Coherent population transfer between weaklycoupled states in a laddertype superconducting qutrit
Abstract
Stimulated Raman adiabatic passage (STIRAP) offers significant advantages for coherent population transfer between un or weaklycoupled states and has the potential of realizing efficient quantum gate, qubit entanglement, and quantum information transfer. Here we report on the realization of STIRAP in a superconducting phase qutrit  a laddertype system in which the ground state population is coherently transferred to the secondexcited state via the dark state subspace. The result agrees well with the numerical simulation of the master equation, which further demonstrates that with the stateoftheart superconducting qutrits the transfer efficiency readily exceeds while keeping the population in the firstexcited state below . We show that population transfer via STIRAP is significantly more robust against variations of the experimental parameters compared to that via the conventional resonant pulse method. Our work opens up a new venue for exploring STIRAP for quantum information processing using the superconducting artificial atoms.
Stimulated Raman adiabatic passage (STIRAP), which combines the processes of stimulated Raman scattering and adiabatic passage, is a powerful tool used for coherent population transfer (CPT) between un or weaklycoupled quantum states ber98 ; sho11 . It has been recognized as an important technique in quantum computing and circuit QED involving superconducting qubits zho02 ; kis02 ; kis04 ; yan03 ; yan04 ; zho04 ; liu05 ; wei08 ; sie09 ; fal13 . For example, qubit rotations can be realized via STIRAP with the two computational states plus an auxiliary state forming a threelevel configuration zho02 ; kis02 . A scheme for generating arbitrary rotation and entanglement in the threelevel type flux qutrits is also proposed kis04 , and the experimental feasibility of realizing quantum information transfer and entanglement between qubits inside microwave cavities has been discussed yan03 ; yan04 . Unlike the conventional resonant pulse method STIRAP is known to be much more robust against variations in experimental parameters, such as the frequency, amplitude, and interaction time of microwave fields, and the environmental noise kis02 ; kis04 ; wei08 ; sie09 .
Recently, multilevel systems (qutrits or qudits) have found important applications in speeding up quantum gates lan09 , realizing quantum algorithms dic09 , simulating quantum systems consisting of spins greater than nee09 , implementing full quantumstate tomography the02 ; bia10 ; sha13 , and testing quantum contextuality cab12 . Unlike the highly anharmonic type flux qutrits the phase and transmon qutrits have the laddertype (type) threelevel configuration which is weakly anharmonic. The dipole coupling between the ground state and the secondexcited state in the phase qutrit is much weaker than those between the firstexcited state and the state or the state. In the case of the transmon qutrit the coupling is simply zero. This unique property makes it difficult to transfer population from to directly using a single pulse tuned to their level spacing . The usual solution is to use the highpower resonant twophoton process or to apply two successive pulses transferring the population first from to and then from to bia10 ; sha13 . These methods often lead to a significant population in the middle level resulting in energy relaxation which degrades the transfer process. In contrast, STIRAP transfers the qutrit population directly from to via the dark state subspace without occupying the middle level .
In this work, we report on the realization of STIRAP in a type superconducting phase qutrit sim04 . As shown schematically in Fig. 1a the qutrit has a loop inductance and a Josephson junction with critical current and capacitance . The potential energy and quantized levels , , and of the qutrit are illustrated in Fig. 1b in which the frequencies of the pump and Stokes fields and their strength (Rabi frequencies) are also indicated. Applying the rotatingwave approximation (RWA) in the doublerotating frame the Hamiltonian of the system can be written as sil09 ; li11 :
(1) 
where the Planck constant is set to unity, and are various detunings, are the qutritmicrowave couplings proportional to the amplitude of the pump and Stokes fields respectively, and for the type configuration with weak anharmonicity. For the fastoscillating terms in equation (1) averages out to zero so the Hamiltonian becomes
(2) 
in which and . For the phase qutrit used here we have . Equation (2) is the wellknown RWA Raman Hamiltonian ber98 ; sho11 . In particular, when the system satisfies the twophoton resonant condition:
(3) 
it has an eigenstate , called the dark state, which corresponds to the eigenvalue of . Here . CPT from the ground state to the secondexcited state without populating the firstexcited state can therefore be realized via STIRAP by initializing the qutrit in the ground state li11 ; nov13 and then slowly increasing the ratio to infinity as long as the following conditions ber98 ; sho11 ; scu97 ; vas09
(4) 
are satisfied so that the qutrit will stay in the dark state
subspace spanned by {}. The first condition
is required to reduce equation (1) to
equation (2) leading to the existence of the dark state
solution while the second ensures adiabatic state following.
