξ (t ) >=  ~  .
 C1 (t ) 
~
(6)
Now, one may choose to make the RWA, corresponding to dropping the fast oscillating term in
α(t). This corresponds to ignoring effects (such as the Bloch-Siegert shift ) of the order of
(go/4ω), which can easily be observable in experiment if go is large [7,8, 13-16]. On the other
hand, by choosing go to be small enough one can make the RWA for any value of ω. We explore
both regimes in this paper. As such, we find the general results without the RWA.
From Eqs. 4 and 6, one gets two coupled differential equations:
g
~•
~
C 0 (t ) = i 0 [1 + exp(−2iωt − 2iφ )] C 1 (t ) ,
2
•
g
~
~
C 1 (t ) = i 0 [1 + exp(+2iωt + 2iφ ))] C 0 (t ) .
2
(7a)
(7b)
We assume that C 0 (t = 0) = 0 is the initial condition, and proceed further to find an approximate
analytical solution of Eq. 7. Given the periodic nature of the effective Hamiltonian, the general
solution to Eq. 7 can be written in the form of Bloch’s periodic functions:
~
| ξ (t ) >=
∞
∑ξ
n = −∞
where β=exp(-i2ωt-i2φ) and
n
βn,
(8)
5
a 
ξn ≡  n 
 bn 
.
(9)
Inserting Eq.8 in Eq.7 and equating the coefficients with the same frequencies, one gets
for all n:
•
a n = i 2nωa n + ig o (bn + bn −1 ) / 2 ,
(10a)
•
b n = i 2nωbn + ig o (a n + a n +1 ) / 2 .
(10b)
In figure 1, we have shown a pictorial representation of the different level interactions. In the
absence of the RWA, the coupling to additional levels results from virtual multi-photon
processes. Here, the coupling between ao and bo is the conventional one present when the RWA
is made. The couplings to the nearest neighbors, a±1 and b±1, are detuned by an amount 2ω, and
so on. To the lowest order in (go/ω), we can ignore terms with |n| >1, thus yielding a truncated
set of six equations :
•
a o = ig o (bo + b−1 ) / 2 ,
•
b o = ig o (a o + a1 ) / 2 ,
•
a1 = i 2ωa1 + ig o (b1 + bo ) / 2 ,
•
b1 = i 2ωb1 + ig o a1 / 2 ,
•
a −1 = −i 2ωa −1 + ig o b−1 / 2 ,
•
b−1 = −i 2ωb−1 + ig o (a −1 + a o ) / 2 .
(11a)
(11b)
(11c)
(11d)
(11e)
(11f)
To solve these equations, one may employ the method of adiabatic elimination, which is valid
for first order in σ ≡ (go/4ω), when the switching time is slow for g(t). Now, in order to
consider a region of parameters where an adiabatic elimination can hold well, we consider g0
to also be a function of time t. We assume a time-dependence of the form
g 0 (t ) = g 0 [1 − exp(−t / τ sw )] , where
the switching time constant τ sw is larger than other
-1
characteristic timescales such as ω . Consider first the last two Eqs.11e and 11f. In order to
simplify these two equations further, one needs to diagonalize the interaction between a-1 and b-1.
Define µ-≡(a-1-b-1) and µ+≡(a-1+b-1), which can now be used to re-express these two equations
in a symmetric form as:
6
•
µ − = −i (2ω + g o (t ) / 2) µ − − ig o (t )a o / 2 ,
•
µ + = −i (2ω − g o (t ) / 2) µ + + ig o (t )a o / 2 .
(12.a)
(12.b)
Adiabatic following then yields (again, to lowest order in σ):
µ − ≈ −σa o ,
µ + ≈ σa o ,
(13)
which in turn yields:
a −1 ≈ 0,
b−1 ≈ σa o .
(14)
In the same manner, we can solve the equations 11c and 11d, yielding:
a1 ≈ −σbo ,
b1 ≈ 0 .
(15)
Note that the amplitudes of a-1 and b1 are vanishing (each proportional to σ2 ) to lowest order in
σ, thereby justifying our truncation of the infinite set of relations in Eq.10. Using Eq.14 and
15 in Eqs.11a and 11b , we get:
•
a o = ig o (t )bo / 2 + i∆(t )a o / 2 ,
•
b o = ig o (t )a o / 2 − i∆ (t )bo / 2 ,
(16a)
(16b)
where ∆=g2o(t)/4ω is essentially the Bloch-Siegert shift. Eq.16 can be thought of as a twolevel system excited by a field detuned by ∆.
