Superconducting phase qubit coupled to a nanomechanical resonator: Beyond the rotatingwave approximation
Abstract
We consider a simple model of a Josephson junction phase qubit coupled to a solidstate nanoelectromechanical resonator. This and many related qubitresonator models are analogous to an atom in an electromagnetic cavity. When the systems are weakly coupled and nearly resonant, the dynamics is accurately described by the rotatingwave approximation (RWA) or the JaynesCummings model of quantum optics. However, the desire to develop faster quantuminformationprocessing protocols necessitates approximate, yet analytic descriptions that are valid for more strongly coupled qubitresonator systems. Here we present a simple theoretical technique, using a basis of dressed states, to perturbatively account for the leadingorder corrections to the RWA. By comparison with exact numerical results, we demonstrate that the method is accurate for moderately strong coupling, and provides a useful theoretical tool for describing fast quantum information processing. The method applies to any quantum twolevel system linearly coupled to a harmonic oscillator or singlemode boson field.
pacs:
03.67.Lx, 85.25.Cp, 85.85.+jI Introduction
Josephson junctions have been shown to be effective qubit elements for solidstate quantum computing architectures.Nakamura 1999 ; Vion etal 2002 ; Yu 2002 ; Martinis 2002 ; Berkley et al 2003 ; Yamamoto etal 2003 Several proposals for multiqubit coupling introduce electromagneticShnirman etal 1997 ; Makhlin etal 1999 ; Buisson and Hekking 2001 ; Smirnov and Zagoskin unpublished 2002 ; Blais etal 2003 ; Plastina and Falci 2003 ; Zhu etal 2003 ; Girvin etal preprint 2003 ; Blais etal preprint 2004 or mechanicalCleland and Geller 2004 ; Geller and Cleland resonators, or other oscillators,Marquardt and Bruder 2001 ; Plastina etal 2001 ; Hekking etal unpublished 2002 to mediate interactions between the qubits. Such resonatorbased coupling schemes have additional functionality resulting from the ability to tune the qubits relative to the resonator frequency, as well as to each other. These qubitresonator systems are analogous to one or more tunable fewlevel atoms in an electromagnetic cavity, and the dynamics is often accurately described by the rotatingwave approximation (RWA) or JaynesCummings model of quantum optics.Scully book
For a qubit with energy level spacing coupled with strength to a resonator with angular frequency and quality factor , the RWA is valid when both and . However, the resonant Rabi frequency, which is proportional to , is then much smaller than the qubit frequency . Therefore, restricting to be in the simpler weak coupling regime leads to quantum information processing that is slower than necessary, allowing fewer operations to be performed during the available quantum coherence lifetime.
The threshold theorem Aharonov and BenOr ; Gottesman ; Kitaev ; Preskill states that if the component failure probability is below some threshold , a computation with an error probability bounded by may be accomplished, provided a sufficient number of quantum gates are used for faulttolerant encoding. In practice, it will be important to have as small as possible. To approach this limit, we wish to study qubitresonator systems with stronger coupling (larger ) than may be correctly described by the RWA. This will allow us to consider faster switching times for qubitresonator gates, and to understand to what extent the coupling may be increased while still retaining good fidelity.
In this paper, we use a basis of dressed statesMeystre and Sargent book to calculate the leadingorder corrections to the RWA for a Josephson junction phase qubit coupled to a solidstate nanoelectromechanical resonator, or for any other model of a twolevel system linearly coupled to a singlemode boson field. By comparison with exact numerical results, we demonstrate that the method is accurate for moderately strong coupling and provides a useful theoretical tool for describing fast quantum information processing.
Ii JunctionResonator Dynamics in the DressedState Basis
ii.1 Qubitresonator Hamiltonian
The Hamiltonian that describes the lowenergy dynamics of a single largearea, currentbiased Josephson junction, coupled to a piezoelectric nanoelectromechanical disk resonator, can be written asCleland and Geller 2004 ; Geller and Cleland
(1) 
where the and denote particle creation and annihilation operators for the Josephson junction states and denote ladder operators for the phonon states of the resonator’s dilatational (thickness oscillation) mode of frequency , is a coupling constant with dimensions of energy, and . The value of depends on material properties and size of the resonator, and can be designed to achieve a wide range of values.Cleland and Geller 2004 ; Geller and Cleland An illustration showing two phase qubits coupled to the same resonator is given in Fig. 1.
For simplicity we will consider only two levels in a single junction; generalization of our method to more than two junction states is cumbersome but straightforward.generalization footnote However, all possible phononnumber states are included. The Hamiltonian may then be written as the sum of two terms, . The first term,
(2)  
is the exactly solvable JaynesCummings Hamiltonian, the eigenfunctions of which are known as dressed states. We will consider the second term,
(3)  
as a perturbation. The RWA applied to the Hamiltonian amounts to neglecting . Therefore, perturbatively including is equivalent to perturbatively going beyond the RWA.
ii.2 Dressed states
It will be useful to define a set of Rabi frequencies according to
(4) 
where
(5) 
are the resonant Rabi frequencies for a qubit coupled to an oscillator containing phonons, and where
(6) 
is the resonatorqubit detuning frequency. The vacuum Rabi frequency on resonance is .
