Quantum eigenstate tomography with qubit tunneling spectroscopy
Measurement of the energy eigenvalues (spectrum) of a multi-qubit system has recently become possible by qubit tunneling spectroscopy (QTS). In the standard QTS experiments, an incoherent probe qubit is strongly coupled to one of the qubits of the system in such a way that its incoherent tunneling r...
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Smirnov, Anatoly Yu. Amin, Mohammad H. 2018-01-19T20:45:03Z 2018-01-19T20:45:03Z 2017 Quantum eigenstate tomography with qubit tunneling spectroscopy / Anatoly Yu. Smirnov Mohammad H. Amin // Физика низких температур. — 2017. — Т. 43, № 7. — С. 969-977. — Бібліогр.: 16 назв. — англ. 0132-6414 PACS: 03.67.–a, 03.67.Lx, 85.25.Am https://nasplib.isofts.kiev.ua/handle/123456789/129528 Measurement of the energy eigenvalues (spectrum) of a multi-qubit system has recently become possible by qubit tunneling spectroscopy (QTS). In the standard QTS experiments, an incoherent probe qubit is strongly coupled to one of the qubits of the system in such a way that its incoherent tunneling rate provides information about the energy eigenvalues of the original (source) system. In this paper, we generalize QTS by coupling the probe qubit to many source qubits. We show that by properly choosing the couplings, one can perform projective measurements of the source system energy eigenstates in an arbitrary basis, thus performing quantum eigenstate tomography. As a practical example of a limited tomography, we apply our scheme to probe the eigenstates of a kink in a frustrated transverse Ising chain. We are thankful to Professor Alexander Omelyanchouk for collaboration and support in early days of D-Wave Systems. We are grateful to Chris Rich for helpful discussions and Fiona Hanington for critical reading of the paper. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Quantum eigenstate tomography with qubit tunneling spectroscopy Article published earlier |
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Quantum eigenstate tomography with qubit tunneling spectroscopy |
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quantum eigenstate tomography with qubit tunneling spectroscopy |
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Measurement of the energy eigenvalues (spectrum) of a multi-qubit system has recently become possible by qubit tunneling spectroscopy (QTS). In the standard QTS experiments, an incoherent probe qubit is strongly coupled to one of the qubits of the system in such a way that its incoherent tunneling rate provides information about the energy eigenvalues of the original (source) system. In this paper, we generalize QTS by coupling the probe qubit to many source qubits. We show that by properly choosing the couplings, one can perform projective measurements of the source system energy eigenstates in an arbitrary basis, thus performing quantum eigenstate tomography. As a practical example of a limited tomography, we apply our scheme to probe the eigenstates of a kink in a frustrated transverse Ising chain.
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https://nasplib.isofts.kiev.ua/handle/123456789/129528 |
| citation_txt |
Quantum eigenstate tomography with qubit tunneling spectroscopy / Anatoly Yu. Smirnov Mohammad H. Amin // Физика низких температур. — 2017. — Т. 43, № 7. — С. 969-977. — Бібліогр.: 16 назв. — англ. |
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| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7, pp. 969–977
Quantum eigenstate tomography
with qubit tunneling spectroscopy
Anatoly Yu. Smirnov and Mohammad H. Amin
D-Wave Systems Inc., 3033 Beta Ave., Burnaby BC V5G 4M9, Canada
E-mail: asmirnov@dwavesys.com; amin@dwavesys.com
Received February 1, 2017, published online May 25, 2017
Measurement of the energy eigenvalues (spectrum) of a multi-qubit system has recently become possible by
qubit tunneling spectroscopy (QTS). In the standard QTS experiments, an incoherent probe qubit is strongly
coupled to one of the qubits of the system in such a way that its incoherent tunneling rate provides information
about the energy eigenvalues of the original (source) system. In this paper, we generalize QTS by coupling the
probe qubit to many source qubits. We show that by properly choosing the couplings, one can perform projective
measurements of the source system energy eigenstates in an arbitrary basis, thus performing quantum eigenstate
tomography. As a practical example of a limited tomography, we apply our scheme to probe the eigenstates of a
kink in a frustrated transverse Ising chain.
PACS: 03.67.–a Quantum information;
03.67.Lx Quantum computation architectures and implementations;
85.25.Am Superconducting device characterization, design, and modeling.
Keywords: superconducting quantum devices, qubit tunneling spectroscopy, quantum information.
1. Introduction
Superconducting qubits are used as the basic building
blocks for practical implementation of scalable quantum
computers [1]. In particular, the existing annealing-based
quantum processing units (QPU) [2] exploit flux qubits
based on superconducting quantum devices (rf-SQUIDs)
[3,4]. The qubits are controlled by a limited number of
low-bandwidth external lines. This feature allows creating
a quantum processor with more than 1000 qubits, while at
the same time keeping a low level of noise in the system.
Experimental technique, termed qubit tunneling spectros-
copy (QTS) [5], has been developed in order to measure
quantum spectra of superconducting qubits (source qubits)
using probe qubits undergoing incoherent tunneling transi-
tions. A similar idea of weakly coupling a probe qubit to
the quantum system with the goal of observing its energy
spectrum was proposed in Ref. 6. In Ref. 5, however, the
coupling between the probe qubit and the source qubits is
strong and the method was experimentally implemented
with rf-SQUID flux qubits. Quantum spectra, characterized
by line splittings of the order of few GHz, were measured
using MHz-range control lines. The same technique was
also employed to demonstrate quantum entanglement in
systems of two and eight flux qubits embedded into an
industrial-scale quantum annealing processor [7]. These
experiments are performed with a unit cell having a linear
size of the order of 0.3 mm. Quantum spectra taken in the
process show very well resolved spectral lines, thus
demonstrating a clear example of macroscopic quantum
coherence [8] in a multi-qubit system.
