Low frequency Rabi spectroscopy for a dissipative two-level system

Abstract

We have analyzed the interaction of a dissipative two level quantum system with high and low frequency excitation. The system is continuously and simultaneously irradiated by these two waves. If the frequency of the first signal is close to the level separation the response of the system exhibits undamped low frequency oscillations whose amplitude has a clear resonance at the Rabi frequency with the width being dependent on the damping rates of the system. The method can be useful for low frequency Rabi spectroscopy in various physical systems which are described by a two level Hamiltonian, such as nuclei spins in NMR, double well quantum dots, superconducting flux and charge qubits, etc. As the examples, the application of the method to a nuclear spin and to the readout of a flux qubit are briefly discussed.

Full text

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arXiv:cond-mat/0507502v2 [cond-mat.mtrl-sci] 3 Dec 2005
Europhysics Letters
PREPRINT
Low frequency Rabi spectroscopy for a dissipative two-
level system
Ya. S. Greenberg1(∗), E. Il’ichev2and A. Izmalkov2
1Novosibirsk State Technical University, 20 K. Marx Ave., 630092 Novosibirsk, Russia
2Institute for Physical High Technology, P.O. Box 100239, D-07702 Jena, Germany
PACS. 03.65.Yz – Decoherence; open systems; quantum statistical methods.
PACS. 85.35.Be – Quantum well devices.
PACS. 85.35.Ds – Quantum interference devices.
Abstract. –
We have analyzed the interaction of a dissipative two level quantum system with high and
low frequency excitation. The system is continuously and simultaneously irradiated by these
two waves. If the frequency of the first signal is close to the level separation the response of the
system exhibits undamped low frequency oscillations whose amplitude has a clear resonance at
the Rabi frequency with the width being dependent on the damping rates of the system. The
method can be useful for low frequency Rabi spectroscopy in various physical systems which
are described by a two level Hamiltonian, such as nuclei spins in NMR, double well quantum
dots, superconducting flux and charge qubits, etc. As the examples, the application of the
method to a nuclear spin and to the readout of a flux qubit are briefly discussed.
It is well known that under resonant irradiation a quantum two level system (TLS) can
undergo coherent (Rabi) oscillations. The frequency of these oscillations is proportional to
the amplitude of the resonant field [1] and is much lower than the gap frequency of TLS. The
effect is widely used in molecular beam spectroscopy [2], and in quantum optics [3].
During the last several years it has been proven experimentally that Rabi spectroscopy can
serve as a valuable tool for the determination of relaxation times in solid state quantum me-
chanical two-level systems, qubits, to be used for quantum information processing [4]. These
systems normally are strongly coupled to the environment, which results in the fast damping
of Rabi oscillations. It prevents the use of conventional continuous measurements schemes for
their detection, though the special schemes for the detection of coherent oscillations through
a weak continuous measurement of a TLS were proposed in [5–7]. That is why Rabi oscilla-
tions are measured with the pulse technique through the statistic of switching events of the
occupation probability between two energy levels with excitation and read out being taken at
the gap frequency of TLS, which normally, lies in GHz range [8–11]. The main drawback of
this technique is that it requires sophisticated high frequency readout electronics.
In this work we propose a new experimental method for a Rabi spectroscopy of a TLS,
when readout electronics is continuously swept across the low (compare to the gap) Rabi
(∗) E-mail: greenbergy@online.nsk.su
c ? EDP Sciences