With the initial condition of all the population in |0> at t=0,
and
t
1
t
g 0′ (t ) = ∫ g o (t )dt = g 0 [1 − ( ) −1 (1 − exp(−t / τ sw ))] , the only non-vanishing (to lowest order
t0
τ sw
in σ ) terms in the solution of Eq.9 are:
a o (t ) ≈ Cos ( g 0′ (t )t / 2); bo (t ) ≈ iSin( g 0′ (t )t / 2) ,
a1 (t ) ≈ −iσSin( g 0′ (t )t / 2); b−1 (t ) ≈ σCos ( g 0′ (t )t / 2) (17)
We have verified this solution via numerical integration of Eq.7 as discussed later. Inserting this
solution in Eq.7 and reversing the rotating wave transformation, we get the following
expressions for the components of Eq.2 :
7
C 0 (t ) = Cos ( g 0′ (t )t / 2) − 2σΣ ⋅ Sin( g 0′ (t )t / 2),
C1 (t ) = ie
− i (ω t +φ )
[ Sin( g 0′ (t )t / 2) + 2σΣ ⋅ Cos( g 0′ (t )t / 2)].
*
(18a)
(18b)
where we have defined Σ ≡ (i / 2) exp[−i (2ωt + 2φ )] . To lowest order in σ, this solution is
normalized at all times. Note that if one wants to produce this excitation on an ensemble of
atoms using a π /2 pulse and measure the population of the state |1> immediately after the
excitation terminates ( at g 0′ (τ )τ /2= π /2), the result would be a signal given by
| C1 ( g 0′ (τ ), φ ) | 2 =
1
[1+2σSin(2ωτ+2φ)],
2
(19)
which contains information of both the amplitude and phase of the driving field. This result
shows that it is possible to determine both the phase and amplitude of a RF signal coupled to a
two-level system by observing the population of one of these levels.
A physical realization of this result can be appreciated best by considering an
experimental arrangement of the type illustrated in Fig. 2. Here, rubidium thermal atoms are
passing through the microwave field. The total passage-time of an atom through the microwave
field is τ , which includes switching-on and switching-off time scale τ switch . The states of the
atoms are measured immediately after the atoms leave the magnetic field.
In Fig.3 (a) we have shown the evolution of the excited state population | C1 (τ ) |2 with
time τ, which is the Rabi oscillation, by plotting the analytical expression in Eq.18(b). The
finer rapid oscillation part of the total Rabi oscillation | C1 (t ) | 2 , i.e. ( g 0 (t ) /4ω) Sin( g 0′ (t ) t)
Sin(2ωt+2φ) is first order in g0/4ω, and is plotted in Fig.3(b). These analytical results agree
closely (within the order of g0/4ω ) to the results which are obtained via direct numerical
integration of Eq.7 with a larger switching time constant, i.e. τ switch >> 1 / ω .
In figure 4 we have plotted numerical calculations of the maximum amplitude of
| C 2 (t ) | 2 with the detuned pumping δ for the cases with RWA and without RWA. The
maximum population has a shift of ∆=g2o/4ω for the case without RWA relative to the case
with the RWA, as described by equation 16.
4. Dependence of the BSO on the Rabi frequency and its effect on a qubit
We have numerically simulated the variation of the maximum BSO amplitude with the
Rabi frequency g. The numerical plots are shown in figure 5. 5a shows the total BSO
amplitude and 5b shows the contribution due to the 2nd harmonic. When all the other
parameters of the problem are fixed, the numerical calculation of the BSO amplitude matches
very well with the theoretical prediction until g/ω =1. For a reasonably larger value of g/ω, the
value of the BSO oscillation can be 25% of the Rabi amplitude. That is quite a strong
oscillation imbedded in the main Rabi oscillation. The main amplitude of the BSO is associated
8
with the 2ω oscillation but its second harmonic at 4ω is still significant. The total BSO
amplitude, the 2ω frequency component, and the 4ω component are shown in figure 5.