In QTS, information about quantum properties of the
source qubits is extracted from the bias dependence of the
incoherent tunneling rate, ( )Γ , of the probe qubit. Here is
the external flux applied to the probe along z-axis. Positions
of the maxima of ( )Γ determine the energy levels of the
source system, whereas its peak amplitude is proportional to
the overlap between the eigenfunctions of the total (probe
plus source) system before and after the tunneling [6,5].
In this paper, we generalize QTS technique to the case
where the probe qubit is coupled to many source qubits in
an arbitrary basis. We show that projective measurements
of energy eigenstates of the source system in an arbitrary
basis can be performed with this approach. Therefore, our
measurement scheme is tomographically complete. As a
practical example, we consider dynamics of a kink in a
frustrated transverse Ising chain, in which the nearby
qubits are coupled ferromagnetically and the first and the
last qubits are biased in the opposite direction. The classi-
cal states with the lowest energy have a kink, which can be
located between any nearby qubits. This kink behaves like
© Anatoly Yu. Smirnov and Mohammad H. Amin, 2017
mailto:asmirnov@dwavesys.com
mailto:amin@dwavesys.com
Anatoly Yu. Smirnov and Mohammad H. Amin
a free particle confined in a potential well. We provide
numerical calculations of the incoherent tunneling rate of
the system taking into account the low frequency environ-
ment. The maxima of the tunneling rate, plotted as a func-
tion of the bias applied to the probe, are shown to be pro-
portional to the modulus squared of the eigenfunctions.
2. Qubit tunneling spectroscopy
Following Ref. 5, here and in Appendix A we derive a
set of formulas describing multi-qubit QTS experiments.
The quantum system under study has N coupled qubits,
with Hamiltonian SH written in terms of of Pauli matrices
{ , , }yx z
i iiσ σ σ where {1, , }i N∈ . We denote eigenstates
and eigenvalues of this Hamiltonian by nΨ and nE , re-
spectively, where {0,1, , (2 1)}Nn ∈ − . There is no need
to specify Hamiltonian SH of the quantum system at this
stage, although later we will consider the transversal Ising
chain as an example. In addition to the above source qubits
operating in a fully quantum regime, we have one probe
qubit characterized by a small tunneling amplitude p∆
and by an external bias . This qubit works in an incoher-
ent regime of macroscopic resonant tunneling (MRT)
[9,10]. We write the total source-probe Hamiltonian as:
0 = ( /2)(1 )z x
S C p p pH H H+ + − σ − ∆ σ . (1)
The probe qubit, which is described by the set of its Pauli
matrices { , , },x y z
p p pσ σ σ has two eigenstates, | p↑ 〉 and
| ,p↓ 〉 of the matrix : | = | ,z z
p p p pσ σ ↑ 〉 ↑ 〉 | =z
p pσ ↓ 〉
| .p= − ↓ 〉 The coupling between the source qubits and the
probe is provided by the term: z
p CH−σ . Once again, the
details of CH does not affect our general description. The
second term in the Hamiltonian vanishes when probe qubit
is in state | .p↑ 〉 Therefore, we can write
0 = | | | ,x
S p p S p p p pH H H↑ ↓⊗ ↑ 〉〈↑ + ⊗ ↓ 〉〈↓ −∆ σ (2)
where =S SH H↑ and = 2S S CH H H↓ + + . We denote
the eigenstates of these Hamiltonians by | n
↓Ψ 〉 and
| :m
↓Ψ 〉
| = | ,S n n nH E↑ ↑ ↑Ψ 〉 Ψ 〉 | = ( ) | .S m m mH E↓ ↓ ↓ ↓Ψ 〉 + Ψ 〉 (3)
Notice that | = |n n
↑Ψ 〉 Ψ 〉 and =n nE E↑ .
In the limit of small p∆ , the eigenstates of 0H are ap-
proximately | = | |n n p
↑ ↑ψ 〉 Ψ 〉 ⊗ ↑ 〉 and | = | | ,m m p
↓ ↓ψ 〉 Ψ 〉 ⊗ ↓ 〉
where , {0,1, , (2 1)}Nn m ∈ − . When the probe qubit is in
its “up” state, | p↑ 〉 , the eigenstates of the source qubits coin-
cide with the eigenstates of the original Hamiltonian SH .
Notice that this is true even when coupling to the probe qubit
is strong. In the opposite case, when the probe is in its
“down” state, | ,p↓ 〉 the coupling to the probe should create a
large bias for the source qubits in such a way that the new
Hamiltonian, SH ↓ , has a well-defined ground state 0| ↓Ψ 〉
isolated from the rest of the eigenstates by a large energy gap.