The eigenstates of , or the dressed states, are labeled by the nonnegative integers and a sign ,
(7) 
where are the eigenstates of the uncoupled system. These states, together with , form a complete basis. The energies are
(8) 
and . On resonance, these reduce to
(9) 
and
(10) 
Below we will restrict ourselves exclusively to the resonant case.
In what follows, we will need the matrix elements of in the dressedstate basis, which are given by
(11)  
and
(12) 
ii.3 Dressed state propagator
In quantum computing applications one will often be interested in calculating transition amplitudes of the form
(13) 
where and are arbitrary initial and final states of the uncoupled qubitresonator system. Expanding and in the dressedstate basis reduces the timeevolution problem to that of calculating the quantity
(14) 
as well as and . is a propagator in the dressedstate basis, and would be equal to if were absent, that is, in the RWA.propagator footnote Although it is possible to directly construct perturbative expressions for the propagator in the basis, the quantity defined in Eq. (14) turns out to be the simplest.
To be specific, we imagine preparing the system at in the state , which corresponds to the qubit in the excited state and the resonator in the ground state . We then calculate the interactionrepresentation probability amplitude
(15) 
for the system at a later time to be in the state . Here . Inserting complete sets of the dressed states leads to
(16) 
and, for ,
(17) 
Using the relations
(18) 
and
(19) 
we obtain
(20) 
and
(21) 
So far everything is exact within the model defined in Eq. (1).
To proceed, we expand the dressedstate propagator in a basis of exact eigenstates of , leading to
(22) 
Here is the energy of stationary state . The propagator is an infinite sum of periodic functions of time. We approximate this quantity by evaluating the and perturbatively in the dressedstate basis.
The leadingorder corrections to the dressedstate energies are of order . We obtain
(23)  
and
(24) 
We will also need the secondorder eigenfunctions, which, for a perturbation having no diagonal dressedstate matrix elements, are
(25) 
and
(26)  
where and the are normalization factors.
Writing out Eq. (22) explicitly as
(27) 
and again making use of the fact that the matrix elements of diagonal in vanish, leads to
(28)  
or
(29)  
where
(30) 
Note that there are no order corrections to the dressedstate propagator. Because of this property, the leading order corrections are of order , and it is therefore necessary to use secondorder perturbative eigenfunctions to obtain all such secondorder terms.
Finally, we note that the normalization constants are simply
(31) 
(32) 
(33) 
and
(34) 
Iii Towards Information Processing With Strong Coupling
In this section, we test our perturbed dressedstate method for the case of a finitedimensional singlequbit, fivephonon system. The junction has parameters and corresponding to that of Ref. Martinis 2002, . The resonator has a frequency of and the interaction strength varies from weak to strong coupling. The bias current is chosen to make the the system exactly in resonance, and this bias is sufficiently smaller than the critical current so that the junction states are well approximated by harmonic oscillator eigenfunctions. The Hamiltonian for this system is diagonalized numerically, and the probability amplitudes are calculated exactly, providing both a test of the accuracy of the analytic perturbative solutions and an estimate of the range of interaction strengths for which it is valid. Setting the initial state to be , as assumed previously, we simulate the transfer of a qubit from the Josephson junction to the resonator, by leaving the systems in resonance for half a vacuum Rabi period Cleland and Geller 2004 ; Geller and Cleland
Figures 2, 3, and 4, show the time evolution of the occupation probabilities and for different values of . In Fig. 2, we plot the results for very weak coupling, . The evolution takes the junction qubit and transfers it to and from the resonator periodically. The exact, RWA, and dressedstate perturbative results are all the same to within the thickness of the lines shown in Fig. 2. Thus, for this value of , the RWA is extremely accurate.
In Fig. 3, we plot the probabilities for stronger coupling, . For this coupling strength, the RWA is observed to fail. For example, the RWA predicts a perfect state transfer between the junction and the resonator, and does not exhibit the oscillations present in the exact solution. The dressedstate perturbative approximation does correctly capture these oscillations. In Fig. 4, we show the same quantities for the case . At this coupling strength, both the RWA and the dressedstate perturbative approximation break down.
Iv State Transfer Fidelity
In this final section, we briefly investigate to what extent we may increase the junctionresonator coupling , and still have an accurate state transfer from the Josephson junction to the resonator. As before, we start at time in the state . In order to define the fidelity of the state transfer operation, we first determine the time of the minimum of the probability Recall that is the probability that the junction is in the excited qubit state and the resonator is in the vacuum state.
