1 [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
Transcript
1
[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.
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 .
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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 two-level 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
=
ε)()(0
tgtg
H)
, (1)
where g(t) = -go [exp(iωt+iφ)+c.c.]/2 and ε=ω, corresponding to resonant excitation. The state vector is written as
=
)()(
)(1
0
tCtC
tξ . (2)
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We now perform a rotating wave transformation by operating on |ξ(t)> with the unitary operator Q given by
+
=)exp(0
01ˆφω iti
Q . (3)
The Schroedinger equation then takes the form (setting h=1):
>−=∂
>∂ )(~|)(~)(~| ttHitt ξξ , (4)
where the effective Hamiltonian is given by
=
0)()(0
*
~
tt
Hα
α, (5)
with α(t)= -go[1 + exp(-i2ωt-i2φ)]/2 and the rotating frame state vector
>=>≡
)(~)(~
)(~|ˆ)(~|1
0
tCtCtQt ξξ . (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:
)(~)]22exp(1[2
)(~1
00 tCiti
gitC φω −−+=
•
, (7a)
)(~))]22exp(1[2
)(~0
01 tCiti
gitC φω +++=
•
. (7b)
We assume that 0)0(0 ==tC 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:
n
nnt βξξ ∑
∞
−∞=
>=)(~| , (8)
where β=exp(-i2ωt-i2φ) and
5
≡
n
nn b
aξ . (9)
Inserting Eq.8 in Eq.7 and equating the coefficients with the same frequencies, one gets for all n:
2/)(2 1−
•
++= nnonn bbigania ω , (10a)
2/)(2 1+
•
++= nnonn aaigbnib ω . (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 :
( ) 2/1−
•
+= bbiga ooo , (11a)
( ) 2/1aaigb ooo +=•
, (11b)
( ) 2/2 111 oo bbigaia ++=•
ω , (11c)
2/2 111 aigbib o+=•
ω , (11d)
2/2 111 −−
•
− +−= bigaia oω , (11e)
( ) 2/2 111 oo aaigbib ++−= −−
•
− ω . (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
)]/exp(1[)( 00 swtgtg τ−−= , where the switching time constant swτ is larger than other characteristic timescales such as ω-1. 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:
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2/)()2/)(2( ooo atigtgi −+−= −
•
− µωµ , (12.a)
2/)()2/)(2( ooo atigtgi +−−= +
•
+ µωµ . (12.b)
Adiabatic following then yields (again, to lowest order in σ):
oo aa σµσµ ≈−≈ +− , , (13) which in turn yields:
oaba σ≈≈ −− 11 ,0 . (14) In the same manner, we can solve the equations 11c and 11d, yielding:
0, 11 ≈−≈ bba oσ . (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:
2/)(2/)( oooo atibtiga ∆+=•
, (16a)
2/)(2/)( oooo btiatigb ∆−=•
, (16b) where ∆=g2
o(t)/4ω is essentially the Bloch-Siegert shift. Eq.16 can be thought of as a two-level system excited by a field detuned by ∆.
With the initial condition of all the population in |0> at t=0, and
))]/exp(1()(1[)(1)( 10
00 sw
sw
t
o ttgdttgt
tg ττ
−−−==′ −∫ , the only non-vanishing (to lowest order
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 :
where we have defined )]22(exp[)2/( φω +−≡Σ tii . 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 ττ )(0g ′ /2= π /2), the result would be a signal given by
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 switchτ . 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 21 |)(| τC 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 2
1 |)(| tC , i.e. ( )(0 tg /4ω) Sin( )(0 tg′ 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. ωτ /1>>switch . In figure 4 we have plotted numerical calculations of the maximum amplitude of
22 |)(| tC 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
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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
(20)
Here, the excited state population varies with the effective dynamical phase of the field 2ωt+2φ. A plot of 2
1 |)(| tC 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 effective-two-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.
).222())(()4/()2/)((|)(| 0022
1 φωωτττωτττ ++′+′=+ tSingSinggSintC
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: ),)(2/( ++= aagH)
(21)
where g is the Rabi frequency, and a+ and a are the creation and the annihilation operators, respectively, of a quantized laser field. 6.1 Eigen States We assume that the quantum state of the atom is ∑ >
ii ic | , where i = 0 and i=1 are the
ground and excited states respectively, and the quantum state of the laser field is of the form ∑ >
nn nc | , where |n> is a quantized state of n photons with energy ωn )1( =h . We can then
write a joint state of the laser and atoms as ∑∑ >⊗>>≡Ψn
ni
i ncic ||| .
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Now the expectation value of the interaction is: >+ +∑∑ '|)( |n< * >i'|i< /2g = >H< naa )1( /2.g = 1,1,1i i, +−± +− nnnn nn δδδ . (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 014/ ωg . 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: >=>↔>=>− 0|| 1|1| inin . The second set of the allowed transitions are at a detuned frequency of ω2 , and are of the type >=>+↔>=> 1|1| 0|| inin , corresponding to the last part of the operator in Eq. 21. Higher order terms, such as ω4 , ω6 , etc, are also allowed transitions detuned by ω4 , ω6 , 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 off-resonant driving fields at frequencies δω +02 for the >↔> 2| 0| transition and δω +12 for the >↔> 1| 2| transition, where δ is the detuning frequency, results in an effective Rabi frequency between the levels |0> and |1>, that is asymptotically equal to δ4/2g=Ω for δ>>g. For simplicity, we have considered that the Rabi frequencies are equal for both the interactions >↔> 2| 0| and >↔> 1| 2| 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
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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
01 g≡Ω 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 02H in Eq. 23 is at a frequency ω2 02, due to the virtual detuned transition at ω2 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> )( 120201 ωωω −= is due to transitions detuned by at least a frequency ω2 02 or ω2 12.
The amplitude of the BSO is then of the order of 20201 )/( ωω , and not of the order of
0101 4/ ωΩ , as in the pure two-level case. When 0201 /ωω is very much smaller than one, the BSO in the effectively two-level lambda system >↔> 1| 0| 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.
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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)
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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 non-zero.
g0
a0b0
g0a-1
b-1
g0
a1b1
g0
a-2b-2
g0
a2b2
g0
g0
g0
0
2ω
-2ω
4ω
-4ω
g0
ENERGY
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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.
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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 time-dependence of the Rabi frequency in the experiment proposed in Fig. 1. Inset: BSO as a function of the absolute phase of the field.
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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).
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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.
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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.
Microwave source
Antenna
r
Quantum Computer
Phase Meter
Target qubit Spatial form of the driving field
r
20
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).
|n>|0> |n-1>|1>
|n+1>|1>
|n>|1>
|n-3>|1>
|n-2>|1>
|n+2>|0>
|n+1>|0>
|n-1>|0>
|n-2>|0>
2ω
2ω
2ω
2ω
2ω
2ω
2ω
δ
δ
δ
δ
δ
δ
δ
δ
|0>
|1>
ω01 g1
(a)
(b)
21
FIG. 9. (a) Schematic picture of a Lambda system. Here g1 and g2 are the Rabi frequencies of the fields coupled to the transitions >↔> 2| 0| and >↔> 2| 1| . The frequencies of the laser fields are 02ω + δ and 12ω + 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 02ω or 2 12ω responsible for the Bloch-Siegert oscillation.
δ
|1> |0>
|2>
g1 g2 ω12 ω02
(b)
(a)
22
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).