The total system is then initialized in 0 0| = | | p
↓ ↓ψ 〉 Ψ 〉 ⊗ ↓ 〉
and tunnels to one of the eigenstates | = | |n n p
↑ ↑ψ 〉 Ψ 〉 ⊗ ↑ 〉 ,
as the probe tunnels from | p↓ 〉 to | p↑ 〉 . The two sets of
states of the total Hamiltonian 0H , with the probe qubit
being in | p↓ 〉 or | p↑ 〉 , can be moved relative to each
other by applying a probe bias . The rate of macroscop-
ic tunneling between the initial and final states can show
well-resolved peaks when the eigenenergies of these
states are in resonance [10].
To have a full description of the dynamics we consider
the system being exposed to an environment. As shown in
Ref. 7, the width of the MRT peaks is predominantly de-
termined by the low frequency noise coupled to the probe
qubit. As such, in our modeling we consider an environ-
ment only interacting with the probe qubit. The dissipative
dynamics of the probe-source system is therefore described
by the master Eq. (A.26) written in our case as (see Ap-
pendix A)
0 0 0 0= ,n n
n
P P P+ Γ Γ∑ (4)
where 0P and are the probabilities of being in state 0| ↓ψ 〉
and | n
↑ψ 〉 , respectively, and
2 2
0 0( ) = | | |p n
n
↑ ↓Γ ε ∆ 〈Ψ Ψ 〉 ×∑
2
0
2 2
( )2 exp ,
2
n pE E
W W
↑ ↓ − − +π × −
(5)
2 2
0 0( ) = | | |n p n
↑ ↓Γ ε ∆ 〈Ψ Ψ 〉 ×
2
0
2 2
( )2 exp .
2
n pE E
W W
↑ ↓ − − −π × −
(6)
The MRT width W and the reorganization energy p are
related by fluctuation dissipation theorem (see Ref. 9 and
Appendix A). At = 0t , we have = 0nP , and therefore the
initial slope of probability decay is only given by 0Γ ,
which is measured in the MRT experiments. At the point
of resonance, where 0= n pE E↑ ↓− + , the peak value of
the escape rate 0Γ is proportional to the overlap of the
initial and final wave functions,
peak 2
0| | | .n
↑ ↓Γ ∝ 〈Ψ Ψ 〉 (7)
In QTS, the probe qubit bias, , is swept within some
range and 0 ( )Γ is measured. The locations of the peaks
of 0Γ give information about the energy spectrum, nE , of
the source system.
3. Quantum eigenstate tomography
For spectroscopy purposes, the details of CH are un-
important as long as the ground state of SH ↓ is well sepa-
rated from excited states and has overlap with the
970 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7
Quantum eigenstate tomography with qubit tunneling spectroscopy
eigenstates of SH ↑ (= )SH that we want to detect. Since
the tunneling rate is proportional to 2
0| | |n
↑ ↓〈Ψ Ψ 〉 , if we
can set 0| ↓Ψ 〉 as an arbitrary state, we can measure the
projection of | n
↑Ψ 〉 on that state and therefore perform
tomography on the source system's eigenstates, | nΨ 〉 .
This can be done by choosing a specific form of CH . For
example, assume that the probe qubit can be coupled to all
source qubits along all three axes. We can therefore write
=1
= ( )
N
y yx x z z
C pi i pi ipi i
i
H J J Jσ + σ + σ∑ . (8)
The coupling constants , yx
pi piJ J and z
piJ create additional
biases for the ith source qubit along x , y , and z direc-
tions. These biases disappear in the case when the probe
qubit is in | p↑ 〉 and, as before, =S SH H↑ . In the opposite
direction of the probe, | ,p↓ 〉 strong coupling constants
{ , , }yx z
pi pipiJ J J can make CH dominate in 0H , thus sup-
pressing the contribution of SH . The standard z–z couplers
[3,4] between the probe and source qubits lead to the follow-
ing set of the constants: {0,0, }z
piJ . This set creates the z-di-
rected initial state 0 =1| = | ,N
i iz↓Ψ 〉 ⊗ 〉 defined in terms of the
eigenstates | iz 〉 of the matrices .z
iσ Again, here the probe
qubit is in the | p↓ 〉 state. The up or down direction of the
specific ith source qubit in the reference state 0| ↓Ψ 〉 de-
pends on the sign of the corresponding coupling coefficient
.z
piJ If instead of z–z couplers we have x–z couplers, i.e.,
0x
piJ ≠ and = = 0,y z
pipiJ J then the reference state will
be 0 =1| = | ,N
i ix↓Ψ 〉 ⊗ 〉 where ix are the eigenfunctions of
x
iσ . We can therefore project the eigenstates | nΨ 〉 onto
the x-basis. Likewise, we can project | nΨ 〉 onto y-basis.
Being able to do projective measurements in all basis
makes our protocol tomographically complete.
In practice, the coupling of the probe qubit to all source
qubits in all bases could be challenging. However, with a
limited number of couplers working in a single basis we
can still do projective measurements in a very limited Hil-
bert subspace and obtain useful information. For example,
in ferromagnetic systems, one can do projective measure-
ments of the eigenstates onto the lowest energy subspace
{| , | }↑↑ ↑〉 ↓↓ ↓〉 with one coupler [7]. In the next
section, we provide another example in which the coupling
of the probe qubit to two source qubits is needed for pro-
jection onto the lowest energy subspace.
4. Wave function of a kink in the frustrated Ising chain
As an example of the source quantum system we con-
sider a frustrated Ising chain described by the following
Hamiltonian SH ,
1 <
= ( ) .