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resonance. The method is rather general and can be applied to a great variety of two level
systems.
We consider a TLS which is irradiated continuously by two external sources. The first
with a frequency ω0, which is close to the energy gap between the two levels, excites the low
frequency Rabi oscillations. Normally, Rabi oscillations are damped out with a rate, which is
dependent on how strongly the system is coupled to the environment. However, if a second low
frequency source is applied simultaneously to TLS it responds with persistent low frequency
oscillations. The amplitude of these low frequency oscillations has a resonance at the Rabi
frequency with the width being dependent on the damping rates of the system. Note, that this
approach has a well known classical analog. Indeed, a damping rate of a classical oscillator
can be easily obtained from its amplitude-frequency characteristics.
We start with a Hamiltonian of a driven TLS , which is subjected to both high and low
frequency excitation:
H =∆
2σx+ε
2σz− σzF cosω0t − σzG(t). (1)
Here the first two terms describe an isolated TLS, which can model a great variety of situations
in physics and chemistry: from a spin 1/2 particle in a magnetic field to superconducting flux
and charge qubits [4], [12]. In order to be exact we consider the first two terms in (1) to
describe a double-well system where only the ground states of the two wells are occupied,
with ∆ being the energy splitting of a symmetric (ε = 0) TLS due to quantum tunnelling
between two wells. The quantity ε is the bias, the external energy parameter which makes
the system asymmetric. The last two terms in (1) describe the interaction with external
time-dependent high frequency, F, and low frequency, G fields which modulate the energy
asymmetry between the two wells.
Hamiltonian (1) is written in the localized state basis, i.e., in the basis of states localized
in each well. In terms of the eigenstates basis, which we denote by upper case subscripts for
the Pauli matrices σX,σY,σZ, Hamiltonian (1) reads:
?∆
?∆ε
2
where ∆ε=√∆2+ ε2is the gap between two energy states.
The inclusion of the dissipative environment in Hamiltonian (1), results in the Bloch-
Redfield equations for the matrix operators σX, σY, σZ, [13], [14]. For weak driving (F ≪ ∆),
and for weak coupling of the TLS to the bath these equations can be approximated by Bloch-
type equations [15]:
H =
∆εF cosω0t +∆
ε
∆εF cosω0t −
∆εG(t)
ε
∆εG(t)
?
?
σX
+
−σZ, (2)
?˙ σZ? = (2f cosω0t + 2g(t))?σY? − ΓZ(?σZ? − Z0), (3)
?˙ σY? = −(2f cosω0t + 2g(t))?σZ?
∆f cosω0t −2ε
+
?∆ε
?
−2ε
∆g(t)
?
?σX? − Γ?σY?, (4)
?˙ σX? = −
?∆ε
?
−2ε
∆f cosω0t −2ε
∆g(t)
?
?σY? − Γ?σX?, (5)