A direct measurement of the BSO can be done by the following arrangement of an
experimental situation. Consider that the interaction time for the Rabi frequency is fixed for a
fixed time τ , and that the initial phase of the driving field is φ. Then, the probability of the
excited state, which varies with the dynamical phase of the field, can be written as
| C1 (t + τ ) | 2 = Sin 2 ( g 0′ (τ )τ / 2) + ( g / 4ω ) Sin ( g 0′ (τ )τ ) Sin ( 2ωτ + 2ωt + 2φ ).
(20)
Here, the excited state population varies with the effective dynamical phase of the field
2ωt+2φ. A plot of | C1 (t ) | 2 with other parameters fixed is shown in the upper panel of figure
6. A Fourier transform on the upper plot is shown in the lower panel of figure 6. The figure
clearly shows that there are effectively two frequency components, the stronger at 2ω and a
weaker higher order term at 4ω.
In figure 7, we show a pictorial diagram of a possible scheme of a quantum computer where
qubit operations are performed by two-level systems. The transitions between the states are
done, as discussed, by Rabi flopping, induced by a microwave field. The operations are
performed on the target qubit, indicated by the arrow, and it is assumed that it is brought to
resonance with the field by somehow changing its energy state levels through a scheme not
relevant to our discussion here. The effective Rabi flopping will depend not only on the
amplitude, but also on the phase of the field at the particular spatial point of the target qubit, at
the moment when the qubit interaction with the microwave field begins. This dependence
complicates the performance of qubit rotations in direct two-level systems when the driving
field is strong.
One usual way to obtain an effective two-level system in the microwave regime, using only
light beams, is to make Raman transitions between two low-lying states of a lambda system
[17-19]. To investigate the effect of the BSO in this qubit made of an effective-two-level
lambda system, we performed some numerical simulations. Fortunately, we observed that the
BSO is negligible in this type of qubit, it has negligible dependence on the driving field if the
qubit is in the microwave regime and the Raman transition is performed by optical fields. In the
following we will explain the appearance of the BSO in a pure two-level and in an effectivetwo-level lambda system, by using pure quantum mechanical arguments.
5. Experimental realization of two level quantum systems.
There are several situations where a two level system can be experimentally realized. We
described here three such scenarios.
9
Example 1: The two levels, |0> and |1>, are realized by components of the two lowest energy
hyperfine levels of the hydrogen atom : |0> ≡ 12S1/2:|F=0,m=0> and |1> ≡12S1/2:|F=1,m=0>.
These two states differs by a frequency of 1.420 GHz. A third auxiliary state |2>
≡22S1/2:|F=0,m=0> which is 1234 THz away from state |1>, can be used to measure the
population of excited state |1>. The transition between the two levels is excited by a π
polarized RF field at 1.420 GHz. Note that polarization selection rules prevent any coupling of
|0> to the other magnetic sublevels of the F=1 manifold, namely 12S1/2:|F=1,m=-1> and
12S1/2:|F=1,m=1>. The next nearest levels (such as |2>) are too far to have any noticeable effect.
Furthermore, in our model the system would be in the |0> state before the start of the interaction.
Under these conditions, this system would qualify as a two level system even as the RWA breaks
down.
Example 2: Another example of an isolated 2-level system can be realized in an 87Rb atom,
using |0>≡52S1/2:|F=1,m=1> and |1> ≡52S1/2:|F=2,m=2>, excited by a right-circularly polarized
RF field at the transition frequency of 6.68347 GHz. A third auxiliary state |2> ≡
52P3/2:|F=3,m=3> (which is 3.844 x1014 Hz apart from state |1>) can be used to monitor the
population of the state |1>. Initially, the system can be pumped optically to be in the state |1>
only (which is possible because under right-circularly polarized laser excitation, |1> couples only
to |2>).
Example 3: Yet another example of an isolated 2-level system can be realized in a 23Na atom
in a manner identical to that for the 87Rb atom described above. The corresponding energy levels
are: |0>≡32S1/2:|F=1,m=1>, |1> ≡32S1/2:|F=2,m=2>, and |2> ≡ 52P3/2:|F=3,m=3>. The transition
frequency between |1> and |2> is 1.7716 GHz, and the |1> to |2> transition frequency is 5.085
x 1014 Hz.
6. A quantum argument to explain the BSO for a two-level atomic system interacting with
an electromagnetic field
The interaction Hamiltonian of an atomic two- level system in the presence of a laser field
can be written as:
)
H = ( g / 2)(a + a + ),
(21)
a+ and a are the creation and the annihilation
where g is the Rabi frequency, and
operators, respectively, of a quantized laser field.