N
z x z z
S i i i i ij i j
i i j
H h J
−
σ − ∆ σ + σ σ∑ ∑ (9)
For this specific case the transverse field is determined by
the set of tunneling amplitudes { }i∆ . The qubits are biased
along z-direction with the strengths { }ih and coupled with
the constants { }ijJ . In principle, the qubits can be biased
and coupled along any direction, x , y , or z . The chain
has N qubits, with a uniform ferromagnetic coupling be-
tween the nearby qubits, 1,= ,z
ij i jJ J ±− δ where > 0.J
Notice that hereafter we work in the z-basis since the probe
qubit is coupled to z
iσ -matrices of the source qubits. This
can be done by means of the standard magnetic couplers
[3,4]. The frustration is created by two additional boundary
qubits, which have fixed spin directions. In Fig. 1 we show
configurations of seven source spins ( = 7)N plus two
boundary qubits. The left boundary qubit is always in the
up-direction, and the right boundary qubit is always di-
rected downwards. These qubits are ferromagnetically
coupled to the first and to the last qubits in the Ising chain,
thus creating a bias 1 =h J− applied to the first qubit and
the bias =Nh J applied to the last qubit in the chain. The
other qubits have zero biases, = 0ih at = 2, , 1.i N − In
the absence of the probe qubit the lowest energy of the N-
spin Ising chain is degenerate. All eigenstates of the source
Hamiltonian SH with the lowest energy have one kink
Fig. 1. (Color online) Eight possible positions of a kink (shown
as a blue star) in a chain with seven source qubits and two bound-
ary spins. A left boundary qubit is always up, and a right bounda-
ry qubit is always down.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7 971
Anatoly Yu. Smirnov and Mohammad H. Amin
located between the source qubits. A kink also can be lo-
cated between the first qubit in the chain and the left
boundary qubit, and between the right boundary qubit and
the last qubit in the chain. The possible locations of the
kink, which is shown as a blue star, are presented in Fig. 1.
It is known that the kink in the Ising chain behaves like a
free quantum particle. The main goal of this part of the
paper is to demonstrate that the QTS measurements allow
us to visualize the quantum distributions of the kink in the
frustrated Ising chain.
In the process of QTS tomography the lowest
eigenstates of the kink's Hamiltonian are projected on the
basis set formed by vectors that have a definite kink loca-
tion. In particular, functions 1 1| , ,| ,N +ψ 〉 ψ 〉 where
1 1 2 1| = | , , , , ,N N−ψ 〉 ↓ ↓ ↓ ↓ 〉
2 1 2 1| = | , , , , , ,N N−ψ 〉 ↑ ↓ ↓ ↓ 〉
1 2 1| = | , , , , ,N N N−ψ 〉 ↑ ↑ ↑ ↓ 〉
1 1 2 1| = | , , , , ,N N N+ −ψ 〉 ↑ ↑ ↑ ↑ 〉 (10)
corresponds to 1N + positions of the kink in the frustrated
Ising chain (see Fig. 1 for the case of seven source qubits,
= 7)N . These functions form the quantum-mechanical
basis. We notice that the basis | lψ 〉 is not complete since
the high-energy states with many kinks are neglected here.
Every state from the set 1 1{| , ,| }N +ψ 〉 ψ 〉 is characteri-
zed by a definite position of the kink. An arbitrary quan-
tum state, for example, the n-eigenstate of the source
qubits, | | ,n n
↑Ψ 〉 ≡ Ψ 〉 can be represented as a superposi-
tion of the basis states | ,lψ 〉
1
=1
| = | |
N
n l n l
l
+
↑ ↑Ψ 〉 〈ψ Ψ 〉 ψ 〉∑ . (11)
The set of amplitudes, ( ) = |n
l nlC ↑〈ψ Ψ 〉 , taken as func-
tions of the quantum number l , describes a wave function
of the nth energy state in a single-kink representation. The
quantum number l serves as a position of the kink in the
frustrated Ising chain. Thus, the l-dependence of the kink
amplitude ( )n
lC is equivalent to the coordinate dependence
of the wave function of the particle in the state correspond-
ing to the n-eigenstate of the source Hamiltonian SH .
Here we have = 0,1, , (2 1).Nn −
We notice that the escape rate 0 ( )Γ (5) is proportional
to the overlap squared, 2
0| | |n
↑ ↓〈Ψ Ψ 〉 , of the n-eigenstate
| n
↑Ψ 〉 of the source Hamiltonian SH and the ground state
0| ↓Ψ 〉 of the biased source qubits. The ground state 0
↓Ψ of
the left manifold can be transformed into a specific basis
state | lψ 〉 , so that 0| = | l
↓Ψ 〉 ψ 〉 , by choosing proper cou-
plings piJ between the probe and the source qubits as it is
shown in Fig. 1. For example, the first state 1| =ψ 〉
1| N= ↓ ↓ 〉 can be generated if the first qubit in the chain
is coupled to the probe with a positive constant
1 = > 0.p pJ J Other source qubits are decoupled from the
probe. The second basis state 2 1 2| = | Nψ 〉 ↑ ↓ ↓ 〉 is cre-
ated when the probe qubit is coupled to the first source qubit
by negative coupling, 1 = ,p pJ J− whereas its coupling to
the second qubit is positive, 2 = > 0.p pJ J To generate the
state 1 2 1| = |l l l N−ψ 〉 ↑ ↑ ↑ ↓ ↓ 〉 , two nearby qubits
1l − and l should be coupled to the probe with opposite
coupling strengths, , 1 =p l pJ J− − and , =p l pJ J . The last
state, 1 1| = | ,N N+ψ 〉 ↑ ↑ 〉 is generated when the N-qubit
is coupled to the probe with a negative coupling strength,
=pN pJ J , and other source qubits do not interact with the
probe.