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Ya. S. Greenberg, E. Il’ichev and A. Izmalkov: Low frequency Rabi spectroscopy 3
where f = ∆F/?∆ε, g(t) = ∆G(t)/?∆ε, and Z0 = −tanh(∆ε/kBT) is the equilibrium
polarization of the system in the absence of external excitation sources (f = 0, g = 0). The
angled brackets in Eqs. (3), (4), and (5) denote the trace over reduced density matrix ρ(t),
which is obtained by tracing out all environment degrees of freedom: ?σX? = Tr(σXρ(t)),
etc. In order to simplify the problem we assume the relaxation, ΓZand dephasing, Γ, rates in
(3)-(5) are time-independent, i.e. the rates are slowly varying functions on the scale of Rabi
period which is of the order of 1/f.
Assuming that a high frequency driving amplitude F is sufficiently small we write the
desired solution of Eqs. (3), (4), and (5) as:
?σZ? = Z(t), (6)
?σY? = Y (t) + A(t)cos(ω0t) + B(t)sin(ω0t), (7)
?σX? = X(t) + C(t)cos(ω0t) + D(t)sin(ω0t), (8)
where X, Y , Z, A, B, C, and D are slowly varying, as compared to the high frequency ω0,
quantities.
As is known, a two-level system resonantly irradiated with a high frequency undergoes
a low frequency Rabi oscillations. However, if the external low frequency excitation, G(t)
is absent (g = 0), the Rabi oscillations are damped out. For this case we obtain at the
degeneracy point (ε = 0) the following solution: X = 0, Y = 0, C = B, D = −A,
Z(t) = z∗+ z0e−Γ1t+ e−Γ2t(z1cosωRt + z2sinωRt), (9)
A(t) = a∗+ a0e−Γ1t+ e−Γ2t(a1cosωRt + a2sinωRt),(10)
B(t) = b∗+ b0e−Γ1t+ e−Γ2t(b1cosωRt + b2sinωRt), (11)
where the relaxation rates, Γ1, Γ2, and the Rabi frequency ωRare determined by the equation
(λ − iΓ)2(λ − iΓZ) − (λ − iΓ)f2− (λ − iΓZ)δ2= 0.
Here λ is the eigenvalue for the oscillation mode, e. g., Z(t) ≈ eiλt. This cubic equation has
simple analytic solutions only for two cases. Firstly, in the absence of a relaxation in Eqs. (3)
- (5) (ΓZ = Γ = 0) we obtain Γ1= Γ2= 0, ωR=
detuning parameter, δ = ω0− ∆ε/?. Secondly, for δ = 0 we obtain Γ1= Γ, Γ2= (Γ +ΓZ)/2,
ωR=
b∗which can be determined as the steady state solution of the equations (3), (4), and (5):
(12)
?δ2+ f2, where δ is the high frequency
?f2− (Γ − ΓZ)2/4. For the subsequent derivation we need only the quantities z∗, a∗,
z∗= PZZ(0), a∗= z∗fΓ/(Γ2+ δ2), b∗= z∗fδ/(Γ2+ δ2), where PZ=
The quantity PZZ0 is the nonequilibrium polarization, i. e., the steady state difference
of occupation probabilities between two energy levels in the case when the high frequency
excitation is applied to the TLS. Therefore, in the absence of the low frequency excitation the
Rabi oscillations decay with a rate Γ2given by Eq. (12). The main goal of our investigation
is to obtain the persistent oscillations of the quantities Z(t) (6), Y (t) (7), X(t) (8) which can
be detected with low frequency (compared to the gap) electronic circuitry. In what follows
we show that a low frequency signal applied to TLS can sustain the persistent low frequency
oscillations of the above mentioned quantities. The amplitude of these oscillations has a clear
ΓZ(Γ2+δ2)
ΓZ(Γ2+δ2)+f2Γ.

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resonance at the Rabi frequency with the width of the resonance being dependent on the
damping rates of the system.
The Eqs. (3), (4), and (5) are analized in the presence of a low frequency excitation G(t).
We insert Eqs. (6), (7), (8) into (3), (4), and (5), and in accordance with the ideology of
rotating wave approximation retain only the low frequency terms. We consider the force G
to be small, and therefore neglect the terms which are of second and higher order in G. In
addition, we keep only the terms which oscillate within the bandwidth of the Rabi frequency
and neglect the terms which are of the order of ?f/∆ε, ?Γ/∆ε, ?ΓZ/∆ε. Consequently we
obtain the following set of equations for low frequency quantities:
˙Z = fA − ΓZZ (13)
˙Y =∆ε
?X − ΓY + 2gz∗−ε
∆fB(14)
˙X = −∆ε
?Y − ΓX +ε
∆fA (15)
¨A + 2Γ˙A + (Ω2
∆gb∗− δ2ε
R+ Γ2)A =
∆ga∗−2ε
f (ΓZ− Γ)Z − Γ2ε
∆˙ gb∗
(16)
¨B + 2Γ˙B +?δ2+ Γ2?B =
−δfZ + Γ2ε
∆ga∗− δ2ε
∆gb∗+2ε
∆˙ ga∗
(17)
From these equations we can obtain the Fourier components for the slowly varying quantities
A(ω) and B(ω), and for the low frequency persistent response of the TLS to a small low
frequency excitation,?Z(ω),?Y (ω), and? X(ω):?C(ω) =? B(ω),? D(ω) = −? A(ω),
?Z(ω) = ? g(ω)2ε
? A(ω) = ? g(ω)2ε
? B(ω) = −? g(ω)2ε
?Y (ω) =
? X(ω) =
where s(ω) = (ΩR−ω +iΓ)(ΩR+ω −iΓ)(iω +ΓZ)−f2(ΓZ− Γ); ΩR=
Rabi frequency in the absence of the damping (ΓZ= γ = 0). (1)
∆f2PZZ0
δ
δ2+ Γ2
2Γ + iω
s(ω)
, (18)
∆f PZZ0
δ
δ2+ Γ2
(2Γ + iω)(iω + ΓZ)
s(ω)
, (19)
∆f PZZ0
δ
δ2+ Γ2
1
(δ2+ (Γ + iω)2)
?f2δ (2Γ + iω)
s(ω)
−δ2− Γ2− iωΓ
δ
?
, (20)
?
∆ε
??
∆ε
ε
∆f? A(ω) +
?2ε
??
∆ε
?2?
2? g(ω)PZZ0−ε
?
?
∆f?B(ω)
?
(Γ + iω),(21)
∆f? A(ω)(Γ + iω) −
∆ε
2? g(ω)PZZ0−ε
∆f?B(ω)
?
?δ2+ f2is the
,(22)
(1)In above expressions we disregard the Fourier components of the damping terms (Eqs. (9), (10), (11))
since for the persistent signal g(t) they are unimportant.