It will be convenient to define two fidelities: is the fidelity (or, more precisely, the fidelity squared) for the junction, and is the squared fidelity for the resonator.fidelity footnote These quantities are different because of leakage to other states; however, in the RWA limit, they are both equal to unity. measure the success of deexciting the qubit, and measures the success of exciting the resonator. In Fig. 5 we plot and as a function of . Typically, the junction fidelity remains close to unity, with some oscillations, for all couplings. This behavior is a consequence of the fact that there is always a time where becomes small, as is evident in Figs. 3 and 4. However, because of leakage to other states, the resonator fidelity decreases significantly (again with oscillations due to the“switching” of with ) with increasing interaction strength. The lower curve in Fig. 5 shows that is possible with , which allows a state transfer in under .
V Discussion
We have developed a theoretical technique to analytically calculate the leadingorder perturbative corrections to the RWA or JaynesCummings Hamiltonian for a quantum twolevel system linearly coupled to a harmonic oscillator or singlemode boson field, a model central to many current quantum computing architectures. Such corrections are necessary to treat the fast informationprocessing regime where the interaction strength approaches the qubit level spacing. The method was applied to a currentbiased Josephson junction coupled to a piezoelectric nanoelectromechanical disk resonator, and good agreement with exact numerical results was obtained.
Acknowledgements.
It is a pleasure to thank Emily Pritchett and Steve Lewis for useful discussions. ANC was supported by the DARPA/DMEA Center for Nanoscience Innovation for Defence. MRG was supported by the National Science Foundation under CAREER Grant No. DMR0093217, and by the Research Corporation.References
 (1) Y. Nakamura, Yu. A. Pashkin, and J. S. Tsai, Nature 398, 786 (1999).
 (2) D. Vion et al., Science 296, 886 (2002).
 (3) Y. Yu, S. Han, X. Chu, S.I. Chu, and Z. Wang, Science 296, 889 (2002).
 (4) J. M. Martinis, S. Nam, J. Aumentado, and C. Urbina, Phys. Rev. Lett. 89, 117901 (2002).
 (5) A. J. Berkley, H. Xu, R. C. Ramos, M. A. Gubrud, F. W. Strauch, P. R. Johnson, J. R. Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, Science 300, 1548 (2003).
 (6) T. Yamamoto, Yu. A. Pashkin, O. Astafiev, Y. Nakamura, and J. S. Tsai, Nature 425, 941 (2003).
 (7) A. Shnirman, G. Schön, and Z. Hermon, Phys. Rev. Lett. 79, 2371 (1997).
 (8) Y. Makhlin, G. Schön, and A. Shnirman, Nature 398, 305 (1999).
 (9) O. Buisson and F. W. J. Hekking, in Macroscopic Quantum Coherence and Quantum Computing, edited by D. V. Averin, B. Ruggiero, and P. Silvestrini (Kluwer, New York, 2001), p. 137.
 (10) A. Yu. Smirnov and A. M. Zagoskin, condmat/0207214.
 (11) A. Blais, A. M. van den Brink, and A. M. Zagoskin, Phys. Rev. Lett. 90, 127901 (2003).
 (12) F. Plastina and G. Falci, Phys. Rev. B 67, 224514 (2003).
 (13) S.L. Zhu, Z. D. Wang, and K. Yang, Phys. Rev. A 68, 34303 (2003).
 (14) S. M. Girvin, R.S. Huang, A. Blais, A. Wallraff, and R. J. Schelkopf, condmat/0310670.
 (15) A. Blais, R.S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schelkopf, condmat/0402216.
 (16) A. N. Cleland and M. R. Geller, condmat/0311007.
 (17) M. R. Geller and A. N. Cleland (unpublished).
 (18) F. Marquardt and C. Bruder, Phys. Rev. B 63, 54514 (2001).
 (19) F. Plastina, R. Fazio, and G. M. Palma, Phys. Rev. B 64, 113306 (2001).
 (20) F. W. J. Hekking, O. Buisson, F. Balestro, and M. G. Vergniory, condmat/0201284.
 (21) M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997).
 (22) D. Aharonov and M. BenOr, in Proceedings of the TwentyNinth Annual ACM Symposium on the Theory of Computing, 176 (1997).
 (23) D. Gottesman, Ph.D. Thesis, California Institute of Technology (1997).
 (24) A. Kitaev, in Quantum Communication, Computing and Measurement (Plenum Press, New York, 1997), p181.
 (25) J. Preskill, Proc. R. Soc. London A, 454 385 (1998).
 (26) P. Meystre and M. Sargent, Elements of Quantum Optics (SpringerVerlag, Berlin, 1990).
 (27) If the higher lying junction states are not initially occupied, and the resonator also starts in the ground state, only the qubit states are coupled by “energy conserving” terms. Therefore, the higher lying junction states can be treated with ordinary perturbation theory in .

(28)
An alternative representation for the propagator is
where , and where is the timeordering operator. However, directly computing by a series or cumulant expansion in leads to a result that is valid only for short times.
 (29) Because the desired final state is , overall squared fidelity of the state transfer operation is simply .