4.1. Quantum distribution of a kink
In order to obtain the quantum distribution of the sys-
tem over all possible positions of the kink in the Ising
chain we have to measure the l-dependence of the function
( ) 2 2| | = | | |n
l nlC 〈ψ Ψ 〉 . To do that, we choose a specific
connection between the probe and source qubits related,
for example, to the state | ,lψ 〉 with a subsequent meas-
Fig. 2. (Color online) A generation of basis states ψl of the left
manifold by selective coupling of the probe qubit (shown as a
hexagon) to the source qubits. A positive probe-source coupling
Jpi is drawn in blue, a negative Jpi is shown in red. A blue star
symbol is related to the kink location.
972 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7
Quantum eigenstate tomography with qubit tunneling spectroscopy
urement of the escape rate 0 ( )Γ for all possible probe
biases . As the next step, we change the initial source-
probe connection and repeat the measurements. Finally, we
obtain the rate 0 ( , )lΓ as a function of the bias and the
kink position, which is characterized by the number l of
the single-kink basis state | lψ 〉 . In Fig. 3 we show the
normalized function 0 ( , )lΓ for the case of seven qubits
in the chain ( = 7)N . Both figures are plotted at
1 = = = 2GHz,N∆ ∆ = 2GHz,J =pJ J , with a tem-
perature of = 12T mK and a MRT linewidth of = 10W
mK. Only four lowest energy levels of source qubits, with
= 0,1,2,3n , are presented. In Fig. 4 we plot the QTS rate
0 ( , )lΓ for the chain that has 16 qubits ( = 16)N and for
the same set of parameters. This figure has a better resolu-
tion than Fig. 3. In both figures, along the -axis we have
the standard QTS peaks corresponding to four energy
eigenstates | nΨ 〉 of the source Hamiltonian SH (9). If we
move along the other axis, we will see a dependence of the
states | nΨ 〉 on the kink position. The probability distribu-
tions of the kink in the frustrated chain shown in Figs. 3
and 4 are similar to the distribution of a quantum particle
in a potential well.
5. Conclusions
We have generalized the qubit tunneling spectroscopy
approach of Ref. 5 to allow performing quantum eigenstate
tomography in a multi-qubit (source) system. An addition-
al (probe) qubit, working in the incoherent regime, has to
be coupled to all source qubits in all bases to make projec-
tive measurement onto an arbitrary basis state possible. A
limited, but practical, version of tomography is described
with an example of a single kink in a frustrated Ising chain.
The lowest energy eigenstates of this system is equivalent
to those of a free quantum particle confined in a potential
well. We have calculated the incoherent tunneling rate of
the system and shown that its peak values correspond to
the modulus squared of the overlap of the eigenstates and a
preselected basis state which is related to a kink position.
Acknowledgements
We are thankful to Professor Alexander Omelyanchouk
for collaboration and support in early days of D-Wave Sys-
tems. We are grateful to Chris Rich for helpful discussions
and Fiona Hanington for critical reading of the paper.
Appendix A: Derivation of the master equation
The system of N source qubits and one probe qubit is
described by the Hamiltonian 0H defined by Eqs. (1) and
(2). The probe qubit is working in a regime of incoherent
tunneling between its wells. The tunneling is introduced by
the small tunneling amplitude p∆ in the Hamiltonian 0H .
We also take into account an interaction of the probe qubit
with its dissipative environment, which is described by the
variable pQ . Weak coupling of source qubits to their envi-
ronments is omitted here. This coupling contributes to the
width of MRT lines of the probe qubit [7]. The main con-
tribution to the linewidth, however, is given by the low-
frequency bath directly coupled to the probe. The total
Hamiltonian H of the source-probe system coupled to the
probe qubit bath has the form
0= ,z
p p BH H Q H− σ + (A.1)
where BH is the free Hamiltonian of the bath, and x
pσ is the
Pauli matrix responsible for the flipping of the probe qubits
between states | p↓ 〉 and | ,p↑ 〉 = | | | | .x
p p p p pσ ↑ 〉 〈↓ + ↓ 〉 〈↑
The bath can be represented as a sum of independent
harmonic oscillators [11,12] with the Hamiltonian
2 2 2
= .
2 2
k k k k
B
kk
p m x
H
m
ω
+
∑ (A.2)
Fig. 3. (Color online) Escape rate 0Γ as a function of the kink
position l and the bias applied to the probe qubit. Here we have
seven source qubits. As in the case of a free particle in the potential
well, the ground state of the kink has a maximum in the middle of
the chain where the first excited state has a node. The second and
third excited states have two and three nodes, respectively.