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Ya. S. Greenberg, E. Il’ichev and A. Izmalkov: Low frequency Rabi spectroscopy 5
0100200 300400 500600 700
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
500550 600650
-0.04
-0.03
-0.02
-0.01
0.00
<σZ>


t (ns)
G/h=0 MHz
G/h=1 MHz
<σZ>
t (ns)
Fig. 1 – Time evolution of < σz >. (thick) G=0, (thin) G/h = 1 MHz. The insert shows the
undamped oscillations of < σz > at G/h = 1 MHz.
As can be concluded from these expressions the persistent low frequency oscillations of
the spin components Z(t), X(t), Y (t) appear as the response to the low frequency external
force, g(t), only in the presence of the high frequency excitation (f ?= 0). Exactly at resonance
(δ = 0)? A(ω) = 0, and?Z(ω) = 0, but?B(ω) ?= 0. At this point the population of the two levels
of the circle being precessed with the frequency ω of the low frequency external source.
As an example we show below the time evolution of the quantity ?σZ?(t) = Z(t), obtained
from the numerical solution of the equations (3), (4), and (5) where we take a low frequency
excitations as G(t) = Gcos(ωLt). The calculations have been performed with initial conditions
?σZ?(0) = 1, ?σX?(0)=?σY?(0) = 0 for the following set of the parameters: F/h = 36 MHz,
∆/h = 1 GHz, Γ/2π = 4 MHz, Γz/2π = 1 MHz, ǫ/∆ = 1, Z0 = −1, δ/2π = 6.366 MHz,
ωL/ΩR = 1. As is seen from Fig.1 in the absence of low frequency signal (G = 0) the
oscillations are damped out, while if G ?= 0 the oscillations persist.
The Fourier spectra of these signals are shown on Fig.2 for different amplitudes of low
frequency excitation. For G = 0 the Rabi frequency is positioned at approximately 26.2 MHz,
which is close to ΩR= 26.24 MHz. With the increase of G the peak becomes higher. It is
worth noting the appearance of the peak at the second harmonic of Rabi frequency. This peak
is due to the contribution of the terms on the order of G2which we omitted in our theoretical
analysis. The comparison of analytical and numerical resonance curves calculated for low
frequency amplitude, G/h = 1 MHz and different dephasing rates, Γ are shown on Fig.3.
The curves at the figure are the peak-to-peak amplitudes of oscillations of Z(t) calculated
from Eq. (18) with ? g(ω) = g(δ(ω + ωL) + δ(ω − ωL))/2, where δ(ω) is Dirac delta function.
curves depend on Γ (see the insert) and the positions of the resonances coincide with the Rabi
frequency. A good agreement between numerics and Eq. 18, as shown at Fig. 3, is observed
only for relative small low frequency amplitude G/h, for which our linear response theory is
valid.
The key point of the method we described above is that it allows for the detection of the
high frequency response of a TLS at a frequency which is much less than the gap frequency.
The low frequency dynamics of the quantities ?σX?,?σY?, ?σZ? bears the information about
are equalized, and spin circularly rotates in the XY plane with frequency ω0, with the center
The point symbols are found from numerical solution of Eqs. (3),(4),(5). The widths of the