6.1 Eigen States
∑
ci | i > , where i = 0 and i=1 are the
We assume that the quantum state of the atom is
i
ground and excited states respectively, and the quantum state of the laser field is of the form
cn | n > , where |n> is a quantized state of n photons with energy nω (h = 1) . We can then
n
| Ψ >≡ ci | i > ⊗ cn | n > .
write a joint state of the laser and atoms as
∑
∑
i
∑
n
10
Now the expectation value of the interaction is:
< H > = g /2
∑
< i | i' > *
∑
< n | ( a + a + ) | n' >
= g /2. δ i, i ±1 ( n − 1δ n,n −1 + nδ n,n +1 ) .
(22)
6.2 The origin of the Bloch-Siegert oscillation at 2ω
In our previous paper [4], using a semi-classical calculation, we showed that the amplitude of
the oscillation within a first order correction to the Rabi oscillation is of the order of g / 4ω 01 .
Here, we show the presence of the BSO by illustrating Eq. 22 in Fig. 8. We can see that the
case where the first set of permitted transitions between the two composite states are at zero
detuning is when the joint states are in the same energy, and these come from the type of
transitions: | n − 1 > |i = 1 > ↔ | n > |i = 0 > . The second set of the allowed transitions are at
a detuned frequency of 2ω , and are of the type | n > |i = 0 > ↔ | n + 1 > |i = 1 > ,
corresponding to the last part of the operator in Eq. 21. Higher order terms, such as 4ω , 6ω ,
etc, are also allowed transitions detuned by 4ω , 6ω , etc, respectively, which not very
significant.
In the next section, using the same type of argument, we will analyze the presence of BSO in
an effectively two-level lambda system.
7. A Raman transition in a Lambda system and Bloch-Siegert oscillation
A schematic picture of a lambda system as we treat it here is described in figure 9. The states
|0> and |1> are the two low-lying nearby states, e.g., the two hyperfine states of the fundamental
state of an alkali atom with an energy difference on the order of several GHz. A third level
|2> is at an optical frequency away from the levels |0> and |1>. Now, applying two offresonant driving fields at frequencies ω 02 + δ for the | 0 > ↔ |2> transition and ω12 + δ for
the | 2 > ↔1|> transition, where δ is the detuning frequency, results in an effective Rabi
frequency between the levels |0> and |1>, that is asymptotically equal to Ω = g 2 / 4δ for
δ>>g. For simplicity, we have considered that the Rabi frequencies are equal for both the
interactions | 0 > ↔|2> and | 2 > ↔1| > and they of magnitude g.
Now we want to study the possibility of BSO in the case of such a Lambda system. In the
following, we have done a quantum analysis of a three level system coupled to two laser
fields, and have shown that the effective BSO between the levels |0> and |1> is effectively
11
negligible in a lambda system of the type described above. Now, following the argument of
the previous discussion for a pure two-level system, we can write the expectation value of the
interaction Hamiltonian of the transition |0> to |1> via an intermediate state |2> as
< H 01 > =< H 02 >< H 21 >
= Ω 01 [ δ i,i ± 2 ( nδ n ,n +1 + n − 1δ n ,n −1 )]02 × [δ j, j±1 ( mδ m,m +1 + m − 1δ m,m −1 )]21 ,
(23)
where Ω 01 ≡ g 2 / 4δ is the effective Rabi frequency between the states |0> and |1> and
m and n are the photon numbers of the two laser field modes, i =0,2 and j=1,2.
Again, following the argument used for the two-level case, we can say that the BSO
in the interaction Hamiltonian part H 02 in Eq. 23 is at a frequency 2ω 02, due to the
virtual detuned transition at 2ω 02.
A schematic level diagram corresponding to Eq.23 for all allowed transitions is given
in Fig.9. Any Bloch-Siegert oscillation which may occur between |0> and |1>
(ω 01 = ω 02 − ω12 ) is due to transitions detuned by at least a frequency 2ω 02 or 2ω 12.
The amplitude of the BSO is then of the order of (ω 01 / ω 02 ) 2 , and not of the order of
Ω 01 / 4ω 01 , as in the pure two-level case. When ω 01 / ω 02 is very much smaller than one,
the BSO in the effectively two-level lambda system | 0 > ↔ 1| > is negligible. This
effect has been shown more clearly by a schematic level diagram in figure 10.