Fig. 4. (Color online) The QTS rate Γ0(,l) reflects the quantum
distribution of a kink position l in the 16-qubit Ising chain for
four lowest energy eigenstates. The peak with the energy ∼ 0
corresponds to the ground state of the kink (particle). The next
peak, with ∼ 1 GHz, is related to the first excited state having
one node, at l = 8, in the wave function. The wave function of the
next state, with the energy ∼ 1,5 GHz, has two nodes, at l = 6
and l = 12. The last state shown in the picture has the energy ∼
∼ 2.25 GHz. This state is described by the wave function having
three nodes located at l = 4, l = 9, and l = 14.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7 973
Anatoly Yu. Smirnov and Mohammad H. Amin
The kth oscillator in the bath is characterized by position
kx , momentum kp , mass km and positive frequency kω .
The bath operator pQ in Eq. (A.1) is given by the formula
2= .p k k kp k
k
Q m z xω∑ (A.3)
A constant kpz determines the strength of coupling be-
tween the probe qubit and the k-mode of the bath. The
Hamiltonian H (A.1) can be written as
= | | | | x
S p p S p p p pH H H↑ ↓⊗ ↑ 〉 〈↑ + ⊗ ↓ 〉 〈↓ −∆ σ +
2 2
2( ) .
2 2
zk k k
k kp p
kk k
p m
x z
m
ω
+ + − σ∑ ∑ (A.4)
Here we have omitted the constant term. The unitary trans-
formation
= e = 1 (e 1)| |
zi ip p p
p p pU
− ξ σ − ξ
+ − ↓ 〉〈↓ +
(e 1)| |
i p
p p
ξ
+ − ↑ 〉 ↑ 〉 , (A.5)
applied to the Hamiltonian H turns this operator to the
form
2 2 2
†= =
2 2
k k k k
p p
kk
p m x
H U HU
m
ω
+ +′
∑
2
| | | | e | |
i p
S p p S p p p p pH H
ξ↑ ↓+ ⊗ ↑ 〉〈↑ + ↓ 〉〈↓ −∆ ↓ 〉〈↑ −
2
e | | .
i p
p p p
− ξ
− ∆ ↑ 〉〈↓ (A.6)
Here =p kp kkz pξ ∑ is a stochastic phase produced by the
bath. We notice that this phase appears only at the tunnel-
ing terms.
The source-probe Hamiltonian has eigenstates µΨ de-
fined by the following equations:
( | | | |) | = | ,S p p S p pH H E↑ ↓
µ µ µ⊗ ↑ 〉〈↑ + ⊗ ↓ 〉〈↓ Ψ 〉 Ψ 〉 (A.7)
where 1{0,1, , (2 1)}.N +µ ∈ − We notice that the set of
eigenstates {| }µΨ 〉 contains two subsets — one, which is
related to the up-state of the probe qubit, and another, which
is related to the down-state of the probe: {| } =µΨ 〉
{| | , | | .n p m p
↑ ↓= Ψ 〉 ⊗ ↑ 〉 Ψ 〉⊗ ↓ 〉 Here the eigenstates | n
↑Ψ 〉
and | m
↓Ψ 〉 can be found from Eqs. (3). The indices m and
n run over 2N states: , {0,1, , (2 1)}.Nm n ∈ − The eigen-
energies { }Eµ also have two subsets: { } =Eµ
{ , },n mE E↑ ↓= + with being the bias applied to the probe
qubit.
Following the approach proposed in Ref. 13 and devel-
oped in Ref. 14 we introduce a time-dependent Heisenberg
operator µνρ of the source-probe system,
= (| |)( ).tµν µ νρ Ψ 〉〈Ψ
In the Heisenberg representation the total Hamiltonian H
(A.6) is given by the formula
= ,BH E Q Hµ µµ µν µν
µ µ≠ν
ρ − ρ +∑ ∑ (A.8)
with the bath operator
2
= e | |
i p
p p pQ
ξ
µν µ ν∆ 〈Ψ ↓ 〉〈↑ Ψ 〉 +
2
e | | .
i p
p p p
− ξ
µ ν+ ∆ 〈Ψ ↑ 〉〈↓ Ψ 〉 (A.9)
Here we drop the prime sign in the Hamiltonian H (A.6).
The diagonal elements of the Hamiltonian (A.8) are deter-
mined by the eigenenergies Eµ of the system-probe Hamil-
tonian 0H (2). The bath Hamiltonian BH is defined by
Eq. (A.2). The operator µνρ obeys the Heisenberg equation
= ( ),i i Q Qµν µ µν νµ µµ µ µ µ ν′ ′ ′ ′
µ′
ρ ω ρ + ρ − ρ∑ ν (A.10)
with = .E Eµν µ νω − Using the approach developed in
Refs. 13 and 14 we derive a set of equations for the qubit
operators averaged over fluctuations of the free bath,
____________________________________________________
(0) (0) (0) (0)
1 1 1 1 1 10 0
= ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
t t
i dt Q t Q t t t dt Q t Q t t tµν µ µν µµ µ ν µ ν µµ′ ′′ ′′ ′′ ′′ ′νµ µ ν µ ν νµ′ ′′ ′′ ′′ ′′ ′ρ − ω ρ − 〈 〉〈ρ ρ 〉 + 〈 〉〈ρ ρ 〉 +∫ ∫ ν
(0) (0) (0) (0)
1 1 1 1 1 10 0
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
t t
dt Q t Q t t t dt Q t Q t t tµ µ ν µ ν µ ν′ ′′ ′′ ′′ ′′ ′µ µ µ ν µ ν µ µ′ ′′ ′′ ′′ ′′ ′+ 〈 〉〈ρ ρ 〉 − 〈 〉〈ρ ρ 〉∫ ∫ν . (A.11)
_______________________________________________
An operator (0) ( )Q tµν is defined by Eq. (20) where the sto-
chastic phases pξ are replaced by their unperturbed values
(0)(0) =p kp kkz pξ ∑ that have free Heisenberg operators
(0)
kp of the bath. A time evolution of the free bath opera-
tors is determined by the Hamiltonian BH (A.2). We also
assume that there are sums over repeated indices
, ,µ µ′ ′′ ′′ν in the right-hand side of Eq. (A.11). The free
bath variables (0) ( )Q tµν are nonlinear functions of the
Gaussian bath operators (0)
pξ . Therefore, non-Markovian
equations (A.11) are valid at the small system-bath cou-
pling only. We consider a regime of incoherent tunneling
for the probe qubit. This regime takes place at the small
tunneling amplitude p∆ . It follows from Eq. (A.9) that the
small p∆ is related to the small system-bath interaction.