Results
Sample parameters and measurements. The sample used in this work is an aluminum phase qutrit sim04 , which is cooled down to mK in an Oxford cryogenfree dilution refrigerator. The qutrit control and measurement circuit includes various filtering, attenuation, and amplification tia12 . For the present experiment, we bias the rfSQUID to have six energy levels in the upper potential well and use the lowest three levels as the qutrit states. The relevant transition frequencies are GHz and GHz, and the relative anharmonicity is . The measured energy relaxation times are ns and ns, respectively, while the dephasing time determined from Ramsey interference experiment is ns. To realize STIRAP, a pair of bellshaped counterintuitive microwave pulses with the Stokes pulse preceding the pump pulse, as illustrated in Fig. 1c, are used. The pulses are defined by and with and sho11 ; vas09 . The pulse width (FWHM) is approximately , and its height is and .
Coherent population transfer. Figure 2a shows the two microwave pulses defined by MHz and ns in their overlapping region. As increases, and start to increase and decrease across at which they are equal. The experimentally measured populations , , and versus time produced by this counterintuitive pulse sequence in the resonant case are plotted in Fig. 2b as symbols. We observe that as time evolves across the population () increases (decreases) rapidly, signifying the occurrence of STIRAP via the dark state of the superconducting qutrit system. The experimentally achieved maximum , or the population transfer efficiency, is about for the present sample under the resonant condition. Notice that in the entire region of , ns, all of the characteristic features of the experimental data, in particular (i) remaining significantly lower than for ns, (ii) the decrease (increase) of () after reaching the maximum (minimum) as well as the gradual rising of , are reproduced well by the numerical simulation (solid lines). The simulated temporal profiles of the populations , and are obtained by solving the master equation using the measured qutrit parameters, where is the Liouvillean containing the relaxation and dephasing processes li11 . The numerical result also confirms that feature (ii) is due primarily to energy relaxation, while the maximum itself is mainly limited by dephasing. Hence by increasing the qutrit decoherence times the undesirable decrease of and the rise of and in the relevant time scale can be suppressed and a higher can be achieved (see below).
In our experiment the conditions imposed by equation (4) are satisfied: in the resonant case is MHz, which is approximately four times that of , and it is easy to verify that the integrated pulse area is greater than . In addition to reducing decoherence the efficiency of the demonstrated STIRAP process can be improved by increasing the relatively small anharmonicity parameter of the present sample up to, say, by optimizing device parameters of the type phase and transmon (or Xmon) qutrits whi14 ; hoi13 . According to equation (4) greater anharmonicity allows the use of larger which would proportionally reduce the duration of the pump and Stokes pulses when the pulse area is kept unchanged to satisfy the adiabatic condition. Shorter pulses also reduce the negative effect of decoherence on the transfer efficiency.
The STIRAP process is often identified in either the time domain or the frequency domain ber98 ; sho11 . The latter is based on equation (3) which specifies the twophoton resonance condition. In Fig. 3, we show the level population versus the pump detuning for four Stokes tone detunings of , and  MHz, respectively. Bright resonance appears as the leftside peak in each curve when the twophoton resonant condition equation (3) is met. It is interesting to see that the maximum value of is reached at the microwave detunings of MHz, which is higher than the value achieved in the resonant case of shown in Fig. 2b. In Fig. 3, the rightside peak in each curve is originated from the twophoton process excited by the single pump microwave tone. Compared to the leftside peaks, although the peak heights are comparable, they are much narrower, indicating that in practice it is less controllable using the twophoton process to perform coherent population transfer from state to state .
Efficiency and robustness. The above results demonstrate clearly CPT from the ground state to the secondexcited state via STIRAP in the type superconducting qutrit. Compared to the usual highpower twophoton process or two nonoverlapping successive resonant pulse excitations shown in Fig. 1d, which involve significant undesired population in the middle level and require precise single photon resonance and pulse area wei08 ; bia10 , CPT via STIRAP demonstrates just the opposite. First, in principle CPT between and can be accomplished without occupying the lossy middle level . More importantly, the process is much more robust against variations in the frequency, duration, and shape of the driving pulses ber98 ; sho11 . In fact, in terms of the conditions equation (3) and equation (4), we see from Fig. 3 that the single photon resonance condition is greatly relaxed. Although are limited by the system anharmonicity, their values, together with , still have much room for variations while maintaining the transfer efficiency.
In Fig. 4a, we show the calculated results in the
versus plane with contours indicating population transfer
efficiency above and , respectively, using qutrits with
decoherence times of = 35.3 s, =
19.6 s, and = 12.4 s, and the
relative anharmonicity of . The qutrit parameters in
the ranges are now attainable with transmon pai11 ; hoi13 ,
Xmon bar13 , flux ste14 , and also possibly phase
whi14 type devices. In Fig. 4b, we show the level populations
versus time (solid lines) for = 100 MHz and =
50 ns, in which a nearly complete transfer above 99 from level
to level is accomplished with the
population in middle level kept below 1. These
results indicate that the transfer efficiency of STIRAP is very
insensitive to , which is limited by systems
anharmonicity, and to , which should be much smaller than the
decoherence time. The allowed variations of a few hundreds of MHz in
and a few tens of ns in for keeping
99, for example, are in sharp contrast with the case of simple
pulse excitations. In fact, using a pulse to transfer
population from the ground state to the firstexcited
state alone, acceptable variations of the Rabi
frequency and pulse width to keep the transfer
efficiency above 99 can be estimated from the pulse area
relation . Thus if we
use = 50 MHz, the pulse width variation must be less
than 0.6 ns. More strict condition is required when successive
excitation from the firstexcited state to the
secondexcited state is considered. From these we see
that the extreme robustness of the STIRAP process is very
advantageous and should be useful in various applications such as
realizing efficient qubit rotation, entanglement, and quantum
information transfer in various superconducting qubit and qutrit systems.