8. Conclusions
We have analyzed the presence of the Bloch-Siegert shift and oscillation caused by
strong fields used to flip qubits. For a direct two-level qubit, we have shown that the
presence of the Bloch-Siegert oscillation (BSO) implies that the final state of a qubit
rotation is a function of the absolute local phase of the driving field. This complicates the
use of strong fields in two-level systems though they are required for fast qubit rotation.
We also analyzed the case in which a qubit is formed from an effectively two-level
lambda system and the rotation is performed via an optical Raman transition. In this
case, we show that the BSO is negligible even when the effective Rabi frequency for the
qubit rotation exceeds the transition frequency (that is in the microwave range). In
conclusion, we suggest that qubits formed by an optically excited microwave transition
may be more controllably rotated than direct microwave two-level qubits. Thereby, the
qubits which are made of lambda systems are more advantageous than the qubits that are
made of direct two level systems for quantum computation and quantum information
processing.
This work was supported by DARPA grant # F30602-01-2-0546 under the QUIST
program, ARO grant # DAAD19-001-0177 under the MURI program, and NRO grant #
NRO-000-00-C-0158.
12
REFERENCES
[1] A. M. Steane, Appl. Phys. B 64, 632 (1997).
[2] M. A. Nielsen I. L. Chuang, Quantum Computation and Quantum Information,
Cambridge, 2002.
[3] D. Bouwmeester, A. Ekert, and A. Zeilinger, Eds., The Physics of Quantum
Information , Springer, 2000.
[4] M.S. Shahriar, Prabhakar Pradhan and Jacob Morzinski, quant-ph/0205120 (With
Phys. Rev. Lett.)
[5] M. S. Shahriar and P. Pradhan " The Proceedings of the 6th International Conference
on Quantum Communication, Measurement and Computing (QCMC'02),” 289
(2003).
[6] Pradhan and M. S. Shahriar, presented at the APS annual meeting, March, 2002.
[7] F. Bloch and A.J.F. Siegert, Phys. Rev. 57, 522(1940).
[8] L. Allen and J. Eberly, Optical Resonance and Two Level Atoms, Wiley, 1975.
[9] J. M. Martinis et al., Phys. Rev. Lett., 89, 117901 (2002).
[10] J. Preskill, quant-ph/9705031.
[11] A.M. Steane et al., quant-ph/0003097
[12] D. Jonathan, M.B. Plenio, and P.L. Knight, quantph/0002092.
[13] A. Corney, Atomic and Laser Spectroscopy , Oxford University Press, 1977.
[14] J.H. Shirley, Phys. Rev. 138, B979 (1965).
[15] S. Stenholm, J. Phys. 64, 1650 (1973).
[16] N.F. Ramsey, Molecular Beams , Clarendon Press, 1956.
[17] J.E. Thomas et al., Phys. Rev. Lett. 48, 867(1982).
[18] M.S. Shahriar and P.R. Hemmer, Phys. Rev. Lett. 65, 1865(1990).
[19] M.S. Shahriar et al., Phys. Rev. A. 55, 2272 (1997)
13
ENERGY
g0
4ω
2ω
0
-2ω
-4ω
a-2
g0
a-1
b-2
g0
g0
a0
b-1
g0
g0
a1
b0
g0
g0
a2
b1
g0
b2
FIG. 1. Schematic illustration of the multiple orders of interaction when the rotating wave
approximation is not made. The strengths of the first higher order interaction, for example,
is weaker than the zeroth order interaction by the ratio of the Rabi frequency, go, and the
effective detuning, 2ω. When the RWA is made, only the terms a0 and b0 may be nonzero.
14
FIG. 2. Schematic illustration of an experimental arrangement for measuring the phase
dependence of the population of the excited state |1>: (a) The microwave field couples the
ground state (|0>) to the excited state (|1>). A third level, |2>, which can be coupled to |1>
optically, is used to measure the population of |1> via fluorescence detection. (b) The
microwave field is turned on adiabatically with a switching time-constant τSW, and the
fluorescence is monitored after a total interaction time of τ.
15
FIG. 3. Illustration of the Bloch-Siegert Oscillation (BSO): (a) The population of state
|1>, as a function of the interaction time τ , showing the BSO perturbation to the
conventional Rabi oscillation. (b) The BSO oscillation alone (amplified scale), produced
by subtracting the Rabi oscillation with RWA from the plot in (a). (c) The timedependence of the Rabi frequency in the experiment proposed in Fig. 1. Inset: BSO as a
function of the absolute phase of the field.