974 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7
Quantum eigenstate tomography with qubit tunneling spectroscopy
Within the perturbation theory in terms of the parameter
p∆ we assume that in Eq. (A.11) the correlation functions
of the qubit operators, such as 1( ) ( )t tµµ µ′ ′′ ′′ρ ρ ν , can be cal-
culated using the free evolution equations. It is convenient
to reduce the operator 1( )tµ′′ ′′ρ ν to the operator ( )tµ′′ ′′ρ ν in
such a way that
( )1
1( ) = e ( )
i t t
t t
− ω −µ ν′′ ′′
µ µ ν′′ ′′ ′′ ′′ρ ρν ,
so that the correlator is given by the equation
( )1
1( ) ( ) = e ( ),
i t t
t t t
− ω −µ′′ ′′
µµ µ µ µ µ′ ′′ ′′ ′ ′′ ′′〈ρ ρ 〉 δ ρν
ν ν
where µ µ′ ′′δ is the Kronecker delta.
A diagonal element =Pµ µµ〈ρ 〉 of the averaged matrix
µ〈ρ 〉ν defines the probability to find the source-probe sys-
tem in the state | µΨ 〉 . It follows from Eqs. (A.11) that these
probabilities are governed by the set of master equations,
= .P P Pµ µ µ µ
ν
+ Γ Γ∑ ν ν (A.12)
Here =µ µΓ Γ∑ ν
ν
is a relaxation rate and µΓ ν is a relax-
ation matrix defined by the equation
( )(0) (0) 1
1 1
0
= ( ) ( ) e h.c.
t i t t
dt Q t Q t
− ω −µ
µ µ µΓ 〈 〉 +∫ ν
ν ν ν (A.13)
The bath operator (0) ( )Q tµν is given by Eq. (A.9) where
(0)
pξ is a free Gaussian operator, (0)(0) = p kp k
k
z pξ ∑ . The
commutator and the correlation function of these Heisen-
berg operators taken at different moments of time are de-
termined by the following expressions
____________________________________________________
2
(0) (0)
2
( )1[ ( ), ( )] = sin ( ) = sin ( ),
2 2 2
k k kp pp
p p k
k
m z dt t i t t i t t−
ω χ ω′′ω
ξ ξ − ω − − ω −′ ′ ′
π ω
∑ ∫
2
(0) (0)
2
( )1[ ( ), ( )] coth cos ( ) = cos ( ),
2 2 2 2
k k kp pk
p p k
k
m z Sdt t t t t t
T+
ω ωω ω ξ ξ = ω − ω −′ ′ ′ π ω
∑ ∫ (A.14)
with T being the equilibrium temperature of the free bath. The dissipative properties of the bath are defined by the imagi-
nary part of its susceptibility ( )ppχ ω′′ and by the spectrum ( )pS ω . They are described by the following formulas
3 2
( ) = [ ( ) ( )],
2
k k kp
pp k k
k
m zω
χ ω π δ ω − ω − δ ω + ω′′ ∑
[ ]
3 2
( ) = ( )coth = coth ( ) ( ) .
2 2 2
k k kp k
p pp k k
k
m z
S
T T
ω ωω ω χ ω π δ ω − ω + δ ω + ω′′ ∑ (A.15)
_______________________________________________
For the correlator of free variables (0)Qµν of the bath we
obtain
(0) (0) 2( ) ( ) = ( ),pQ t Q t t tµ µ µ〈 〉 ∆ Φ −′ ′ν ν ν (A.16)
where the prefactor 2
µ∆ ν is defined as
2 2 2 2= (| | | | | | | | ).p p p p pµν µ ν µ ν∆ ∆ 〈Ψ ↓ 〉〈↑ Ψ 〉 + 〈Ψ ↑ 〉〈↓ Ψ 〉
(A.17)
The characteristic functional of the bath is given by the
formula:
(0) (0)2 ( ) 2 ( )
( ) = e e =
i t i tp p
p t t
ξ − ξ ′
Φ − 〈 〉′
2
1 cos ( )exp 4 ( )
2 p
d t tSω − ω − ′= − ω − π ω ∫
2
sin ( )4 ( )
2 p
d t ti ω ω − ′ − χ ω′′ π ω ∫ . (A.18)
The heat bath acting on the probe qubit may have both, low-
frequency and high-frequency, components [15]. In this case
the dissipative function ( )pχ ω′′ is represented as a sum of
the low-frequency susceptibility, ( )LFχ ω′′ , and the high-
frequency function ( )HFχ ω′′ : ( ) = ( ) ( ).p LF HFχ ω χ ω + χ ω′′ ′′ ′′
The functional ( , )p t tΦ ′ is equal to the product of the
low-frequency and high-frequency parts: ( ) =pΦ τ
( ) ( ).LF HF= Φ τ Φ τ Here, the low-frequency factor is deter-
mined by the formula
2 2
( ) = e exp ,
2
i p
LF
W− τ τ
Φ τ −
(A.19)
with the reorganization energy p and with the width W
defined by the following equations,
( )( )
= 4 ,
2
LF
p
d χ ω ω′′ω
π ω∫
2 = 4 ( ) = 2 .