Discussion
We have experimentally demonstrated coherent population transfer between two weaklycoupled states, and , of a superconducting phase qutrit having the type ladder configuration via STIRAP. The qutrit had a small relative anharmonicity of , and moderate decoherence times of ns, ns, and ns, respectively. We demonstrated that by applying a pair of counterintuitive microwave pulses in which the Stokes tone precedes the pump tone, coherent population transfer from to with a efficiency can be achieved with a much smaller population in the firstexcited state . Using the measured qutrit parameters, including decoherence times, we simulated the STIRAP process by numerically solving the master equation. The result agrees well with the experimental data. We showed that by increasing the decoherence times of the qutrits to the order of a few tens of microseconds, currently attainable experimentally, the transfer efficiency can be increased to greater than while keeping the population of the firstexcited state below .
We have also shown that coherent population transfer via STIRAP is
much more robust against variations of the experimental parameters,
including the amplitude, detuning, and time duration of the
microwave fields, and the environmental noise over the conventional
methods such as using highpower twophoton excitation and two
resonant pulses tuned to and ,
respectively. Therefore STIRAP is advantageous for achieving robust
coherent population transfer in the laddertype superconducting
artificial atoms that play increasingly important roles in various
fields ranging from fundamental physics to quantum information
processing. Furthermore, the method can be readily extended to the
type systems such as the superconducting flux qutrits, in
which the initial and target states locate in different potential
wells representing circulating currents in opposite directions. Our
work paves the way for further progress in
these directions.
Methods
Determination of level populations. The qutrit level populations () at a given time are determined using two carefully calibrated nanosecondscale measurement flux pulses A and B, which reduce the potential barrier to two different levels so that tunneling probabilities and in each case are measured. Pulse A leads to low tunneling probability for state , high tunneling probability for state and, of course, even higher tunneling probability for state . Pulse B results in a slightly deeper potential well than pulse A does so that , , and a much larger for state , and respectively. Denoting the density operator of the qutrit as , we have
(5) 
where , and can be found from the experimentally determined tunneling probabilities of the th energy level for given amplitudes of pulses A and B sha13 . Combining the normalization condition
(6) 
we obtain
(7) 
where = 0, 1, 2 with in circulative order like , , and , and is the determinant
(8) 
Hence the level populations , ,and can be obtained by measuring and .
Numerical simulations. We numerically calculate the level populations = , = , and = at any given time by solving the master equation
(9) 
where is the system’s 33 density matrix, is the Hamiltonian given by equation (1), and is the Liouvillean containing various relaxation and dephasing processes. Since experimentally the pump and Stokes microwaves are not correlated, we introduce a phase difference between the two microwaves in the actual calculations li12 . In this case, the doublerotating reference frame is described by the operator , and the rotatingwave approximation leads to a Hamiltonian in the following form:
(10) 
where the Liouvillean operator in equation (9) is given by li11 :
(11) 
We use the parameters = 2.8310 sec, = 5.1010 sec, and = 8.0610 sec measured directly from experiment, while from the measured dc Stark shift of the twophoton spectral lines we estimate 2 and . In our calculations is obtained by solving equation (9) using the fourthorder RungeKutta method. Since the phase difference of the two microwaves in our experiment is random, we average the result over and finally arrive at:
(12) 
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Acknowledgements
We thank J. M. Martinis (UCSB) for providing us with the sample used
in this work. This work was supported by the Ministry of Science and
Technology of China (Grant Nos. 2011CBA00106, 2014CB921202, and
2015CB921104) and the National Natural Science Foundation of China
(Grant Nos. 91321208 and 11161130519). S. Han acknowledges support
by the US NSF (PHY1314861).
Author contributions
H.K.X, S.H., and S.P.Z. designed the experiment. H.K.X., W.Y.L.,
G.M.X., and F.F.S. performed the measurement and numerical
simulation. Y.T., H.D., D.N.Z., Y.P.Z., and H.W. contributed to the
experiment in the lowtemperature achievement, sample mounting and
characterization. Y.X.L. provided theoretical support. S.P.Z. and
S.H. wrote the manuscript in cooperation with all the authors.
Additional information
Competing financial interests: The authors declare no competing financial interests.