16
FIG. 4. Two-level system. Numerical calculation of the Rabi amplitude with a finite
detuning for the case with RWA (dotted line) and without RWA (solid line). It is clear
that the peak amplitude of the Rabi oscillation is shifted by an amount g2 /(2ω) for the
case without RWA relative to the case with RWA. This is consistent with the analytical
expression for small g (Eq. 16).
17
FIG. 5. Two-level system. Top: Total BSO amplitude versus the Rabi frequency g for
(a) total BSO amplitude, (b) the 2ω part of the BSO, and (c) theoretical prediction
(g/4ω) Sin(gτ)Sin(2ωτ+2φ). Bottom: BSO amplitude for the 4ω component of the
oscillation.
18
FIG. 6. (Top) Excited state population of a two-level system as a function of the phase
argument of the driving field. Because of a high Rabi frequency, the BSO amplitude
shows a visible 2ω and 4ω peak. (Bottom) The Fourier transform of the BSO shows the
presence of two prominent frequencies 2ω and 4ω where ω is the frequency of the
driving field.
19
Phase Meter
Microwave source
Antenna
r
r
Quantum
Computer
Spatial form of
the driving field
Target qubit
FIG. 7. In this schematic picture we have shown fixed qubits (spins) and their energy
levels inside a quantum computer. The target qubit is shown by an arrow. The microwave
field that interact with the target qubit is shown by a Sin curve varying spatially in space.
The BSO effect makes the transition probability depend not only on the amplitude, but
also on the phase of the microwave field at the spatial point of the target qubit.
20
2ω
δ
|1>
δ
2ω
g1
ω01
|n+2>|0>
|0>
|n+1>|0>
|n+1>|1>
δ
2ω
(a)
|n>|1>
δ
2ω
|n>|0>
δ
2ω
|n-1>|0>
δ
2ω
|n-2>|0>
δ
2ω
|n-1>|1>
|n-2>|1>
|n-3>|1>
δ
(b)
FIG. 8. (a) Schematic picture of a two level system. ω01 is the resonance frequency and
δ is the detuning. (b) Joint states of atoms and laser of a two level system and their
allowed transitions. The left side of the figure shows infinite manifolds of laser with the
ground state, i.e. |0>|n> (n=n, n+1,…), subsequent states are separated by an energy ω.
Similarly, the excited state manifolds are shown in the right side of the figure. The arrows
indicate the allowed transitions according to Eq.(21). The transitions indicated by the
horizontal arrows are due to joint state transitions with zero detuning, which correspond
to the usual Rabi oscillation. The inclined arrows indicate transitions that are detuned by
a frequency 2ω and that are responsible for the Bloch-Siegert oscillation (BSO).
δ
(a)
g1
21
|2>
ω02
ω12
g2
|1>
|0>
(b)
FIG. 9. (a) Schematic picture of a Lambda system. Here g1 and g2 are the Rabi
frequencies of the fields coupled to the transitions | 0 > ↔| 2> and | 1 > ↔| 2> . The
frequencies of the laser fields are ω 02 + δ and ω12 + δ, and δ is the detuning. (b)
Schematic picture of the energy level of a lambda-atom plus fields composite system, and
the allowed transitions among the composite levels. The three columns represent the
composite energy levels manifolds of each of the three atomic states with the field
photons. The allowed transitions are according to Eq. (2). The horizontal arrows represent
the resonant or quasi-resonant one-photon transitions responsible for the usual Rabi
flopping. The inclined arrows represent non-resonant two-photon transitions, i.e.,
transitions virtually detuned by 2 ω 02 or 2 ω12 responsible for the Bloch-Siegert
oscillation.
22
δ
|2>
g1
BSO
g2
ω02
ω12
BSO
|1>
|0>
Negligible BSO
FIG. 10. Extension of figure 9. The only mechanism for the BSO type multi-photon
transition coupling |0> to |1> is through the corresponding BSO type transition in the |0>
to |1> (or |2> to |1>) transition. However, the BSO type transition in |0> to |2> (for
example) is negligible for most practical situations (since g optical << ωoptical ~1015 Hz ).
This is true even when geff ( 0 →1) > ω ( 0→1).