2 LF p
dW S Tω
ω
π∫ (A.20)
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7 975
Anatoly Yu. Smirnov and Mohammad H. Amin
The high-frequency noise acting on the probe qubit is usu-
ally described by the Ohmic spectral density
( ) = exp( | | / )HF cχ ω ηω − ω ω′′
where η is a small dimensionless coupling constant and
cω is the cutoff frequency [12]. In this case the high-
frequency factor ( )HFΦ τ of the functional ( )pΦ τ is giv-
en by the expression
4 /
1( ) = .
1 sinh( )HF
c
T
i T
η π π τ
Φ τ + ω τ π τ
(A.21)
The relaxation matrix (A.24) can be written as
____________________________________________________
4 /
2 2( )2 /2
0
1= e e h.c.
1 sinh( )
i Wp
c
Td
i T
η π∞ − ω + τ − τµν
µν µν
π τ
Γ ∆ τ + + ω τ π τ
∫
(A.22)
_______________________________________________
A more comprehensive description of the dissipative dy-
namics of the open quantum system has been carried out in
Ref. 16. We notice that in the case of the very weak cou-
pling of the slow probe qubit to the high-frequency bath,
when 4 / 1,η π the relaxation matrix is given by the Mar-
cus formula [9],
2
2
2 2
( )2= exp .
2
p
W W
µν
µν µν
ω +π Γ ∆ −
(A.23)
For indices µ and ν we have two possible cases:
(a) | = | | , | = | | ,n p m p
↑ ↓
µ νΨ 〉 Ψ 〉 ⊗ ↑ 〉 Ψ 〉 Ψ 〉 ⊗ ↓ 〉
and
(b) | = | | , | = | | .m p n p
↓ ↑
µ νΨ 〉 Ψ 〉 ⊗ ↓ 〉 Ψ 〉 Ψ 〉 ⊗ ↑ 〉
These cases correspond to two sets of eigensenergies and
frequencies:
(a) = , = , = ,n m n mE E E E E E↑ ↓ ↑ ↓
µ ν µν+ ω − −
and
(b) = , = , = .m n m nE E E E E E↓ ↑ ↓ ↑
µ ν µν+ ω − +
For these two sets we obtain the following relaxation
matrices:
____________________________________________________
2
( ) 2 2
2 2
( )2(a) = | | | exp ,
2
n m pa
p n m
E E
W W
↑ ↓
↑ ↓
µν
− − +π Γ ∆ 〈Ψ Ψ 〉 −
2
( ) 2 2
2 2
( )2(b) = | | | exp .
2
n m pb
p n m
E E
W W
↑ ↓
↑ ↓
µν
− − −π Γ ∆ 〈Ψ Ψ 〉 −
(A.24)
_______________________________________________
If we start the QTS experiment with the probe qubit being
in its | p↓ 〉 state and allow the qubit to tunnel into the | p↑ 〉
state, the situation is described by the master equation
(A.12) where mP Pµ ≡ is the probability to find the system
in state | = | | .m p
↓
µΨ 〉 Ψ 〉 ⊗ ↓ 〉 The system tunnels into the
state | = | | ,n p
↑
νΨ 〉 Ψ 〉 ⊗ ↑ 〉 so that here we have the case
(b) described by the relaxation matrix ( )b
mnµνΓ ≡ Γ defined
in Eq. (A.24).
The probability to find the system in the state νΨ is
given by the variable nP Pν ≡ . It follows from Eq. (A.12)
that the escape rate µΓ is determined by the transposed
matrix νµΓ , since =µ νµ
ν
Γ Γ∑ . In our case the matrix
νµΓ is described by the case ( )a , so that the relaxation
rate µΓ is given by the formula
____________________________________________________
2
2 2
2 2
( )2= | | | exp .
2
n m p
m p n m
n
E E
W W
↑ ↓
↑ ↓
µ
− − +π Γ ≡ Γ ∆ 〈Ψ Ψ 〉 −
∑
(A.25)
______________________________________________
As a result, the time evolution of the probability mP to
find the source-probe system in the state | |m p
↓Ψ 〉 ⊗ ↓ 〉 is
governed by the equation
= ,m m m mn n
n
P P P+ Γ Γ∑ (A.26)
where nP is the probability for the system to be in state
| |n p
↑Ψ 〉 ⊗ ↑ 〉 .
976 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7
Quantum eigenstate tomography with qubit tunneling spectroscopy
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7 977
1. Introduction
2. Qubit tunneling spectroscopy
3. Quantum eigenstate tomography
4. Wave function of a kink in the frustrated Ising chain
4.1. Quantum distribution of a kink
5. Conclusions
Acknowledgements
Appendix A: Derivation of the master equation
|